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Mathlib/Analysis/Calculus/FDeriv/Mul.lean | DifferentiableOn.mul | [] | [
334,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
333,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean | le_add_tsub' | [] | [
233,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
1
] |
Mathlib/Order/Lattice.lean | Prod.fst_sup | [] | [
1269,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1268,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean | GroupCat.mono_iff_ker_eq_bot | [] | [
90,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | pow_strictMono_right | [
{
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"state_before": "β : Type ?u.226813\nA : Type ?u.226816\nG : Type ?u.226819\nM : Type ?u.226822\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\na x y : R\nn✝ m : ℕ\nh : 1 < a\nthis : 0 < a\nn : ℕ\n⊢ a ^ n < a ^ (n + 1)",
"tactic": "simpa only [one_mul, pow_succ] using mul_lt_mul h (le_refl (a ^ n)) (pow_pos this _) this.le"
}
] | [
498,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
495,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean | Measurable.cos | [] | [
153,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
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Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | Submodule.isOrtho_orthogonal_left | [] | [
316,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
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Mathlib/Data/Polynomial/Cardinal.lean | Polynomial.cardinal_mk_le_max | [
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},
{
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"state_before": "case inl\nR : Type u\ninst✝ : Semiring R\nh✝ : Subsingleton R\n⊢ (#R[X]) ≤ max (#R) ℵ₀",
"tactic": "exact (mk_eq_one _).trans_le (le_max_of_le_right one_le_aleph0)"
},
{
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"state_before": "case inr\nR : Type u\ninst✝ : Semiring R\nh✝ : Nontrivial R\n⊢ (#R[X]) ≤ max (#R) ℵ₀",
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}
] | [
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/LinearAlgebra/LinearIndependent.lean | exists_maximal_independent' | [
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"tactic": "let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)"
},
{
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"tactic": "let X := { I : Set ι // indep I }"
},
{
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"tactic": "let r : X → X → Prop := fun I J => I.1 ⊆ J.1"
},
{
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"state_before": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nkey : ∀ (c : Set X), IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), ↑I)\n⊢ ∃ I, (LinearIndependent R fun x => s ↑x) ∧ ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun x => s ↑x) → I = J",
"tactic": "have trans : Transitive r := fun I J K => Set.Subset.trans"
},
{
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"tactic": "obtain ⟨⟨I, hli : indep I⟩, hmax : ∀ a, r ⟨I, hli⟩ a → r a ⟨I, hli⟩⟩ :=\n @exists_maximal_of_chains_bounded _ r\n (fun c hc => ⟨⟨⋃ I ∈ c, (I : Set ι), key c hc⟩, fun I => Set.subset_biUnion_of_mem⟩) @trans"
},
{
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"tactic": "exact ⟨I, hli, fun J hsub hli => Set.Subset.antisymm hsub (hmax ⟨J, hli⟩ hsub)⟩"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\n⊢ indep (⋃ (I : X) (_ : I ∈ c), ↑I)",
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"tactic": "intro c hc"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\n⊢ LinearIndependent R (s ∘ Subtype.val)",
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"tactic": "dsimp"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\n⊢ ∀ (l : ι →₀ R),\n l ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I) →\n ↑(Finsupp.total ι M R s) l = 0 → l = 0",
"state_before": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\n⊢ LinearIndependent R (s ∘ Subtype.val)",
"tactic": "rw [linearIndependent_comp_subtype]"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\n⊢ f = 0",
"state_before": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\n⊢ ∀ (l : ι →₀ R),\n l ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I) →\n ↑(Finsupp.total ι M R s) l = 0 → l = 0",
"tactic": "intro f hsupport hsum"
},
{
"state_after": "case inl\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nf : ι →₀ R\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhc : IsChain r ∅\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ ∅), ↑I)\n⊢ f = 0\n\ncase inr\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhn : Set.Nonempty c\n⊢ f = 0",
"state_before": "ι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\n⊢ f = 0",
"tactic": "rcases eq_empty_or_nonempty c with (rfl | hn)"
},
{
"state_after": "case inr\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhn : Set.Nonempty c\nthis : IsRefl X r\n⊢ f = 0",
"state_before": "case inr\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhn : Set.Nonempty c\n⊢ f = 0",
"tactic": "haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩"
},
{
"state_after": "case inr.intro.intro\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhn : Set.Nonempty c\nthis : IsRefl X r\nI : X\n_I_mem : I ∈ c\nhI : ↑f.support ⊆ ↑I\n⊢ f = 0",
"state_before": "case inr\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhn : Set.Nonempty c\nthis : IsRefl X r\n⊢ f = 0",
"tactic": "obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support : Set ι) ⊆ I :=\n hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\nf : ι →₀ R\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ c), ↑I)\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhn : Set.Nonempty c\nthis : IsRefl X r\nI : X\n_I_mem : I ∈ c\nhI : ↑f.support ⊆ ↑I\n⊢ f = 0",
"tactic": "exact linearIndependent_comp_subtype.mp I.2 f hI hsum"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u'\nι' : Type ?u.501528\nR : Type u_1\nK : Type ?u.501534\nM : Type u_2\nM' : Type ?u.501540\nM'' : Type ?u.501543\nV : Type u\nV' : Type ?u.501548\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : ι → M\nindep : Set ι → Prop := fun I => LinearIndependent R (s ∘ Subtype.val)\nX : Type u' := { I // indep I }\nr : X → X → Prop := fun I J => ↑I ⊆ ↑J\nf : ι →₀ R\nhsum : ↑(Finsupp.total ι M R s) f = 0\nhc : IsChain r ∅\nhsupport : f ∈ Finsupp.supported R R (⋃ (I : { I // LinearIndependent R (s ∘ Subtype.val) }) (_ : I ∈ ∅), ↑I)\n⊢ f = 0",
"tactic": "simpa using hsupport"
}
] | [
894,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
872,
1
] |
Mathlib/Data/Polynomial/Eval.lean | Polynomial.comp_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\n⊢ comp p 0 = ↑C (eval 0 p)",
"tactic": "rw [← C_0, comp_C]"
}
] | [
564,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
564,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.direction_inf_of_mem | [
{
"state_after": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns₁ s₂ : AffineSubspace k P\np : P\nh₁ : p ∈ s₁\nh₂ : p ∈ s₂\nv : V\n⊢ v ∈ direction (s₁ ⊓ s₂) ↔ v ∈ direction s₁ ⊓ direction s₂",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns₁ s₂ : AffineSubspace k P\np : P\nh₁ : p ∈ s₁\nh₂ : p ∈ s₂\n⊢ direction (s₁ ⊓ s₂) = direction s₁ ⊓ direction s₂",
"tactic": "ext v"
},
{
"state_after": "no goals",
"state_before": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns₁ s₂ : AffineSubspace k P\np : P\nh₁ : p ∈ s₁\nh₂ : p ∈ s₂\nv : V\n⊢ v ∈ direction (s₁ ⊓ s₂) ↔ v ∈ direction s₁ ⊓ direction s₂",
"tactic": "rw [Submodule.mem_inf, ← vadd_mem_iff_mem_direction v h₁, ← vadd_mem_iff_mem_direction v h₂, ←\n vadd_mem_iff_mem_direction v ((mem_inf_iff p s₁ s₂).2 ⟨h₁, h₂⟩), mem_inf_iff]"
}
] | [
913,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
909,
1
] |
Mathlib/Topology/Order/Basic.lean | disjoint_nhds_atTop | [
{
"state_after": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : NoMaxOrder α\nx y : α\nhy : x < y\n⊢ Disjoint (𝓝 x) atTop",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : NoMaxOrder α\nx : α\n⊢ Disjoint (𝓝 x) atTop",
"tactic": "rcases exists_gt x with ⟨y, hy : x < y⟩"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : NoMaxOrder α\nx y : α\nhy : x < y\n⊢ Disjoint (Iio y) {b | y ≤ b}",
"state_before": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : NoMaxOrder α\nx y : α\nhy : x < y\n⊢ Disjoint (𝓝 x) atTop",
"tactic": "refine' disjoint_of_disjoint_of_mem _ (Iio_mem_nhds hy) (mem_atTop y)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : NoMaxOrder α\nx y : α\nhy : x < y\n⊢ Disjoint (Iio y) {b | y ≤ b}",
"tactic": "exact disjoint_left.mpr fun z => not_le.2"
}
] | [
1535,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1532,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.forall_iff | [] | [
252,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
251,
1
] |
Mathlib/Data/Matrix/Block.lean | Matrix.blockDiag_transpose | [] | [
520,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
518,
1
] |
Mathlib/Order/Ideal.lean | Order.Ideal.lt_sup_principal_of_not_mem | [
{
"state_after": "no goals",
"state_before": "P : Type u_1\ninst✝¹ : SemilatticeSup P\ninst✝ : IsDirected P fun x x_1 => x ≥ x_1\nx : P\nI J K s t : Ideal P\nhx : ¬x ∈ I\nh : I = I ⊔ principal x\n⊢ x ∈ I",
"tactic": "simpa only [left_eq_sup, principal_le_iff] using h"
}
] | [
427,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
426,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean | Nat.ArithmeticFunction.natCoe_one | [
{
"state_after": "case h\nR : Type u_1\ninst✝ : AddMonoidWithOne R\nn : ℕ\n⊢ ↑↑1 n = ↑1 n",
"state_before": "R : Type u_1\ninst✝ : AddMonoidWithOne R\n⊢ ↑1 = 1",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝ : AddMonoidWithOne R\nn : ℕ\n⊢ ↑↑1 n = ↑1 n",
"tactic": "simp [one_apply]"
}
] | [
198,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
195,
1
] |
Mathlib/Probability/Kernel/Basic.lean | ProbabilityTheory.kernel.lintegral_deterministic | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.398136\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nf : β → ℝ≥0∞\ng : α → β\na : α\nhg : Measurable g\ninst✝ : MeasurableSingletonClass β\n⊢ (∫⁻ (x : β), f x ∂↑(deterministic g hg) a) = f (g a)",
"tactic": "rw [kernel.deterministic_apply, lintegral_dirac (g a) f]"
}
] | [
379,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
377,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.lift_spec | [] | [
961,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
960,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.filter_inter | [] | [
2070,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2063,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean | mul_inf | [] | [
88,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean | FiniteField.X_pow_card_sub_X_natDegree_eq | [
{
"state_after": "K : Type ?u.726863\nR : Type ?u.726866\ninst✝² : Field K\ninst✝¹ : Fintype K\nK' : Type u_1\ninst✝ : Field K'\np n : ℕ\nhp : 1 < p\nh1 : degree X < degree (X ^ p)\n⊢ natDegree (X ^ p - X) = p",
"state_before": "K : Type ?u.726863\nR : Type ?u.726866\ninst✝² : Field K\ninst✝¹ : Fintype K\nK' : Type u_1\ninst✝ : Field K'\np n : ℕ\nhp : 1 < p\n⊢ natDegree (X ^ p - X) = p",
