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Mathlib/Order/Synonym.lean | ofLex_symm_eq | [] | [
183,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/Data/Set/Finite.lean | Set.finite_isBot | [] | [
1427,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1426,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.adjoin_zero | [] | [
687,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
686,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.isCaratheodory_union | [
{
"state_after": "α : Type u\nm : OuterMeasure α\ns s₁ s₂ : Set α\nh₁ : IsCaratheodory m s₁\nh₂ : IsCaratheodory m s₂\nt : Set α\n⊢ ↑m (t ∩ s₁ ∩ s₂) + ↑m ((t ∩ s₁) \\ s₂) + (↑m (t \\ s₁ ∩ s₂) + ↑m ((t \\ s₁) \\ s₂)) =\n ↑m (t ∩ s₁ ∩ s₂) + ↑m ((t ∩ s₁) \\ s₂) + ↑m (t ∩ (s₂ \\ s₁)) + ↑m (t \\ (s₁ ∪ s₂))",
"state_before": "α : Type u\nm : OuterMeasure α\ns s₁ s₂ : Set α\nh₁ : IsCaratheodory m s₁\nh₂ : IsCaratheodory m s₂\nt : Set α\n⊢ ↑m t = ↑m (t ∩ (s₁ ∪ s₂)) + ↑m (t \\ (s₁ ∪ s₂))",
"tactic": "rw [h₁ t, h₂ (t ∩ s₁), h₂ (t \\ s₁), h₁ (t ∩ (s₁ ∪ s₂)), inter_diff_assoc _ _ s₁,\n Set.inter_assoc _ _ s₁, inter_eq_self_of_subset_right (Set.subset_union_left _ _),\n union_diff_left, h₂ (t ∩ s₁)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nm : OuterMeasure α\ns s₁ s₂ : Set α\nh₁ : IsCaratheodory m s₁\nh₂ : IsCaratheodory m s₂\nt : Set α\n⊢ ↑m (t ∩ s₁ ∩ s₂) + ↑m ((t ∩ s₁) \\ s₂) + (↑m (t \\ s₁ ∩ s₂) + ↑m ((t \\ s₁) \\ s₂)) =\n ↑m (t ∩ s₁ ∩ s₂) + ↑m ((t ∩ s₁) \\ s₂) + ↑m (t ∩ (s₂ \\ s₁)) + ↑m (t \\ (s₁ ∪ s₂))",
"tactic": "simp [diff_eq, add_assoc]"
}
] | [
971,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
966,
1
] |
Mathlib/Data/List/Basic.lean | List.nthLe_reverse' | [
{
"state_after": "ι : Type ?u.97720\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\nhn : n < length (reverse l)\nhn' : length l - 1 - n < length l\n⊢ nthLe l (length l - 1 - n) hn' = nthLe (reverse l) n hn",
"state_before": "ι : Type ?u.97720\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\nhn : n < length (reverse l)\nhn' : length l - 1 - n < length l\n⊢ nthLe (reverse l) n hn = nthLe l (length l - 1 - n) hn'",
"tactic": "rw [eq_comm]"
},
{
"state_after": "case h.e'_2\nι : Type ?u.97720\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\nhn : n < length (reverse l)\nhn' : length l - 1 - n < length l\n⊢ nthLe l (length l - 1 - n) hn' =\n nthLe (reverse (reverse l)) (length (reverse l) - 1 - n)\n (_ : length (reverse l) - 1 - n < length (reverse (reverse l)))",
"state_before": "ι : Type ?u.97720\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\nhn : n < length (reverse l)\nhn' : length l - 1 - n < length l\n⊢ nthLe l (length l - 1 - n) hn' = nthLe (reverse l) n hn",
"tactic": "convert nthLe_reverse l.reverse n (by simpa) hn using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nι : Type ?u.97720\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\nhn : n < length (reverse l)\nhn' : length l - 1 - n < length l\n⊢ nthLe l (length l - 1 - n) hn' =\n nthLe (reverse (reverse l)) (length (reverse l) - 1 - n)\n (_ : length (reverse l) - 1 - n < length (reverse (reverse l)))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.97720\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\nn : ℕ\nhn : n < length (reverse l)\nhn' : length l - 1 - n < length l\n⊢ length (reverse l) - 1 - n < length (reverse (reverse l))",
"tactic": "simpa"
}
] | [
1470,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1466,
1
] |
Mathlib/Data/Nat/Prime.lean | Nat.minFac_sq_le_self | [] | [
417,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
412,
1
] |
Mathlib/CategoryTheory/Monoidal/End.lean | CategoryTheory.μ_naturality | [] | [
132,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Topology/Sober.lean | IsGenericPoint.inseparable | [] | [
84,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
11
] |
Std/Data/List/Lemmas.lean | List.isPrefix.sublist | [] | [
1607,
20
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1606,
11
] |
Mathlib/GroupTheory/Subsemigroup/Center.lean | Set.inv_mem_center₀ | [
{
"state_after": "case inl\nM : Type u_1\ninst✝ : GroupWithZero M\nha : 0 ∈ center M\n⊢ 0⁻¹ ∈ center M\n\ncase inr\nM : Type u_1\ninst✝ : GroupWithZero M\na : M\nha : a ∈ center M\nha0 : a ≠ 0\n⊢ a⁻¹ ∈ center M",
"state_before": "M : Type u_1\ninst✝ : GroupWithZero M\na : M\nha : a ∈ center M\n⊢ a⁻¹ ∈ center M",
"tactic": "obtain rfl | ha0 := eq_or_ne a 0"
},
{
"state_after": "case inr.intro\nM : Type u_1\ninst✝ : GroupWithZero M\na : Mˣ\nha : ↑a ∈ center M\nha0 : ↑a ≠ 0\n⊢ (↑a)⁻¹ ∈ center M",
"state_before": "case inr\nM : Type u_1\ninst✝ : GroupWithZero M\na : M\nha : a ∈ center M\nha0 : a ≠ 0\n⊢ a⁻¹ ∈ center M",
"tactic": "rcases IsUnit.mk0 _ ha0 with ⟨a, rfl⟩"
},
{
"state_after": "case inr.intro\nM : Type u_1\ninst✝ : GroupWithZero M\na : Mˣ\nha : ↑a ∈ center M\nha0 : ↑a ≠ 0\n⊢ ↑a⁻¹ ∈ center M",
"state_before": "case inr.intro\nM : Type u_1\ninst✝ : GroupWithZero M\na : Mˣ\nha : ↑a ∈ center M\nha0 : ↑a ≠ 0\n⊢ (↑a)⁻¹ ∈ center M",
"tactic": "rw [← Units.val_inv_eq_inv_val]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nM : Type u_1\ninst✝ : GroupWithZero M\na : Mˣ\nha : ↑a ∈ center M\nha0 : ↑a ≠ 0\n⊢ ↑a⁻¹ ∈ center M",
"tactic": "exact center_units_subset (inv_mem_center (subset_center_units ha))"
},
{
"state_after": "case inl\nM : Type u_1\ninst✝ : GroupWithZero M\nha : 0 ∈ center M\n⊢ 0 ∈ center M",
"state_before": "case inl\nM : Type u_1\ninst✝ : GroupWithZero M\nha : 0 ∈ center M\n⊢ 0⁻¹ ∈ center M",
"tactic": "rw [inv_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl\nM : Type u_1\ninst✝ : GroupWithZero M\nha : 0 ∈ center M\n⊢ 0 ∈ center M",
"tactic": "exact zero_mem_center M"
}
] | [
112,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/Topology/Instances/AddCircle.lean | toIcoMod_eventuallyEq_toIocMod | [
{
"state_after": "𝕜 : Type u_1\nB : Type ?u.15815\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ↑x ≠ ↑a\n⊢ IsOpen (⋃ (z : ℤ), Ioo (a + z • p) (a + p + z • p))",
"state_before": "𝕜 : Type u_1\nB : Type ?u.15815\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ↑x ≠ ↑a\n⊢ IsOpen {x | (fun x => toIcoMod hp a x = toIocMod hp a x) x}",
"tactic": "rw [Ico_eq_locus_Ioc_eq_iUnion_Ioo]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nB : Type ?u.15815\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ↑x ≠ ↑a\n⊢ IsOpen (⋃ (z : ℤ), Ioo (a + z • p) (a + p + z • p))",
"tactic": "exact isOpen_iUnion fun i => isOpen_Ioo"
}
] | [
103,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.comap_monotone | [
{
"state_after": "ι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\n⊢ Subgraph.comap f H ≤ Subgraph.comap f H'",
"state_before": "ι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\n⊢ Monotone (Subgraph.comap f)",
"tactic": "intro H H' h"
},
{
"state_after": "case left\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\n⊢ (Subgraph.comap f H).verts ⊆ (Subgraph.comap f H').verts\n\ncase right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\n⊢ ∀ ⦃v w : V⦄, Adj (Subgraph.comap f H) v w → Adj (Subgraph.comap f H') v w",
"state_before": "ι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\n⊢ Subgraph.comap f H ≤ Subgraph.comap f H'",
"tactic": "constructor"
},
{
"state_after": "case left\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\na✝ : V\n⊢ a✝ ∈ (Subgraph.comap f H).verts → a✝ ∈ (Subgraph.comap f H').verts",
"state_before": "case left\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\n⊢ (Subgraph.comap f H).verts ⊆ (Subgraph.comap f H').verts",
"tactic": "intro"
},
{
"state_after": "case left\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\na✝ : V\n⊢ ↑f a✝ ∈ H.verts → ↑f a✝ ∈ H'.verts",
"state_before": "case left\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\na✝ : V\n⊢ a✝ ∈ (Subgraph.comap f H).verts → a✝ ∈ (Subgraph.comap f H').verts",
"tactic": "simp only [comap_verts, Set.mem_preimage]"
},
{
"state_after": "no goals",
"state_before": "case left\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\na✝ : V\n⊢ ↑f a✝ ∈ H.verts → ↑f a✝ ∈ H'.verts",
"tactic": "apply h.1"
},
{
"state_after": "case right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\nv w : V\n⊢ Adj (Subgraph.comap f H) v w → Adj (Subgraph.comap f H') v w",
"state_before": "case right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\n⊢ ∀ ⦃v w : V⦄, Adj (Subgraph.comap f H) v w → Adj (Subgraph.comap f H') v w",
"tactic": "intro v w"
},
{
"state_after": "case right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\nv w : V\n⊢ SimpleGraph.Adj G v w → Adj H (↑f v) (↑f w) → Adj H' (↑f v) (↑f w)",
