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Mathlib/LinearAlgebra/LinearIndependent.lean | LinearIndependent.extend_subset | [] | [
1309,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1306,
1
] |
Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.add_mem | [] | [
182,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
11
] |
Mathlib/Algebra/Tropical/Basic.lean | Tropical.untrop_le_iff | [] | [
177,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
] |
Mathlib/Data/List/Perm.lean | List.Perm.filterMap | [
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → Option β\nl₁ l₂ : List α\np : l₁ ~ l₂\n⊢ List.filterMap f l₁ ~ List.filterMap f l₂",
"tactic": "induction p with\n| nil => simp\n| cons x _p IH =>\n simp only [filterMap]\n cases h : f x\n <;> simp [h, filterMap, IH, Perm.cons]\n| swap x y l₂ =>\n simp only [filterMap]\n cases hx : f x\n <;> cases hy : f y\n <;> simp [hx, hy, filterMap, swap]\n| trans _p₁ _p₂ IH₁ IH₂ =>\n exact IH₁.trans IH₂"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nf : α → Option β\nl₁ l₂ : List α\n⊢ List.filterMap f [] ~ List.filterMap f []",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\nf : α → Option β\nl₁ l₂ : List α\nx : α\nl₁✝ l₂✝ : List α\n_p : l₁✝ ~ l₂✝\nIH : List.filterMap f l₁✝ ~ List.filterMap f l₂✝\n⊢ List.filterMap f (x :: l₁✝) ~ List.filterMap f (x :: l₂✝)",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\nf : α → Option β\nl₁ l₂ : List α\nx : α\nl₁✝ l₂✝ : List α\n_p : l₁✝ ~ l₂✝\nIH : List.filterMap f l₁✝ ~ List.filterMap f l₂✝\n⊢ List.filterMap f (x :: l₁✝) ~ List.filterMap f (x :: l₂✝)",
"tactic": "simp only [filterMap]"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\nf : α → Option β\nl₁ l₂ : List α\nx : α\nl₁✝ l₂✝ : List α\n_p : l₁✝ ~ l₂✝\nIH : List.filterMap f l₁✝ ~ List.filterMap f l₂✝\n⊢ List.filterMap f (x :: l₁✝) ~ List.filterMap f (x :: l₂✝)",
"tactic": "cases h : f x\n <;> simp [h, filterMap, IH, Perm.cons]"
},
{
"state_after": "case swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ : List α\nf : α → Option β\nl₁ l₂✝ : List α\nx y : α\nl₂ : List α\n⊢ List.filterMap f (y :: x :: l₂) ~ List.filterMap f (x :: y :: l₂)",
"state_before": "case swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ : List α\nf : α → Option β\nl₁ l₂✝ : List α\nx y : α\nl₂ : List α\n⊢ List.filterMap f (y :: x :: l₂) ~ List.filterMap f (x :: y :: l₂)",
"tactic": "simp only [filterMap]"
},
{
"state_after": "no goals",
"state_before": "case swap\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ : List α\nf : α → Option β\nl₁ l₂✝ : List α\nx y : α\nl₂ : List α\n⊢ List.filterMap f (y :: x :: l₂) ~ List.filterMap f (x :: y :: l₂)",
"tactic": "cases hx : f x\n <;> cases hy : f y\n <;> simp [hx, hy, filterMap, swap]"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁✝¹ l₂✝¹ : List α\nf : α → Option β\nl₁ l₂ l₁✝ l₂✝ l₃✝ : List α\n_p₁ : l₁✝ ~ l₂✝\n_p₂ : l₂✝ ~ l₃✝\nIH₁ : List.filterMap f l₁✝ ~ List.filterMap f l₂✝\nIH₂ : List.filterMap f l₂✝ ~ List.filterMap f l₃✝\n⊢ List.filterMap f l₁✝ ~ List.filterMap f l₃✝",
"tactic": "exact IH₁.trans IH₂"
}
] | [
254,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
240,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | InnerProductSpace.Core.inner_self_eq_zero | [
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.538902\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\n⊢ inner 0 0 = 0",
"state_before": "𝕜 : Type u_1\nE : Type ?u.538902\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx : F\n⊢ x = 0 → inner x x = 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.538902\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\n⊢ inner 0 0 = 0",
"tactic": "exact inner_zero_left _"
}
] | [
261,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean | BoxIntegral.Box.splitUpper_eq_self | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : Type ?u.15762\nn : ℕ\nI : Box ι\ni : ι\nx : ℝ\ny : ι → ℝ\n⊢ splitUpper I i x = ↑I ↔ x ≤ lower I i",
"tactic": "simp [splitUpper, update_eq_iff]"
}
] | [
128,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_inv_nhdsWithin_Ioi | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalGroup G\ninst✝³ : TopologicalSpace α\nf : α → G\ns : Set α\nx : α\ninst✝² : TopologicalSpace H\ninst✝¹ : OrderedCommGroup H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto (fun a => a⁻¹) (𝓟 (Ioi a)) (𝓟 (Iio a⁻¹))",
"tactic": "simp [tendsto_principal_principal]"
}
] | [
561,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
560,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean | BilinForm.IsAlt.ortho_comm | [] | [
977,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
976,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ico_inter_Iio | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.204664\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ ∀ (x : α), x ∈ Ico a b ∩ Iio c ↔ x ∈ Ico a (min b c)",
"tactic": "simp (config := { contextual := true }) [iff_def]"
}
] | [
1832,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1831,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.isOpen_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.96096\nι : Type ?u.96099\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\n⊢ IsOpen s ↔ ∀ (x : α), x ∈ s → ∃ ε, ε > 0 ∧ ball x ε ⊆ s",
"tactic": "simp only [isOpen_iff_mem_nhds, mem_nhds_iff]"
}
] | [
1005,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1004,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean | AffineSubspace.pointwise_vadd_span | [] | [
78,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_quotient_center | [
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = n\n⊢ nilpotencyClass (G ⧸ center G) = n - 1",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\n⊢ nilpotencyClass (G ⧸ center G) = nilpotencyClass G - 1",
"tactic": "generalize hn : Group.nilpotencyClass G = n"
},
{
"state_after": "case zero\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nhn : nilpotencyClass G = Nat.zero\n⊢ nilpotencyClass (G ⧸ center G) = Nat.zero - 1\n\ncase succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ nilpotencyClass (G ⧸ center G) = Nat.succ n - 1",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = n\n⊢ nilpotencyClass (G ⧸ center G) = n - 1",
"tactic": "rcases n with (rfl | n)"
},
{
"state_after": "case zero\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nhn : Subsingleton G\n⊢ Subsingleton (G ⧸ center G)",
"state_before": "case zero\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nhn : nilpotencyClass G = Nat.zero\n⊢ nilpotencyClass (G ⧸ center G) = Nat.zero - 1",
"tactic": "simp [nilpotencyClass_zero_iff_subsingleton] at *"
},
{
"state_after": "no goals",
"state_before": "case zero\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nhn : Subsingleton G\n⊢ Subsingleton (G ⧸ center G)",
"tactic": "exact Quotient.instSubsingletonQuotient (leftRel (center G))"
},
{
"state_after": "case succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ nilpotencyClass (G ⧸ center G) = n",
"state_before": "case succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ nilpotencyClass (G ⧸ center G) = Nat.succ n - 1",
"tactic": "suffices Group.nilpotencyClass (G ⧸ center G) = n by simpa"
},
{
"state_after": "case succ.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ nilpotencyClass (G ⧸ center G) ≤ n\n\ncase succ.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ n ≤ nilpotencyClass (G ⧸ center G)",
"state_before": "case succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ nilpotencyClass (G ⧸ center G) = n",
"tactic": "apply le_antisymm"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\nthis : nilpotencyClass (G ⧸ center G) = n\n⊢ nilpotencyClass (G ⧸ center G) = Nat.succ n - 1",
"tactic": "simpa"
},
{
"state_after": "case succ.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ upperCentralSeries (G ⧸ center G) n = ⊤",
"state_before": "case succ.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ nilpotencyClass (G ⧸ center G) ≤ n",
"tactic": "apply upperCentralSeries_eq_top_iff_nilpotencyClass_le.mp"
},
{
"state_after": "case succ.a.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = comap (mk' (center G)) ⊤",
"state_before": "case succ.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ upperCentralSeries (G ⧸ center G) n = ⊤",
"tactic": "apply @comap_injective G _ _ _ (mk' (center G)) (surjective_quot_mk _)"
},
{
"state_after": "case succ.a.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ upperCentralSeries G (nilpotencyClass G) = ⊤",
"state_before": "case succ.a.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = comap (mk' (center G)) ⊤",
"tactic": "rw [comap_upperCentralSeries_quotient_center, comap_top, ← hn]"
},
{
"state_after": "no goals",
"state_before": "case succ.a.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ upperCentralSeries G (nilpotencyClass G) = ⊤",
"tactic": "exact upperCentralSeries_nilpotencyClass"
},
{
"state_after": "case succ.a.bc\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ n + ?succ.a.a ≤ nilpotencyClass (G ⧸ center G) + ?succ.a.a\n\ncase succ.a.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ ℕ",
"state_before": "case succ.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ n ≤ nilpotencyClass (G ⧸ center G)",
"tactic": "apply le_of_add_le_add_right"
},
{
"state_after": "no goals",
"state_before": "case succ.a.bc\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ n + ?succ.a.a ≤ nilpotencyClass (G ⧸ center G) + ?succ.a.a\n\ncase succ.a.a\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nhH : Group.IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = Nat.succ n\n⊢ ℕ",
"tactic": "calc\n n + 1 = n.succ := rfl\n _ = Group.nilpotencyClass G := (symm hn)\n _ ≤ Group.nilpotencyClass (G ⧸ center G) + 1 :=\n nilpotencyClass_le_of_ker_le_center _ (le_of_eq (ker_mk' _)) _"
}
] | [
639,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
622,
1
] |
Mathlib/Data/List/Sigma.lean | List.nodupKeys_of_nodupKeys_cons | [] | [
117,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.EventuallyLE.compl | [] | [
1727,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1725,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean | Submodule.iInf_coe | [
{
"state_after": "R : Type u_2\nS : Type ?u.139553\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Sort u_1\np : ι → Submodule R M\n⊢ (⋂ (p_1 : Submodule R M) (_ : p_1 ∈ Set.range fun i => p i), ↑p_1) = ⋂ (i : ι), ↑(p i)",
"state_before": "R : Type u_2\nS : Type ?u.139553\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Sort u_1\np : ι → Submodule R M\n⊢ ↑(⨅ (i : ι), p i) = ⋂ (i : ι), ↑(p i)",
"tactic": "rw [iInf, sInf_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type ?u.139553\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Sort u_1\np : ι → Submodule R M\n⊢ (⋂ (p_1 : Submodule R M) (_ : p_1 ∈ Set.range fun i => p i), ↑p_1) = ⋂ (i : ι), ↑(p i)",
"tactic": "simp only [Set.mem_range, Set.iInter_exists, Set.iInter_iInter_eq']"
}
] | [
259,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/UV.lean | UV.shadow_compression_subset_compression_shadow | [
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\n⊢ (∂ ) 𝒜' ⊆ 𝓒 u v ((∂ ) 𝒜)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n⊢ (∂ ) (𝓒 u v 𝒜) ⊆ 𝓒 u v ((∂ ) 𝒜)",
"tactic": "set 𝒜' := 𝓒 u v 𝒜"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\n⊢ (∂ ) 𝒜' ⊆ 𝓒 u v ((∂ ) 𝒜)\n\ncase H\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\n⊢ ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\n⊢ (∂ ) 𝒜' ⊆ 𝓒 u v ((∂ ) 𝒜)",
"tactic": "suffices H : ∀ s ∈ (∂ ) 𝒜',\n s ∉ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ (s ∪ v) \\ u ∉ (∂ ) 𝒜'"
},
{
"state_after": "case H\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\n⊢ ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "intro s hs𝒜' hs𝒜"
},
{
"state_after": "case H\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have m : ∀ y, y ∉ s → insert y s ∉ 𝒜 := fun y h a => hs𝒜 (mem_shadow_iff_insert_mem.2 ⟨y, h, a⟩)"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "obtain ⟨x, _, _⟩ := mem_shadow_iff_insert_mem.1 hs𝒜'"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have hus : u ⊆ insert x s := le_of_mem_compression_of_not_mem ‹_ ∈ 𝒜'› (m _ ‹x ∉ s›)"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have hvs : Disjoint v (insert x s) := disjoint_of_mem_compression_of_not_mem ‹_› (m _ ‹x ∉ s›)"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have : (insert x s ∪ v) \\ u ∈ 𝒜 := sup_sdiff_mem_of_mem_compression_of_not_mem ‹_› (m _ ‹x ∉ s›)"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have hsv : Disjoint s v := hvs.symm.mono_left (subset_insert _ _)"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have hvu : Disjoint v u := disjoint_of_subset_right hus hvs"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have hxv : x ∉ v := disjoint_right.1 hvs (mem_insert_self _ _)"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have : v \\ u = v := ‹Disjoint v u›.sdiff_eq_left"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have : x ∉ u := by\n intro hxu\n obtain ⟨y, hyv, hxy⟩ := huv x hxu\n apply m y (disjoint_right.1 hsv hyv)\n have : ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y ∈ 𝒜 := by\n refine'\n sup_sdiff_mem_of_mem_compression (by rwa [hxy.eq]) _\n (disjoint_of_subset_left (erase_subset _ _) disjoint_sdiff)\n rw [union_sdiff_distrib, ‹v \\ u = v›]\n exact (erase_subset _ _).trans (subset_union_right _ _)\n convert this using 1\n rw [sdiff_union_erase_cancel (hus.trans <| subset_union_left _ _) ‹x ∈ u›, erase_union_distrib,\n erase_insert ‹x ∉ s›, erase_eq_of_not_mem ‹x ∉ v›, sdiff_erase (mem_union_right _ hyv),\n union_sdiff_cancel_right hsv]"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "have hus : u ⊆ s := by rwa [← erase_eq_of_not_mem ‹x ∉ u›, ← subset_insert_iff]"