"tactic": "have h1 : (X : K'[X]).degree < (X ^ p : K'[X]).degree := by\n rw [degree_X_pow, degree_X]\n exact WithBot.coe_lt_coe.2 hp"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.726863\nR : Type ?u.726866\ninst✝² : Field K\ninst✝¹ : Fintype K\nK' : Type u_1\ninst✝ : Field K'\np n : ℕ\nhp : 1 < p\nh1 : degree X < degree (X ^ p)\n⊢ natDegree (X ^ p - X) = p",
"tactic": "rw [natDegree_eq_of_degree_eq (degree_sub_eq_left_of_degree_lt h1), natDegree_X_pow]"
},
{
"state_after": "K : Type ?u.726863\nR : Type ?u.726866\ninst✝² : Field K\ninst✝¹ : Fintype K\nK' : Type u_1\ninst✝ : Field K'\np n : ℕ\nhp : 1 < p\n⊢ 1 < ↑p",
"state_before": "K : Type ?u.726863\nR : Type ?u.726866\ninst✝² : Field K\ninst✝¹ : Fintype K\nK' : Type u_1\ninst✝ : Field K'\np n : ℕ\nhp : 1 < p\n⊢ degree X < degree (X ^ p)",
"tactic": "rw [degree_X_pow, degree_X]"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.726863\nR : Type ?u.726866\ninst✝² : Field K\ninst✝¹ : Fintype K\nK' : Type u_1\ninst✝ : Field K'\np n : ℕ\nhp : 1 < p\n⊢ 1 < ↑p",
"tactic": "exact WithBot.coe_lt_coe.2 hp"
}
] | [
238,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
233,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | Class.sInter_apply | [
{
"state_after": "x : Class\ny : ZFSet\n⊢ (∀ (z : ZFSet), x z → y ∈ z) → (⋂₀ x) y",
"state_before": "x : Class\ny : ZFSet\n⊢ (⋂₀ x) y ↔ ∀ (z : ZFSet), x z → y ∈ z",
"tactic": "refine' ⟨fun hxy z hxz => hxy _ ⟨z, rfl, hxz⟩, _⟩"
},
{
"state_after": "case intro.intro\nx : Class\ny : ZFSet\nH : ∀ (z : ZFSet), x z → y ∈ z\nz : ZFSet\nhxz : x z\n⊢ y ∈ ↑z",
"state_before": "x : Class\ny : ZFSet\n⊢ (∀ (z : ZFSet), x z → y ∈ z) → (⋂₀ x) y",
"tactic": "rintro H - ⟨z, rfl, hxz⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nx : Class\ny : ZFSet\nH : ∀ (z : ZFSet), x z → y ∈ z\nz : ZFSet\nhxz : x z\n⊢ y ∈ ↑z",
"tactic": "exact H _ hxz"
}
] | [
1703,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1700,
1
] |
Mathlib/NumberTheory/ADEInequality.lean | ADEInequality.lt_four | [
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\nh4 : 0 < 4\n⊢ q < 4",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\n⊢ q < 4",
"tactic": "have h4 : (0 : ℚ) < 4 := by norm_num"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nH : ¬q < 4\n⊢ sumInv {2, q, r} ≤ 1",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\nh4 : 0 < 4\n⊢ q < 4",
"tactic": "contrapose! H"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nH : ¬q < 4\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nH : ¬q < 4\n⊢ sumInv {2, q, r} ≤ 1",
"tactic": "rw [sumInv_pqr]"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nH : ¬q < 4\nh4r : 4 ≤ r\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nH : ¬q < 4\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"tactic": "have h4r := (not_lt.mp H).trans hqr"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nH : ¬q < 4\nh4r : 4 ≤ r\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"tactic": "simp at H"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"tactic": "have hq: (q : ℚ)⁻¹ ≤ 4⁻¹ := by\n rw [inv_le_inv _ h4]\n assumption_mod_cast\n norm_num"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\nhr : (↑↑r)⁻¹ ≤ 4⁻¹\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"tactic": "have hr: (r : ℚ)⁻¹ ≤ 4⁻¹ := by\n rw [inv_le_inv _ h4]\n assumption_mod_cast\n norm_num"
},
{
"state_after": "no goals",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\nhr : (↑↑r)⁻¹ ≤ 4⁻¹\n⊢ (↑↑2)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"tactic": "calc\n (2⁻¹ + (q : ℚ)⁻¹ + (r : ℚ)⁻¹) ≤ 2⁻¹ + 4⁻¹ + 4⁻¹ := add_le_add (add_le_add le_rfl hq) hr\n _ = 1 := by norm_num"
},
{
"state_after": "no goals",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nH : 1 < sumInv {2, q, r}\n⊢ 0 < 4",
"tactic": "norm_num"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ 4 ≤ ↑↑q\n\nq r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ 0 < ↑↑q",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ (↑↑q)⁻¹ ≤ 4⁻¹",
"tactic": "rw [inv_le_inv _ h4]"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ 0 < ↑↑q",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ 4 ≤ ↑↑q\n\nq r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ 0 < ↑↑q",
"tactic": "assumption_mod_cast"
},
{
"state_after": "no goals",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\n⊢ 0 < ↑↑q",
"tactic": "norm_num"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ 4 ≤ ↑↑r\n\nq r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ 0 < ↑↑r",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ (↑↑r)⁻¹ ≤ 4⁻¹",
"tactic": "rw [inv_le_inv _ h4]"
},
{
"state_after": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ 0 < ↑↑r",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ 4 ≤ ↑↑r\n\nq r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ 0 < ↑↑r",
"tactic": "assumption_mod_cast"
},
{
"state_after": "no goals",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\n⊢ 0 < ↑↑r",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "q r : ℕ+\nhqr : q ≤ r\nh4 : 0 < 4\nh4r : 4 ≤ r\nH : 4 ≤ q\nhq : (↑↑q)⁻¹ ≤ 4⁻¹\nhr : (↑↑r)⁻¹ ≤ 4⁻¹\n⊢ 2⁻¹ + 4⁻¹ + 4⁻¹ = 1",
"tactic": "norm_num"
}
] | [
216,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
200,
1
] |
Mathlib/CategoryTheory/Sites/Plus.lean | CategoryTheory.GrothendieckTopology.diagramNatTrans_comp | [
{
"state_after": "case w.h\nC : Type u\ninst✝² : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ P Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nX : C\nx✝ : (Cover J X)ᵒᵖ\n⊢ (diagramNatTrans J (η ≫ γ) X).app x✝ = (diagramNatTrans J η X ≫ diagramNatTrans J γ X).app x✝",
"state_before": "C : Type u\ninst✝² : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ P Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nX : C\n⊢ diagramNatTrans J (η ≫ γ) X = diagramNatTrans J η X ≫ diagramNatTrans J γ X",
"tactic": "ext : 2"
},
{
"state_after": "case w.h\nC : Type u\ninst✝² : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ P Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nX : C\nx✝ : (Cover J X)ᵒᵖ\ni : (Cover.index x✝.unop R).L\n⊢ (diagramNatTrans J (η ≫ γ) X).app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop R) i =\n (diagramNatTrans J η X ≫ diagramNatTrans J γ X).app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop R) i",
"state_before": "case w.h\nC : Type u\ninst✝² : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ P Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nX : C\nx✝ : (Cover J X)ᵒᵖ\n⊢ (diagramNatTrans J (η ≫ γ) X).app x✝ = (diagramNatTrans J η X ≫ diagramNatTrans J γ X).app x✝",
"tactic": "refine' Multiequalizer.hom_ext _ _ _ (fun i => _)"
},
{
"state_after": "case w.h\nC : Type u\ninst✝² : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ P Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nX : C\nx✝ : (Cover J X)ᵒᵖ\ni : (Cover.index x✝.unop R).L\n⊢ Multiequalizer.lift (Cover.index x✝.unop R) (multiequalizer (Cover.index x✝.unop P))\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op ≫ γ.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index x✝.unop R).R),\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ (η ≫ γ).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.fst (Cover.index x✝.unop R) i =\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ (η ≫ γ).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.snd (Cover.index x✝.unop R) i) ≫\n Multiequalizer.ι (Cover.index x✝.unop R) i =\n (Multiequalizer.lift (Cover.index x✝.unop Q) (multiequalizer (Cover.index x✝.unop P))\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index x✝.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index x✝.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index x✝.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index x✝.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index x✝.unop Q) i) ≫\n Multiequalizer.lift (Cover.index x✝.unop R) (multiequalizer (Cover.index x✝.unop Q))\n (fun i => Multiequalizer.ι (Cover.index x✝.unop Q) i ≫ γ.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index x✝.unop R).R),\n (fun i => Multiequalizer.ι (Cover.index x✝.unop Q) i ≫ γ.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.fst (Cover.index x✝.unop R) i =\n (fun i => Multiequalizer.ι (Cover.index x✝.unop Q) i ≫ γ.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.snd (Cover.index x✝.unop R) i)) ≫\n Multiequalizer.ι (Cover.index x✝.unop R) i",
"state_before": "case w.h\nC : Type u\ninst✝² : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ P Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nX : C\nx✝ : (Cover J X)ᵒᵖ\ni : (Cover.index x✝.unop R).L\n⊢ (diagramNatTrans J (η ≫ γ) X).app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop R) i =\n (diagramNatTrans J η X ≫ diagramNatTrans J γ X).app x✝ ≫ Multiequalizer.ι (Cover.index x✝.unop R) i",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u\ninst✝² : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹ : Category D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ P Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nX : C\nx✝ : (Cover J X)ᵒᵖ\ni : (Cover.index x✝.unop R).L\n⊢ Multiequalizer.lift (Cover.index x✝.unop R) (multiequalizer (Cover.index x✝.unop P))\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op ≫ γ.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index x✝.unop R).R),\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ (η ≫ γ).app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.fst (Cover.index x✝.unop R) i =\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ (η ≫ γ).app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.snd (Cover.index x✝.unop R) i) ≫\n Multiequalizer.ι (Cover.index x✝.unop R) i =\n (Multiequalizer.lift (Cover.index x✝.unop Q) (multiequalizer (Cover.index x✝.unop P))\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index x✝.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index x✝.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index x✝.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index x✝.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index x✝.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index x✝.unop Q) i) ≫\n Multiequalizer.lift (Cover.index x✝.unop R) (multiequalizer (Cover.index x✝.unop Q))\n (fun i => Multiequalizer.ι (Cover.index x✝.unop Q) i ≫ γ.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index x✝.unop R).R),\n (fun i => Multiequalizer.ι (Cover.index x✝.unop Q) i ≫ γ.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.fst (Cover.index x✝.unop R) i =\n (fun i => Multiequalizer.ι (Cover.index x✝.unop Q) i ≫ γ.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index x✝.unop R) i) ≫\n MulticospanIndex.snd (Cover.index x✝.unop R) i)) ≫\n Multiequalizer.ι (Cover.index x✝.unop R) i",
"tactic": "simp"
}
] | [
100,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.mk_range_le | [] | [
2008,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2007,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPullback.inl_snd' | [
{
"state_after": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPullback (b.inl ≫ b.fst) 0 0 (0 ≫ 0)",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPullback b.inl 0 b.snd 0",
"tactic": "refine' of_right _ (by simp) (of_isBilimit h)"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPullback (b.inl ≫ b.fst) 0 0 (0 ≫ 0)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ b.inl ≫ b.snd = 0 ≫ 0",
"tactic": "simp"
}
] | [