"state_before": "case right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\nv w : V\n⊢ Adj (Subgraph.comap f H) v w → Adj (Subgraph.comap f H') v w",
"tactic": "simp (config := { contextual := true }) only [comap_Adj, and_imp, true_and_iff]"
},
{
"state_after": "case right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\nv w : V\na✝ : SimpleGraph.Adj G v w\n⊢ Adj H (↑f v) (↑f w) → Adj H' (↑f v) (↑f w)",
"state_before": "case right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\nv w : V\n⊢ SimpleGraph.Adj G v w → Adj H (↑f v) (↑f w) → Adj H' (↑f v) (↑f w)",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case right\nι : Sort ?u.167405\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G'\nh : H ≤ H'\nv w : V\na✝ : SimpleGraph.Adj G v w\n⊢ Adj H (↑f v) (↑f w) → Adj H' (↑f v) (↑f w)",
"tactic": "apply h.2"
}
] | [
703,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
694,
1
] |
Mathlib/Topology/Basic.lean | not_mem_of_not_mem_closure | [] | [
424,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/Deprecated/Subgroup.lean | Multiplicative.isSubgroup_iff | [
{
"state_after": "case mk.mk\nG : Type ?u.3681\nH : Type ?u.3684\nA : Type u_1\na a₁ a₂ b c : G\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns : Set A\nh₃ : ∀ {a : Multiplicative A}, a ∈ s → a⁻¹ ∈ s\nh₁ : 1 ∈ s\nh₂ : ∀ {a b : Multiplicative A}, a ∈ s → b ∈ s → a * b ∈ s\n⊢ IsAddSubgroup s",
"state_before": "G : Type ?u.3681\nH : Type ?u.3684\nA : Type u_1\na a₁ a₂ b c : G\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns : Set A\n⊢ IsSubgroup s → IsAddSubgroup s",
"tactic": "rintro ⟨⟨h₁, h₂⟩, h₃⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nG : Type ?u.3681\nH : Type ?u.3684\nA : Type u_1\na a₁ a₂ b c : G\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns : Set A\nh₃ : ∀ {a : Multiplicative A}, a ∈ s → a⁻¹ ∈ s\nh₁ : 1 ∈ s\nh₂ : ∀ {a b : Multiplicative A}, a ∈ s → b ∈ s → a * b ∈ s\n⊢ IsAddSubgroup s",
"tactic": "exact @IsAddSubgroup.mk A _ _ ⟨h₁, @h₂⟩ @h₃"
}
] | [
82,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
] |
src/lean/Init/Data/Nat/Linear.lean | Nat.Linear.PolyCnstr.eq_true_of_isValid | [
{
"state_after": "case mk\nctx : Context\neq✝ : Bool\nlhs✝ rhs✝ : Poly\n⊢ isValid { eq := eq✝, lhs := lhs✝, rhs := rhs✝ } = true → denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ } = True",
"state_before": "ctx : Context\nc : PolyCnstr\n⊢ isValid c = true → denote ctx c = True",
"tactic": "cases c"
},
{
"state_after": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n⊢ isValid { eq := eq, lhs := lhs, rhs := rhs } = true → denote ctx { eq := eq, lhs := lhs, rhs := rhs } = True",
"state_before": "case mk\nctx : Context\neq✝ : Bool\nlhs✝ rhs✝ : Poly\n⊢ isValid { eq := eq✝, lhs := lhs✝, rhs := rhs✝ } = true → denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ } = True",
"tactic": "rename_i eq lhs rhs"
},
{
"state_after": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n⊢ (bif eq then Poly.isZero lhs && Poly.isZero rhs else Poly.isZero lhs) = true →\n denote ctx { eq := eq, lhs := lhs, rhs := rhs } = True",
"state_before": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n⊢ isValid { eq := eq, lhs := lhs, rhs := rhs } = true → denote ctx { eq := eq, lhs := lhs, rhs := rhs } = True",
"tactic": "simp [isValid]"
},
{
"state_after": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\n⊢ Poly.isZero lhs = true ∧ Poly.isZero rhs = true → (Poly.denote ctx lhs = Poly.denote ctx rhs) = True\n\ncase mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : ¬eq = true\n⊢ Poly.isZero lhs = true → (Poly.denote ctx lhs ≤ Poly.denote ctx rhs) = True",
"state_before": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n⊢ (bif eq then Poly.isZero lhs && Poly.isZero rhs else Poly.isZero lhs) = true →\n denote ctx { eq := eq, lhs := lhs, rhs := rhs } = True",
"tactic": "by_cases he : eq = true <;> simp [he, denote, Poly.denote_eq, Poly.denote_le]"
},
{
"state_after": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nh₁ : Poly.isZero lhs = true\nh₂ : Poly.isZero rhs = true\n⊢ (Poly.denote ctx lhs = Poly.denote ctx rhs) = True",
"state_before": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\n⊢ Poly.isZero lhs = true ∧ Poly.isZero rhs = true → (Poly.denote ctx lhs = Poly.denote ctx rhs) = True",
"tactic": "intro ⟨h₁, h₂⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nh₁ : Poly.isZero lhs = true\nh₂ : Poly.isZero rhs = true\n⊢ (Poly.denote ctx lhs = Poly.denote ctx rhs) = True",
"tactic": "simp [Poly.of_isZero, h₁, h₂]"
},
{
"state_after": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : ¬eq = true\nh : Poly.isZero lhs = true\n⊢ (Poly.denote ctx lhs ≤ Poly.denote ctx rhs) = True",
"state_before": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : ¬eq = true\n⊢ Poly.isZero lhs = true → (Poly.denote ctx lhs ≤ Poly.denote ctx rhs) = True",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : ¬eq = true\nh : Poly.isZero lhs = true\n⊢ (Poly.denote ctx lhs ≤ Poly.denote ctx rhs) = True",
"tactic": "simp [Poly.of_isZero, h]"
}
] | [
659,
29
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
652,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.norm_sub_map_le_sub | [] | [
426,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
425,
1
] |
Mathlib/Topology/Basic.lean | isOpen_interior | [] | [
290,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
] |
Mathlib/CategoryTheory/Monoidal/Mon_.lean | Mon_.id_hom' | [] | [
127,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/GroupTheory/Perm/ViaEmbedding.lean | Equiv.Perm.viaEmbeddingHom_apply | [] | [
48,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.EventuallyLE.le_iff_eq | [] | [
1689,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1687,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean | tendstoLocallyUniformlyOn_iff_forall_isCompact | [] | [
753,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
751,
1
] |
Mathlib/Topology/Support.lean | HasCompactSupport.smul_left | [
{
"state_after": "X : Type ?u.20432\nα : Type u_1\nα' : Type ?u.20438\nβ : Type ?u.20441\nγ : Type ?u.20444\nδ : Type ?u.20447\nM : Type u_2\nE : Type ?u.20453\nR : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : MonoidWithZero R\ninst✝¹ : AddMonoid M\ninst✝ : DistribMulAction R M\nf : α → R\nf' : α → M\nx : α\nhf : f' =ᶠ[coclosedCompact α] 0\n⊢ f • f' =ᶠ[coclosedCompact α] 0",
"state_before": "X : Type ?u.20432\nα : Type u_1\nα' : Type ?u.20438\nβ : Type ?u.20441\nγ : Type ?u.20444\nδ : Type ?u.20447\nM : Type u_2\nE : Type ?u.20453\nR : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : MonoidWithZero R\ninst✝¹ : AddMonoid M\ninst✝ : DistribMulAction R M\nf : α → R\nf' : α → M\nx : α\nhf : HasCompactSupport f'\n⊢ HasCompactSupport (f • f')",
"tactic": "rw [hasCompactSupport_iff_eventuallyEq] at hf ⊢"
},
{
"state_after": "no goals",
"state_before": "X : Type ?u.20432\nα : Type u_1\nα' : Type ?u.20438\nβ : Type ?u.20441\nγ : Type ?u.20444\nδ : Type ?u.20447\nM : Type u_2\nE : Type ?u.20453\nR : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : MonoidWithZero R\ninst✝¹ : AddMonoid M\ninst✝ : DistribMulAction R M\nf : α → R\nf' : α → M\nx : α\nhf : f' =ᶠ[coclosedCompact α] 0\n⊢ f • f' =ᶠ[coclosedCompact α] 0",
"tactic": "exact hf.mono fun x hx => by simp_rw [Pi.smul_apply', hx, Pi.zero_apply, smul_zero]"
},
{
"state_after": "no goals",
"state_before": "X : Type ?u.20432\nα : Type u_1\nα' : Type ?u.20438\nβ : Type ?u.20441\nγ : Type ?u.20444\nδ : Type ?u.20447\nM : Type u_2\nE : Type ?u.20453\nR : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : MonoidWithZero R\ninst✝¹ : AddMonoid M\ninst✝ : DistribMulAction R M\nf : α → R\nf' : α → M\nx✝ : α\nhf : f' =ᶠ[coclosedCompact α] 0\nx : α\nhx : f' x = OfNat.ofNat 0 x\n⊢ (f • f') x = OfNat.ofNat 0 x",
"tactic": "simp_rw [Pi.smul_apply', hx, Pi.zero_apply, smul_zero]"
}
] | [
259,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Mathlib/Analysis/Calculus/Series.lean | hasFDerivAt_tsum | [
{
"state_after": "case this\nα : Type u_1\nβ : Type ?u.43853\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhu : Summable u\nhf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n\nhf0 : Summable fun n => f n x₀\nx : E\n⊢ NormedSpace ℝ E\n\nα : Type u_1\nβ : Type ?u.43853\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhu : Summable u\nhf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n\nhf0 : Summable fun n => f n x₀\nx : E\nthis : NormedSpace ℝ E := ?this\n⊢ HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x",
"state_before": "α : Type u_1\nβ : Type ?u.43853\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhu : Summable u\nhf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n\nhf0 : Summable fun n => f n x₀\nx : E\n⊢ HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x",