},
{
"state_after": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ u ⊆ s ∧ Disjoint v s ∧ (∃ a x, insert a ((s ∪ v) \\ u) ∈ 𝒜) ∧ ¬∃ a x, insert a ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'",
"tactic": "simp_rw [mem_shadow_iff_insert_mem]"
},
{
"state_after": "case H.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ ¬x ∈ (s ∪ v) \\ u\n\ncase H.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ insert x ((s ∪ v) \\ u) ∈ 𝒜\n\ncase H.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ ¬∃ a x, insert a ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜",
"state_before": "case H.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ u ⊆ s ∧ Disjoint v s ∧ (∃ a x, insert a ((s ∪ v) \\ u) ∈ 𝒜) ∧ ¬∃ a x, insert a ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜",
"tactic": "refine' ⟨hus, hsv.symm, ⟨x, _, _⟩, _⟩"
},
{
"state_after": "case H.intro.intro.refine'_3.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\n⊢ False",
"state_before": "case H.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ ¬∃ a x, insert a ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜",
"tactic": "rintro ⟨w, hwB, hw𝒜'⟩"
},
{
"state_after": "case H.intro.intro.refine'_3.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\n⊢ False",
"state_before": "case H.intro.intro.refine'_3.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\n⊢ False",
"tactic": "have : v ⊆ insert w ((s ∪ v) \\ u) :=\n (subset_sdiff.2 ⟨subset_union_right _ _, hvu⟩).trans (subset_insert _ _)"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\n⊢ False\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\n⊢ False",
"state_before": "case H.intro.intro.refine'_3.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\n⊢ False",
"tactic": "by_cases hwu : w ∈ u"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\n⊢ False",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\n⊢ False",
"tactic": "rw [mem_sdiff, ← not_imp, Classical.not_not] at hwB"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\n⊢ insert w s ∈ 𝒜",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\n⊢ False",
"tactic": "apply m w (hwu ∘ hwB ∘ mem_union_left _)"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\nthis : (insert w ((s ∪ v) \\ u) ∪ u) \\ v ∈ 𝒜\n⊢ insert w s ∈ 𝒜",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\n⊢ insert w s ∈ 𝒜",
"tactic": "have : (insert w ((s ∪ v) \\ u) ∪ u) \\ v ∈ 𝒜 :=\n sup_sdiff_mem_of_mem_compression ‹insert w ((s ∪ v) \\ u) ∈ 𝒜'› ‹_›\n (disjoint_insert_right.2 ⟨‹_›, disjoint_sdiff⟩)"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\nthis : (insert w ((s ∪ v) \\ u) ∪ u) \\ v ∈ 𝒜\n⊢ insert w s = (insert w ((s ∪ v) \\ u) ∪ u) \\ v",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\nthis : (insert w ((s ∪ v) \\ u) ∪ u) \\ v ∈ 𝒜\n⊢ insert w s ∈ 𝒜",
"tactic": "convert this using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : w ∈ s ∪ v → w ∈ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : ¬w ∈ u\nthis : (insert w ((s ∪ v) \\ u) ∪ u) \\ v ∈ 𝒜\n⊢ insert w s = (insert w ((s ∪ v) \\ u) ∪ u) \\ v",
"tactic": "rw [insert_union, sdiff_union_of_subset (hus.trans <| subset_union_left _ _),\n insert_sdiff_of_not_mem _ (hwu ∘ hwB ∘ mem_union_right _), union_sdiff_cancel_right hsv]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\n⊢ s ∈ 𝓒 u v ((∂ ) 𝒜)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\n⊢ (∂ ) 𝒜' ⊆ 𝓒 u v ((∂ ) 𝒜)",
"tactic": "rintro s hs'"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\n⊢ s ∈ 𝓒 u v ((∂ ) 𝒜)",
"tactic": "rw [mem_compression]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : ¬s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"tactic": "by_cases hs : s ∈ 𝒜.shadow"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : ¬s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s\n\ncase pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : ¬s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"tactic": "swap"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ compress u v s ∈ (∂ ) 𝒜",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"tactic": "refine' Or.inl ⟨hs, _⟩"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ (if Disjoint u s ∧ v ≤ s then (s ⊔ u) \\ v else s) ∈ (∂ ) 𝒜",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ compress u v s ∈ (∂ ) 𝒜",
"tactic": "rw [compress]"
},
{
"state_after": "case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : Disjoint u s ∧ v ≤ s\n⊢ (s ⊔ u) \\ v ∈ (∂ ) 𝒜\n\ncase pos.inr\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : ¬(Disjoint u s ∧ v ≤ s)\n⊢ s ∈ (∂ ) 𝒜",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\n⊢ (if Disjoint u s ∧ v ≤ s then (s ⊔ u) \\ v else s) ∈ (∂ ) 𝒜",
"tactic": "split_ifs with huvs"
},
{
"state_after": "case pos.inr\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : ¬(Disjoint u s ∧ v ≤ s)\n⊢ s ∈ (∂ ) 𝒜\n\ncase pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : Disjoint u s ∧ v ≤ s\n⊢ (s ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : Disjoint u s ∧ v ≤ s\n⊢ (s ⊔ u) \\ v ∈ (∂ ) 𝒜\n\ncase pos.inr\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : ¬(Disjoint u s ∧ v ≤ s)\n⊢ s ∈ (∂ ) 𝒜",
"tactic": "swap"
},
{
"state_after": "case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : ∃ t, t ∈ 𝒜' ∧ ∃ a, a ∈ t ∧ erase t a = s\nhs : s ∈ (∂ ) 𝒜\nhuvs : Disjoint u s ∧ v ≤ s\n⊢ (s ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : Disjoint u s ∧ v ≤ s\n⊢ (s ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "rw [mem_shadow_iff] at hs'"
},
{
"state_after": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : ∃ t, t ∈ 𝒜' ∧ ∃ a, a ∈ t ∧ erase t a = s\nhs : s ∈ (∂ ) 𝒜\nhuvs : Disjoint u s ∧ v ≤ s\n⊢ (s ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "obtain ⟨t, Ht, a, hat, rfl⟩ := hs'"
},
{
"state_after": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "have hav : a ∉ v := not_mem_mono huvs.2 (not_mem_erase a t)"
},
{
"state_after": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "have hvt : v ≤ t := huvs.2.trans (erase_subset _ t)"
},
{
"state_after": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "have ht : t ∈ 𝒜 := mem_of_mem_compression Ht hvt (aux huv)"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : a ∈ u\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos.inl.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "by_cases hau : a ∈ u"
},
{
"state_after": "case neg.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : ¬s ∈ (∂ ) 𝒜\nhus : u ⊆ s\nhvs : Disjoint v s\nh : (s ∪ v) \\ u ∈ (∂ ) 𝒜\nright✝ : ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : ¬s ∈ (∂ ) 𝒜\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"tactic": "obtain ⟨hus, hvs, h, _⟩ := H _ hs' hs"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : ¬s ∈ (∂ ) 𝒜\nhus : u ⊆ s\nhvs : Disjoint v s\nh : (s ∪ v) \\ u ∈ (∂ ) 𝒜\nright✝ : ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\n⊢ s ∈ (∂ ) 𝒜 ∧ compress u v s ∈ (∂ ) 𝒜 ∨ ¬s ∈ (∂ ) 𝒜 ∧ ∃ b, b ∈ (∂ ) 𝒜 ∧ compress u v b = s",
"tactic": "exact Or.inr ⟨hs, _, h, compress_of_disjoint_of_le' hvs hus⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.inr\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\ns : Finset α\nhs' : s ∈ (∂ ) 𝒜'\nhs : s ∈ (∂ ) 𝒜\nhuvs : ¬(Disjoint u s ∧ v ≤ s)\n⊢ s ∈ (∂ ) 𝒜",
"tactic": "exact hs"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : a ∈ u\nb : α\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : a ∈ u\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "obtain ⟨b, hbv, Hcomp⟩ := huv a hau"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : a ∈ u\nb : α\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\n⊢ insert b ((erase t a ⊔ u) \\ v) ∈ 𝒜",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : a ∈ u\nb : α\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "refine' mem_shadow_iff_insert_mem.2 ⟨b, not_mem_sdiff_of_mem_right hbv, _⟩"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nhau : a ∈ u\nb : α\nht : t ∈ 𝓒 (erase u a) (erase v b) 𝒜\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\n⊢ insert b ((erase t a ⊔ u) \\ v) ∈ 𝒜",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : a ∈ u\nb : α\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\n⊢ insert b ((erase t a ⊔ u) \\ v) ∈ 𝒜",
"tactic": "rw [← Hcomp.eq] at ht"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nhau : a ∈ u\nb : α\nht : t ∈ 𝓒 (erase u a) (erase v b) 𝒜\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\nhsb : (t ⊔ erase u a) \\ erase v b ∈ 𝒜\n⊢ insert b ((erase t a ⊔ u) \\ v) ∈ 𝒜",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nhau : a ∈ u\nb : α\nht : t ∈ 𝓒 (erase u a) (erase v b) 𝒜\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\n⊢ insert b ((erase t a ⊔ u) \\ v) ∈ 𝒜",
"tactic": "have hsb :=\n sup_sdiff_mem_of_mem_compression ht ((erase_subset _ _).trans hvt)\n (disjoint_erase_comm.2 huvs.1)"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nhau : a ∈ u\nb : α\nht : t ∈ 𝓒 (erase u a) (erase v b) 𝒜\nhbv : b ∈ v\nHcomp : IsCompressed (erase u a) (erase v b) 𝒜\nhsb : (t ⊔ erase u a) \\ erase v b ∈ 𝒜\n⊢ insert b ((erase t a ⊔ u) \\ v) ∈ 𝒜",
"tactic": "rwa [sup_eq_union, sdiff_erase (mem_union_left _ <| hvt hbv), union_erase_of_mem hat, ←\n erase_union_of_mem hau] at hsb"
},
{
"state_after": "case neg.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ a ∈ (t ⊔ u) \\ v\n\ncase neg.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ erase ((t ⊔ u) \\ v) a = (erase t a ⊔ u) \\ v",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ (erase t a ⊔ u) \\ v ∈ (∂ ) 𝒜",
"tactic": "refine'\n mem_shadow_iff.2\n ⟨(t ⊔ u) \\ v,\n sup_sdiff_mem_of_mem_compression Ht hvt <| disjoint_of_erase_right hau huvs.1, a, _, _⟩"
},
{
"state_after": "case neg.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ (a ∈ t ∨ a ∈ u) ∧ ¬a ∈ v",
"state_before": "case neg.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ a ∈ (t ⊔ u) \\ v",
"tactic": "rw [sup_eq_union, mem_sdiff, mem_union]"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ (a ∈ t ∨ a ∈ u) ∧ ¬a ∈ v",
"tactic": "exact ⟨Or.inl hat, hav⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a✝ u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ (s : Finset α), s ∈ (∂ ) 𝒜' → ¬s ∈ (∂ ) 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ (∂ ) 𝒜 ∧ ¬(s ∪ v) \\ u ∈ (∂ ) 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na : α\nhat : a ∈ t\nhs : erase t a ∈ (∂ ) 𝒜\nhuvs : Disjoint u (erase t a) ∧ v ≤ erase t a\nhav : ¬a ∈ v\nhvt : v ≤ t\nht : t ∈ 𝒜\nhau : ¬a ∈ u\n⊢ erase ((t ⊔ u) \\ v) a = (erase t a ⊔ u) \\ v",
"tactic": "rw [← erase_sdiff_comm, sup_eq_union, erase_union_distrib, erase_eq_of_not_mem hau]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\n⊢ False",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\n⊢ ¬x ∈ u",
"tactic": "intro hxu"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ False",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\n⊢ False",
"tactic": "obtain ⟨y, hyv, hxy⟩ := huv x hxu"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ insert y s ∈ 𝒜",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ False",
"tactic": "apply m y (disjoint_right.1 hsv hyv)"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\nthis : ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y ∈ 𝒜\n⊢ insert y s ∈ 𝒜",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ insert y s ∈ 𝒜",
"tactic": "have : ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y ∈ 𝒜 := by\n refine'\n sup_sdiff_mem_of_mem_compression (by rwa [hxy.eq]) _\n (disjoint_of_subset_left (erase_subset _ _) disjoint_sdiff)\n rw [union_sdiff_distrib, ‹v \\ u = v›]\n exact (erase_subset _ _).trans (subset_union_right _ _)"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\nthis : ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y ∈ 𝒜\n⊢ insert y s = ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\nthis : ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y ∈ 𝒜\n⊢ insert y s ∈ 𝒜",
"tactic": "convert this using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\nthis : ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y ∈ 𝒜\n⊢ insert y s = ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y",
"tactic": "rw [sdiff_union_erase_cancel (hus.trans <| subset_union_left _ _) ‹x ∈ u›, erase_union_distrib,\n erase_insert ‹x ∉ s›, erase_eq_of_not_mem ‹x ∉ v›, sdiff_erase (mem_union_right _ hyv),\n union_sdiff_cancel_right hsv]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ erase v y ≤ (insert x s ∪ v) \\ u",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ ((insert x s ∪ v) \\ u ∪ erase u x) \\ erase v y ∈ 𝒜",
"tactic": "refine'\n sup_sdiff_mem_of_mem_compression (by rwa [hxy.eq]) _\n (disjoint_of_subset_left (erase_subset _ _) disjoint_sdiff)"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ erase v y ≤ insert x s \\ u ∪ v",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ erase v y ≤ (insert x s ∪ v) \\ u",
"tactic": "rw [union_sdiff_distrib, ‹v \\ u = v›]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ erase v y ≤ insert x s \\ u ∪ v",
"tactic": "exact (erase_subset _ _).trans (subset_union_right _ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis : v \\ u = v\nhxu : x ∈ u\ny : α\nhyv : y ∈ v\nhxy : IsCompressed (erase u x) (erase v y) 𝒜\n⊢ (insert x s ∪ v) \\ u ∈ 𝓒 (erase u x) (erase v y) 𝒜",