574,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
571,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean | QuotientGroup.quotientQuotientEquivQuotientAux_mk_mk | [] | [
640,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
638,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigO_one_nat_atTop_iff | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.689842\nβ : Type ?u.689845\nE : Type ?u.689848\nF : Type ?u.689851\nG : Type ?u.689854\nE' : Type ?u.689857\nF' : Type ?u.689860\nG' : Type ?u.689863\nE'' : Type u_1\nF'' : Type ?u.689869\nG'' : Type ?u.689872\nR : Type ?u.689875\nR' : Type ?u.689878\n𝕜 : Type ?u.689881\n𝕜' : Type ?u.689884\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : ℕ → E''\n⊢ (∃ C, ∀ (x : ℕ), ‖f x‖ ≤ C * ‖1‖) ↔ ∃ C, ∀ (n : ℕ), ‖f n‖ ≤ C",
"tactic": "simp only [norm_one, mul_one]"
}
] | [
2131,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2128,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.IsThreeCycle.isCycle | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nh : IsThreeCycle σ\n⊢ IsCycle σ",
"tactic": "rw [← card_cycleType_eq_one, h.cycleType, card_singleton]"
}
] | [
594,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
593,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean | NormedAddGroupHom.mkNormedAddGroupHom_norm_le' | [
{
"state_after": "case h\nV : Type ?u.279839\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.279848\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g : NormedAddGroupHom V₁ V₂\nf : V₁ →+ V₂\nC : ℝ\nh : ∀ (x : V₁), ‖↑f x‖ ≤ C * ‖x‖\nx : V₁\n⊢ C ≤ max C 0",
"state_before": "V : Type ?u.279839\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.279848\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g : NormedAddGroupHom V₁ V₂\nf : V₁ →+ V₂\nC : ℝ\nh : ∀ (x : V₁), ‖↑f x‖ ≤ C * ‖x‖\nx : V₁\n⊢ C * ‖x‖ ≤ max C 0 * ‖x‖",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\nV : Type ?u.279839\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.279848\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g : NormedAddGroupHom V₁ V₂\nf : V₁ →+ V₂\nC : ℝ\nh : ∀ (x : V₁), ‖↑f x‖ ≤ C * ‖x‖\nx : V₁\n⊢ C ≤ max C 0",
"tactic": "apply le_max_left"
}
] | [
318,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
] |
Mathlib/Topology/Algebra/Order/Group.lean | Filter.Tendsto.abs | [] | [
68,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
11
] |
Mathlib/Order/Bounds/Basic.lean | MonotoneOn.map_isLeast | [] | [
1197,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1196,
1
] |
Mathlib/Algebra/Order/WithZero.lean | ne_zero_of_lt | [] | [
111,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/MeasureTheory/Measure/MutuallySingular.lean | MeasureTheory.Measure.MutuallySingular.smul | [] | [
122,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean | FreeAbelianGroup.map_comp | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ng : β → γ\nx✝ : α\n⊢ ↑(AddMonoidHom.comp (map g) (map f)) (of x✝) = ↑(map (g ∘ f)) (of x✝)",
"tactic": "simp [map]"
}
] | [
385,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
1
] |
Mathlib/Data/Multiset/Range.lean | Multiset.range_zero | [] | [
34,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
33,
1
] |
Mathlib/Algebra/Invertible.lean | mul_mul_invOf_self_cancel' | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na b : α\nx✝ : Invertible b\n⊢ a * b * ⅟b = a",
"tactic": "simp [mul_assoc]"
}
] | [
144,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
] |
Mathlib/MeasureTheory/PiSystem.lean | piiUnionInter_singleton | [
{
"state_after": "case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\n⊢ s ∈ piiUnionInter π {i} ↔ s ∈ π i ∪ {univ}",
"state_before": "α : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\n⊢ piiUnionInter π {i} = π i ∪ {univ}",
"tactic": "ext1 s"
},
{
"state_after": "case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x} ↔ s ∈ π i ∨ s ∈ {univ}",
"state_before": "case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\n⊢ s ∈ piiUnionInter π {i} ↔ s ∈ π i ∪ {univ}",
"tactic": "simp only [piiUnionInter, exists_prop, mem_union]"
},
{
"state_after": "case h.refine'_1\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x} → s ∈ π i ∨ s ∈ {univ}\n\ncase h.refine'_2\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nh : s ∈ π i ∨ s ∈ {univ}\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x}",
"state_before": "case h\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x} ↔ s ∈ π i ∨ s ∈ {univ}",
"tactic": "refine' ⟨_, fun h => _⟩"
},
{
"state_after": "case h.refine'_1.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nhti : ↑t ⊆ {i}\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"state_before": "case h.refine'_1\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x} → s ∈ π i ∨ s ∈ {univ}",
"tactic": "rintro ⟨t, hti, f, hfπ, rfl⟩"
},
{
"state_after": "case h.refine'_1.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"state_before": "case h.refine'_1.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nhti : ↑t ⊆ {i}\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"tactic": "simp only [subset_singleton_iff, Finset.mem_coe] at hti"
},
{
"state_after": "case pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}\n\ncase neg\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"state_before": "case h.refine'_1.intro.intro.intro.intro\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"tactic": "by_cases hi : i ∈ t"
},
{
"state_after": "case pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nht_eq_i : t = {i}\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"state_before": "case pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"tactic": "have ht_eq_i : t = {i} := by\n ext1 x\n rw [Finset.mem_singleton]\n exact ⟨fun h => hti x h, fun h => h.symm ▸ hi⟩"
},
{
"state_after": "case pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nht_eq_i : t = {i}\n⊢ f i ∈ π i ∨ f i ∈ {univ}",
"state_before": "case pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nht_eq_i : t = {i}\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"tactic": "simp only [ht_eq_i, Finset.mem_singleton, iInter_iInter_eq_left]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nht_eq_i : t = {i}\n⊢ f i ∈ π i ∨ f i ∈ {univ}",
"tactic": "exact Or.inl (hfπ i hi)"
},
{
"state_after": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nx : ι\n⊢ x ∈ t ↔ x ∈ {i}",
"state_before": "α : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\n⊢ t = {i}",
"tactic": "ext1 x"
},
{
"state_after": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nx : ι\n⊢ x ∈ t ↔ x = i",
"state_before": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nx : ι\n⊢ x ∈ t ↔ x ∈ {i}",
"tactic": "rw [Finset.mem_singleton]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : i ∈ t\nx : ι\n⊢ x ∈ t ↔ x = i",
"tactic": "exact ⟨fun h => hti x h, fun h => h.symm ▸ hi⟩"
},
{
"state_after": "case neg\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\nht_empty : t = ∅\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"state_before": "case neg\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"tactic": "have ht_empty : t = ∅ := by\n ext1 x\n simp only [Finset.not_mem_empty, iff_false_iff]\n exact fun hx => hi (hti x hx ▸ hx)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\nht_empty : t = ∅\n⊢ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ π i ∨ (⋂ (x : ι) (_ : x ∈ t), f x) ∈ {univ}",
"tactic": "simp [ht_empty, Finset.not_mem_empty, iInter_false, iInter_univ, Set.mem_singleton univ,\n or_true_iff]"
},
{
"state_after": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\nx : ι\n⊢ x ∈ t ↔ x ∈ ∅",
"state_before": "α : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\n⊢ t = ∅",
"tactic": "ext1 x"
},
{
"state_after": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\nx : ι\n⊢ ¬x ∈ t",
"state_before": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\nx : ι\n⊢ x ∈ t ↔ x ∈ ∅",
"tactic": "simp only [Finset.not_mem_empty, iff_false_iff]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ (x : ι), x ∈ t → f x ∈ π x\nhti : ∀ (y : ι), y ∈ t → y = i\nhi : ¬i ∈ t\nx : ι\n⊢ ¬x ∈ t",
"tactic": "exact fun hx => hi (hti x hx ▸ hx)"
},
{
"state_after": "case h.refine'_2.inl\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x}\n\ncase h.refine'_2.inr\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ {univ}\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x}",
"state_before": "case h.refine'_2\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nh : s ∈ π i ∨ s ∈ {univ}\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x}",
"tactic": "cases' h with hs hs"
},
{
"state_after": "case h.refine'_2.inl.refine'_1\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\n⊢ ↑{i} ⊆ {i}\n\ncase h.refine'_2.inl.refine'_2\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\nx : ι\nhx : x ∈ {i}\n⊢ (fun x => s) x ∈ π x\n\ncase h.refine'_2.inl.refine'_3\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\n⊢ s = ⋂ (x : ι) (_ : x ∈ {i}), (fun x => s) x",
"state_before": "case h.refine'_2.inl\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x}",
"tactic": "refine' ⟨{i}, _, fun _ => s, ⟨fun x hx => _, _⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2.inl.refine'_1\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\n⊢ ↑{i} ⊆ {i}",
"tactic": "rw [Finset.coe_singleton]"
},
{
"state_after": "case h.refine'_2.inl.refine'_2\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\nx : ι\nhx : x = i\n⊢ (fun x => s) x ∈ π x",
"state_before": "case h.refine'_2.inl.refine'_2\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\nx : ι\nhx : x ∈ {i}\n⊢ (fun x => s) x ∈ π x",
"tactic": "rw [Finset.mem_singleton] at hx"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2.inl.refine'_2\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\nx : ι\nhx : x = i\n⊢ (fun x => s) x ∈ π x",
"tactic": "rwa [hx]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2.inl.refine'_3\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ π i\n⊢ s = ⋂ (x : ι) (_ : x ∈ {i}), (fun x => s) x",
"tactic": "simp only [Finset.mem_singleton, iInter_iInter_eq_left]"
},
{
"state_after": "case h.refine'_2.inr\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ {univ}\n⊢ ↑∅ ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ ∅ → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ ∅), f x",
"state_before": "case h.refine'_2.inr\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ {univ}\n⊢ s ∈ {s | ∃ t, ↑t ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ t → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ t), f x}",
"tactic": "refine' ⟨∅, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2.inr\nα : Type u_1\nι : Type u_2\nπ : ι → Set (Set α)\ni : ι\ns : Set α\nhs : s ∈ {univ}\n⊢ ↑∅ ⊆ {i} ∧ ∃ f, (∀ (x : ι), x ∈ ∅ → f x ∈ π x) ∧ s = ⋂ (x : ι) (_ : x ∈ ∅), f x",
"tactic": "simpa only [Finset.coe_empty, subset_singleton_iff, mem_empty_iff_false, IsEmpty.forall_iff,\n imp_true_iff, Finset.not_mem_empty, iInter_false, iInter_univ, true_and_iff,\n exists_const] using hs"
}
] | [
396,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
366,
1
] |
Mathlib/Order/Filter/Prod.lean | Filter.Eventually.prod_inl | [] | [
159,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Order/Heyting/Basic.lean | le_sup_sdiff | [] | [
534,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