"tactic": "let : NormedSpace ℝ E"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.43853\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhu : Summable u\nhf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n\nhf0 : Summable fun n => f n x₀\nx : E\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x",
"state_before": "case this\nα : Type u_1\nβ : Type ?u.43853\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhu : Summable u\nhf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n\nhf0 : Summable fun n => f n x₀\nx : E\n⊢ NormedSpace ℝ E\n\nα : Type u_1\nβ : Type ?u.43853\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhu : Summable u\nhf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n\nhf0 : Summable fun n => f n x₀\nx : E\nthis : NormedSpace ℝ E := ?this\n⊢ HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x",
"tactic": "exact NormedSpace.restrictScalars ℝ 𝕜 _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.43853\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhu : Summable u\nhf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n\nhf0 : Summable fun n => f n x₀\nx : E\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\n⊢ HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x",
"tactic": "exact hasFDerivAt_tsum_of_isPreconnected hu isOpen_univ isPreconnected_univ\n (fun n x _ => hf n x) (fun n x _ => hf' n x) (mem_univ _) hf0 (mem_univ _)"
}
] | [
160,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Data/Num/Lemmas.lean | PosNum.le_iff_cmp | [
{
"state_after": "m n : PosNum\n⊢ Ordering.swap (cmp m n) = Ordering.lt ↔ cmp m n = Ordering.gt",
"state_before": "m n : PosNum\n⊢ cmp n m = Ordering.lt ↔ cmp m n = Ordering.gt",
"tactic": "rw [← cmp_swap]"
},
{
"state_after": "no goals",
"state_before": "m n : PosNum\n⊢ Ordering.swap (cmp m n) = Ordering.lt ↔ cmp m n = Ordering.gt",
"tactic": "cases cmp m n <;> exact by decide"
},
{
"state_after": "no goals",
"state_before": "m n : PosNum\n⊢ Ordering.swap Ordering.gt = Ordering.lt ↔ Ordering.gt = Ordering.gt",
"tactic": "decide"
}
] | [
830,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
829,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean | hasFDerivWithinAt_pi' | [] | [
422,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
420,
1
] |
Mathlib/Computability/Primrec.lean | Primrec.ulower_up | [] | [
1267,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1265,
1
] |
Mathlib/Analysis/Convex/Complex.lean | convex_halfspace_re_ge | [] | [
35,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
34,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.disjoint_sdiff_inter | [] | [
2356,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2355,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ioo_eq_empty_of_le | [] | [
398,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
397,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.closure_empty | [] | [
1242,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1241,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean | mem_zpowers_iff_mem_range_orderOf | [
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝³ : Group G\ninst✝² : AddGroup A\ninst✝¹ : Finite G\ninst✝ : DecidableEq G\n⊢ y ∈ zpowers x ↔ y ∈ Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (orderOf x))",
"tactic": "rw [← mem_powers_iff_mem_zpowers, mem_powers_iff_mem_range_orderOf]"
}
] | [
832,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
830,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.zero_lt_one | [] | [
415,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
414,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | CategoryTheory.Limits.prod.diag_map_fst_snd | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX✝ Y✝ X Y : C\ninst✝¹ : HasBinaryProduct X Y\ninst✝ : HasBinaryProduct (X ⨯ Y) (X ⨯ Y)\n⊢ diag (X ⨯ Y) ≫ map fst snd = 𝟙 (X ⨯ Y)",
"tactic": "simp"
}
] | [
809,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
808,
1
] |
Mathlib/Algebra/Algebra/Basic.lean | RingHom.algebraMap_toAlgebra | [] | [
276,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
] |
Mathlib/Data/Real/ConjugateExponents.lean | Real.IsConjugateExponent.sub_one_pos | [] | [
59,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
1
] |
Mathlib/Order/LocallyFinite.lean | Finset.subtype_Ioc_eq | [] | [
1276,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1275,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.adjoin_rootSet_isSplittingField | [] | [
543,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Complex.tan_pi_sub | [] | [
1311,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1310,
1
] |
Mathlib/Data/Real/Hyperreal.lean | Hyperreal.infinitesimal_neg | [] | [
706,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
705,
9
] |
Mathlib/Algebra/Order/Ring/Defs.lean | bit1_lt_bit1 | [] | [
905,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
904,
1
] |
Std/Data/List/Lemmas.lean | List.exists_of_mem_join | [] | [
186,
75
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
186,
1
] |
Mathlib/Topology/QuasiSeparated.lean | IsQuasiSeparated.of_quasiSeparatedSpace | [] | [
128,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/Data/Set/Image.lean | Set.range_comp | [] | [
722,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
720,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.map_snd_darts | [
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\n⊢ List.map (fun x => x.snd) (darts p) = List.tail (support p)",
"tactic": "simpa using congr_arg List.tail (cons_map_snd_darts p)"
}
] | [
727,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
726,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | Matrix.SpecialLinearGroup.SL2_inv_expl_det | [
{
"state_after": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\n⊢ vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 1 1 * vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 0 0 -\n vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 0 1 * vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 1 0 =\n 1",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\n⊢ det ![![↑A 1 1, -↑A 0 1], ![-↑A 1 0, ↑A 0 0]] = 1",
"tactic": "rw [Matrix.det_fin_two, mul_comm]"
},
{
"state_after": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\n⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\n⊢ vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 1 1 * vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 0 0 -\n vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 0 1 * vecCons ![↑A 1 1, -↑A 0 1] ![![-↑A 1 0, ↑A 0 0]] 1 0 =\n 1",
"tactic": "simp only [cons_val_zero, cons_val_one, head_cons, mul_neg, neg_mul, neg_neg]"
},
{
"state_after": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\nthis : det ↑A = 1\n⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\n⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1",
"tactic": "have := A.2"
},
{
"state_after": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\nthis : ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1\n⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\nthis : det ↑A = 1\n⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1",
"tactic": "rw [Matrix.det_fin_two] at this"
},
{
"state_after": "no goals",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\nS : Type ?u.737904\ninst✝ : CommRing S\nA : SL(2, R)\nthis : ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1\n⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1",
"tactic": "convert this"
}
] | [
294,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
288,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.bot_add | [] | [
644,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
643,
1
] |
Mathlib/Order/Hom/Basic.lean | OrderIso.isCompl_iff | [] | [
1373,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1372,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean | Real.summable_pow_div_factorial | [
{
"state_after": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\n⊢ Summable fun n => x ^ n / ↑n !",
"state_before": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\n⊢ Summable fun n => x ^ n / ↑n !",
"tactic": "have A : (0 : ℝ) < ⌊‖x‖⌋₊ + 1 := zero_lt_one.trans_le (by simp)"
},
{
"state_after": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\n⊢ Summable fun n => x ^ n / ↑n !",
"state_before": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\n⊢ Summable fun n => x ^ n / ↑n !",
"tactic": "have B : ‖x‖ / (⌊‖x‖⌋₊ + 1) < 1 := (div_lt_one A).2 (Nat.lt_floor_add_one _)"
},
{