"tactic": "rwa [hxy.eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\n⊢ u ⊆ s",
"tactic": "rwa [← erase_eq_of_not_mem ‹x ∉ u›, ← subset_insert_iff]"
},
{
"state_after": "no goals",
"state_before": "case H.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ ¬x ∈ (s ∪ v) \\ u",
"tactic": "exact not_mem_sdiff_of_not_mem_left (not_mem_union.2 ⟨‹x ∉ s›, ‹x ∉ v›⟩)"
},
{
"state_after": "no goals",
"state_before": "case H.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝¹ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝ : v \\ u = v\nthis : ¬x ∈ u\nhus : u ⊆ s\n⊢ insert x ((s ∪ v) \\ u) ∈ 𝒜",
"tactic": "rwa [← insert_sdiff_of_not_mem _ ‹x ∉ u›, ← insert_union]"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\n⊢ False",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\n⊢ False",
"tactic": "obtain ⟨z, hz, hxy⟩ := huv w hwu"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\n⊢ insert z s ∈ 𝒜",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\n⊢ False",
"tactic": "apply m z (disjoint_right.1 hsv hz)"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ insert z s ∈ 𝒜",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝² : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝¹ : v \\ u = v\nthis✝ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\n⊢ insert z s ∈ 𝒜",
"tactic": "have : insert w ((s ∪ v) \\ u) ∈ 𝒜 := mem_of_mem_compression hw𝒜' ‹_› (aux huv)"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝⁴ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝³ : v \\ u = v\nthis✝² : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝¹ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis✝ : insert w ((s ∪ v) \\ u) ∈ 𝒜\nthis : (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z ∈ 𝒜\n⊢ insert z s ∈ 𝒜",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ insert z s ∈ 𝒜",
"tactic": "have : (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z ∈ 𝒜 := by\n refine' sup_sdiff_mem_of_mem_compression (by rwa [hxy.eq]) ((erase_subset _ _).trans ‹_›) _\n rw [← sdiff_erase (mem_union_left _ <| hus hwu)]\n exact disjoint_sdiff"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝⁴ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝³ : v \\ u = v\nthis✝² : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝¹ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis✝ : insert w ((s ∪ v) \\ u) ∈ 𝒜\nthis : (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z ∈ 𝒜\n⊢ insert z s = (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝⁴ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝³ : v \\ u = v\nthis✝² : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝¹ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis✝ : insert w ((s ∪ v) \\ u) ∈ 𝒜\nthis : (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z ∈ 𝒜\n⊢ insert z s ∈ 𝒜",
"tactic": "convert this using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝⁴ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝³ : v \\ u = v\nthis✝² : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝¹ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis✝ : insert w ((s ∪ v) \\ u) ∈ 𝒜\nthis : (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z ∈ 𝒜\n⊢ insert z s = (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z",
"tactic": "rw [insert_union_comm, insert_erase ‹w ∈ u›,\n sdiff_union_of_subset (hus.trans $ subset_union_left _ _),\n sdiff_erase (mem_union_right _ ‹z ∈ v›), union_sdiff_cancel_right hsv]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ Disjoint (erase u w) (insert w ((s ∪ v) \\ u))",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ (insert w ((s ∪ v) \\ u) ∪ erase u w) \\ erase v z ∈ 𝒜",
"tactic": "refine' sup_sdiff_mem_of_mem_compression (by rwa [hxy.eq]) ((erase_subset _ _).trans ‹_›) _"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ Disjoint (erase u w) ((s ∪ v) \\ erase u w)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ Disjoint (erase u w) (insert w ((s ∪ v) \\ u))",
"tactic": "rw [← sdiff_erase (mem_union_left _ <| hus hwu)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ Disjoint (erase u w) ((s ∪ v) \\ erase u w)",
"tactic": "exact disjoint_sdiff"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu✝ v✝ a u v : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ (∂ ) 𝒜'\nhs𝒜 : ¬s ∈ (∂ ) 𝒜\nm : ∀ (y : α), ¬y ∈ s → ¬insert y s ∈ 𝒜\nx : α\nw✝ : ¬x ∈ s\nh✝ : insert x s ∈ 𝒜'\nhus✝ : u ⊆ insert x s\nhvs : Disjoint v (insert x s)\nthis✝³ : (insert x s ∪ v) \\ u ∈ 𝒜\nhsv : Disjoint s v\nhvu : Disjoint v u\nhxv : ¬x ∈ v\nthis✝² : v \\ u = v\nthis✝¹ : ¬x ∈ u\nhus : u ⊆ s\nw : α\nhwB : ¬w ∈ (s ∪ v) \\ u\nhw𝒜' : insert w ((s ∪ v) \\ u) ∈ 𝓒 u v 𝒜\nthis✝ : v ⊆ insert w ((s ∪ v) \\ u)\nhwu : w ∈ u\nz : α\nhz : z ∈ v\nhxy : IsCompressed (erase u w) (erase v z) 𝒜\nthis : insert w ((s ∪ v) \\ u) ∈ 𝒜\n⊢ insert w ((s ∪ v) \\ u) ∈ 𝓒 (erase u w) (erase v z) 𝒜",
"tactic": "rwa [hxy.eq]"
}
] | [
426,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
324,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean | HasFDerivAt.const_smul | [] | [
83,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
8
] |
Mathlib/Data/Rat/Defs.lean | Rat.num_den | [] | [
106,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | PrimeSpectrum.comap_singleton_isClosed_of_isIntegral | [] | [
643,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
638,
1
] |
Mathlib/Data/ZMod/Basic.lean | ZMod.valMinAbs_mul_two_eq_iff | [
{
"state_after": "case zero\na : ZMod Nat.zero\n⊢ valMinAbs a * 2 = ↑Nat.zero ↔ 2 * val a = Nat.zero\n\ncase succ\nn : ℕ\na : ZMod (Nat.succ n)\n⊢ valMinAbs a * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n",
"state_before": "n : ℕ\na : ZMod n\n⊢ valMinAbs a * 2 = ↑n ↔ 2 * val a = n",
"tactic": "cases' n with n"
},
{
"state_after": "case pos\nn : ℕ\na : ZMod (Nat.succ n)\nh : val a ≤ Nat.succ n / 2\n⊢ valMinAbs a * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n\n\ncase neg\nn : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ valMinAbs a * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n",
"state_before": "case succ\nn : ℕ\na : ZMod (Nat.succ n)\n⊢ valMinAbs a * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n",
"tactic": "by_cases h : a.val ≤ n.succ / 2"
},
{
"state_after": "n : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ ¬valMinAbs a * 2 = ↑(Nat.succ n)\n\nn : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ 2 * val a = Nat.succ n → val a ≤ Nat.succ n / 2",
"state_before": "case neg\nn : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ valMinAbs a * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n",
"tactic": "apply iff_of_false _ (mt _ h)"
},
{
"state_after": "no goals",
"state_before": "case zero\na : ZMod Nat.zero\n⊢ valMinAbs a * 2 = ↑Nat.zero ↔ 2 * val a = Nat.zero",
"tactic": "simp"
},
{
"state_after": "case pos\nn : ℕ\na : ZMod (Nat.succ n)\nh : val a ≤ Nat.succ n / 2\n⊢ (if val a ≤ Nat.succ n / 2 then ↑(val a) else ↑(val a) - ↑(Nat.succ n)) * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n",
"state_before": "case pos\nn : ℕ\na : ZMod (Nat.succ n)\nh : val a ≤ Nat.succ n / 2\n⊢ valMinAbs a * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n",
"tactic": "dsimp [valMinAbs]"
},
{
"state_after": "no goals",
"state_before": "case pos\nn : ℕ\na : ZMod (Nat.succ n)\nh : val a ≤ Nat.succ n / 2\n⊢ (if val a ≤ Nat.succ n / 2 then ↑(val a) else ↑(val a) - ↑(Nat.succ n)) * 2 = ↑(Nat.succ n) ↔ 2 * val a = Nat.succ n",
"tactic": "rw [if_pos h, ← Int.coe_nat_inj', Nat.cast_mul, Nat.cast_two, mul_comm]"
},
{
"state_after": "n : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\nhe : valMinAbs a * 2 = ↑(Nat.succ n)\n⊢ False",
"state_before": "n : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ ¬valMinAbs a * 2 = ↑(Nat.succ n)",
"tactic": "intro he"
},
{
"state_after": "n : ℕ\na : ZMod (Nat.succ n)\nh✝ : ¬0 ≤ valMinAbs a\nh : ¬0 ≤ ↑(Nat.succ n)\nhe : valMinAbs a * 2 = ↑(Nat.succ n)\n⊢ False\n\nn : ℕ\na : ZMod (Nat.succ n)\nh : ¬0 ≤ valMinAbs a\nhe : valMinAbs a * 2 = ↑(Nat.succ n)\n⊢ 0 < 2",
"state_before": "n : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\nhe : valMinAbs a * 2 = ↑(Nat.succ n)\n⊢ False",
"tactic": "rw [← a.valMinAbs_nonneg_iff, ← mul_nonneg_iff_left_nonneg_of_pos, he] at h"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ZMod (Nat.succ n)\nh✝ : ¬0 ≤ valMinAbs a\nh : ¬0 ≤ ↑(Nat.succ n)\nhe : valMinAbs a * 2 = ↑(Nat.succ n)\n⊢ False\n\nn : ℕ\na : ZMod (Nat.succ n)\nh : ¬0 ≤ valMinAbs a\nhe : valMinAbs a * 2 = ↑(Nat.succ n)\n⊢ 0 < 2",
"tactic": "exacts [h (Nat.cast_nonneg _), zero_lt_two]"
},
{
"state_after": "n : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ val a * 2 = Nat.succ n → val a ≤ Nat.succ n / 2",
"state_before": "n : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ 2 * val a = Nat.succ n → val a ≤ Nat.succ n / 2",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ZMod (Nat.succ n)\nh : ¬val a ≤ Nat.succ n / 2\n⊢ val a * 2 = Nat.succ n → val a ≤ Nat.succ n / 2",
"tactic": "exact fun h => (Nat.le_div_iff_mul_le zero_lt_two).2 h.le"
}
] | [
952,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
941,
1
] |
Mathlib/RingTheory/Ideal/Over.lean | Ideal.coeff_zero_mem_comap_of_root_mem | [] | [
56,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean | MeasureTheory.integral_norm_eq_pos_sub_neg | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ (∫ (x : α), ‖f x‖ ∂μ) = (∫ (x : α) in {x | 0 ≤ f x}, ‖f x‖ ∂μ) + ∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ",
"tactic": "rw [← integral_add_compl₀ h_meas hfi.norm]"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ (∫ (x : α) in {x | 0 ≤ f x}, ‖f x‖ ∂μ) = ∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ ((∫ (x : α) in {x | 0 ≤ f x}, ‖f x‖ ∂μ) + ∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ) =\n (∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) + ∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}\n⊢ ‖f x‖ = f x",
"state_before": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ (∫ (x : α) in {x | 0 ≤ f x}, ‖f x‖ ∂μ) = ∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ",
"tactic": "refine' set_integral_congr₀ h_meas fun x hx => _"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}\n⊢ ‖f x‖ = f x",
"state_before": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}\n⊢ ‖f x‖ = f x",
"tactic": "dsimp only"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}\n⊢ 0 ≤ f x",
"state_before": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}\n⊢ ‖f x‖ = f x",
"tactic": "rw [Real.norm_eq_abs, abs_eq_self.mpr _]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}\n⊢ 0 ≤ f x",
"tactic": "exact hx"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ (∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ) = -∫ (x : α) in {x | 0 ≤ f x}ᶜ, f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ ((∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) + ∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ) =\n (∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) - ∫ (x : α) in {x | 0 ≤ f x}ᶜ, f x ∂μ",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ (∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ) = ∫ (a : α) in {x | 0 ≤ f x}ᶜ, -f a ∂μ",
"state_before": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ (∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ) = -∫ (x : α) in {x | 0 ≤ f x}ᶜ, f x ∂μ",
"tactic": "rw [← integral_neg]"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}ᶜ\n⊢ ‖f x‖ = -f x",
"state_before": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ (∫ (x : α) in {x | 0 ≤ f x}ᶜ, ‖f x‖ ∂μ) = ∫ (a : α) in {x | 0 ≤ f x}ᶜ, -f a ∂μ",
"tactic": "refine' set_integral_congr₀ h_meas.compl fun x hx => _"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}ᶜ\n⊢ ‖f x‖ = -f x",
"state_before": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}ᶜ\n⊢ ‖f x‖ = -f x",
"tactic": "dsimp only"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}ᶜ\n⊢ f x ≤ 0",
"state_before": "case e_a\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}ᶜ\n⊢ ‖f x‖ = -f x",
"tactic": "rw [Real.norm_eq_abs, abs_eq_neg_self.mpr _]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : ¬0 ≤ f x\n⊢ f x ≤ 0",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : x ∈ {x | 0 ≤ f x}ᶜ\n⊢ f x ≤ 0",
"tactic": "rw [Set.mem_compl_iff, Set.nmem_setOf_iff] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\nhx : ¬0 ≤ f x\n⊢ f x ≤ 0",
"tactic": "linarith"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ ((∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) - ∫ (x : α) in {x | 0 ≤ f x}ᶜ, f x ∂μ) =\n (∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) - ∫ (x : α) in {x | f x < 0}, f x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ ((∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) - ∫ (x : α) in {x | 0 ≤ f x}ᶜ, f x ∂μ) =\n (∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) - ∫ (x : α) in {x | f x ≤ 0}, f x ∂μ",
"tactic": "rw [← set_integral_neg_eq_set_integral_nonpos hfi.1]"
},
{
"state_after": "case e_a.e_μ.e_s\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ {x | 0 ≤ f x}ᶜ = {x | f x < 0}",
"state_before": "α : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ ((∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) - ∫ (x : α) in {x | 0 ≤ f x}ᶜ, f x ∂μ) =\n (∫ (x : α) in {x | 0 ≤ f x}, f x ∂μ) - ∫ (x : α) in {x | f x < 0}, f x ∂μ",
"tactic": "congr"
},
{
"state_after": "case e_a.e_μ.e_s.h\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\n⊢ x ∈ {x | 0 ≤ f x}ᶜ ↔ x ∈ {x | f x < 0}",