533,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean | MulLECancellable.inj | [] | [
1611,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1609,
11
] |
Mathlib/Algebra/Star/SelfAdjoint.lean | isSelfAdjoint_iff | [] | [
79,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.infEdist_ne_top | [
{
"state_after": "case intro\nι : Sort ?u.57852\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y✝ : α\nΦ : α → β\ny : α\nhy : y ∈ s\n⊢ infEdist x s ≠ ⊤",
"state_before": "ι : Sort ?u.57852\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nh : Set.Nonempty s\n⊢ infEdist x s ≠ ⊤",
"tactic": "rcases h with ⟨y, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Sort ?u.57852\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y✝ : α\nΦ : α → β\ny : α\nhy : y ∈ s\n⊢ infEdist x s ≠ ⊤",
"tactic": "exact ne_top_of_le_ne_top (edist_ne_top _ _) (infEdist_le_edist_of_mem hy)"
}
] | [
491,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
489,
1
] |
Mathlib/Algebra/Order/UpperLower.lean | IsUpperSet.smul_subset | [] | [
31,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
30,
1
] |
Mathlib/Topology/Sober.lean | quasiSober_of_open_cover | [
{
"state_after": "α : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\n⊢ ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S",
"state_before": "α : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\n⊢ QuasiSober α",
"tactic": "rw [quasiSober_iff]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\n⊢ ∃ x, IsGenericPoint x t",
"state_before": "α : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\n⊢ ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S",
"tactic": "intro t h h'"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\n⊢ ∃ x, IsGenericPoint x t",
"state_before": "α : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\n⊢ ∃ x, IsGenericPoint x t",
"tactic": "obtain ⟨x, hx⟩ := h.1"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\n⊢ ∃ x, IsGenericPoint x t",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\n⊢ ∃ x, IsGenericPoint x t",
"tactic": "obtain ⟨U, hU, hU'⟩ : x ∈ ⋃₀ S := by\n rw [hS'']\n trivial"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\n⊢ ∃ x, IsGenericPoint x t",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\n⊢ ∃ x, IsGenericPoint x t",
"tactic": "haveI : QuasiSober U := hS' ⟨U, hU⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\nH : IsPreirreducible (Subtype.val ⁻¹' t)\n⊢ ∃ x, IsGenericPoint x t",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\n⊢ ∃ x, IsGenericPoint x t",
"tactic": "have H : IsPreirreducible ((↑) ⁻¹' t : Set U) :=\n h.2.preimage (hS ⟨U, hU⟩).openEmbedding_subtype_val"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\n⊢ ∃ x, IsGenericPoint x t",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\nH : IsPreirreducible (Subtype.val ⁻¹' t)\n⊢ ∃ x, IsGenericPoint x t",
"tactic": "replace H : IsIrreducible ((↑) ⁻¹' t : Set U) := ⟨⟨⟨x, hU'⟩, by simpa using hx⟩, H⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\n⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\n⊢ ∃ x, IsGenericPoint x t",
"tactic": "use H.genericPoint"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' closure t\n⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\n⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t",
"tactic": "have := continuous_subtype_val.closure_preimage_subset _ H.genericPoint_spec.mem"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' closure t\n⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t",
"tactic": "rw [h'.closure_eq] at this"
},
{
"state_after": "case intro.intro.intro.a\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ closure {↑(IsIrreducible.genericPoint H)} ≤ t\n\ncase intro.intro.intro.a\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ t ≤ closure {↑(IsIrreducible.genericPoint H)}",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ IsGenericPoint (↑(IsIrreducible.genericPoint H)) t",
"tactic": "apply le_antisymm"
},
{
"state_after": "case intro.intro.intro.a\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ t ≤ closure (Subtype.val '' closure (Subtype.val ⁻¹' t))",
"state_before": "case intro.intro.intro.a\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ t ≤ closure {↑(IsIrreducible.genericPoint H)}",
"tactic": "rw [← image_singleton, ← closure_image_closure continuous_subtype_val, H.genericPoint_spec.def]"
},
{
"state_after": "case intro.intro.intro.a\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ t ∩ ↑{ val := U, property := hU } ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)",
"state_before": "case intro.intro.intro.a\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ t ≤ closure (Subtype.val '' closure (Subtype.val ⁻¹' t))",
"tactic": "refine' (subset_closure_inter_of_isPreirreducible_of_isOpen h.2 (hS ⟨U, hU⟩) ⟨x, hx, hU'⟩).trans\n (closure_mono _)"
},
{
"state_after": "case intro.intro.intro.a\nα : Type u_1\nβ : Type ?u.12585\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\nhS' : ∀ (s : ↑S), QuasiSober ↑↑s\nhS'' : ⋃₀ S = ⊤\nt : Set α\nh : IsIrreducible t\nh' : IsClosed t\nx : α\nhx : x ∈ t\nU : Set α\nhU : U ∈ S\nhU' : x ∈ U\nthis✝ : QuasiSober ↑U\nH : IsIrreducible (Subtype.val ⁻¹' t)\nthis : IsIrreducible.genericPoint H ∈ Subtype.val ⁻¹' t\n⊢ Subtype.val '' (Subtype.val ⁻¹' t) ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)",
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Mathlib/Probability/ProbabilityMassFunction/Uniform.lean | Pmf.toOuterMeasure_ofMultiset_apply | [
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"tactic": "by_cases hx : x ∈ t <;> simp [Set.indicator, hx, div_eq_mul_inv]"
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208,
67
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Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.image_closedBall | [] | [
1060,
41
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Mathlib/LinearAlgebra/Finsupp.lean | Finsupp.supported_eq_span_single | [
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"tactic": "apply Set.mem_image_of_mem _ (hl il)"
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] | [
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Mathlib/Algebra/Homology/QuasiIso.lean | HomologicalComplex.Hom.fromSingle₀KernelAtZeroIso_inv_eq | [
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"tactic": "rw [f.2 0 1 rfl]"
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{
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"state_before": "ι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type ?u.70126\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ d ((CochainComplex.single₀ W).obj Y) 0 1 ≫ HomologicalComplex.Hom.f f 1 = 0",
"tactic": "exact zero_comp"
},
{
"state_after": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ (fromSingle₀KernelAtZeroIso f).inv ≫ equalizer.ι (d X 0 1) 0 =\n kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) (_ : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0) ≫\n equalizer.ι (d X 0 1) 0",
"state_before": "ι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ (fromSingle₀KernelAtZeroIso f).inv =\n kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) (_ : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0)",
"tactic": "apply equalizer.hom_ext"
},
{
"state_after": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ ((Iso.mk\n (homology.map (_ : dTo X 0 ≫ dFrom X 0 = 0) (_ : 0 ≫ d X 0 1 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).hom\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).hom\n (_ : 𝟙 (HomologicalComplex.X X 0) = 𝟙 (HomologicalComplex.X X 0)))\n (homology.map (_ : 0 ≫ d X 0 1 = 0) (_ : dTo X 0 ≫ dFrom X 0 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv\n (_ :\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv.right =\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv.left)) ≪≫\n (cokernelIsoOfEq (_ : imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) = 0) ≪≫ cokernelZeroIsoTarget) ≪≫\n kernelSubobjectIso (d X 0 1)).symm ≪≫\n (asIso ((homologyFunctor W (ComplexShape.up ℕ) 0).map f)).symm ≪≫\n (NatIso.ofComponents fun X =>\n homology.congr\n (_ : dTo ((CochainComplex.single₀ W).obj X) 0 ≫ dFrom ((CochainComplex.single₀ W).obj X) 0 = 0)\n (_ : 0 ≫ 0 = 0)\n (_ : d ((CochainComplex.single₀ W).obj X) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0)\n (_ : 0 = 0) ≪≫\n homologyZeroZero).app\n Y).inv ≫\n equalizer.ι (d X 0 1) 0 =\n kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) (_ : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0) ≫\n equalizer.ι (d X 0 1) 0",
"state_before": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ (fromSingle₀KernelAtZeroIso f).inv ≫ equalizer.ι (d X 0 1) 0 =\n kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) (_ : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0) ≫\n equalizer.ι (d X 0 1) 0",
"tactic": "dsimp only [fromSingle₀KernelAtZeroIso, CochainComplex.homologyZeroIso, homologyOfZeroLeft,\n homology.mapIso, CochainComplex.homologyFunctor0Single₀, kernel.map]"
},
{
"state_after": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ (NatIso.ofComponents fun X =>\n homology.congr\n (_ : dTo ((CochainComplex.single₀ W).obj X) 0 ≫ dFrom ((CochainComplex.single₀ W).obj X) 0 = 0)\n (_ : 0 ≫ 0 = 0)\n (_ : d ((CochainComplex.single₀ W).obj X) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0)\n (_ : 0 = 0) ≪≫\n homologyZeroZero).inv.app\n Y ≫\n (asIso ((homologyFunctor W (ComplexShape.up ℕ) 0).map f)).hom ≫\n (Iso.mk\n (homology.map (_ : dTo X 0 ≫ dFrom X 0 = 0) (_ : 0 ≫ d X 0 1 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).hom\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).hom\n (_ : 𝟙 (HomologicalComplex.X X 0) = 𝟙 (HomologicalComplex.X X 0)))\n (homology.map (_ : 0 ≫ d X 0 1 = 0) (_ : dTo X 0 ≫ dFrom X 0 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv\n (_ :\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv.right =\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv.left)) ≪≫\n (cokernelIsoOfEq (_ : imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) = 0) ≪≫ cokernelZeroIsoTarget) ≪≫\n kernelSubobjectIso (d X 0 1)).hom ≫\n kernel.ι (d X 0 1) =\n HomologicalComplex.Hom.f f 0",
"state_before": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ ((Iso.mk\n (homology.map (_ : dTo X 0 ≫ dFrom X 0 = 0) (_ : 0 ≫ d X 0 1 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).hom\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).hom\n (_ : 𝟙 (HomologicalComplex.X X 0) = 𝟙 (HomologicalComplex.X X 0)))\n (homology.map (_ : 0 ≫ d X 0 1 = 0) (_ : dTo X 0 ≫ dFrom X 0 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv\n (_ :\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv.right =\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv.left)) ≪≫\n (cokernelIsoOfEq (_ : imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) = 0) ≪≫ cokernelZeroIsoTarget) ≪≫\n kernelSubobjectIso (d X 0 1)).symm ≪≫\n (asIso ((homologyFunctor W (ComplexShape.up ℕ) 0).map f)).symm ≪≫\n (NatIso.ofComponents fun X =>\n homology.congr\n (_ : dTo ((CochainComplex.single₀ W).obj X) 0 ≫ dFrom ((CochainComplex.single₀ W).obj X) 0 = 0)\n (_ : 0 ≫ 0 = 0)\n (_ : d ((CochainComplex.single₀ W).obj X) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0)\n (_ : 0 = 0) ≪≫\n homologyZeroZero).app\n Y).inv ≫\n equalizer.ι (d X 0 1) 0 =\n kernel.lift (d X 0 1) (HomologicalComplex.Hom.f f 0) (_ : HomologicalComplex.Hom.f f 0 ≫ d X 0 1 = 0) ≫\n equalizer.ι (d X 0 1) 0",