"state_after": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\nthis : ∀ (n : ℕ), n ≥ ⌊‖x‖⌋₊ → ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖\n⊢ Summable fun n => x ^ n / ↑n !\n\ncase this\nα : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\n⊢ ∀ (n : ℕ), n ≥ ⌊‖x‖⌋₊ → ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖",
"state_before": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\n⊢ Summable fun n => x ^ n / ↑n !",
"tactic": "suffices : ∀ n ≥ ⌊‖x‖⌋₊, ‖x ^ (n + 1) / (n + 1)!‖ ≤ ‖x‖ / (⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖"
},
{
"state_after": "case this\nα : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\n⊢ ∀ (n : ℕ), n ≥ ⌊‖x‖⌋₊ → ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖",
"state_before": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\nthis : ∀ (n : ℕ), n ≥ ⌊‖x‖⌋₊ → ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖\n⊢ Summable fun n => x ^ n / ↑n !\n\ncase this\nα : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\n⊢ ∀ (n : ℕ), n ≥ ⌊‖x‖⌋₊ → ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖",
"tactic": "exact summable_of_ratio_norm_eventually_le B (eventually_atTop.2 ⟨⌊‖x‖⌋₊, this⟩)"
},
{
"state_after": "case this\nα : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\nn : ℕ\nhn : n ≥ ⌊‖x‖⌋₊\n⊢ ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖",
"state_before": "case this\nα : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\n⊢ ∀ (n : ℕ), n ≥ ⌊‖x‖⌋₊ → ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖",
"tactic": "intro n hn"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\nn : ℕ\nhn : n ≥ ⌊‖x‖⌋₊\n⊢ ‖x ^ (n + 1) / ↑(n + 1)!‖ ≤ ‖x‖ / (↑⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖",
"tactic": "calc\n ‖x ^ (n + 1) / (n + 1)!‖ = ‖x‖ / (n + 1) * ‖x ^ n / (n !)‖ := by\n rw [_root_.pow_succ, Nat.factorial_succ, Nat.cast_mul, ← _root_.div_mul_div_comm, norm_mul,\n norm_div, Real.norm_coe_nat, Nat.cast_succ]\n _ ≤ ‖x‖ / (⌊‖x‖⌋₊ + 1) * ‖x ^ n / (n !)‖ :=\n mul_le_mul_of_nonneg_right\n (div_le_div (norm_nonneg x) (le_refl ‖x‖) A (add_le_add (mono_cast hn) (le_refl 1)))\n (norm_nonneg (x ^ n / n !))"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\n⊢ 1 ≤ ↑⌊‖x‖⌋₊ + 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.2025885\nβ : Type ?u.2025888\nι : Type ?u.2025891\nx : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\nn : ℕ\nhn : n ≥ ⌊‖x‖⌋₊\n⊢ ‖x ^ (n + 1) / ↑(n + 1)!‖ = ‖x‖ / (↑n + 1) * ‖x ^ n / ↑n !‖",
"tactic": "rw [_root_.pow_succ, Nat.factorial_succ, Nat.cast_mul, ← _root_.div_mul_div_comm, norm_mul,\n norm_div, Real.norm_coe_nat, Nat.cast_succ]"
}
] | [
675,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
657,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearEquiv.arrowCongr_apply | [] | [
2329,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2325,
1
] |
Mathlib/Order/Compare.lean | cmp_self_eq_eq | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.19132\ninst✝ : LinearOrder α\nx y : α\n⊢ cmp x x = eq",
"tactic": "rw [cmp_eq_eq_iff]"
}
] | [
244,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
244,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.range_toAddSubmonoid | [] | [
1207,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1205,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.subtypeDomain_zero | [] | [
1043,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1042,
1
] |
Mathlib/Topology/Bornology/Constructions.lean | Bornology.cobounded_pi | [] | [
104,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Topology/MetricSpace/CauSeqFilter.lean | isCauSeq_iff_cauchySeq | [] | [
90,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean | CategoryTheory.Sieve.pullback_id | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\n⊢ pullback (𝟙 X) S = S",
"tactic": "simp [Sieve.ext_iff]"
}
] | [
464,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
464,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | Complex.log_im | [
{
"state_after": "no goals",
"state_before": "x : ℂ\n⊢ (log x).im = arg x",
"tactic": "simp [log]"
}
] | [
39,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean | Units.eq_or_eq_neg_of_sq_eq_sq | [] | [
312,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
11
] |
Mathlib/Data/Set/Intervals/Group.lean | Set.add_mem_Ioc_iff_right | [] | [
93,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
] |
Mathlib/Algebra/Category/Mon/FilteredColimits.lean | MonCat.FilteredColimits.colimit_one_eq | [
{
"state_after": "case h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj : J\n⊢ ∃ k f g, ↑(F.map f) { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.snd = ↑(F.map g) { fst := j, snd := 1 }.snd",
"state_before": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj : J\n⊢ 1 = M.mk F { fst := j, snd := 1 }",
"tactic": "apply M.mk_eq"
},
{
"state_after": "case h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj : J\n⊢ ↑(F.map (IsFiltered.leftToMax { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.fst j))\n { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.snd =\n ↑(F.map (IsFiltered.rightToMax { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.fst j))\n { fst := j, snd := 1 }.snd",
"state_before": "case h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj : J\n⊢ ∃ k f g, ↑(F.map f) { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.snd = ↑(F.map g) { fst := j, snd := 1 }.snd",
"tactic": "refine' ⟨max' _ j, IsFiltered.leftToMax _ j, IsFiltered.rightToMax _ j, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj : J\n⊢ ↑(F.map (IsFiltered.leftToMax { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.fst j))\n { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.snd =\n ↑(F.map (IsFiltered.rightToMax { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.fst j))\n { fst := j, snd := 1 }.snd",
"tactic": "simp"
}
] | [
110,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.rightMoves_mk | [] | [
152,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.BoundedFormula.IsQF.induction_on_sup_not | [] | [
589,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
581,
1
] |
Mathlib/NumberTheory/Bernoulli.lean | bernoulli'_four | [
{
"state_after": "A : Type ?u.188932\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nthis : Nat.choose 4 2 = 6\n⊢ bernoulli' 4 = -1 / 30",
"state_before": "A : Type ?u.188932\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ bernoulli' 4 = -1 / 30",
"tactic": "have : Nat.choose 4 2 = 6 := by decide"
},
{
"state_after": "A : Type ?u.188932\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nthis : Nat.choose 4 2 = 6\n⊢ 1 -\n (0 + ↑(Nat.choose 4 0) / (↑4 - ↑0 + 1) * bernoulli' 0 + ↑(Nat.choose 4 1) / (↑4 - ↑1 + 1) * bernoulli' 1 +\n ↑6 / (↑4 - ↑2 + 1) * bernoulli' 2 +\n ↑(Nat.choose 4 3) / (↑4 - ↑3 + 1) * bernoulli' 3) =\n -1 / 30",
"state_before": "A : Type ?u.188932\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nthis : Nat.choose 4 2 = 6\n⊢ bernoulli' 4 = -1 / 30",
"tactic": "rw [bernoulli'_def, sum_range_succ, sum_range_succ, sum_range_succ,\n sum_range_succ, sum_range_zero, this]"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.188932\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nthis : Nat.choose 4 2 = 6\n⊢ 1 -\n (0 + ↑(Nat.choose 4 0) / (↑4 - ↑0 + 1) * bernoulli' 0 + ↑(Nat.choose 4 1) / (↑4 - ↑1 + 1) * bernoulli' 1 +\n ↑6 / (↑4 - ↑2 + 1) * bernoulli' 2 +\n ↑(Nat.choose 4 3) / (↑4 - ↑3 + 1) * bernoulli' 3) =\n -1 / 30",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.188932\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\n⊢ Nat.choose 4 2 = 6",
"tactic": "decide"
}
] | [
135,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.map_join | [
{
"state_after": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\nS : WSeq (WSeq α)\n⊢ Seq.IsBisimulation fun s1 s2 => ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))\n\ncase r\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\nS : WSeq (WSeq α)\n⊢ ∃ s S_1, map f (join S) = append s (map f (join S_1)) ∧ join (map (map f) S) = append s (join (map (map f) S_1))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\nS : WSeq (WSeq α)\n⊢ map f (join S) = join (map (map f) S)",
"tactic": "apply\n Seq.eq_of_bisim fun s1 s2 =>\n ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))"
},
{
"state_after": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\nS : WSeq (WSeq α)\ns1 s2 : Seq (Option β)\nh : ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))\n⊢ Seq.BisimO (fun s1 s2 => ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S)))\n (Seq.destruct s1) (Seq.destruct s2)",