"state_before": "case e_a.e_μ.e_s\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\n⊢ {x | 0 ≤ f x}ᶜ = {x | f x < 0}",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_μ.e_s.h\nα : Type u_1\nβ : Type ?u.110031\nE : Type ?u.110034\nF : Type ?u.110037\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf✝ g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ\nhfi : Integrable f\nh_meas : NullMeasurableSet {x | 0 ≤ f x}\nx : α\n⊢ x ∈ {x | 0 ≤ f x}ᶜ ↔ x ∈ {x | f x < 0}",
"tactic": "simp"
}
] | [
444,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
422,
1
] |
src/lean/Init/Data/List/Basic.lean | List.reverse_append | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\n⊢ reverse (as ++ bs) = reverse bs ++ reverse as",
"tactic": "induction as generalizing bs with\n| nil => simp\n| cons a as ih => simp [ih]; rw [append_assoc]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nbs : List α\n⊢ reverse (nil ++ bs) = reverse bs ++ reverse nil",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas : List α\nih : ∀ (bs : List α), reverse (as ++ bs) = reverse bs ++ reverse as\nbs : List α\n⊢ reverse bs ++ reverse as ++ a :: nil = reverse bs ++ (reverse as ++ a :: nil)",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas : List α\nih : ∀ (bs : List α), reverse (as ++ bs) = reverse bs ++ reverse as\nbs : List α\n⊢ reverse (a :: as ++ bs) = reverse bs ++ reverse (a :: as)",
"tactic": "simp [ih]"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nas : List α\nih : ∀ (bs : List α), reverse (as ++ bs) = reverse bs ++ reverse as\nbs : List α\n⊢ reverse bs ++ reverse as ++ a :: nil = reverse bs ++ (reverse as ++ a :: nil)",
"tactic": "rw [append_assoc]"
}
] | [
181,
49
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
178,
9
] |
Mathlib/Data/List/Nodup.lean | List.Nodup.not_mem_erase | [] | [
329,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
328,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | strictConcaveOn_sqrt_mul_log_Ioi | [
{
"state_after": "⊢ ContinuousOn (fun x => sqrt x * log x) (Ioi 1)\n\nx : ℝ\nhx : x ∈ Ioi 1\n⊢ (deriv^[2]) (fun x => sqrt x * log x) x < 0",
"state_before": "⊢ StrictConcaveOn ℝ (Ioi 1) fun x => sqrt x * log x",
"tactic": "apply strictConcaveOn_of_deriv2_neg' (convex_Ioi 1) _ fun x hx => ?_"
},
{
"state_after": "no goals",
"state_before": "⊢ ContinuousOn (fun x => sqrt x * log x) (Ioi 1)",
"tactic": "exact continuous_sqrt.continuousOn.mul\n (continuousOn_log.mono fun x hx => ne_of_gt (zero_lt_one.trans hx))"
},
{
"state_after": "x : ℝ\nhx : x ∈ Ioi 1\n⊢ -log x / (↑4 * sqrt x ^ 3) < 0",
"state_before": "x : ℝ\nhx : x ∈ Ioi 1\n⊢ (deriv^[2]) (fun x => sqrt x * log x) x < 0",
"tactic": "rw [deriv2_sqrt_mul_log x]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ∈ Ioi 1\n⊢ -log x / (↑4 * sqrt x ^ 3) < 0",
"tactic": "exact div_neg_of_neg_of_pos (neg_neg_of_pos (log_pos hx))\n (mul_pos four_pos (pow_pos (sqrt_pos.mpr (zero_lt_one.trans hx)) 3))"
}
] | [
163,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean | DifferentiableAt.inner | [] | [
135,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | tsum_pi_single | [] | [
542,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/Algebra/Periodic.lean | Function.Periodic.const_add | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82300\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝ : AddSemigroup α\nh : Periodic f c\na x : α\n⊢ (fun x => f (a + x)) (x + c) = (fun x => f (a + x)) x",
"tactic": "simpa [add_assoc] using h (a + x)"
}
] | [
183,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.congr_fun | [] | [
158,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Analysis/SpecialFunctions/Stirling.lean | Stirling.stirlingSeq'_bounded_by_pos_constant | [
{
"state_after": "case intro\nc : ℝ\nh : ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))\n⊢ ∃ a, 0 < a ∧ ∀ (n : ℕ), a ≤ stirlingSeq (succ n)",
"state_before": "⊢ ∃ a, 0 < a ∧ ∀ (n : ℕ), a ≤ stirlingSeq (succ n)",
"tactic": "cases' log_stirlingSeq_bounded_by_constant with c h"
},
{
"state_after": "case intro\nc : ℝ\nh : ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))\nn : ℕ\n⊢ exp c ≤ stirlingSeq (succ n)",
"state_before": "case intro\nc : ℝ\nh : ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))\n⊢ ∃ a, 0 < a ∧ ∀ (n : ℕ), a ≤ stirlingSeq (succ n)",
"tactic": "refine' ⟨exp c, exp_pos _, fun n => _⟩"
},
{
"state_after": "case intro\nc : ℝ\nh : ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))\nn : ℕ\n⊢ c ≤ Real.log (stirlingSeq (succ n))",
"state_before": "case intro\nc : ℝ\nh : ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))\nn : ℕ\n⊢ exp c ≤ stirlingSeq (succ n)",
"tactic": "rw [← le_log_iff_exp_le (stirlingSeq'_pos n)]"
},
{
"state_after": "no goals",
"state_before": "case intro\nc : ℝ\nh : ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (succ n))\nn : ℕ\n⊢ c ≤ Real.log (stirlingSeq (succ n))",
"tactic": "exact h n"
}
] | [
196,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/Data/List/Basic.lean | List.bind_eq_bind | [] | [
532,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
531,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | uniformity_prod_eq_prod | [
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.154039\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\n⊢ 𝓤 (α × β) = map (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) (𝓤 α ×ˢ 𝓤 β)",
"tactic": "rw [map_swap4_eq_comap, uniformity_prod_eq_comap_prod]"
}
] | [
1583,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1581,
1
] |
Mathlib/Data/Set/Sups.lean | Set.infs_comm | [] | [
354,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
1
] |
Mathlib/ModelTheory/Types.lean | FirstOrder.Language.Theory.CompleteType.mem_typeOf | [] | [
194,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.mul_div_cancel_left | [
{
"state_after": "no goals",
"state_before": "m n : Nat\nH : 0 < n\n⊢ n * m / n = m",
"tactic": "rw [Nat.mul_comm, Nat.mul_div_cancel _ H]"
}
] | [
593,
45
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
592,
11
] |
Mathlib/CategoryTheory/Abelian/Exact.lean | CategoryTheory.Abelian.exact_tfae | [
{
"state_after": "case tfae_1_iff_2\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ntfae_1_iff_2 : Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ TFAE [Exact f g, f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0, imageSubobject f = kernelSubobject g]",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ TFAE [Exact f g, f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0, imageSubobject f = kernelSubobject g]",
"tactic": "tfae_have 1 ↔ 2"
},
{
"state_after": "case tfae_1_iff_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ntfae_1_iff_2 : Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ Exact f g ↔ imageSubobject f = kernelSubobject g\n\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ntfae_1_iff_2 : Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\ntfae_1_iff_3 : Exact f g ↔ imageSubobject f = kernelSubobject g\n⊢ TFAE [Exact f g, f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0, imageSubobject f = kernelSubobject g]",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ntfae_1_iff_2 : Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ TFAE [Exact f g, f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0, imageSubobject f = kernelSubobject g]",
"tactic": "tfae_have 1 ↔ 3"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ntfae_1_iff_2 : Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\ntfae_1_iff_3 : Exact f g ↔ imageSubobject f = kernelSubobject g\n⊢ TFAE [Exact f g, f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0, imageSubobject f = kernelSubobject g]",
"tactic": "tfae_finish"
},
{
"state_after": "no goals",
"state_before": "case tfae_1_iff_2\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0",
"tactic": "apply exact_iff"
},
{
"state_after": "no goals",
"state_before": "case tfae_1_iff_3\nC : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ntfae_1_iff_2 : Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0\n⊢ Exact f g ↔ imageSubobject f = kernelSubobject g",
"tactic": "apply exact_iff_image_eq_kernel"
}
] | [
106,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Std/Data/Int/Lemmas.lean | Int.le_neg_of_le_neg | [
{
"state_after": "a b : Int\nh✝ : a ≤ -b\nh : - -b ≤ -a\n⊢ b ≤ -a",
"state_before": "a b : Int\nh : a ≤ -b\n⊢ b ≤ -a",
"tactic": "have h := Int.neg_le_neg h"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh✝ : a ≤ -b\nh : - -b ≤ -a\n⊢ b ≤ -a",
"tactic": "rwa [Int.neg_neg] at h"
}
] | [
904,
25
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
902,
11
] |
Mathlib/Data/List/Basic.lean | List.insertNth_of_length_lt | [
{
"state_after": "case nil\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn✝ : ℕ\nh✝ : length l < n✝\nn : ℕ\nh : length [] < n\n⊢ insertNth n x [] = []\n\ncase cons\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn✝ : ℕ\nh✝ : length l < n✝\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nn : ℕ\nh : length (hd :: tl) < n\n⊢ insertNth n x (hd :: tl) = hd :: tl",
"state_before": "ι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh : length l < n\n⊢ insertNth n x l = l",
"tactic": "induction' l with hd tl IH generalizing n"
},
{
"state_after": "case nil.zero\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nh : length [] < zero\n⊢ insertNth zero x [] = []\n\ncase nil.succ\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nn✝ : ℕ\nh : length [] < succ n✝\n⊢ insertNth (succ n✝) x [] = []",
"state_before": "case nil\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn✝ : ℕ\nh✝ : length l < n✝\nn : ℕ\nh : length [] < n\n⊢ insertNth n x [] = []",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case nil.zero\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nh : length [] < zero\n⊢ insertNth zero x [] = []",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "case nil.succ\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nn✝ : ℕ\nh : length [] < succ n✝\n⊢ insertNth (succ n✝) x [] = []",
"tactic": "simp"
},
{
"state_after": "case cons.zero\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nh : length (hd :: tl) < zero\n⊢ insertNth zero x (hd :: tl) = hd :: tl\n\ncase cons.succ\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nn✝ : ℕ\nh : length (hd :: tl) < succ n✝\n⊢ insertNth (succ n✝) x (hd :: tl) = hd :: tl",
"state_before": "case cons\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn✝ : ℕ\nh✝ : length l < n✝\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nn : ℕ\nh : length (hd :: tl) < n\n⊢ insertNth n x (hd :: tl) = hd :: tl",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case cons.zero\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nh : length (hd :: tl) < zero\n⊢ insertNth zero x (hd :: tl) = hd :: tl",
"tactic": "simp at h"
},
{
"state_after": "case cons.succ\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nn✝ : ℕ\nh : length tl < n✝\n⊢ insertNth (succ n✝) x (hd :: tl) = hd :: tl",
"state_before": "case cons.succ\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nn✝ : ℕ\nh : length (hd :: tl) < succ n✝\n⊢ insertNth (succ n✝) x (hd :: tl) = hd :: tl",
"tactic": "simp only [Nat.succ_lt_succ_iff, length] at h"
},
{
"state_after": "no goals",
"state_before": "case cons.succ\nι : Type ?u.122867\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nh✝ : length l < n\nhd : α\ntl : List α\nIH : ∀ (n : ℕ), length tl < n → insertNth n x tl = tl\nn✝ : ℕ\nh : length tl < n✝\n⊢ insertNth (succ n✝) x (hd :: tl) = hd :: tl",
"tactic": "simpa using IH _ h"
}
] | [
1670,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1661,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineMap.homothety_neg_one_apply | [
{
"state_after": "no goals",
"state_before": "k : Type ?u.707847\nP₁ : Type u_1\nP₂ : Type ?u.707853\nP₃ : Type ?u.707856\nP₄ : Type ?u.707859\nV₁ : Type u_2\nV₂ : Type ?u.707865\nV₃ : Type ?u.707868\nV₄ : Type ?u.707871\ninst✝¹⁴ : Ring k\ninst✝¹³ : AddCommGroup V₁\ninst✝¹² : Module k V₁\ninst✝¹¹ : AffineSpace V₁ P₁\ninst✝¹⁰ : AddCommGroup V₂\ninst✝⁹ : Module k V₂\ninst✝⁸ : AffineSpace V₂ P₂\ninst✝⁷ : AddCommGroup V₃\ninst✝⁶ : Module k V₃\ninst✝⁵ : AffineSpace V₃ P₃\ninst✝⁴ : AddCommGroup V₄\ninst✝³ : Module k V₄\ninst✝² : AffineSpace V₄ P₄\nR' : Type u_3\ninst✝¹ : CommRing R'\ninst✝ : Module R' V₁\nc p : P₁\n⊢ ↑(homothety c (-1)) p = ↑(pointReflection R' c) p",
"tactic": "simp [(homothety_apply), pointReflection_apply _, (neg_vsub_eq_vsub_rev)]"
}
] | [
657,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
655,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.distribMulActionHom_ext | [] | [
1673,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1671,
1
] |
Mathlib/Init/Data/Int/Order.lean | Int.lt.elim | [] | [
38,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
37,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | csSup_empty | [] | [
1015,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1014,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one | [
{
"state_after": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), finrank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥ ↔ y ∈ ⊤",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), finrank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\n⊢ ⊥ = ⊤",
"tactic": "ext y"
},
{
"state_after": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), finrank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥",
"state_before": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), finrank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥ ↔ y ∈ ⊤",