"tactic": "simp only [Iso.trans_inv, Iso.app_inv, Iso.symm_inv, Category.assoc, equalizer_as_kernel,\n kernel.lift_ι]"
},
{
"state_after": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ ((inv (Subobject.arrow (kernelSubobject 0)) ≫ homology.π 0 0 (_ : 0 ≫ 0 = 0)) ≫\n homology.map (_ : 0 ≫ 0 = 0)\n (_ : dTo ((CochainComplex.single₀ W).obj Y) 0 ≫ dFrom ((CochainComplex.single₀ W).obj Y) 0 = 0)\n (CommaMorphism.mk (𝟙 (xPrev ((CochainComplex.single₀ W).obj Y) 0)) (𝟙 Y))\n (CommaMorphism.mk (𝟙 Y) (𝟙 (xNext ((CochainComplex.single₀ W).obj Y) 0)))\n (_ :\n (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right =\n (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right)) ≫\n homology.map (_ : dTo ((CochainComplex.single₀ W).obj Y) 0 ≫ dFrom ((CochainComplex.single₀ W).obj Y) 0 = 0)\n (_ : dTo X 0 ≫ dFrom X 0 = 0) (sqTo f 0) (sqFrom f 0) (_ : (sqTo f 0).right = (sqTo f 0).right) ≫\n (homology.map (_ : dTo X 0 ≫ dFrom X 0 = 0) (_ : 0 ≫ d X 0 1 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (HomologicalComplex.X X 0))).hom\n (Arrow.isoMk (Iso.refl (HomologicalComplex.X X 0))\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).hom\n (_ : 𝟙 (HomologicalComplex.X X 0) = 𝟙 (HomologicalComplex.X X 0)) ≫\n ((cokernelIsoOfEq (_ : imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) = 0)).hom ≫\n cokernelZeroIsoTarget.hom) ≫\n (kernelSubobjectIso (d X 0 1)).hom) ≫\n kernel.ι (d X 0 1) =\n HomologicalComplex.Hom.f f 0",
"state_before": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ (NatIso.ofComponents fun X =>\n homology.congr\n (_ : dTo ((CochainComplex.single₀ W).obj X) 0 ≫ dFrom ((CochainComplex.single₀ W).obj X) 0 = 0)\n (_ : 0 ≫ 0 = 0)\n (_ : d ((CochainComplex.single₀ W).obj X) (ComplexShape.prev (ComplexShape.up ℕ) 0) 0 = 0)\n (_ : 0 = 0) ≪≫\n homologyZeroZero).inv.app\n Y ≫\n (asIso ((homologyFunctor W (ComplexShape.up ℕ) 0).map f)).hom ≫\n (Iso.mk\n (homology.map (_ : dTo X 0 ≫ dFrom X 0 = 0) (_ : 0 ≫ d X 0 1 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).hom\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).hom\n (_ : 𝟙 (HomologicalComplex.X X 0) = 𝟙 (HomologicalComplex.X X 0)))\n (homology.map (_ : 0 ≫ d X 0 1 = 0) (_ : dTo X 0 ≫ dFrom X 0 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv\n (_ :\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (Arrow.mk (dTo X 0)).right)).inv.right =\n (Arrow.isoMk (Iso.refl (Arrow.mk (dFrom X 0)).left)\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).inv.left)) ≪≫\n (cokernelIsoOfEq (_ : imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) = 0) ≪≫ cokernelZeroIsoTarget) ≪≫\n kernelSubobjectIso (d X 0 1)).hom ≫\n kernel.ι (d X 0 1) =\n HomologicalComplex.Hom.f f 0",
"tactic": "dsimp [asIso]"
},
{
"state_after": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ kernelSubobjectMap (CommaMorphism.mk (𝟙 Y) (𝟙 (xNext ((CochainComplex.single₀ W).obj Y) 0))) ≫\n kernelSubobjectMap (sqFrom f 0) ≫\n kernelSubobjectMap\n (Arrow.isoMk (Iso.refl (HomologicalComplex.X X 0))\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).hom ≫\n homology.π 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) ≫\n cokernel.desc (imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0))\n (𝟙 (Subobject.underlying.obj (kernelSubobject (d X 0 1))))\n (_ :\n imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) ≫\n 𝟙 (Subobject.underlying.obj (kernelSubobject (d X 0 1))) =\n 0) ≫\n Subobject.arrow (kernelSubobject (d X 0 1)) =\n Subobject.arrow (kernelSubobject 0) ≫ HomologicalComplex.Hom.f f 0",
"state_before": "case h\nι : Type ?u.69952\nV : Type u\ninst✝⁸ : Category V\ninst✝⁷ : HasZeroMorphisms V\ninst✝⁶ : HasZeroObject V\ninst✝⁵ : HasEqualizers V\ninst✝⁴ : HasImages V\ninst✝³ : HasImageMaps V\ninst✝² : HasCokernels V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nW : Type u_2\ninst✝¹ : Category W\ninst✝ : Abelian W\nX : CochainComplex W ℕ\nY : W\nf : (CochainComplex.single₀ W).obj Y ⟶ X\nhf : QuasiIso f\n⊢ ((inv (Subobject.arrow (kernelSubobject 0)) ≫ homology.π 0 0 (_ : 0 ≫ 0 = 0)) ≫\n homology.map (_ : 0 ≫ 0 = 0)\n (_ : dTo ((CochainComplex.single₀ W).obj Y) 0 ≫ dFrom ((CochainComplex.single₀ W).obj Y) 0 = 0)\n (CommaMorphism.mk (𝟙 (xPrev ((CochainComplex.single₀ W).obj Y) 0)) (𝟙 Y))\n (CommaMorphism.mk (𝟙 Y) (𝟙 (xNext ((CochainComplex.single₀ W).obj Y) 0)))\n (_ :\n (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right =\n (CommaMorphism.mk (𝟙 (Arrow.mk 0).left) (𝟙 (Arrow.mk 0).right)).right)) ≫\n homology.map (_ : dTo ((CochainComplex.single₀ W).obj Y) 0 ≫ dFrom ((CochainComplex.single₀ W).obj Y) 0 = 0)\n (_ : dTo X 0 ≫ dFrom X 0 = 0) (sqTo f 0) (sqFrom f 0) (_ : (sqTo f 0).right = (sqTo f 0).right) ≫\n (homology.map (_ : dTo X 0 ≫ dFrom X 0 = 0) (_ : 0 ≫ d X 0 1 = 0)\n (Arrow.isoMk (xPrevIsoSelf X CochainComplex.homologyZeroIso.proof_9)\n (Iso.refl (HomologicalComplex.X X 0))).hom\n (Arrow.isoMk (Iso.refl (HomologicalComplex.X X 0))\n (xNextIso X CochainComplex.homologyZeroIso.proof_11)).hom\n (_ : 𝟙 (HomologicalComplex.X X 0) = 𝟙 (HomologicalComplex.X X 0)) ≫\n ((cokernelIsoOfEq (_ : imageToKernel 0 (d X 0 1) (_ : 0 ≫ d X 0 1 = 0) = 0)).hom ≫\n cokernelZeroIsoTarget.hom) ≫\n (kernelSubobjectIso (d X 0 1)).hom) ≫\n kernel.ι (d X 0 1) =\n HomologicalComplex.Hom.f f 0",
"tactic": "simp only [Category.assoc, homology.π_map, cokernelZeroIsoTarget_hom,\n cokernelIsoOfEq_hom_comp_desc, kernelSubobject_arrow, homology.π_map_assoc, IsIso.inv_comp_eq]"
},
{
"state_after": "no goals",
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"tactic": "simp [homology.π, kernelSubobjectMap_comp, Iso.refl_hom (X.X 0), Category.comp_id]"
}
] | [
175,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.subgroupOf_inj | [
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.291162\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.291171\ninst✝² : AddGroup A\nH K✝ : Subgroup G\nk : Set G\nN : Type ?u.291192\ninst✝¹ : Group N\nP : Type ?u.291198\ninst✝ : Group P\nH₁ H₂ K : Subgroup G\n⊢ subgroupOf H₁ K = subgroupOf H₂ K ↔ H₁ ⊓ K = H₂ ⊓ K",
"tactic": "simpa only [SetLike.ext_iff, mem_inf, mem_subgroupOf, and_congr_left_iff] using Subtype.forall"
}
] | [
1670,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1668,
1
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Mathlib/FieldTheory/Subfield.lean | Subfield.coe_toSubring | [] | [
227,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
226,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.X_pow_eq | [] | [
1452,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1451,
1
] |
Mathlib/Algebra/Periodic.lean | Function.Antiperiodic.mul_const_inv | [] | [
500,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
498,
1
] |
Mathlib/RingTheory/Ideal/Prod.lean | Ideal.map_snd_prod | [
{
"state_after": "case h\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ x ∈ map (RingHom.snd R S) (prod I J) ↔ x ∈ J",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\n⊢ map (RingHom.snd R S) (prod I J) = J",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ (∃ x_1, x_1 ∈ prod I J ∧ ↑(RingHom.snd R S) x_1 = x) ↔ x ∈ J",
"state_before": "case h\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ x ∈ map (RingHom.snd R S) (prod I J) ↔ x ∈ J",
"tactic": "rw [mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ (∃ x_1, x_1 ∈ prod I J ∧ ↑(RingHom.snd R S) x_1 = x) ↔ x ∈ J",
"tactic": "exact\n ⟨by\n rintro ⟨x, ⟨h, rfl⟩⟩\n exact h.2, fun h => ⟨⟨0, x⟩, ⟨⟨Ideal.zero_mem _, h⟩, rfl⟩⟩⟩"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : R × S\nh : x ∈ prod I J\n⊢ ↑(RingHom.snd R S) x ∈ J",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ (∃ x_1, x_1 ∈ prod I J ∧ ↑(RingHom.snd R S) x_1 = x) → x ∈ J",
"tactic": "rintro ⟨x, ⟨h, rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx : R × S\nh : x ∈ prod I J\n⊢ ↑(RingHom.snd R S) x ∈ J",
"tactic": "exact h.2"
}
] | [
81,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean | Set.ordConnected_pi | [] | [
126,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/Algebra/Algebra/Hom.lean | AlgHom.coe_mk | [] | [
159,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/Data/Finset/Prod.lean | Finset.nonempty_product | [] | [
207,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
206,
1
] |
Mathlib/Combinatorics/Configuration.lean | Configuration.HasLines.lineCount_eq_pointCount | [
{
"state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\n⊢ lineCount L p = pointCount P l",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\n⊢ lineCount L p = pointCount P l",
"tactic": "obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL"
},
{
"state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\n⊢ lineCount L p = pointCount P l",
"state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\n⊢ lineCount L p = pointCount P l",
"tactic": "let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }"
},
{
"state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ lineCount L p = pointCount P l",
"state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\n⊢ lineCount L p = pointCount P l",
"tactic": "have step1 : (∑ i : P × L, lineCount L i.1) = ∑ i : P × L, pointCount P i.2 := by\n rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]\n simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]"
},
{
"state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nstep2 : ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd\nstep3 : ∑ i in sᶜ, lineCount L i.fst = ∑ i in sᶜ, pointCount P i.snd\n⊢ lineCount L p = pointCount P l",
"state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nstep2 : ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd\n⊢ lineCount L p = pointCount P l",
"tactic": "have step3 : (∑ i in sᶜ, lineCount L i.1) = ∑ i in sᶜ, pointCount P i.2 := by\n rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1"
},
{
"state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nstep2 : ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd\nstep3 :\n ∑ i in Set.toFinset ({i | i.fst ∈ i.snd}ᶜ), lineCount L i.fst =\n ∑ i in Set.toFinset ({i | i.fst ∈ i.snd}ᶜ), pointCount P i.snd\n⊢ lineCount L p = pointCount P l",
"state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nstep2 : ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd\nstep3 : ∑ i in sᶜ, lineCount L i.fst = ∑ i in sᶜ, pointCount P i.snd\n⊢ lineCount L p = pointCount P l",
"tactic": "rw [← Set.toFinset_compl] at step3"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nstep2 : ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd\nstep3 :\n ∑ i in Set.toFinset ({i | i.fst ∈ i.snd}ᶜ), lineCount L i.fst =\n ∑ i in Set.toFinset ({i | i.fst ∈ i.snd}ᶜ), pointCount P i.snd\n⊢ lineCount L p = pointCount P l",
"tactic": "exact\n ((Finset.sum_eq_sum_iff_of_le fun i hi =>\n HasLines.pointCount_le_lineCount (by exact Set.mem_toFinset.mp hi)).mp\n step3.symm (p, l) (Set.mem_toFinset.mpr hpl)).symm"
},
{
"state_after": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\n⊢ ∑ y : L, ∑ x : P, lineCount L (x, y).fst = ∑ x : P, ∑ y : L, pointCount P (x, y).snd",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\n⊢ ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd",
"tactic": "rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]"