"state_before": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\nS : WSeq (WSeq α)\n⊢ Seq.IsBisimulation fun s1 s2 => ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))",
"tactic": "intro s1 s2 h"
},
{
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{
"state_after": "case h1.h2\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\nS✝ : WSeq (WSeq α)\ns1 s2 : Seq (Option β)\nh : ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))\ns : WSeq α\nS : WSeq (WSeq α)\n⊢ ∃ s_1 S_1,\n append (map f s) (map f (join S)) = append s_1 (map f (join S_1)) ∧\n append (map f s) (join (map (map f) S)) = append s_1 (join (map (map f) S_1))\n\ncase h1.h3\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\nS✝ : WSeq (WSeq α)\ns1 s2 : Seq (Option β)\nh : ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))\nS : WSeq (WSeq α)\n⊢ ∃ s S_1, map f (join S) = append s (map f (join S_1)) ∧ join (map (map f) S) = append s (join (map (map f) S_1))",
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{
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},
{
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{
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},
{
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{
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] | [
1761,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1747,
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Mathlib/Order/Zorn.lean | zorn_partialOrder | [] | [
164,
44
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161,
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Mathlib/LinearAlgebra/Basic.lean | Submodule.comap_top | [] | [
855,
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Mathlib/Data/Set/Sups.lean | Set.sups_union_left | [] | [
161,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
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Mathlib/MeasureTheory/Group/Integration.lean | MeasureTheory.integral_div_left_eq_self | [
{
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"tactic": "simp_rw [div_eq_mul_inv]"
},
{
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"tactic": "rw [integral_inv_eq_self (fun x => f (x' * x)) μ, integral_mul_left_eq_self f x']"
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] | [
194,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
190,
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Mathlib/Data/Real/Hyperreal.lean | Hyperreal.Infinitesimal.neg | [
{
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] | [
701,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
700,
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Mathlib/Data/Matrix/Kronecker.lean | Matrix.mul_kronecker_mul | [] | [
362,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
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Mathlib/RingTheory/Adjoin/Basic.lean | Algebra.adjoin_span | [] | [
212,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
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Mathlib/Data/Set/NAry.lean | Set.image2_image_left | [
{
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{
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},
{
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{
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{
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{
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"tactic": "refine' ⟨_, b, ⟨a, ha, rfl⟩, hb, rfl⟩"
}
] | [
288,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Analysis/Normed/Group/Basic.lean | comap_norm_nhds_one | [
{
"state_after": "no goals",
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"tactic": "simpa only [dist_one_right] using nhds_comap_dist (1 : E)"
}
] | [
1041,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1040,
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Mathlib/Data/Finset/Basic.lean | Finset.val_le_iff | [] | [
385,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
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Mathlib/Data/List/Sort.lean | List.mergeSort_eq_insertionSort | [] | [
460,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
457,
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Mathlib/Deprecated/Subgroup.lean | Group.closure_mono | [] | [
562,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
561,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.Subsingleton.atTop_eq | [
{
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"tactic": "refine' top_unique fun s hs x => _"
},
{
"state_after": "ι : Type ?u.24415\nι' : Type ?u.24418\nα✝ : Type ?u.24421\nβ : Type ?u.24424\nγ : Type ?u.24427\nα : Type u_1\ninst✝¹ : Subsingleton α\ninst✝ : Preorder α\ns : Set α\nx : α\nhs : Ici x ⊆ s\n⊢ x ∈ s",
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"tactic": "rw [atTop, ciInf_subsingleton x, mem_principal] at hs"
},
{
"state_after": "no goals",
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"tactic": "exact hs left_mem_Ici"
}
] | [
295,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
] |
Mathlib/RingTheory/Finiteness.lean | Module.Finite.of_restrictScalars_finite | [
{
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"tactic": "rw [finite_def, Submodule.fg_def] at hM⊢"
},
{
"state_after": "case intro.intro\nR✝ : Type ?u.395725\nA✝ : Type ?u.395728\nB : Type ?u.395731\nM✝ : Type ?u.395734\nN : Type ?u.395737\ninst✝¹¹ : Semiring R✝\ninst✝¹⁰ : AddCommMonoid M✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : AddCommMonoid N\ninst✝⁷ : Module R✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module A M\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A M\nS : Set M\nhSfin : Set.Finite S\nhSgen : Submodule.span R S = ⊤\n⊢ ∃ S, Set.Finite S ∧ Submodule.span A S = ⊤",
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"tactic": "obtain ⟨S, hSfin, hSgen⟩ := hM"
},
{
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"tactic": "refine' ⟨S, hSfin, eq_top_iff.2 _⟩"
},
{
"state_after": "case intro.intro\nR✝ : Type ?u.395725\nA✝ : Type ?u.395728\nB : Type ?u.395731\nM✝ : Type ?u.395734\nN : Type ?u.395737\ninst✝¹¹ : Semiring R✝\ninst✝¹⁰ : AddCommMonoid M✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : AddCommMonoid N\ninst✝⁷ : Module R✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module A M\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A M\nS : Set M\nhSfin : Set.Finite S\nhSgen : Submodule.span R S = ⊤\nthis : Submodule.span R S ≤ Submodule.restrictScalars R (Submodule.span A S)\n⊢ ⊤ ≤ Submodule.span A S",
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"tactic": "have := Submodule.span_le_restrictScalars R A S"
},
{
"state_after": "case intro.intro\nR✝ : Type ?u.395725\nA✝ : Type ?u.395728\nB : Type ?u.395731\nM✝ : Type ?u.395734\nN : Type ?u.395737\ninst✝¹¹ : Semiring R✝\ninst✝¹⁰ : AddCommMonoid M✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : AddCommMonoid N\ninst✝⁷ : Module R✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module A M\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A M\nS : Set M\nhSfin : Set.Finite S\nhSgen : Submodule.span R S = ⊤\nthis : ⊤ ≤ Submodule.restrictScalars R (Submodule.span A S)\n⊢ ⊤ ≤ Submodule.span A S",
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"tactic": "rw [hSgen] at this"
},
{
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"tactic": "exact this"
}
] | [
593,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
585,
1
] |
Mathlib/Data/Quot.lean | Quotient.lift_mk | [] | [
311,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/Data/PFunctor/Multivariate/W.lean | MvPFunctor.w_cases | [] | [
237,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/Algebra/Module/Injective.lean | Module.injective_object_of_injective_module | [
{
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"tactic": "rcases Module.Injective.out _ _ f ((ModuleCat.mono_iff_injective f).mp mn) g with ⟨h, eq1⟩"
},
{
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"tactic": "exact ⟨h, LinearMap.ext eq1⟩"
}
] | [
71,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
67,
1
] |
Mathlib/GroupTheory/Congruence.lean | Con.induction_on | [] | [
342,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
341,
11
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.BoundedFormula.realize_relabelEquiv | [
{
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"tactic": "simp only [relabelEquiv, mapTermRelEquiv_apply, Equiv.coe_refl]"
},
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.163438\nP : Type ?u.163441\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ng : α ≃ β\nk : ℕ\nφ : BoundedFormula L α k\nv : β → M\nxs✝ : Fin k → M\nn : ℕ\nt : Term L (α ⊕ Fin n)\nxs : Fin n → M\n⊢ realize (Sum.elim v xs) (↑(Term.relabelEquiv (Equiv.sumCongr g (_root_.Equiv.refl (Fin n)))) t) =\n realize (Sum.elim (v ∘ ↑g) xs) t",