"tactic": "rw [iff_true_right IntermediateField.mem_top]"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\nh : ∀ (x : E), finrank F { x_1 // x_1 ∈ F⟮x⟯ } = 1\ny : E\n⊢ y ∈ ⊥",
"tactic": "exact finrank_adjoin_simple_eq_one_iff.mp (h y)"
}
] | [
760,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
756,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.pow_apply_mem_of_forall_mem | [
{
"state_after": "case zero\nR : Type u_1\nR₁ : Type ?u.246811\nR₂ : Type ?u.246814\nR₃ : Type ?u.246817\nR₄ : Type ?u.246820\nS : Type ?u.246823\nK : Type ?u.246826\nK₂ : Type ?u.246829\nM : Type u_2\nM' : Type ?u.246835\nM₁ : Type ?u.246838\nM₂ : Type ?u.246841\nM₃ : Type ?u.246844\nM₄ : Type ?u.246847\nN : Type ?u.246850\nN₂ : Type ?u.246853\nι : Type ?u.246856\nV : Type ?u.246859\nV₂ : Type ?u.246862\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf' : M →ₗ[R] M\np : Submodule R M\nh : ∀ (x : M), x ∈ p → ↑f' x ∈ p\nx✝ : M\nhx✝ : x✝ ∈ p\nx : M\nhx : x ∈ p\n⊢ ↑(f' ^ Nat.zero) x ∈ p\n\ncase succ\nR : Type u_1\nR₁ : Type ?u.246811\nR₂ : Type ?u.246814\nR₃ : Type ?u.246817\nR₄ : Type ?u.246820\nS : Type ?u.246823\nK : Type ?u.246826\nK₂ : Type ?u.246829\nM : Type u_2\nM' : Type ?u.246835\nM₁ : Type ?u.246838\nM₂ : Type ?u.246841\nM₃ : Type ?u.246844\nM₄ : Type ?u.246847\nN : Type ?u.246850\nN₂ : Type ?u.246853\nι : Type ?u.246856\nV : Type ?u.246859\nV₂ : Type ?u.246862\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf' : M →ₗ[R] M\np : Submodule R M\nh : ∀ (x : M), x ∈ p → ↑f' x ∈ p\nx✝ : M\nhx✝ : x✝ ∈ p\nn : ℕ\nih : ∀ (x : M), x ∈ p → ↑(f' ^ n) x ∈ p\nx : M\nhx : x ∈ p\n⊢ ↑(f' ^ Nat.succ n) x ∈ p",
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Mathlib/LinearAlgebra/TensorProduct.lean | TensorProduct.congr_symm_tmul | [] | [
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Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.cycleType_mul_inv_mem_cycleFactorsFinset_eq_sub | [
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Mathlib/Order/Filter/Pointwise.lean | Filter.smul_le_smul_right | [] | [
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Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.parallel_bot_iff_eq_bot | [
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Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_mul_const' | [
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Mathlib/Data/Finset/Basic.lean | Finset.piecewise_congr | [] | [
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Mathlib/Data/Set/Basic.lean | Set.mem_powerset | [] | [
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Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.VectorMeasure.AbsolutelyContinuous.map | [
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"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nL : Type ?u.642970\nM : Type u_3\nN : Type u_4\ninst✝⁶ : AddCommMonoid L\ninst✝⁵ : TopologicalSpace L\ninst✝⁴ : AddCommMonoid M\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommMonoid N\ninst✝¹ : TopologicalSpace N\nv : VectorMeasure α M\nw : VectorMeasure α N\ninst✝ : MeasureSpace β\nh : v ≪ᵥ w\nf : α → β\nhf : ¬Measurable f\ns : Set β\na✝ : ↑(VectorMeasure.map w f) s = 0\n⊢ ↑(VectorMeasure.map v f) s = 0",
"tactic": "rw [map_not_measurable v hf, zero_apply]"
}
] | [
1152,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1146,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | GroupTopology.continuous_mul' | [
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis : TopologicalSpace α := g.toTopologicalSpace\n⊢ Continuous fun p => p.fst * p.snd",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "letI := g.toTopologicalSpace"
},
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis✝ : TopologicalSpace α := g.toTopologicalSpace\nthis : TopologicalGroup α\n⊢ Continuous fun p => p.fst * p.snd",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis : TopologicalSpace α := g.toTopologicalSpace\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "haveI := g.toTopologicalGroup"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝ : Group α\ng : GroupTopology α\nthis✝ : TopologicalSpace α := g.toTopologicalSpace\nthis : TopologicalGroup α\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "exact continuous_mul"
}
] | [
1857,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1852,
1
] |
Mathlib/CategoryTheory/ConnectedComponents.lean | CategoryTheory.inclusion_comp_decomposedTo | [] | [
132,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Computability/Primrec.lean | Primrec.and | [] | [
720,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
719,
11
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.mem_of_closed' | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.305337\nι : Type ?u.305340\ninst✝ : PseudoMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ\ns✝ s : Set α\nhs : IsClosed s\na : α\n⊢ a ∈ s ↔ ∀ (ε : ℝ), ε > 0 → ∃ b, b ∈ s ∧ dist a b < ε",
"tactic": "simpa only [hs.closure_eq] using @mem_closure_iff _ _ s a"
}
] | [
1915,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1913,
1
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Mathlib/Topology/Algebra/Group/Basic.lean | Continuous.inv | [] | [
243,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean | Set.IccExtend_eq_self | [
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\n⊢ IccExtend h (f ∘ Subtype.val) = f",
"tactic": "ext x"
},
{
"state_after": "case h.inl\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhxa : x < a\n⊢ IccExtend h (f ∘ Subtype.val) x = f x\n\ncase h.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"tactic": "cases' lt_or_le x a with hxa hax"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhxa : x < a\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"tactic": "simp [IccExtend_of_le_left _ _ hxa.le, ha x hxa]"
},
{
"state_after": "case h.inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\nhxb : x ≤ b\n⊢ IccExtend h (f ∘ Subtype.val) x = f x\n\ncase h.inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\nhbx : b < x\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"state_before": "case h.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"tactic": "cases' le_or_lt x b with hxb hbx"
},
{
"state_after": "case h.inr.inl.intro\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : { x // x ∈ Icc a b }\nhax : a ≤ ↑x\nhxb : ↑x ≤ b\n⊢ IccExtend h (f ∘ Subtype.val) ↑x = f ↑x",
"state_before": "case h.inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\nhxb : x ≤ b\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"tactic": "lift x to Icc a b using ⟨hax, hxb⟩"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inl.intro\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : { x // x ∈ Icc a b }\nhax : a ≤ ↑x\nhxb : ↑x ≤ b\n⊢ IccExtend h (f ∘ Subtype.val) ↑x = f ↑x",
"tactic": "rw [IccExtend_val, comp_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx✝ : α\nf : α → β\nha : ∀ (x : α), x < a → f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\nhbx : b < x\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"tactic": "simp [IccExtend_of_right_le _ _ hbx.le, hb x hbx]"
}
] | [
152,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.closure_eq | [] | [
984,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
983,
1
] |
Mathlib/Data/MvPolynomial/Rename.lean | MvPolynomial.rename_eq | [
{
"state_after": "σ : Type u_1\nτ : Type u_3\nα : Type ?u.245747\nR : Type u_2\nS : Type ?u.245753\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\np : MvPolynomial σ R\n⊢ (sum p fun s a => ↑(monomial (sum s fun a b => Finsupp.single (f a) b)) a) =\n sum p fun a => Finsupp.single (sum a fun a => Finsupp.single (f a))",
"state_before": "σ : Type u_1\nτ : Type u_3\nα : Type ?u.245747\nR : Type u_2\nS : Type ?u.245753\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\np : MvPolynomial σ R\n⊢ ↑(rename f) p = Finsupp.mapDomain (Finsupp.mapDomain f) p",
"tactic": "simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,\n X_pow_eq_monomial, ← monomial_finsupp_sum_index]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nτ : Type u_3\nα : Type ?u.245747\nR : Type u_2\nS : Type ?u.245753\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\np : MvPolynomial σ R\n⊢ (sum p fun s a => ↑(monomial (sum s fun a b => Finsupp.single (f a) b)) a) =\n sum p fun a => Finsupp.single (sum a fun a => Finsupp.single (f a))",
"tactic": "rfl"
}
] | [
114,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Order/CompleteLattice.lean | iInf_const_mono | [] | [
885,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
884,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | MeasureTheory.Lp.simpleFunc.denseInducing | [] | [
791,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
789,
11
] |
Mathlib/Order/LiminfLimsup.lean | Filter.OrderIso.apply_bliminf | [] | [
959,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
957,
1
] |
Std/Logic.lean | or_comm | [] | [
263,
43
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
263,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | NNReal.hasSum_iff_tendsto_nat | [
{
"state_after": "α : Type ?u.312215\nβ : Type ?u.312218\nγ : Type ?u.312221\nf : ℕ → ℝ≥0\nr : ℝ≥0\n⊢ Tendsto (fun n => ∑ i in Finset.range n, ↑(f i)) atTop (𝓝 ↑r) ↔\n Tendsto (fun n => ∑ i in Finset.range n, f i) atTop (𝓝 r)",
"state_before": "α : Type ?u.312215\nβ : Type ?u.312218\nγ : Type ?u.312221\nf : ℕ → ℝ≥0\nr : ℝ≥0\n⊢ HasSum f r ↔ Tendsto (fun n => ∑ i in Finset.range n, f i) atTop (𝓝 r)",
"tactic": "rw [← ENNReal.hasSum_coe, ENNReal.hasSum_iff_tendsto_nat]"
},
{
"state_after": "α : Type ?u.312215\nβ : Type ?u.312218\nγ : Type ?u.312221\nf : ℕ → ℝ≥0\nr : ℝ≥0\n⊢ Tendsto (fun n => ↑(∑ a in Finset.range n, f a)) atTop (𝓝 ↑r) ↔\n Tendsto (fun n => ∑ a in Finset.range n, f a) atTop (𝓝 r)",
"state_before": "α : Type ?u.312215\nβ : Type ?u.312218\nγ : Type ?u.312221\nf : ℕ → ℝ≥0\nr : ℝ≥0\n⊢ Tendsto (fun n => ∑ i in Finset.range n, ↑(f i)) atTop (𝓝 ↑r) ↔\n Tendsto (fun n => ∑ i in Finset.range n, f i) atTop (𝓝 r)",
"tactic": "simp only [← ENNReal.coe_finset_sum]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.312215\nβ : Type ?u.312218\nγ : Type ?u.312221\nf : ℕ → ℝ≥0\nr : ℝ≥0\n⊢ Tendsto (fun n => ↑(∑ a in Finset.range n, f a)) atTop (𝓝 ↑r) ↔\n Tendsto (fun n => ∑ a in Finset.range n, f a) atTop (𝓝 r)",
"tactic": "exact ENNReal.tendsto_coe"
}
] | [
1129,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1125,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.VectorMeasure.measurable_of_not_restrict_le_zero | [] | [
1022,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1021,
1
] |
Mathlib/Data/Fin/Tuple/Sort.lean | Tuple.self_comp_sort | [
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ graph.proj ∘ (↑(graphEquiv₁ f) ∘ ↑(graphEquiv₁ f).symm) ∘ ↑(graphEquiv₂ f).toEquiv = graph.proj ∘ ↑(graphEquiv₂ f)",
"tactic": "simp"
}
] | [
99,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
] |
Mathlib/Order/Lattice.lean | Lattice.ext | [
{
"state_after": "case mk\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\nB : Lattice α\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝ inf_le_right✝ le_inf✝ = B",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\nA B : Lattice α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ A = B",
"tactic": "cases A"
},
{
"state_after": "case mk.mk\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\ntoSemilatticeSup✝¹ : SemilatticeSup α\ntoInf✝¹ : Inf α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"state_before": "case mk\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\nB : Lattice α\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝ inf_le_right✝ le_inf✝ = B",
"tactic": "cases B"
},
{
"state_after": "case mk.mk.refl\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝¹ : Inf α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"state_before": "case mk.mk\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\ntoSemilatticeSup✝¹ : SemilatticeSup α\ntoInf✝¹ : Inf α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"tactic": "cases SemilatticeSup.ext H"
},
{
"state_after": "case mk.mk.refl.refl\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝ : Inf α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"state_before": "case mk.mk.refl\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝¹ : Inf α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ntoInf✝ : Inf α\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"tactic": "cases SemilatticeInf.ext H"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.refl.refl\nα✝ : Type u\nβ : Type v\ninst✝ : Lattice α✝\na b c d : α✝\nα : Type u_1\ntoSemilatticeSup✝ : SemilatticeSup α\ntoInf✝ : Inf α\ninf_le_left✝¹ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝¹ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝¹ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\ninf_le_left✝ : ∀ (a b : α), a ⊓ b ≤ a\ninf_le_right✝ : ∀ (a b : α), a ⊓ b ≤ b\nle_inf✝ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ mk inf_le_left✝¹ inf_le_right✝¹ le_inf✝¹ = mk inf_le_left✝ inf_le_right✝ le_inf✝",
"tactic": "congr"
}
] | [
726,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
720,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snorm_le_snorm_top_mul_snorm | [
{
"state_after": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ\n\ncase neg\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ",