},
{
"state_after": "no goals",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\n⊢ ∑ y : L, ∑ x : P, lineCount L (x, y).fst = ∑ x : P, ∑ y : L, pointCount P (x, y).snd",
"tactic": "simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]"
},
{
"state_after": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∑ c : P, ∑ a in Set.toFinset {l | c ∈ l}, lineCount L (c, a).fst = ∑ i in s, pointCount P i.snd\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.fst ∈ Finset.univ ∧ p.snd ∈ Set.toFinset {l | p.fst ∈ l}",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd",
"tactic": "rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]"
},
{
"state_after": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∑ c : P, ∑ a in Set.toFinset {l | c ∈ l}, lineCount L (c, a).fst =\n ∑ c : L, ∑ a in Set.toFinset {p | p ∈ c}, pointCount P (a, c).snd\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.snd ∈ Finset.univ ∧ p.fst ∈ Set.toFinset {p_1 | p_1 ∈ p.snd}\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.fst ∈ Finset.univ ∧ p.snd ∈ Set.toFinset {l | p.fst ∈ l}",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∑ c : P, ∑ a in Set.toFinset {l | c ∈ l}, lineCount L (c, a).fst = ∑ i in s, pointCount P i.snd\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.fst ∈ Finset.univ ∧ p.snd ∈ Set.toFinset {l | p.fst ∈ l}",
"tactic": "rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]"
},
{
"state_after": "case refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl✝ : L\nhpl : ¬p ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nl : L\nhl : l ∈ Finset.univ\n⊢ ∑ a in Set.toFinset {p | p ∈ l}, pointCount P (a, l).snd =\n ∑ a in Set.toFinset {l_1 | (fun l x => f l) l hl ∈ l_1}, lineCount L ((fun l x => f l) l hl, a).fst\n\ncase refine'_2\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np✝ : P\nl : L\nhpl : ¬p✝ ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\np : P\nx✝ : p ∈ Finset.univ\n⊢ ∃ a ha, p = (fun l x => f l) a ha\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.snd ∈ Finset.univ ∧ p.fst ∈ Set.toFinset {p_1 | p_1 ∈ p.snd}\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.fst ∈ Finset.univ ∧ p.snd ∈ Set.toFinset {l | p.fst ∈ l}",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∑ c : P, ∑ a in Set.toFinset {l | c ∈ l}, lineCount L (c, a).fst =\n ∑ c : L, ∑ a in Set.toFinset {p | p ∈ c}, pointCount P (a, c).snd\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.snd ∈ Finset.univ ∧ p.fst ∈ Set.toFinset {p_1 | p_1 ∈ p.snd}\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.fst ∈ Finset.univ ∧ p.snd ∈ Set.toFinset {l | p.fst ∈ l}",
"tactic": "refine'\n (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)\n (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm"
},
{
"state_after": "no goals",
"state_before": "case h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.snd ∈ Finset.univ ∧ p.fst ∈ Set.toFinset {p_1 | p_1 ∈ p.snd}\n\ncase h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.fst ∈ Finset.univ ∧ p.snd ∈ Set.toFinset {l | p.fst ∈ l}",
"tactic": "all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl"
},
{
"state_after": "case refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl✝ : L\nhpl : ¬p ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nl : L\nhl : l ∈ Finset.univ\n⊢ Nat.card ↑{p | p ∈ l} • pointCount P l = Nat.card ↑{l_1 | f l ∈ l_1} • lineCount L (f l)",
"state_before": "case refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl✝ : L\nhpl : ¬p ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nl : L\nhl : l ∈ Finset.univ\n⊢ ∑ a in Set.toFinset {p | p ∈ l}, pointCount P (a, l).snd =\n ∑ a in Set.toFinset {l_1 | (fun l x => f l) l hl ∈ l_1}, lineCount L ((fun l x => f l) l hl, a).fst",
"tactic": "simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]"
},
{
"state_after": "case refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl✝ : L\nhpl : ¬p ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nl : L\nhl : l ∈ Finset.univ\n⊢ pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)",
"state_before": "case refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl✝ : L\nhpl : ¬p ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nl : L\nhl : l ∈ Finset.univ\n⊢ Nat.card ↑{p | p ∈ l} • pointCount P l = Nat.card ↑{l_1 | f l ∈ l_1} • lineCount L (f l)",
"tactic": "change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl✝ : L\nhpl : ¬p ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nl : L\nhl : l ∈ Finset.univ\n⊢ pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)",
"tactic": "rw [hf2]"
},
{
"state_after": "case refine'_2.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np✝ : P\nl✝ : L\nhpl : ¬p✝ ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\np : P\nx✝ : p ∈ Finset.univ\nl : L\nhl : f l = p\n⊢ ∃ a ha, p = (fun l x => f l) a ha",
"state_before": "case refine'_2\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np✝ : P\nl : L\nhpl : ¬p✝ ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\np : P\nx✝ : p ∈ Finset.univ\n⊢ ∃ a ha, p = (fun l x => f l) a ha",
"tactic": "obtain ⟨l, hl⟩ := hf1.2 p"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np✝ : P\nl✝ : L\nhpl : ¬p✝ ∈ l✝\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\np : P\nx✝ : p ∈ Finset.univ\nl : L\nhl : f l = p\n⊢ ∃ a ha, p = (fun l x => f l) a ha",
"tactic": "exact ⟨l, Finset.mem_univ l, hl.symm⟩"
},
{
"state_after": "case h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ {i | i.fst ∈ i.snd} ↔ p.snd ∈ {l | p.fst ∈ l}",
"state_before": "case h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ s ↔ p.fst ∈ Finset.univ ∧ p.snd ∈ Set.toFinset {l | p.fst ∈ l}",
"tactic": "simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]"
},
{
"state_after": "no goals",
"state_before": "case h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\n⊢ ∀ (p : P × L), p ∈ {i | i.fst ∈ i.snd} ↔ p.snd ∈ {l | p.fst ∈ l}",
"tactic": "exact fun p => Iff.rfl"
},
{
"state_after": "no goals",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nstep2 : ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd\n⊢ ∑ i in sᶜ, lineCount L i.fst = ∑ i in sᶜ, pointCount P i.snd",
"tactic": "rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1"
},
{
"state_after": "no goals",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : ¬p ∈ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := Set.toFinset {i | i.fst ∈ i.snd}\nstep1 : ∑ i : P × L, lineCount L i.fst = ∑ i : P × L, pointCount P i.snd\nstep2 : ∑ i in s, lineCount L i.fst = ∑ i in s, pointCount P i.snd\nstep3 :\n ∑ i in Set.toFinset ({i | i.fst ∈ i.snd}ᶜ), lineCount L i.fst =\n ∑ i in Set.toFinset ({i | i.fst ∈ i.snd}ᶜ), pointCount P i.snd\ni : P × L\nhi : i ∈ Set.toFinset ({i | i.fst ∈ i.snd}ᶜ)\n⊢ ¬i.fst ∈ i.snd",
"tactic": "exact Set.mem_toFinset.mp hi"
}
] | [
312,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
] |
Mathlib/Data/Fin/VecNotation.lean | Matrix.head_sub | [] | [
517,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
516,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean | Ideal.pow_succ_lt_pow | [] | [
772,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
769,
1
] |
Mathlib/Data/Bool/Basic.lean | Bool.and_intro | [
{
"state_after": "no goals",
"state_before": "⊢ ∀ {a b : Bool}, a = true → b = true → (a && b) = true",
"tactic": "decide"
}
] | [
176,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
] |
Mathlib/Order/Filter/NAry.lean | Filter.map₂_left_identity | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.45119\nβ : Type u_1\nβ' : Type ?u.45125\nγ : Type ?u.45128\nγ' : Type ?u.45131\nδ : Type ?u.45134\nδ' : Type ?u.45137\nε : Type ?u.45140\nε' : Type ?u.45143\nm : α → β → γ\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh✝ h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na✝ : α\nb : β\nc : γ\nf : α → β → β\na : α\nh : ∀ (b : β), f a b = b\nl : Filter β\n⊢ map₂ f (pure a) l = l",
"tactic": "rw [map₂_pure_left, show f a = id from funext h, map_id]"
}
] | [
414,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
] |
Mathlib/Analysis/Calculus/DiffContOnCl.lean | IsClosed.diffContOnCl_iff | [] | [
48,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/Algebra/Squarefree.lean | UniqueFactorizationMonoid.dvd_pow_iff_dvd_of_squarefree | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\nx y : R\nn : ℕ\nhsq : Squarefree x\nh0 : n ≠ 0\n⊢ x ∣ y ^ n ↔ x ∣ y",
"tactic": "classical\n haveI := UniqueFactorizationMonoid.toGCDMonoid R\n exact ⟨hsq.isRadical n y, fun h => h.pow h0⟩"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\nx y : R\nn : ℕ\nhsq : Squarefree x\nh0 : n ≠ 0\nthis : GCDMonoid R\n⊢ x ∣ y ^ n ↔ x ∣ y",
"state_before": "R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\nx y : R\nn : ℕ\nhsq : Squarefree x\nh0 : n ≠ 0\n⊢ x ∣ y ^ n ↔ x ∣ y",
"tactic": "haveI := UniqueFactorizationMonoid.toGCDMonoid R"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\nx y : R\nn : ℕ\nhsq : Squarefree x\nh0 : n ≠ 0\nthis : GCDMonoid R\n⊢ x ∣ y ^ n ↔ x ∣ y",
"tactic": "exact ⟨hsq.isRadical n y, fun h => h.pow h0⟩"
}
] | [
264,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
260,
1
] |
Mathlib/NumberTheory/Fermat4.lean | Fermat42.minimal_comm | [] | [
114,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
113,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean | ofMul_pow | [] | [
458,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
457,
1
] |
Mathlib/Order/Filter/Bases.lean | Filter.disjoint_principal_principal | [
{
"state_after": "no goals",
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"tactic": "rw [← subset_compl_iff_disjoint_left, disjoint_principal_left, mem_principal]"
}
] | [
709,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
708,
1
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Mathlib/Algebra/GCDMonoid/Basic.lean | dvd_gcd_mul_of_dvd_mul | [] | [
512,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
511,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Products.lean | CategoryTheory.Limits.Pi.reindex_inv_π | [
{
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"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasProduct f\ninst✝ : HasProduct (f ∘ ↑ε)\nb : β\n⊢ (reindex ε f).inv ≫ π (f ∘ ↑ε) b = π f (↑ε b)",
"tactic": "simp [Iso.inv_comp_eq]"
}
] | [
416,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
415,
1
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Mathlib/Data/Nat/GCD/Basic.lean | Nat.coprime_add_mul_right_left | [
{
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"tactic": "rw [coprime, coprime, gcd_add_mul_right_left]"
}
] | [
174,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean | LinearMap.toMatrix'_toLin' | [] | [
330,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
329,
1
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Mathlib/Data/Finset/Preimage.lean | Finset.preimage_empty | [
{
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"tactic": "simp [InjOn]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\n⊢ ↑(preimage ∅ f (_ : ∀ (a : α), a ∈ f ⁻¹' ↑∅ → ∀ (a_2 : α), a_2 ∈ f ⁻¹' ↑∅ → f a = f a_2 → a = a_2)) = ↑∅",
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}
] | [
50,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
49,
1