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"tactic": "refine' realize_mapTermRel_id (fun n t xs => _) fun _ _ _ => rfl"
},
{
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"tactic": "simp only [relabelEquiv_apply, Term.realize_relabel]"
},
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.163438\nP : Type ?u.163441\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ng : α ≃ β\nk : ℕ\nφ : BoundedFormula L α k\nv : β → M\nxs✝ : Fin k → M\nn : ℕ\nt : Term L (α ⊕ Fin n)\nxs : Fin n → M\n⊢ Sum.elim v xs ∘ ↑(Equiv.sumCongr g (_root_.Equiv.refl (Fin n))) = Sum.elim (v ∘ ↑g) xs",
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"tactic": "refine' congr (congr rfl _) rfl"
},
{
"state_after": "no goals",
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"tactic": "ext (i | i) <;> rfl"
}
] | [
496,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
490,
1
] |
Mathlib/Topology/Semicontinuous.lean | upperSemicontinuous_const | [] | [
728,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
727,
1
] |
Mathlib/Order/SymmDiff.lean | IsCompl.symmDiff_eq_top | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.56328\nα : Type u_1\nβ : Type ?u.56334\nπ : ι → Type ?u.56339\ninst✝ : CoheytingAlgebra α\na✝ a b : α\nh : IsCompl a b\n⊢ a ∆ b = ⊤",
"tactic": "rw [h.eq_hnot, hnot_symmDiff_self]"
}
] | [
359,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
358,
1
] |
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.closure_induction | [] | [
353,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
] |
Mathlib/Algebra/Quaternion.lean | Quaternion.normSq_star | [
{
"state_after": "no goals",
"state_before": "S : Type ?u.588276\nT : Type ?u.588279\nR : Type u_1\ninst✝ : CommRing R\nr x y z : R\na b c : ℍ[R]\n⊢ ↑normSq (star a) = ↑normSq a",
"tactic": "simp [normSq_def']"
}
] | [
1211,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1211,
1
] |
Mathlib/Analysis/MeanInequalitiesPow.lean | Real.zpow_arith_mean_le_arith_mean_zpow | [] | [
96,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Algebra/Opposites.lean | MulOpposite.op_sub | [] | [
309,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
308,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | Complex.deriv_sinh | [] | [
132,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean | mulSalemSpencer_insert | [
{
"state_after": "F : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\n⊢ (MulSalemSpencer s ∧\n (∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b) ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c) →\n MulSalemSpencer (insert a s)",
"state_before": "F : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\n⊢ MulSalemSpencer (insert a s) ↔\n MulSalemSpencer s ∧\n (∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b) ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c",
"tactic": "refine' ⟨fun hs => ⟨hs.mono (subset_insert _ _),\n fun b c hb hc => hs (Or.inl rfl) (Or.inr hb) (Or.inr hc),\n fun b c hb hc => hs (Or.inr hb) (Or.inr hc) (Or.inl rfl)⟩, _⟩"
},
{
"state_after": "case intro.intro\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nhb : b ∈ insert a s\nhc : c ∈ insert a s\nhd : d ∈ insert a s\nh : b * c = d * d\n⊢ b = c",
"state_before": "F : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\n⊢ (MulSalemSpencer s ∧\n (∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b) ∧ ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c) →\n MulSalemSpencer (insert a s)",
"tactic": "rintro ⟨hs, ha, ha'⟩ b c d hb hc hd h"
},
{
"state_after": "case intro.intro\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nhb : b = a ∨ b ∈ s\nhc : c = a ∨ c ∈ s\nhd : d = a ∨ d ∈ s\nh : b * c = d * d\n⊢ b = c",
"state_before": "case intro.intro\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nhb : b ∈ insert a s\nhc : c ∈ insert a s\nhd : d ∈ insert a s\nh : b * c = d * d\n⊢ b = c",
"tactic": "rw [mem_insert_iff] at hb hc hd"
},
{
"state_after": "case intro.intro.inl.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nc d : α\nh : c * c = d * d\nha : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → c * b = c_1 * c_1 → c = b\nha' : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → b * c_1 = c * c → b = c_1\nhd : d = c ∨ d ∈ s\n⊢ c = c\n\ncase intro.intro.inl.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nha : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b * b_1 = c * c → b = b_1\nha' : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b_1 * c = b * b → b_1 = c\nhd : d = b ∨ d ∈ s\nhc : c ∈ s\n⊢ b = c\n\ncase intro.intro.inr.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nha : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → c * b = c_1 * c_1 → c = b\nha' : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → b * c_1 = c * c → b = c_1\nhd : d = c ∨ d ∈ s\n⊢ b = c\n\ncase intro.intro.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nhd : d = a ∨ d ∈ s\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\n⊢ b = c",
"state_before": "case intro.intro\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nhb : b = a ∨ b ∈ s\nhc : c = a ∨ c ∈ s\nhd : d = a ∨ d ∈ s\nh : b * c = d * d\n⊢ b = c",
"tactic": "obtain rfl | hb := hb <;> obtain rfl | hc := hc"
},
{
"state_after": "case intro.intro.inl.inr.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nc d : α\nhc : c ∈ s\nh : d * c = d * d\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → d * b = c * c → d = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = d * d → b = c\n⊢ d = c\n\ncase intro.intro.inl.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nha : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b * b_1 = c * c → b = b_1\nha' : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b_1 * c = b * b → b_1 = c\nhc : c ∈ s\nhd : d ∈ s\n⊢ b = c\n\ncase intro.intro.inr.inl.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb d : α\nhb : b ∈ s\nh : b * d = d * d\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → d * b = c * c → d = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = d * d → b = c\n⊢ b = d\n\ncase intro.intro.inr.inl.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nha : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → c * b = c_1 * c_1 → c = b\nha' : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → b * c_1 = c * c → b = c_1\nhd : d ∈ s\n⊢ b = c\n\ncase intro.intro.inr.inr.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → d * b = c * c → d = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = d * d → b = c\n⊢ b = c\n\ncase intro.intro.inr.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\nhd : d ∈ s\n⊢ b = c",
"state_before": "case intro.intro.inl.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nha : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b * b_1 = c * c → b = b_1\nha' : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b_1 * c = b * b → b_1 = c\nhd : d = b ∨ d ∈ s\nhc : c ∈ s\n⊢ b = c\n\ncase intro.intro.inr.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nha : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → c * b = c_1 * c_1 → c = b\nha' : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → b * c_1 = c * c → b = c_1\nhd : d = c ∨ d ∈ s\n⊢ b = c\n\ncase intro.intro.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nhd : d = a ∨ d ∈ s\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\n⊢ b = c",
"tactic": "all_goals obtain rfl | hd := hd"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nc d : α\nh : c * c = d * d\nha : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → c * b = c_1 * c_1 → c = b\nha' : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → b * c_1 = c * c → b = c_1\nhd : d = c ∨ d ∈ s\n⊢ c = c",
"tactic": "rfl"
},
{
"state_after": "case intro.intro.inr.inr.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → d * b = c * c → d = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = d * d → b = c\n⊢ b = c\n\ncase intro.intro.inr.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\nhd : d ∈ s\n⊢ b = c",
"state_before": "case intro.intro.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nhd : d = a ∨ d ∈ s\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\n⊢ b = c",
"tactic": "obtain rfl | hd := hd"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl.inr.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nc d : α\nhc : c ∈ s\nh : d * c = d * d\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → d * b = c * c → d = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = d * d → b = c\n⊢ d = c",