"tactic": "by_cases hp_top : p = ∞"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : p = 0\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ\n\ncase neg\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ",
"state_before": "case neg\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ",
"tactic": "by_cases hp_zero : p = 0"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖b (f x) (g x)‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ≤\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"state_before": "case neg\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ",
"tactic": "simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top, snorm_exponent_top, snormEssSup]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\n⊢ snormEssSup (fun x => b (f x) (g x)) μ ≤ snormEssSup f μ * snormEssSup g μ",
"state_before": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ",
"tactic": "simp_rw [hp_top, snorm_exponent_top]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ (fun x => ↑‖(fun x => b (f x) (g x)) x‖₊) a ≤ ((fun x => ↑‖f x‖₊) * fun x => ↑‖g x‖₊) a",
"state_before": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\n⊢ snormEssSup (fun x => b (f x) (g x)) μ ≤ snormEssSup f μ * snormEssSup g μ",
"tactic": "refine' le_trans (essSup_mono_ae <| h.mono fun a ha => _) (ENNReal.essSup_mul_le _ _)"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊",
"state_before": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ (fun x => ↑‖(fun x => b (f x) (g x)) x‖₊) a ≤ ((fun x => ↑‖f x‖₊) * fun x => ↑‖g x‖₊) a",
"tactic": "simp_rw [Pi.mul_apply, ← ENNReal.coe_mul, ENNReal.coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : p = ⊤\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊",
"tactic": "exact ha"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : p = 0\n⊢ snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ⊤ μ * snorm g p μ",
"tactic": "simp only [hp_zero, snorm_exponent_zero, MulZeroClass.mul_zero, le_zero_iff]"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖b (f x) (g x)‖₊ ^ ENNReal.toReal p ∂μ) ≤\n ∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖b (f x) (g x)‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ≤\n (∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"tactic": "refine' ENNReal.rpow_le_rpow _ (one_div_nonneg.mpr ENNReal.toReal_nonneg)"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ↑‖b (f a) (g a)‖₊ ^ ENNReal.toReal p ≤ ↑‖f a‖₊ ^ ENNReal.toReal p * ↑‖g a‖₊ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖b (f x) (g x)‖₊ ^ ENNReal.toReal p ∂μ) ≤\n ∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ",
"tactic": "refine' lintegral_mono_ae (h.mono fun a ha => _)"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ↑‖b (f a) (g a)‖₊ ^ ENNReal.toReal p ≤ (↑‖f a‖₊ * ↑‖g a‖₊) ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ↑‖b (f a) (g a)‖₊ ^ ENNReal.toReal p ≤ ↑‖f a‖₊ ^ ENNReal.toReal p * ↑‖g a‖₊ ^ ENNReal.toReal p",
"tactic": "rw [← ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg]"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ↑‖b (f a) (g a)‖₊ ≤ ↑‖f a‖₊ * ↑‖g a‖₊",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ↑‖b (f a) (g a)‖₊ ^ ENNReal.toReal p ≤ (↑‖f a‖₊ * ↑‖g a‖₊) ^ ENNReal.toReal p",
"tactic": "refine' ENNReal.rpow_le_rpow _ ENNReal.toReal_nonneg"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ↑‖b (f a) (g a)‖₊ ≤ ↑‖f a‖₊ * ↑‖g a‖₊",
"tactic": "rw [← ENNReal.coe_mul, ENNReal.coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\na : α\nha : ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊\n⊢ ‖b (f a) (g a)‖₊ ≤ ‖f a‖₊ * ‖g a‖₊",
"tactic": "exact ha"
},
{
"state_after": "case refine'_1\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ≤\n ∫⁻ (x : α), essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ\n\ncase refine'_2\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ≤\n (∫⁻ (x : α), essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^\n (1 / ENNReal.toReal p)",
"tactic": "refine' ENNReal.rpow_le_rpow _ _"
},
{
"state_after": "case refine'_2\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p\n\ncase refine'_1\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ≤\n ∫⁻ (x : α), essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ",
"state_before": "case refine'_1\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ≤\n ∫⁻ (x : α), essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ\n\ncase refine'_2\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p",
"tactic": "swap"
},
{
"state_after": "case refine'_1\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ ∀ᵐ (a : α) ∂μ,\n ↑‖f a‖₊ ^ ENNReal.toReal p * ↑‖g a‖₊ ^ ENNReal.toReal p ≤\n essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g a‖₊ ^ ENNReal.toReal p",
"state_before": "case refine'_1\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ≤\n ∫⁻ (x : α), essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ",
"tactic": "refine' lintegral_mono_ae _"
},
{
"state_after": "case h\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\nx : α\nhx : ↑‖f x‖₊ ≤ essSup (fun x => ↑‖f x‖₊) μ\n⊢ ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ≤\n essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p",
"state_before": "case refine'_1\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ ∀ᵐ (a : α) ∂μ,\n ↑‖f a‖₊ ^ ENNReal.toReal p * ↑‖g a‖₊ ^ ENNReal.toReal p ≤\n essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g a‖₊ ^ ENNReal.toReal p",
"tactic": "filter_upwards [@ENNReal.ae_le_essSup _ _ μ fun x => (‖f x‖₊ : ℝ≥0∞)]with x hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\nx : α\nhx : ↑‖f x‖₊ ≤ essSup (fun x => ↑‖f x‖₊) μ\n⊢ ↑‖f x‖₊ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ≤\n essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p",
"tactic": "exact mul_le_mul_right' (ENNReal.rpow_le_rpow hx ENNReal.toReal_nonneg) _"
},
{
"state_after": "case refine'_2\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ ENNReal.toReal p",
"state_before": "case refine'_2\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p",
"tactic": "rw [one_div_nonneg]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ ENNReal.toReal p",
"tactic": "exact ENNReal.toReal_nonneg"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)\n\ncase hf\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ AEMeasurable fun x => ↑‖g x‖₊ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (∫⁻ (x : α), essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"tactic": "rw [lintegral_const_mul'']"
},
{
"state_after": "case hf\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ AEMeasurable fun x => ↑‖g x‖₊ ^ ENNReal.toReal p\n\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)\n\ncase hf\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ AEMeasurable fun x => ↑‖g x‖₊ ^ ENNReal.toReal p",
"tactic": "swap"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n (∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)\n\ncase hz\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p * ∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^\n (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"tactic": "rw [ENNReal.mul_rpow_of_nonneg]"
},
{
"state_after": "case hz\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p\n\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n (∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n (∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)\n\ncase hz\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p",
"tactic": "swap"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ ENNReal.toReal p ≠ 0",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ (essSup (fun x => ↑‖f x‖₊) μ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) *\n (∫⁻ (a : α), ↑‖g a‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) =\n essSup (fun x => ↑‖f x‖₊) μ * (∫⁻ (x : α), ↑‖g x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p)",
"tactic": "rw [← ENNReal.rpow_mul, one_div, mul_inv_cancel, ENNReal.rpow_one]"
},
{
"state_after": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ ¬p = 0 ∧ ¬p = ⊤",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ ENNReal.toReal p ≠ 0",
"tactic": "rw [Ne.def, ENNReal.toReal_eq_zero_iff, not_or]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ ¬p = 0 ∧ ¬p = ⊤",
"tactic": "exact ⟨hp_zero, hp_top⟩"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ AEMeasurable fun x => ↑‖g x‖₊ ^ ENNReal.toReal p",
"tactic": "exact hg.nnnorm.aemeasurable.coe_nnreal_ennreal.pow aemeasurable_const"
},
{
"state_after": "case hz\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ ENNReal.toReal p",
"state_before": "case hz\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ 1 / ENNReal.toReal p",
"tactic": "rw [one_div_nonneg]"
},
{
"state_after": "no goals",
"state_before": "case hz\nα : Type u_1\nE : Type u_4\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → E\ng : α → F\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nh : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊\nhp_top : ¬p = ⊤\nhp_zero : ¬p = 0\n⊢ 0 ≤ ENNReal.toReal p",
"tactic": "exact ENNReal.toReal_nonneg"
}
] | [
1388,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1347,
1
] |
Mathlib/LinearAlgebra/Determinant.lean | AlternatingMap.eq_smul_basis_det | [
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2583010\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nf : AlternatingMap R M R ι\ni : ι → ι\nh : Injective i\n⊢ (↑f fun i_1 => ↑e (i i_1)) = ↑(↑f ↑e • Basis.det e) fun i_1 => ↑e (i i_1)",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2583010\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nf : AlternatingMap R M R ι\n⊢ f = ↑f ↑e • Basis.det e",
"tactic": "refine' Basis.ext_alternating e fun i h => _"
},
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2583010\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nf : AlternatingMap R M R ι\ni : ι → ι\nh : Injective i\nσ : Equiv.Perm ι := Equiv.ofBijective i (_ : Bijective i)\n⊢ (↑f fun i_1 => ↑e (i i_1)) = ↑(↑f ↑e • Basis.det e) fun i_1 => ↑e (i i_1)",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2583010\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nf : AlternatingMap R M R ι\ni : ι → ι\nh : Injective i\n⊢ (↑f fun i_1 => ↑e (i i_1)) = ↑(↑f ↑e • Basis.det e) fun i_1 => ↑e (i i_1)",
"tactic": "let σ : Equiv.Perm ι := Equiv.ofBijective i (Finite.injective_iff_bijective.1 h)"
},
{
"state_after": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2583010\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nf : AlternatingMap R M R ι\ni : ι → ι\nh : Injective i\nσ : Equiv.Perm ι := Equiv.ofBijective i (_ : Bijective i)\n⊢ ↑f (↑e ∘ ↑σ) = ↑(↑f ↑e • Basis.det e) (↑e ∘ ↑σ)",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2583010\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nf : AlternatingMap R M R ι\ni : ι → ι\nh : Injective i\nσ : Equiv.Perm ι := Equiv.ofBijective i (_ : Bijective i)\n⊢ (↑f fun i_1 => ↑e (i i_1)) = ↑(↑f ↑e • Basis.det e) fun i_1 => ↑e (i i_1)",
"tactic": "change f (e ∘ σ) = (f e • e.det) (e ∘ σ)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2583010\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nf : AlternatingMap R M R ι\ni : ι → ι\nh : Injective i\nσ : Equiv.Perm ι := Equiv.ofBijective i (_ : Bijective i)\n⊢ ↑f (↑e ∘ ↑σ) = ↑(↑f ↑e • Basis.det e) (↑e ∘ ↑σ)",
"tactic": "simp [AlternatingMap.map_perm, Basis.det_self]"
}
] | [
580,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
576,
1
] |
Mathlib/Topology/Sober.lean | genericPoint_specializes | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5792\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : QuasiSober α\ninst✝ : IrreducibleSpace α\nx : α\n⊢ x ∈ closure univ",
"tactic": "simp"
}
] | [
164,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
] |
Mathlib/RingTheory/Ideal/Over.lean | Ideal.IntegralClosure.comap_lt_comap | [] | [
343,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.typein_surj | [] | [
490,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
488,
1
] |
Mathlib/Order/SuccPred/Relation.lean | reflTransGen_of_pred_of_ge | [] | [
103,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/FieldTheory/Separable.lean | Polynomial.separable_X_pow_sub_C | [] | [
404,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
402,
1
] |
Mathlib/Data/ZMod/Basic.lean | ZMod.valMinAbs_nonneg_iff | [
{
"state_after": "n : ℕ\ninst✝ : NeZero n\nx : ZMod n\n⊢ (0 ≤ if val x ≤ n / 2 then ↑(val x) else ↑(val x) - ↑n) ↔ val x ≤ n / 2",
"state_before": "n : ℕ\ninst✝ : NeZero n\nx : ZMod n\n⊢ 0 ≤ valMinAbs x ↔ val x ≤ n / 2",
"tactic": "rw [valMinAbs_def_pos]"
},
{
"state_after": "case inl\nn : ℕ\ninst✝ : NeZero n\nx : ZMod n\nh : val x ≤ n / 2\n⊢ 0 ≤ ↑(val x) ↔ val x ≤ n / 2\n\ncase inr\nn : ℕ\ninst✝ : NeZero n\nx : ZMod n\nh : ¬val x ≤ n / 2\n⊢ 0 ≤ ↑(val x) - ↑n ↔ val x ≤ n / 2",
"state_before": "n : ℕ\ninst✝ : NeZero n\nx : ZMod n\n⊢ (0 ≤ if val x ≤ n / 2 then ↑(val x) else ↑(val x) - ↑n) ↔ val x ≤ n / 2",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nn : ℕ\ninst✝ : NeZero n\nx : ZMod n\nh : val x ≤ n / 2\n⊢ 0 ≤ ↑(val x) ↔ val x ≤ n / 2",
"tactic": "exact iff_of_true (Nat.cast_nonneg _) h"
},
{
"state_after": "no goals",
"state_before": "case inr\nn : ℕ\ninst✝ : NeZero n\nx : ZMod n\nh : ¬val x ≤ n / 2\n⊢ 0 ≤ ↑(val x) - ↑n ↔ val x ≤ n / 2",
"tactic": "exact iff_of_false (sub_lt_zero.2 <| Int.ofNat_lt.2 x.val_lt).not_le h"
}
] | [
938,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
935,
1
] |
Mathlib/Algebra/GeomSum.lean | geom_sum_Ico_mul_neg | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Ring α\nx : α\nm n : ℕ\nhmn : m ≤ n\n⊢ (∑ i in Ico m n, x ^ i) * (1 - x) = x ^ m - x ^ n",