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Mathlib/CategoryTheory/MorphismProperty.lean | CategoryTheory.MorphismProperty.StableUnderBaseChange.diagonal | [
{
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"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.79385\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\n⊢ ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S), MorphismProperty.diagonal P g → MorphismProperty.diagonal P pullback.fst",
"tactic": "introv h"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.79385\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nX Y S : C\nf : X ⟶ S\ng : Y ⟶ S\nh : MorphismProperty.diagonal P g\n⊢ P ((baseChange f).map (Over.homMk (pullback.diagonal g))).left",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.79385\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nX Y S : C\nf : X ⟶ S\ng : Y ⟶ S\nh : MorphismProperty.diagonal P g\n⊢ MorphismProperty.diagonal P pullback.fst",
"tactic": "rw [diagonal_iff, diagonal_pullback_fst, hP'.cancel_left_isIso, hP'.cancel_right_isIso]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.79385\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nX Y S : C\nf : X ⟶ S\ng : Y ⟶ S\nh : MorphismProperty.diagonal P g\n⊢ P ((baseChange f).map (Over.homMk (pullback.diagonal g))).left",
"tactic": "exact hP.baseChange_map f _ (by simpa)"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.79385\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nX Y S : C\nf : X ⟶ S\ng : Y ⟶ S\nh : MorphismProperty.diagonal P g\n⊢ P (Over.homMk (pullback.diagonal g)).left",
"tactic": "simpa"
}
] | [
555,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
549,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean | Subsemigroup.mem_iInf | [
{
"state_after": "no goals",
"state_before": "M : Type u_2\nN : Type ?u.9228\nA : Type ?u.9231\ninst✝¹ : Mul M\ns : Set M\ninst✝ : Add A\nt : Set A\nS✝ : Subsemigroup M\nι : Sort u_1\nS : ι → Subsemigroup M\nx : M\n⊢ (x ∈ ⨅ (i : ι), S i) ↔ ∀ (i : ι), x ∈ S i",
"tactic": "simp only [iInf, mem_sInf, Set.forall_range_iff]"
}
] | [
256,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
255,
1
] |
Mathlib/Data/Nat/Order/Basic.lean | Nat.not_dvd_of_between_consec_multiples | [
{
"state_after": "case intro\nn k l d : ℕ\nh1 : n * k < n * d\nh2 : n * d < n * (k + 1)\n⊢ False",
"state_before": "m n k l : ℕ\nh1 : n * k < m\nh2 : m < n * (k + 1)\n⊢ ¬n ∣ m",
"tactic": "rintro ⟨d, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nn k l d : ℕ\nh1 : n * k < n * d\nh2 : n * d < n * (k + 1)\n⊢ False",
"tactic": "exact Monotone.ne_of_lt_of_lt_nat (Covariant.monotone_of_const n) k h1 h2 d rfl"
}
] | [
564,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
562,
1
] |
Mathlib/RingTheory/Polynomial/Pochhammer.lean | pochhammer_mul | [
{
"state_after": "case zero\nS : Type u\ninst✝ : Semiring S\nn : ℕ\n⊢ pochhammer S n * comp (pochhammer S Nat.zero) (X + ↑n) = pochhammer S (n + Nat.zero)\n\ncase succ\nS : Type u\ninst✝ : Semiring S\nn m : ℕ\nih : pochhammer S n * comp (pochhammer S m) (X + ↑n) = pochhammer S (n + m)\n⊢ pochhammer S n * comp (pochhammer S (Nat.succ m)) (X + ↑n) = pochhammer S (n + Nat.succ m)",
"state_before": "S : Type u\ninst✝ : Semiring S\nn m : ℕ\n⊢ pochhammer S n * comp (pochhammer S m) (X + ↑n) = pochhammer S (n + m)",
"tactic": "induction' m with m ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nS : Type u\ninst✝ : Semiring S\nn : ℕ\n⊢ pochhammer S n * comp (pochhammer S Nat.zero) (X + ↑n) = pochhammer S (n + Nat.zero)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nS : Type u\ninst✝ : Semiring S\nn m : ℕ\nih : pochhammer S n * comp (pochhammer S m) (X + ↑n) = pochhammer S (n + m)\n⊢ pochhammer S n * comp (pochhammer S (Nat.succ m)) (X + ↑n) = pochhammer S (n + Nat.succ m)",
"tactic": "rw [pochhammer_succ_right, Polynomial.mul_X_add_nat_cast_comp, ← mul_assoc, ih,\n Nat.succ_eq_add_one, ← add_assoc, pochhammer_succ_right, Nat.cast_add, add_assoc]"
}
] | [
143,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
] |
Mathlib/Data/Real/Basic.lean | Real.sSup_le | [
{
"state_after": "case inl\nx y a : ℝ\nha : 0 ≤ a\nhS : ∀ (x : ℝ), x ∈ ∅ → x ≤ a\n⊢ sSup ∅ ≤ a\n\ncase inr\nx y : ℝ\nS : Set ℝ\na : ℝ\nhS : ∀ (x : ℝ), x ∈ S → x ≤ a\nha : 0 ≤ a\nhS₂ : Set.Nonempty S\n⊢ sSup S ≤ a",
"state_before": "x y : ℝ\nS : Set ℝ\na : ℝ\nhS : ∀ (x : ℝ), x ∈ S → x ≤ a\nha : 0 ≤ a\n⊢ sSup S ≤ a",
"tactic": "rcases S.eq_empty_or_nonempty with (rfl | hS₂)"
},
{
"state_after": "no goals",
"state_before": "case inl\nx y a : ℝ\nha : 0 ≤ a\nhS : ∀ (x : ℝ), x ∈ ∅ → x ≤ a\n⊢ sSup ∅ ≤ a\n\ncase inr\nx y : ℝ\nS : Set ℝ\na : ℝ\nhS : ∀ (x : ℝ), x ∈ S → x ≤ a\nha : 0 ≤ a\nhS₂ : Set.Nonempty S\n⊢ sSup S ≤ a",
"tactic": "exacts [sSup_empty.trans_le ha, csSup_le hS₂ hS]"
}
] | [
878,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
876,
11
] |
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | Asymptotics.IsEquivalent.tendsto_nhds_iff | [] | [
165,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
] |
Mathlib/Data/Nat/Size.lean | Nat.size_le_size | [] | [
175,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.inv_unique | [
{
"state_after": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.575607\ninst✝ : CommMonoid P\nf : M →* N\nh : ∀ (y : { x // x ∈ S }), IsUnit (↑f ↑y)\ny : { x // x ∈ S }\nz : N\nH : ↑f ↑y * z = 1\n⊢ ↑1 = ↑f ↑y * z",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.575607\ninst✝ : CommMonoid P\nf : M →* N\nh : ∀ (y : { x // x ∈ S }), IsUnit (↑f ↑y)\ny : { x // x ∈ S }\nz : N\nH : ↑f ↑y * z = 1\n⊢ ↑(↑(IsUnit.liftRight (MonoidHom.restrict f S) h) y)⁻¹ = z",
"tactic": "rw [← one_mul _⁻¹, Units.val_mul, mul_inv_left]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.575607\ninst✝ : CommMonoid P\nf : M →* N\nh : ∀ (y : { x // x ∈ S }), IsUnit (↑f ↑y)\ny : { x // x ∈ S }\nz : N\nH : ↑f ↑y * z = 1\n⊢ ↑1 = ↑f ↑y * z",
"tactic": "exact H.symm"
}
] | [
674,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
671,
1
] |
Mathlib/Algebra/Hom/Group.lean | MulHom.comp_mul | [
{
"state_after": "case h\nα : Type ?u.204587\nβ : Type ?u.204590\nM : Type u_1\nN : Type u_2\nP : Type u_3\nG : Type ?u.204602\nH : Type ?u.204605\nF : Type ?u.204608\ninst✝² : Mul M\ninst✝¹ : CommSemigroup N\ninst✝ : CommSemigroup P\ng : N →ₙ* P\nf₁ f₂ : M →ₙ* N\nx✝ : M\n⊢ ↑(comp g (f₁ * f₂)) x✝ = ↑(comp g f₁ * comp g f₂) x✝",
"state_before": "α : Type ?u.204587\nβ : Type ?u.204590\nM : Type u_1\nN : Type u_2\nP : Type u_3\nG : Type ?u.204602\nH : Type ?u.204605\nF : Type ?u.204608\ninst✝² : Mul M\ninst✝¹ : CommSemigroup N\ninst✝ : CommSemigroup P\ng : N →ₙ* P\nf₁ f₂ : M →ₙ* N\n⊢ comp g (f₁ * f₂) = comp g f₁ * comp g f₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.204587\nβ : Type ?u.204590\nM : Type u_1\nN : Type u_2\nP : Type u_3\nG : Type ?u.204602\nH : Type ?u.204605\nF : Type ?u.204608\ninst✝² : Mul M\ninst✝¹ : CommSemigroup N\ninst✝ : CommSemigroup P\ng : N →ₙ* P\nf₁ f₂ : M →ₙ* N\nx✝ : M\n⊢ ↑(comp g (f₁ * f₂)) x✝ = ↑(comp g f₁ * comp g f₂) x✝",
"tactic": "simp only [mul_apply, Function.comp_apply, map_mul, coe_comp]"
}
] | [
1475,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1472,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.blimsup_congr | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\n⊢ sInf {a | ∀ᶠ (x : β) in f, p x → u x ≤ a} = blimsup v f p",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\n⊢ blimsup u f p = blimsup v f p",
"tactic": "rw [blimsup_eq]"
},
{
"state_after": "case e_a.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\n⊢ b ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a} ↔ b ∈ {a | ∀ᶠ (x : β) in f, p x → v x ≤ a}",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\n⊢ sInf {a | ∀ᶠ (x : β) in f, p x → u x ≤ a} = blimsup v f p",
"tactic": "congr with b"
},
{
"state_after": "case e_a.h.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → u x ≤ b\nh₂ : p x\n⊢ v x ≤ b\n\ncase e_a.h.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → v x ≤ b\nh₂ : p x\n⊢ u x ≤ b",
"state_before": "case e_a.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\n⊢ b ∈ {a | ∀ᶠ (x : β) in f, p x → u x ≤ a} ↔ b ∈ {a | ∀ᶠ (x : β) in f, p x → v x ≤ a}",
"tactic": "refine' eventually_congr (h.mono fun x hx => ⟨fun h₁ h₂ => _, fun h₁ h₂ => _⟩)"
},
{
"state_after": "case e_a.h.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → u x ≤ b\nh₂ : p x\n⊢ u x ≤ b",
"state_before": "case e_a.h.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → u x ≤ b\nh₂ : p x\n⊢ v x ≤ b",
"tactic": "rw [← hx h₂]"
},
{
"state_after": "no goals",
"state_before": "case e_a.h.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → u x ≤ b\nh₂ : p x\n⊢ u x ≤ b",
"tactic": "exact h₁ h₂"
},
{
"state_after": "case e_a.h.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → v x ≤ b\nh₂ : p x\n⊢ v x ≤ b",
"state_before": "case e_a.h.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → v x ≤ b\nh₂ : p x\n⊢ u x ≤ b",
"tactic": "rw [hx h₂]"
},
{
"state_after": "no goals",
"state_before": "case e_a.h.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.81206\nι : Type ?u.81209\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\nb : α\nx : β\nhx : p x → u x = v x\nh₁ : p x → v x ≤ b\nh₂ : p x\n⊢ v x ≤ b",
"tactic": "exact h₁ h₂"
}
] | [
586,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
578,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.shiftLeft_eq | [
{
"state_after": "no goals",
"state_before": "a b✝ b : Nat\n⊢ 2 * a * 2 ^ b = a * 2 ^ (b + 1)",
"tactic": "simp [pow_succ, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm]"
}
] | [
803,
68
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
799,
9
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean | UniformCauchySeqOnFilter.mono_left | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\np'' : Filter ι\nhf : UniformCauchySeqOnFilter F p p'\nhp : p'' ≤ p\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∀ᶠ (m : (ι × ι) × α) in (p'' ×ˢ p'') ×ˢ p', (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\np'' : Filter ι\nhf : UniformCauchySeqOnFilter F p p'\nhp : p'' ≤ p\n⊢ UniformCauchySeqOnFilter F p'' p'",
"tactic": "intro u hu"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\np'' : Filter ι\nhf : UniformCauchySeqOnFilter F p p'\nhp : p'' ≤ p\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nthis : ∀ᶠ (x : (ι × ι) × α) in (p'' ×ˢ p'') ×ˢ p', (F x.fst.fst x.snd, F x.fst.snd x.snd) ∈ u\n⊢ ∀ᶠ (m : (ι × ι) × α) in (p'' ×ˢ p'') ×ˢ p', (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\np'' : Filter ι\nhf : UniformCauchySeqOnFilter F p p'\nhp : p'' ≤ p\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∀ᶠ (m : (ι × ι) × α) in (p'' ×ˢ p'') ×ˢ p', (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"tactic": "have := (hf u hu).filter_mono (p'.prod_mono_left (Filter.prod_mono hp hp))"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\np'' : Filter ι\nhf : UniformCauchySeqOnFilter F p p'\nhp : p'' ≤ p\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nthis : ∀ᶠ (x : (ι × ι) × α) in (p'' ×ˢ p'') ×ˢ p', (F x.fst.fst x.snd, F x.fst.snd x.snd) ∈ u\n⊢ ∀ᶠ (m : (ι × ι) × α) in (p'' ×ˢ p'') ×ˢ p', (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"tactic": "exact this.mono (by simp)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\np'' : Filter ι\nhf : UniformCauchySeqOnFilter F p p'\nhp : p'' ≤ p\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nthis : ∀ᶠ (x : (ι × ι) × α) in (p'' ×ˢ p'') ×ˢ p', (F x.fst.fst x.snd, F x.fst.snd x.snd) ∈ u\n⊢ ∀ (x : (ι × ι) × α), (F x.fst.fst x.snd, F x.fst.snd x.snd) ∈ u → (F x.fst.fst x.snd, F x.fst.snd x.snd) ∈ u",