"tactic": "exact (mul_left_cancel h).symm"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nha : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b * b_1 = c * c → b = b_1\nha' : ∀ ⦃b_1 c : α⦄, b_1 ∈ s → c ∈ s → b_1 * c = b * b → b_1 = c\nhc : c ∈ s\nhd : d ∈ s\n⊢ b = c",
"tactic": "exact ha hc hd h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inl.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb d : α\nhb : b ∈ s\nh : b * d = d * d\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → d * b = c * c → d = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = d * d → b = c\n⊢ b = d",
"tactic": "exact mul_right_cancel h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inl.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nha : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → c * b = c_1 * c_1 → c = b\nha' : ∀ ⦃b c_1 : α⦄, b ∈ s → c_1 ∈ s → b * c_1 = c * c → b = c_1\nhd : d ∈ s\n⊢ b = c",
"tactic": "exact (ha hb hd <| (mul_comm _ _).trans h).symm"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inr.inl\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\nhs : MulSalemSpencer s\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → d * b = c * c → d = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = d * d → b = c\n⊢ b = c",
"tactic": "exact ha' hb hc h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inr.inr\nF : Type ?u.68556\nα : Type u_1\nβ : Type ?u.68562\n𝕜 : Type ?u.68565\nE : Type ?u.68568\ninst✝ : CancelCommMonoid α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → a * b = c * c → a = b\nha' : ∀ ⦃b c : α⦄, b ∈ s → c ∈ s → b * c = a * a → b = c\nb c d : α\nh : b * c = d * d\nhb : b ∈ s\nhc : c ∈ s\nhd : d ∈ s\n⊢ b = c",
"tactic": "exact hs hb hc hd h"
}
] | [
164,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Data/Polynomial/Splits.lean | Polynomial.splits_of_splits_id | [
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf x✝¹ : K[X]\nhu : IsUnit x✝¹\nx✝ : Splits (RingHom.id K) x✝¹\n⊢ 0 ≤ 1",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf a p : K[X]\nha0 : a ≠ 0\nhp : Prime p\nih : Splits (RingHom.id K) a → Splits i a\nhfi : Splits (RingHom.id K) (p * a)\n⊢ p ∣ map (RingHom.id K) p",
"tactic": "rw [map_id]"
}
] | [
418,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
410,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean | PadicSeq.add_eq_max_of_ne | [
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis : LimZero (f - 0)\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have : LimZero (f - 0) := hf"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis✝ : LimZero (f - 0)\nthis : f + g ≈ g\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis : LimZero (f - 0)\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have : f + g ≈ g := show LimZero (f + g - g) by simpa only [sub_zero, add_sub_cancel]"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis✝ : LimZero (f - 0)\nthis : f + g ≈ g\nh1 : norm (f + g) = norm g\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis✝ : LimZero (f - 0)\nthis : f + g ≈ g\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have h1 : (f + g).norm = g.norm := norm_equiv this"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis✝ : LimZero (f - 0)\nthis : f + g ≈ g\nh1 : norm (f + g) = norm g\nh2 : norm f = 0\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis✝ : LimZero (f - 0)\nthis : f + g ≈ g\nh1 : norm (f + g) = norm g\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have h2 : f.norm = 0 := (norm_zero_iff _).2 hf"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis✝ : LimZero (f - 0)\nthis : f + g ≈ g\nh1 : norm (f + g) = norm g\nh2 : norm f = 0\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "rw [h1, h2, max_eq_right (norm_nonneg _)]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : f ≈ 0\nthis : LimZero (f - 0)\n⊢ LimZero (f + g - g)",
"tactic": "simpa only [sub_zero, add_sub_cancel]"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis : LimZero (g - 0)\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have : LimZero (g - 0) := hg"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis✝ : LimZero (g - 0)\nthis : f + g ≈ f\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis : LimZero (g - 0)\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have : f + g ≈ f := show LimZero (f + g - f) by rw [add_sub_cancel']; simpa only [sub_zero]"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis✝ : LimZero (g - 0)\nthis : f + g ≈ f\nh1 : norm (f + g) = norm f\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis✝ : LimZero (g - 0)\nthis : f + g ≈ f\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have h1 : (f + g).norm = f.norm := norm_equiv this"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis✝ : LimZero (g - 0)\nthis : f + g ≈ f\nh1 : norm (f + g) = norm f\nh2 : norm g = 0\n⊢ norm (f + g) = max (norm f) (norm g)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis✝ : LimZero (g - 0)\nthis : f + g ≈ f\nh1 : norm (f + g) = norm f\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "have h2 : g.norm = 0 := (norm_zero_iff _).2 hg"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis✝ : LimZero (g - 0)\nthis : f + g ≈ f\nh1 : norm (f + g) = norm f\nh2 : norm g = 0\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "rw [h1, h2, max_eq_left (norm_nonneg _)]"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis : LimZero (g - 0)\n⊢ LimZero g",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis : LimZero (g - 0)\n⊢ LimZero (f + g - f)",
"tactic": "rw [add_sub_cancel']"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : g ≈ 0\nthis : LimZero (g - 0)\n⊢ LimZero g",
"tactic": "simpa only [sub_zero]"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne :\n (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) ≠\n if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf))\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ (if hf : f + g ≈ 0 then 0 else padicNorm p (↑(f + g) (stationaryPoint hf))) =\n max (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf)))\n (if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf)))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f ≠ norm g\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ norm (f + g) = max (norm f) (norm g)",
"tactic": "unfold norm at hfgne⊢"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) =\n max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne :\n (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) ≠\n if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf))\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ (if hf : f + g ≈ 0 then 0 else padicNorm p (↑(f + g) (stationaryPoint hf))) =\n max (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf)))\n (if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf)))",
"tactic": "split_ifs at hfgne⊢"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) =\n max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) =\n max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))",
"tactic": "rw [lift_index_left hf, lift_index_right hg] at hfgne"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ padicNorm p (↑(f + g) (max (stationaryPoint hfg) (max ?v2 ?v3))) =\n max (padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))))\n (padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg)))))\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) =\n max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ",
"tactic": "rw [lift_index_left_left hfg, lift_index_left hf, lift_index_right hg]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ padicNorm p (↑(f + g) (max (stationaryPoint hfg) (max ?v2 ?v3))) =\n max (padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))))\n (padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg)))))\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝¹ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne✝ : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne :\n padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg))))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne✝ : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\nhfgne : padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nhfgne : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))\n⊢ ℕ",
"tactic": "exact padicNorm.add_eq_max_of_ne hfgne"
}
] | [
457,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
436,
1
] |
Mathlib/Data/Multiset/Fintype.lean | Multiset.coe_mem | [] | [
92,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean | HomologicalComplex.Hom.next_eq | [