"tactic": "rw [sum_Ico_eq_sub _ hmn, sub_mul, geom_sum_mul_neg, geom_sum_mul_neg, sub_sub_sub_cancel_left]"
}
] | [
340,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | PrimeSpectrum.basicOpen_eq_bot_iff | [
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R\n⊢ zeroLocus {f}ᶜ = ↑⊥ ↔ IsNilpotent f",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R\n⊢ basicOpen f = ⊥ ↔ IsNilpotent f",
"tactic": "rw [← TopologicalSpace.Opens.coe_inj, basicOpen_eq_zeroLocus_compl]"
},
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R\n⊢ (∀ (x : PrimeSpectrum R), f ∈ x.asIdeal) ↔ ∀ (J : Ideal R), Ideal.IsPrime J → f ∈ J",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R\n⊢ zeroLocus {f}ᶜ = ↑⊥ ↔ IsNilpotent f",
"tactic": "simp only [Set.eq_univ_iff_forall, Set.singleton_subset_iff, TopologicalSpace.Opens.coe_bot,\n nilpotent_iff_mem_prime, Set.compl_empty_iff, mem_zeroLocus, SetLike.mem_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R\n⊢ (∀ (x : PrimeSpectrum R), f ∈ x.asIdeal) ↔ ∀ (J : Ideal R), Ideal.IsPrime J → f ∈ J",
"tactic": "exact ⟨fun h I hI => h ⟨I, hI⟩, fun h ⟨I, hI⟩ => h I hI⟩"
}
] | [
874,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
870,
1
] |
Mathlib/Order/Filter/Extr.lean | IsMinFilter.sup | [] | [
534,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
531,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean | finprod_mem_insert | [] | [
865,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
863,
1
] |
Mathlib/InformationTheory/Hamming.lean | Hamming.toHamming_neg | [] | [
360,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
1
] |
Mathlib/Data/Nat/Bits.lean | Nat.bit1_mod_two | [
{
"state_after": "n : ℕ\n⊢ (bif bodd (bit1 n) then 1 else 0) = 1",
"state_before": "n : ℕ\n⊢ bit1 n % 2 = 1",
"tactic": "rw [Nat.mod_two_of_bodd]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (bif bodd (bit1 n) then 1 else 0) = 1",
"tactic": "simp"
}
] | [
113,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | Matrix.IsAdjMatrix.adjMatrix_toGraph_eq | [
{
"state_after": "case a.h\nV : Type u_2\nα : Type u_1\nβ : Type ?u.79711\ninst✝² : MulZeroOneClass α\ninst✝¹ : Nontrivial α\nA : Matrix V V α\nh : IsAdjMatrix A\ninst✝ : DecidableEq α\ni j : V\n⊢ adjMatrix α (toGraph h) i j = A i j",
"state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.79711\ninst✝² : MulZeroOneClass α\ninst✝¹ : Nontrivial α\nA : Matrix V V α\nh : IsAdjMatrix A\ninst✝ : DecidableEq α\n⊢ adjMatrix α (toGraph h) = A",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case a.h\nV : Type u_2\nα : Type u_1\nβ : Type ?u.79711\ninst✝² : MulZeroOneClass α\ninst✝¹ : Nontrivial α\nA : Matrix V V α\nh : IsAdjMatrix A\ninst✝ : DecidableEq α\ni j : V\n⊢ adjMatrix α (toGraph h) i j = A i j",
"tactic": "obtain h' | h' := h.zero_or_one i j <;> simp [h']"
}
] | [
297,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
295,
1
] |
Mathlib/Data/List/Perm.lean | List.perm_replicate_append_replicate | [
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\n⊢ (count a l = count a (replicate m a ++ replicate n b) ∧\n (b ≠ a → count b l = count b (replicate m a ++ replicate n b)) ∧\n ∀ (b_1 : α), b_1 ≠ b → b_1 ≠ a → count b_1 l = count b_1 (replicate m a ++ replicate n b)) ↔\n count a l = m ∧ count b l = n ∧ l ⊆ [a, b]",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\n⊢ l ~ replicate m a ++ replicate n b ↔ count a l = m ∧ count b l = n ∧ l ⊆ [a, b]",
"tactic": "rw [perm_iff_count, ← Decidable.and_forall_ne a, ← Decidable.and_forall_ne b]"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\nthis : l ⊆ [a, b] ↔ ∀ (c : α), c ≠ b → c ≠ a → ¬c ∈ l\n⊢ (count a l = count a (replicate m a ++ replicate n b) ∧\n (b ≠ a → count b l = count b (replicate m a ++ replicate n b)) ∧\n ∀ (b_1 : α), b_1 ≠ b → b_1 ≠ a → count b_1 l = count b_1 (replicate m a ++ replicate n b)) ↔\n count a l = m ∧ count b l = n ∧ l ⊆ [a, b]\n\ncase this\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\n⊢ l ⊆ [a, b] ↔ ∀ (c : α), c ≠ b → c ≠ a → ¬c ∈ l",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\n⊢ (count a l = count a (replicate m a ++ replicate n b) ∧\n (b ≠ a → count b l = count b (replicate m a ++ replicate n b)) ∧\n ∀ (b_1 : α), b_1 ≠ b → b_1 ≠ a → count b_1 l = count b_1 (replicate m a ++ replicate n b)) ↔\n count a l = m ∧ count b l = n ∧ l ⊆ [a, b]",
"tactic": "suffices : l ⊆ [a, b] ↔ ∀ c, c ≠ b → c ≠ a → c ∉ l"
},
{
"state_after": "case this\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\n⊢ l ⊆ [a, b] ↔ ∀ (c : α), c ≠ b → c ≠ a → ¬c ∈ l",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\nthis : l ⊆ [a, b] ↔ ∀ (c : α), c ≠ b → c ≠ a → ¬c ∈ l\n⊢ (count a l = count a (replicate m a ++ replicate n b) ∧\n (b ≠ a → count b l = count b (replicate m a ++ replicate n b)) ∧\n ∀ (b_1 : α), b_1 ≠ b → b_1 ≠ a → count b_1 l = count b_1 (replicate m a ++ replicate n b)) ↔\n count a l = m ∧ count b l = n ∧ l ⊆ [a, b]\n\ncase this\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\n⊢ l ⊆ [a, b] ↔ ∀ (c : α), c ≠ b → c ≠ a → ¬c ∈ l",
"tactic": "{ simp (config := { contextual := true }) [count_replicate, h, h.symm, this] }"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl : List α\na b : α\nm n : ℕ\nh : a ≠ b\n⊢ l ⊆ [a, b] ↔ ∀ (c : α), c ≠ b → c ≠ a → ¬c ∈ l",
"tactic": "simp_rw [Ne.def, ← and_imp, ← not_or, Decidable.not_imp_not, subset_def, mem_cons,\n not_mem_nil, or_false, or_comm]"
}
] | [
890,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
884,
1
] |
Mathlib/Topology/Order/Basic.lean | nhdsWithin_Iic_basis | [] | [
1127,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1125,
1
] |
Mathlib/Data/Rat/Order.lean | Rat.nonneg_total | [
{
"state_after": "case mk'\nb c : ℚ\nn : ℤ\nden✝ : ℕ\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs n) den✝\n⊢ Rat.Nonneg (mk' n den✝) ∨ Rat.Nonneg (-mk' n den✝)",
"state_before": "a b c : ℚ\n⊢ Rat.Nonneg a ∨ Rat.Nonneg (-a)",
"tactic": "cases' a with n"
},
{
"state_after": "no goals",
"state_before": "case mk'\nb c : ℚ\nn : ℤ\nden✝ : ℕ\nden_nz✝ : den✝ ≠ 0\nreduced✝ : Nat.coprime (Int.natAbs n) den✝\n⊢ Rat.Nonneg (mk' n den✝) ∨ Rat.Nonneg (-mk' n den✝)",
"tactic": "exact Or.imp_right neg_nonneg_of_nonpos (le_total 0 n)"
}
] | [
86,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
11
] |
src/lean/Init/Control/Lawful.lean | ExceptT.run_pure | [] | [
108,
102
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
108,
9
] |
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.toNonUnitalRingHom_apply_symm_toNonUnitalRingHom_apply | [] | [
636,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
634,
1
] |
Mathlib/Algebra/Symmetrized.lean | SymAlg.sym_one | [] | [
163,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/Probability/Kernel/Basic.lean | ProbabilityTheory.kernel.comapRight_apply | [] | [
563,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
561,
1
] |
Mathlib/Analysis/Convex/Segment.lean | openSegment_symm | [] | [
88,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Order/Bounds/Basic.lean | isLeast_Ico | [] | [
724,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
723,
1
] |
Mathlib/Data/List/Basic.lean | List.indexOf_eq_length | [
{
"state_after": "case nil\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\n⊢ indexOf a [] = length [] ↔ ¬a ∈ []\n\ncase cons\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\n⊢ indexOf a (b :: l) = length (b :: l) ↔ ¬a ∈ b :: l",
"state_before": "ι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\nl : List α\n⊢ indexOf a l = length l ↔ ¬a ∈ l",
"tactic": "induction' l with b l ih"
},
{
"state_after": "case cons\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\n⊢ (if a = b then 0 else succ (indexOf a l)) = length l + 1 ↔ ¬(a = b ∨ a ∈ l)",
"state_before": "case cons\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\n⊢ indexOf a (b :: l) = length (b :: l) ↔ ¬a ∈ b :: l",
"tactic": "simp only [length, mem_cons, indexOf_cons]"
},
{
"state_after": "case cons.inl\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : a = b\n⊢ 0 = length l + 1 ↔ ¬(a = b ∨ a ∈ l)\n\ncase cons.inr\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : ¬a = b\n⊢ succ (indexOf a l) = length l + 1 ↔ ¬(a = b ∨ a ∈ l)",
"state_before": "case cons\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\n⊢ (if a = b then 0 else succ (indexOf a l)) = length l + 1 ↔ ¬(a = b ∨ a ∈ l)",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\n⊢ indexOf a [] = length [] ↔ ¬a ∈ []",
"tactic": "exact iff_of_true rfl (not_mem_nil _)"
},
{
"state_after": "no goals",
"state_before": "case cons.inl\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : a = b\n⊢ 0 = length l + 1 ↔ ¬(a = b ∨ a ∈ l)",
"tactic": "exact iff_of_false (by rintro ⟨⟩) fun H => H <| Or.inl h"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : a = b\n⊢ ¬0 = length l + 1",
"tactic": "rintro ⟨⟩"
},
{
"state_after": "case cons.inr\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : ¬a = b\n⊢ succ (indexOf a l) = length l + 1 ↔ ¬a ∈ l",
"state_before": "case cons.inr\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : ¬a = b\n⊢ succ (indexOf a l) = length l + 1 ↔ ¬(a = b ∨ a ∈ l)",
"tactic": "simp only [h, false_or_iff]"
},
{
"state_after": "case cons.inr\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : ¬a = b\n⊢ succ (indexOf a l) = length l + 1 ↔ indexOf a l = length l",
"state_before": "case cons.inr\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : ¬a = b\n⊢ succ (indexOf a l) = length l + 1 ↔ ¬a ∈ l",
"tactic": "rw [← ih]"
},
{
"state_after": "no goals",
"state_before": "case cons.inr\nι : Type ?u.79233\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l = length l ↔ ¬a ∈ l\nh : ¬a = b\n⊢ succ (indexOf a l) = length l + 1 ↔ indexOf a l = length l",
"tactic": "exact succ_inj'"
}
] | [
1190,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1183,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | ContDiffAt.of_le | [] | [
1349,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1348,
1
] |
Mathlib/Data/Real/GoldenRatio.lean | goldConj_irrational | [
{
"state_after": "this : Irrational (sqrt ↑5)\n⊢ Irrational ψ",
"state_before": "⊢ Irrational ψ",
"tactic": "have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)"
},
{
"state_after": "this✝ : Irrational (sqrt ↑5)\nthis : Irrational (↑1 - sqrt ↑5)\n⊢ Irrational ψ",
"state_before": "this : Irrational (sqrt ↑5)\n⊢ Irrational ψ",
"tactic": "have := this.rat_sub 1"
},
{
"state_after": "this✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 - sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 - sqrt ↑5))\n⊢ Irrational ψ",
"state_before": "this✝ : Irrational (sqrt ↑5)\nthis : Irrational (↑1 - sqrt ↑5)\n⊢ Irrational ψ",
"tactic": "have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)"
},
{
"state_after": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 - sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 - sqrt ↑5))\n⊢ ψ = ↑0.5 * (↑1 - sqrt ↑5)",
"state_before": "this✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 - sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 - sqrt ↑5))\n⊢ Irrational ψ",
"tactic": "convert this"
},
{
"state_after": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 - sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 - sqrt ↑5))\n⊢ ψ = 1 / 2 * (1 - sqrt 5)",
"state_before": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 - sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 - sqrt ↑5))\n⊢ ψ = ↑0.5 * (↑1 - sqrt ↑5)",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1\nthis✝¹ : Irrational (sqrt ↑5)\nthis✝ : Irrational (↑1 - sqrt ↑5)\nthis : Irrational (↑0.5 * (↑1 - sqrt ↑5))\n⊢ ψ = 1 / 2 * (1 - sqrt 5)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "⊢ Nat.Prime 5",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "this✝ : Irrational (sqrt ↑5)\nthis : Irrational (↑1 - sqrt ↑5)\n⊢ 0.5 ≠ 0",
"tactic": "norm_num"
}
] | [
157,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_liminf_le | [] | [
1019,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1017,
1
] |
Mathlib/Data/Sym/Basic.lean | Sym.mem_map | [] | [
360,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
358,
1
] |
Mathlib/Data/List/NodupEquivFin.lean | List.sublist_iff_exists_fin_orderEmbedding_get_eq | [
{
"state_after": "α : Type u_1\nl l' : List α\n⊢ (∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)) ↔ ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"state_before": "α : Type u_1\nl l' : List α\n⊢ l <+ l' ↔ ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"tactic": "rw [sublist_iff_exists_orderEmbedding_get?_eq]"
},
{
"state_after": "case mp\nα : Type u_1\nl l' : List α\n⊢ (∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)) → ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\n\ncase mpr\nα : Type u_1\nl l' : List α\n⊢ (∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)) → ∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)",
"state_before": "α : Type u_1\nl l' : List α\n⊢ (∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)) ↔ ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\n⊢ ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"state_before": "case mp\nα : Type u_1\nl l' : List α\n⊢ (∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)) → ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"tactic": "rintro ⟨f, hf⟩"
},
{
"state_after": "case mp.intro\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\n⊢ ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"state_before": "case mp.intro\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\n⊢ ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"tactic": "have h : ∀ {i : ℕ} (_ : i < l.length), f i < l'.length := by\n intro i hi\n specialize hf i\n rw [get?_eq_get hi, eq_comm, get?_eq_some] at hf\n obtain ⟨h, -⟩ := hf\n exact h"
},
{
"state_after": "case mp.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\n⊢ ∀ (a b : Fin (length l)),\n (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') }) a ≤\n (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') }) b ↔\n a ≤ b\n\ncase mp.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\n⊢ ∀ (ix : Fin (length l)),\n get l ix =\n get l'\n (↑(OrderEmbedding.ofMapLEIff (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') })\n ?mp.intro.refine'_1)\n ix)",
"state_before": "case mp.intro\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\n⊢ ∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)",
"tactic": "refine' ⟨OrderEmbedding.ofMapLEIff (fun ix => ⟨f ix, h ix.is_lt⟩) _, _⟩"
},
{
"state_after": "α : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\ni : ℕ\nhi : i < length l\n⊢ ↑f i < length l'",
"state_before": "α : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\n⊢ ∀ {i : ℕ}, i < length l → ↑f i < length l'",
"tactic": "intro i hi"
},
{
"state_after": "α : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\ni : ℕ\nhi : i < length l\nhf : get? l i = get? l' (↑f i)\n⊢ ↑f i < length l'",
"state_before": "α : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\ni : ℕ\nhi : i < length l\n⊢ ↑f i < length l'",
"tactic": "specialize hf i"
},
{
"state_after": "α : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\ni : ℕ\nhi : i < length l\nhf : ∃ h, get l' { val := ↑f i, isLt := h } = get l { val := i, isLt := hi }\n⊢ ↑f i < length l'",
"state_before": "α : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\ni : ℕ\nhi : i < length l\nhf : get? l i = get? l' (↑f i)\n⊢ ↑f i < length l'",
"tactic": "rw [get?_eq_get hi, eq_comm, get?_eq_some] at hf"
},
{
"state_after": "case intro\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\ni : ℕ\nhi : i < length l\nh : ↑f i < length l'\n⊢ ↑f i < length l'",
"state_before": "α : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\ni : ℕ\nhi : i < length l\nhf : ∃ h, get l' { val := ↑f i, isLt := h } = get l { val := i, isLt := hi }\n⊢ ↑f i < length l'",
"tactic": "obtain ⟨h, -⟩ := hf"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\ni : ℕ\nhi : i < length l\nh : ↑f i < length l'\n⊢ ↑f i < length l'",
"tactic": "exact h"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\n⊢ ∀ (a b : Fin (length l)),\n (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') }) a ≤\n (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') }) b ↔\n a ≤ b",
"tactic": "simp"
},
{
"state_after": "case mp.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\ni : Fin (length l)\n⊢ get l i =\n get l'\n (↑(OrderEmbedding.ofMapLEIff (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') })\n (_ :\n ∀ (a a_1 : Fin (length l)),\n { val := ↑f ↑a, isLt := (_ : ↑f ↑a < length l') } ≤\n { val := ↑f ↑a_1, isLt := (_ : ↑f ↑a_1 < length l') } ↔\n a ≤ a_1))\n i)",
"state_before": "case mp.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\n⊢ ∀ (ix : Fin (length l)),\n get l ix =\n get l'\n (↑(OrderEmbedding.ofMapLEIff (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') })\n (_ :\n ∀ (a a_1 : Fin (length l)),\n { val := ↑f ↑a, isLt := (_ : ↑f ↑a < length l') } ≤\n { val := ↑f ↑a_1, isLt := (_ : ↑f ↑a_1 < length l') } ↔\n a ≤ a_1))\n ix)",
"tactic": "intro i"
},
{
"state_after": "case mp.intro.refine'_2.a\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\ni : Fin (length l)\n⊢ some (get l i) =\n some\n (get l'\n (↑(OrderEmbedding.ofMapLEIff (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') })\n (_ :\n ∀ (a a_1 : Fin (length l)),\n { val := ↑f ↑a, isLt := (_ : ↑f ↑a < length l') } ≤\n { val := ↑f ↑a_1, isLt := (_ : ↑f ↑a_1 < length l') } ↔\n a ≤ a_1))\n i))",
"state_before": "case mp.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\ni : Fin (length l)\n⊢ get l i =\n get l'\n (↑(OrderEmbedding.ofMapLEIff (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') })\n (_ :\n ∀ (a a_1 : Fin (length l)),\n { val := ↑f ↑a, isLt := (_ : ↑f ↑a < length l') } ≤\n { val := ↑f ↑a_1, isLt := (_ : ↑f ↑a_1 < length l') } ↔\n a ≤ a_1))\n i)",
"tactic": "apply Option.some_injective"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.refine'_2.a\nα : Type u_1\nl l' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)\nh : ∀ {i : ℕ}, i < length l → ↑f i < length l'\ni : Fin (length l)\n⊢ some (get l i) =\n some\n (get l'\n (↑(OrderEmbedding.ofMapLEIff (fun ix => { val := ↑f ↑ix, isLt := (_ : ↑f ↑ix < length l') })\n (_ :\n ∀ (a a_1 : Fin (length l)),\n { val := ↑f ↑a, isLt := (_ : ↑f ↑a < length l') } ≤\n { val := ↑f ↑a_1, isLt := (_ : ↑f ↑a_1 < length l') } ↔\n a ≤ a_1))\n i))",
"tactic": "simpa [get?_eq_get i.2, get?_eq_get (h i.2)] using hf i"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\n⊢ ∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)",
"state_before": "case mpr\nα : Type u_1\nl l' : List α\n⊢ (∃ f, ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)) → ∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)",
"tactic": "rintro ⟨f, hf⟩"
},
{
"state_after": "case mpr.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\n⊢ StrictMono fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l'\n\ncase mpr.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\n⊢ ∀ (ix : ℕ),\n get? l ix =\n get? l'\n (↑(OrderEmbedding.ofStrictMono\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l')\n ?mpr.intro.refine'_1)\n ix)",
"state_before": "case mpr.intro\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\n⊢ ∃ f, ∀ (ix : ℕ), get? l ix = get? l' (↑f ix)",
"tactic": "refine'\n ⟨OrderEmbedding.ofStrictMono (fun i => if hi : i < l.length then f ⟨i, hi⟩ else i + l'.length)\n _,\n _⟩"
},
{
"state_after": "case mpr.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\n⊢ (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') j",
"state_before": "case mpr.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\n⊢ StrictMono fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l'",
"tactic": "intro i j h"
},
{
"state_after": "case mpr.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\n⊢ (if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') <\n if hi : j < length l then ↑(↑f { val := j, isLt := hi }) else j + length l'",
"state_before": "case mpr.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\n⊢ (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') j",
"tactic": "dsimp only"
},
{
"state_after": "case mpr.intro.refine'_1.inl.inl\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : i < length l\nhj : j < length l\n⊢ ↑(↑f { val := i, isLt := hi }) < ↑(↑f { val := j, isLt := hj })\n\ncase mpr.intro.refine'_1.inl.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : i < length l\nhj : ¬j < length l\n⊢ ↑(↑f { val := i, isLt := hi }) < j + length l'\n\ncase mpr.intro.refine'_1.inr.inl\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : ¬i < length l\nhj : j < length l\n⊢ i + length l' < ↑(↑f { val := j, isLt := hj })\n\ncase mpr.intro.refine'_1.inr.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : ¬i < length l\nhj : ¬j < length l\n⊢ i + length l' < j + length l'",
"state_before": "case mpr.intro.refine'_1\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\n⊢ (if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') <\n if hi : j < length l then ↑(↑f { val := j, isLt := hi }) else j + length l'",
"tactic": "split_ifs with hi hj hj"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.refine'_1.inl.inl\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : i < length l\nhj : j < length l\n⊢ ↑(↑f { val := i, isLt := hi }) < ↑(↑f { val := j, isLt := hj })",
"tactic": "rwa [Fin.val_fin_lt, f.lt_iff_lt]"
},
{
"state_after": "case mpr.intro.refine'_1.inl.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : i < length l\nhj : ¬j < length l\n⊢ ↑(↑f { val := i, isLt := hi }) < length l' + j",
"state_before": "case mpr.intro.refine'_1.inl.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : i < length l\nhj : ¬j < length l\n⊢ ↑(↑f { val := i, isLt := hi }) < j + length l'",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.refine'_1.inl.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : i < length l\nhj : ¬j < length l\n⊢ ↑(↑f { val := i, isLt := hi }) < length l' + j",
"tactic": "exact lt_add_of_lt_of_pos (Fin.is_lt _) (i.zero_le.trans_lt h)"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.refine'_1.inr.inl\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : ¬i < length l\nhj : j < length l\n⊢ i + length l' < ↑(↑f { val := j, isLt := hj })",
"tactic": "exact absurd (h.trans hj) hi"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.refine'_1.inr.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni j : ℕ\nh : i < j\nhi : ¬i < length l\nhj : ¬j < length l\n⊢ i + length l' < j + length l'",
"tactic": "simpa using h"
},
{
"state_after": "case mpr.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\n⊢ get? l i =\n get? l'\n (↑(OrderEmbedding.ofStrictMono\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l')\n (_ :\n ∀ ⦃i j : ℕ⦄,\n i < j →\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') j))\n i)",
"state_before": "case mpr.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\n⊢ ∀ (ix : ℕ),\n get? l ix =\n get? l'\n (↑(OrderEmbedding.ofStrictMono\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l')\n (_ :\n ∀ ⦃i j : ℕ⦄,\n i < j →\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') j))\n ix)",
"tactic": "intro i"
},
{
"state_after": "case mpr.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\n⊢ get? l i = get? l' (if h : i < length l then ↑(↑f { val := i, isLt := (_ : i < length l) }) else i + length l')",
"state_before": "case mpr.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\n⊢ get? l i =\n get? l'\n (↑(OrderEmbedding.ofStrictMono\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l')\n (_ :\n ∀ ⦃i j : ℕ⦄,\n i < j →\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then ↑(↑f { val := i, isLt := hi }) else i + length l') j))\n i)",
"tactic": "simp only [OrderEmbedding.coe_ofStrictMono]"
},
{
"state_after": "case mpr.intro.refine'_2.inl\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : i < length l\n⊢ get? l i = get? l' ↑(↑f { val := i, isLt := (_ : i < length l) })\n\ncase mpr.intro.refine'_2.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : ¬i < length l\n⊢ get? l i = get? l' (i + length l')",
"state_before": "case mpr.intro.refine'_2\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\n⊢ get? l i = get? l' (if h : i < length l then ↑(↑f { val := i, isLt := (_ : i < length l) }) else i + length l')",
"tactic": "split_ifs with hi"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.refine'_2.inl\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : i < length l\n⊢ get? l i = get? l' ↑(↑f { val := i, isLt := (_ : i < length l) })",
"tactic": "rw [get?_eq_get hi, get?_eq_get, ← hf]"
},
{
"state_after": "case mpr.intro.refine'_2.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : ¬i < length l\n⊢ length l' ≤ i + length l'\n\ncase mpr.intro.refine'_2.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : ¬i < length l\n⊢ length l ≤ i",
"state_before": "case mpr.intro.refine'_2.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : ¬i < length l\n⊢ get? l i = get? l' (i + length l')",
"tactic": "rw [get?_eq_none.mpr, get?_eq_none.mpr]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.refine'_2.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : ¬i < length l\n⊢ length l' ≤ i + length l'",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.refine'_2.inr\nα : Type u_1\nl l' : List α\nf : Fin (length l) ↪o Fin (length l')\nhf : ∀ (ix : Fin (length l)), get l ix = get l' (↑f ix)\ni : ℕ\nhi : ¬i < length l\n⊢ length l ≤ i",
"tactic": "simpa using hi"
}
] | [
209,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
172,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean | StieltjesFunction.measure_Iic | [
{
"state_after": "f : StieltjesFunction\nl : ℝ\nhf : Tendsto (↑f) atBot (𝓝 l)\nx : ℝ\n⊢ Tendsto (fun x_1 => ↑↑(StieltjesFunction.measure f) (Ioc x_1 x)) atBot (𝓝 (ofReal (↑f x - l)))",
"state_before": "f : StieltjesFunction\nl : ℝ\nhf : Tendsto (↑f) atBot (𝓝 l)\nx : ℝ\n⊢ ↑↑(StieltjesFunction.measure f) (Iic x) = ofReal (↑f x - l)",
"tactic": "refine' tendsto_nhds_unique (tendsto_measure_Ioc_atBot _ _) _"
},
{
"state_after": "f : StieltjesFunction\nl : ℝ\nhf : Tendsto (↑f) atBot (𝓝 l)\nx : ℝ\n⊢ Tendsto (fun x_1 => ofReal (↑f x - ↑f x_1)) atBot (𝓝 (ofReal (↑f x - l)))",
"state_before": "f : StieltjesFunction\nl : ℝ\nhf : Tendsto (↑f) atBot (𝓝 l)\nx : ℝ\n⊢ Tendsto (fun x_1 => ↑↑(StieltjesFunction.measure f) (Ioc x_1 x)) atBot (𝓝 (ofReal (↑f x - l)))",
"tactic": "simp_rw [measure_Ioc]"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\nl : ℝ\nhf : Tendsto (↑f) atBot (𝓝 l)\nx : ℝ\n⊢ Tendsto (fun x_1 => ofReal (↑f x - ↑f x_1)) atBot (𝓝 (ofReal (↑f x - l)))",
"tactic": "exact ENNReal.tendsto_ofReal (Tendsto.const_sub _ hf)"
}
] | [
590,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
586,
1
] |
Mathlib/Algebra/BigOperators/Order.lean | Finset.prod_nonneg | [] | [
589,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
588,
1
] |
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