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC C₁ C₂ C₃ : HomologicalComplex V c\nf : Hom C₁ C₂\ni : ι\nw : ComplexShape.Rel c i (ComplexShape.next c i)\n⊢ next f i = (xNextIso C₁ w).hom ≫ HomologicalComplex.Hom.f f (ComplexShape.next c i) ≫ (xNextIso C₂ w).inv",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC C₁ C₂ C₃ : HomologicalComplex V c\nf : Hom C₁ C₂\ni j : ι\nw : ComplexShape.Rel c i j\n⊢ next f i = (xNextIso C₁ w).hom ≫ HomologicalComplex.Hom.f f j ≫ (xNextIso C₂ w).inv",
"tactic": "obtain rfl := c.next_eq' w"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC C₁ C₂ C₃ : HomologicalComplex V c\nf : Hom C₁ C₂\ni : ι\nw : ComplexShape.Rel c i (ComplexShape.next c i)\n⊢ next f i = (xNextIso C₁ w).hom ≫ HomologicalComplex.Hom.f f (ComplexShape.next c i) ≫ (xNextIso C₂ w).inv",
"tactic": "simp only [xNextIso, eqToIso_refl, Iso.refl_hom, Iso.refl_inv, comp_id, id_comp]"
}
] | [
541,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
538,
1
] |
Mathlib/Logic/Equiv/Basic.lean | Equiv.prodComm_symm | [] | [
134,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
] |
Mathlib/Topology/SubsetProperties.lean | OpenEmbedding.locallyCompactSpace | [
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nthis : ∀ (x : α), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s\n⊢ LocallyCompactSpace α",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\n⊢ LocallyCompactSpace α",
"tactic": "have : ∀ x : α, (𝓝 x).HasBasis\n (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s := by\n intro x\n rw [hf.toInducing.nhds_eq_comap]\n exact\n ((compact_basis_nhds _).restrict_subset <| hf.open_range.mem_nhds <| mem_range_self _).comap _"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nthis : ∀ (x : α), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s\nx : α\ns : Set β\nhs : (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f\n⊢ IsCompact (f ⁻¹' s)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nthis : ∀ (x : α), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s\n⊢ LocallyCompactSpace α",
"tactic": "refine' locallyCompactSpace_of_hasBasis this fun x s hs => _"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nthis : ∀ (x : α), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s\nx : α\ns : Set β\nhs : (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f\n⊢ IsCompact s",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nthis : ∀ (x : α), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s\nx : α\ns : Set β\nhs : (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f\n⊢ IsCompact (f ⁻¹' s)",
"tactic": "rw [← hf.toInducing.isCompact_iff, image_preimage_eq_of_subset hs.2]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns✝ t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nthis : ∀ (x : α), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s\nx : α\ns : Set β\nhs : (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f\n⊢ IsCompact s",
"tactic": "exact hs.1.2"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nx : α\n⊢ HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\n⊢ ∀ (x : α), HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s",
"tactic": "intro x"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nx : α\n⊢ HasBasis (comap f (𝓝 (f x))) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nx : α\n⊢ HasBasis (𝓝 x) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s",
"tactic": "rw [hf.toInducing.nhds_eq_comap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150029\nπ : ι → Type ?u.150034\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ns t : Set α\ninst✝ : LocallyCompactSpace β\nf : α → β\nhf : OpenEmbedding f\nx : α\n⊢ HasBasis (comap f (𝓝 (f x))) (fun s => (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) fun s => f ⁻¹' s",
"tactic": "exact\n ((compact_basis_nhds _).restrict_subset <| hf.open_range.mem_nhds <| mem_range_self _).comap _"
}
] | [
1210,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1200,
11
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.isConj_of_support_equiv | [
{
"state_after": "ι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\n⊢ extendSubtype f * σ * (extendSubtype f)⁻¹ = τ",
"state_before": "ι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\n⊢ IsConj σ τ",
"tactic": "refine' isConj_iff.2 ⟨Equiv.extendSubtype f, _⟩"
},
{
"state_after": "ι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\n⊢ extendSubtype f * σ = τ * extendSubtype f",
"state_before": "ι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\n⊢ extendSubtype f * σ * (extendSubtype f)⁻¹ = τ",
"tactic": "rw [mul_inv_eq_iff_eq_mul]"
},
{
"state_after": "case H\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\n⊢ ↑(extendSubtype f * σ) x = ↑(τ * extendSubtype f) x",
"state_before": "ι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\n⊢ extendSubtype f * σ = τ * extendSubtype f",
"tactic": "ext x"
},
{
"state_after": "case H\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\n⊢ ↑(extendSubtype f) (↑σ x) = ↑τ (↑(extendSubtype f) x)",
"state_before": "case H\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\n⊢ ↑(extendSubtype f * σ) x = ↑(τ * extendSubtype f) x",
"tactic": "simp only [Perm.mul_apply]"
},
{
"state_after": "case pos\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : x ∈ support σ\n⊢ ↑(extendSubtype f) (↑σ x) = ↑τ (↑(extendSubtype f) x)\n\ncase neg\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : ¬x ∈ support σ\n⊢ ↑(extendSubtype f) (↑σ x) = ↑τ (↑(extendSubtype f) x)",
"state_before": "case H\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\n⊢ ↑(extendSubtype f) (↑σ x) = ↑τ (↑(extendSubtype f) x)",
"tactic": "by_cases hx : x ∈ σ.support"
},
{
"state_after": "case pos\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : x ∈ support σ\n⊢ ↑(↑f { val := ↑σ x, property := ?pos.hx✝ }) = ↑τ ↑(↑f { val := x, property := ?pos.hx✝ })\n\ncase pos.hx\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : x ∈ support σ\n⊢ x ∈ ↑(support σ)\n\ncase pos.hx\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : x ∈ support σ\n⊢ ↑σ x ∈ ↑(support σ)\n\ncase pos.hx\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : x ∈ support σ\n⊢ ↑σ x ∈ ↑(support σ)",
"state_before": "case pos\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : x ∈ support σ\n⊢ ↑(extendSubtype f) (↑σ x) = ↑τ (↑(extendSubtype f) x)",
"tactic": "rw [Equiv.extendSubtype_apply_of_mem, Equiv.extendSubtype_apply_of_mem]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : x ∈ support σ\n⊢ ↑(↑f { val := ↑σ x, property := ?pos.hx✝ }) = ↑τ ↑(↑f { val := x, property := ?pos.hx✝ })",
"tactic": "exact hf x (Finset.mem_coe.2 hx)"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.3024744\nα : Type u_1\nβ : Type ?u.3024750\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ τ : Perm α\nf : { x // x ∈ ↑(support σ) } ≃ { x // x ∈ ↑(support τ) }\nhf :\n ∀ (x : α) (hx : x ∈ ↑(support σ)),\n ↑(↑f { val := ↑σ x, property := (_ : ↑σ x ∈ support σ) }) = ↑τ ↑(↑f { val := x, property := hx })\nx : α\nhx : ¬x ∈ support σ\n⊢ ↑(extendSubtype f) (↑σ x) = ↑τ (↑(extendSubtype f) x)",
"tactic": "rwa [Classical.not_not.1 ((not_congr mem_support).1 (Equiv.extendSubtype_not_mem f _ _)),\n Classical.not_not.1 ((not_congr mem_support).mp hx)]"
}
] | [
1720,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1705,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.lf_of_lf_of_lt | [] | [
588,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
587,
1
] |
Mathlib/Data/Prod/Basic.lean | Prod.swap_swap_eq | [] | [
183,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/Combinatorics/SimpleGraph/Clique.lean | SimpleGraph.is3Clique_triple_iff | [
{
"state_after": "α : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\n⊢ ((Set.Pairwise (↑{c}) G.Adj ∧ ∀ (b_1 : α), b_1 ∈ ↑{c} → b ≠ b_1 → Adj G b b_1) ∧\n ∀ (b_1 : α), b_1 ∈ insert b ↑{c} → a ≠ b_1 → Adj G a b_1) ∧\n Finset.card {a, b, c} = 3 ↔\n Adj G a b ∧ Adj G a c ∧ Adj G b c",
"state_before": "α : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\n⊢ IsNClique G 3 {a, b, c} ↔ Adj G a b ∧ Adj G a c ∧ Adj G b c",
"tactic": "simp only [isNClique_iff, isClique_iff, Set.pairwise_insert_of_symmetric G.symm, coe_insert]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\n⊢ ((Set.Pairwise (↑{c}) G.Adj ∧ ∀ (b_1 : α), b_1 ∈ ↑{c} → b ≠ b_1 → Adj G b b_1) ∧\n ∀ (b_1 : α), b_1 ∈ insert b ↑{c} → a ≠ b_1 → Adj G a b_1) ∧\n Finset.card {a, b, c} = 3 ↔\n Adj G a b ∧ Adj G a c ∧ Adj G b c",
"tactic": "by_cases hab : a = b <;> by_cases hbc : b = c <;> by_cases hac : a = c <;> subst_vars <;>\n simp [G.ne_of_adj, and_rotate, *]"
}
] | [
137,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
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