"tactic": "simp"
}
] | [
481,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
477,
1
] |
Mathlib/Analysis/Convex/Extreme.lean | IsExtreme.inter | [
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\n⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → ∀ ⦃x : E⦄, x ∈ B ∩ C → x ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\n⊢ IsExtreme 𝕜 A (B ∩ C)",
"tactic": "use Subset.trans (inter_subset_left _ _) hAB.1"
},
{
"state_after": "case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx✝ : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\nx : E\nhxB : x ∈ B\nhxC : x ∈ C\nhx : x ∈ openSegment 𝕜 x₁ x₂\n⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\n⊢ ∀ ⦃x₁ : E⦄, x₁ ∈ A → ∀ ⦃x₂ : E⦄, x₂ ∈ A → ∀ ⦃x : E⦄, x ∈ B ∩ C → x ∈ openSegment 𝕜 x₁ x₂ → x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"tactic": "rintro x₁ hx₁A x₂ hx₂A x ⟨hxB, hxC⟩ hx"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx✝ : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\nx : E\nhxB : x ∈ B\nhxC : x ∈ C\nhx : x ∈ openSegment 𝕜 x₁ x₂\nhx₁B : x₁ ∈ B\nhx₂B : x₂ ∈ B\n⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"state_before": "case intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx✝ : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\nx : E\nhxB : x ∈ B\nhxC : x ∈ C\nhx : x ∈ openSegment 𝕜 x₁ x₂\n⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"tactic": "obtain ⟨hx₁B, hx₂B⟩ := hAB.2 hx₁A hx₂A hxB hx"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx✝ : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\nx : E\nhxB : x ∈ B\nhxC : x ∈ C\nhx : x ∈ openSegment 𝕜 x₁ x₂\nhx₁B : x₁ ∈ B\nhx₂B : x₂ ∈ B\nhx₁C : x₁ ∈ C\nhx₂C : x₂ ∈ C\n⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"state_before": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx✝ : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\nx : E\nhxB : x ∈ B\nhxC : x ∈ C\nhx : x ∈ openSegment 𝕜 x₁ x₂\nhx₁B : x₁ ∈ B\nhx₂B : x₂ ∈ B\n⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"tactic": "obtain ⟨hx₁C, hx₂C⟩ := hAC.2 hx₁A hx₂A hxC hx"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2321\nι : Type ?u.2324\nπ : ι → Type ?u.2329\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx✝ : E\nhAB : IsExtreme 𝕜 A B\nhAC : IsExtreme 𝕜 A C\nx₁ : E\nhx₁A : x₁ ∈ A\nx₂ : E\nhx₂A : x₂ ∈ A\nx : E\nhxB : x ∈ B\nhxC : x ∈ C\nhx : x ∈ openSegment 𝕜 x₁ x₂\nhx₁B : x₁ ∈ B\nhx₂B : x₂ ∈ B\nhx₁C : x₁ ∈ C\nhx₂C : x₂ ∈ C\n⊢ x₁ ∈ B ∩ C ∧ x₂ ∈ B ∩ C",
"tactic": "exact ⟨⟨hx₁B, hx₁C⟩, hx₂B, hx₂C⟩"
}
] | [
105,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometry.map_eq_iff | [] | [
288,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/Algebra/Order/Ring/WithTop.lean | WithTop.coe_mul | [
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : a = 0\n⊢ ↑(a * b) = ↑a * ↑b\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\n⊢ ↑(a * b) = ↑a * ↑b",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\n⊢ ↑(a * b) = ↑a * ↑b",
"tactic": "by_cases ha : a = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : a = 0\n⊢ ↑(a * b) = ↑a * ↑b",
"tactic": "simp [ha]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\nhb : b = 0\n⊢ ↑(a * b) = ↑a * ↑b\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\nhb : ¬b = 0\n⊢ ↑(a * b) = ↑a * ↑b",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\n⊢ ↑(a * b) = ↑a * ↑b",
"tactic": "by_cases hb : b = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\nhb : b = 0\n⊢ ↑(a * b) = ↑a * ↑b",
"tactic": "simp [hb]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\nhb : ¬b = 0\n⊢ ↑(a * b) = Option.map₂ (fun x x_1 => x * x_1) ↑a ↑b",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\nhb : ¬b = 0\n⊢ ↑(a * b) = ↑a * ↑b",
"tactic": "simp [*, mul_def]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na b : α\nha : ¬a = 0\nhb : ¬b = 0\n⊢ ↑(a * b) = Option.map₂ (fun x x_1 => x * x_1) ↑a ↑b",
"tactic": "rfl"
}
] | [
98,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.some_eq_coe' | [] | [
155,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
9
] |
Mathlib/Data/Matrix/Basic.lean | Matrix.neg_mulVec | [
{
"state_after": "case h\nl : Type ?u.908864\nm : Type u_2\nn : Type u_1\no : Type ?u.908873\nm' : o → Type ?u.908878\nn' : o → Type ?u.908883\nR : Type ?u.908886\nS : Type ?u.908889\nα : Type v\nβ : Type w\nγ : Type ?u.908896\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : Fintype n\nv : n → α\nA : Matrix m n α\nx✝ : m\n⊢ mulVec (-A) v x✝ = (-mulVec A v) x✝",
"state_before": "l : Type ?u.908864\nm : Type u_2\nn : Type u_1\no : Type ?u.908873\nm' : o → Type ?u.908878\nn' : o → Type ?u.908883\nR : Type ?u.908886\nS : Type ?u.908889\nα : Type v\nβ : Type w\nγ : Type ?u.908896\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : Fintype n\nv : n → α\nA : Matrix m n α\n⊢ mulVec (-A) v = -mulVec A v",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.908864\nm : Type u_2\nn : Type u_1\no : Type ?u.908873\nm' : o → Type ?u.908878\nn' : o → Type ?u.908883\nR : Type ?u.908886\nS : Type ?u.908889\nα : Type v\nβ : Type w\nγ : Type ?u.908896\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : Fintype n\nv : n → α\nA : Matrix m n α\nx✝ : m\n⊢ mulVec (-A) v x✝ = (-mulVec A v) x✝",
"tactic": "apply neg_dotProduct"
}
] | [
1908,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1906,
1
] |
Mathlib/Topology/Order/Basic.lean | nhds_basis_Ioo | [] | [
1335,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1333,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_map' | [
{
"state_after": "case e_μ\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1148988\nδ : Type ?u.1148991\nm : MeasurableSpace α\nμ ν : Measure α\nmβ : MeasurableSpace β\nf : β → ℝ≥0∞\ng : α → β\nhf : AEMeasurable f\nhg : AEMeasurable g\n⊢ Measure.map g μ = Measure.map (AEMeasurable.mk g hg) μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1148988\nδ : Type ?u.1148991\nm : MeasurableSpace α\nμ ν : Measure α\nmβ : MeasurableSpace β\nf : β → ℝ≥0∞\ng : α → β\nhf : AEMeasurable f\nhg : AEMeasurable g\n⊢ (∫⁻ (a : β), AEMeasurable.mk f hf a ∂Measure.map g μ) =\n ∫⁻ (a : β), AEMeasurable.mk f hf a ∂Measure.map (AEMeasurable.mk g hg) μ",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_μ\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.1148988\nδ : Type ?u.1148991\nm : MeasurableSpace α\nμ ν : Measure α\nmβ : MeasurableSpace β\nf : β → ℝ≥0∞\ng : α → β\nhf : AEMeasurable f\nhg : AEMeasurable g\n⊢ Measure.map g μ = Measure.map (AEMeasurable.mk g hg) μ",
"tactic": "exact Measure.map_congr hg.ae_eq_mk"
}
] | [
1279,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1268,
1
] |
Mathlib/Algebra/Quaternion.lean | Quaternion.smul_imJ | [] | [
1041,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1041,
9
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean | SimplexCategory.mono_iff_injective | [
{
"state_after": "n m : SimplexCategory\nf : n ⟶ m\n⊢ Mono (skeletalEquivalence.functor.map f) ↔ Function.Injective ↑(Hom.toOrderHom f)",
"state_before": "n m : SimplexCategory\nf : n ⟶ m\n⊢ Mono f ↔ Function.Injective ↑(Hom.toOrderHom f)",
"tactic": "rw [← Functor.mono_map_iff_mono skeletalEquivalence.functor.{0}]"
},
{
"state_after": "n m : SimplexCategory\nf : n ⟶ m\n⊢ Mono (skeletalFunctor.map f) ↔ Function.Injective ↑(Hom.toOrderHom f)",
"state_before": "n m : SimplexCategory\nf : n ⟶ m\n⊢ Mono (skeletalEquivalence.functor.map f) ↔ Function.Injective ↑(Hom.toOrderHom f)",
"tactic": "dsimp only [skeletalEquivalence, Functor.asEquivalence_functor]"
},
{
"state_after": "no goals",
"state_before": "n m : SimplexCategory\nf : n ⟶ m\n⊢ Mono (skeletalFunctor.map f) ↔ Function.Injective ↑(Hom.toOrderHom f)",
"tactic": "rw [NonemptyFinLinOrdCat.mono_iff_injective, skeletalFunctor.coe_map,\n Function.Injective.of_comp_iff ULift.up_injective,\n Function.Injective.of_comp_iff' _ ULift.down_bijective]"
}
] | [
483,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
477,
1
] |
Mathlib/Topology/Connected.lean | isConnected_singleton | [] | [
89,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean | tsub_tsub_tsub_cancel_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : c ≤ b\n⊢ a - c - (b - c) = a - b",
"tactic": "rw [tsub_tsub, add_tsub_cancel_of_le h]"
}
] | [
71,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/RingTheory/QuotientNilpotent.lean | Ideal.isRadical_iff_quotient_reduced | [
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ IsRadical (RingHom.ker (Quotient.mk I)) ↔ IsReduced (R ⧸ I)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ IsRadical I ↔ IsReduced (R ⧸ I)",
"tactic": "conv_lhs => rw [← @Ideal.mk_ker R _ I]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ IsRadical (RingHom.ker (Quotient.mk I)) ↔ IsReduced (R ⧸ I)",
"tactic": "exact RingHom.ker_isRadical_iff_reduced_of_surjective (@Ideal.Quotient.mk_surjective R _ I)"
}
] | [
21,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
18,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean | Associates.out_dvd_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\nb : Associates α\n⊢ ∀ (a_1 : α),\n Associates.out (Quotient.mk (Associated.setoid α) a_1) ∣ a ↔ Quotient.mk (Associated.setoid α) a_1 ≤ Associates.mk a",
"tactic": "simp [Associates.out_mk, Associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]"
}
] | [
243,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
241,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean | Nat.ascFactorial_le_pow_add | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ascFactorial n 0 ≤ (n + 0) ^ 0",
"tactic": "rw [ascFactorial_zero, pow_zero]"
},
{
"state_after": "n k : ℕ\n⊢ succ (n + k) * ascFactorial n k ≤ succ (n + k) * succ (n + k) ^ k",
"state_before": "n k : ℕ\n⊢ ascFactorial n (k + 1) ≤ (n + (k + 1)) ^ (k + 1)",
"tactic": "rw [ascFactorial_succ, pow_succ, ← add_assoc,\n← Nat.succ_eq_add_one (n + k), mul_comm _ (succ (n + k))]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ succ (n + k) * ascFactorial n k ≤ succ (n + k) * succ (n + k) ^ k",
"tactic": "exact\n Nat.mul_le_mul_of_nonneg_left\n ((ascFactorial_le_pow_add _ k).trans (Nat.pow_le_pow_of_le_left (le_succ _) _))"
}
] | [
312,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
] |
Mathlib/Data/Vector/Basic.lean | Vector.get_zero | [] | [
258,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Std/Data/List/Init/Lemmas.lean | List.append_bind | [
{
"state_after": "case nil\nα : Type u_1\nβ : Type u_2\nys : List α\nf : α → List β\n⊢ List.bind ([] ++ ys) f = List.bind [] f ++ List.bind ys f\n\ncase cons\nα : Type u_1\nβ : Type u_2\nys : List α\nf : α → List β\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : List.bind (tail✝ ++ ys) f = List.bind tail✝ f ++ List.bind ys f\n⊢ List.bind (head✝ :: tail✝ ++ ys) f = List.bind (head✝ :: tail✝) f ++ List.bind ys f",
"state_before": "α : Type u_1\nβ : Type u_2\nxs ys : List α\nf : α → List β\n⊢ List.bind (xs ++ ys) f = List.bind xs f ++ List.bind ys f",
"tactic": "induction xs"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type u_2\nys : List α\nf : α → List β\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : List.bind (tail✝ ++ ys) f = List.bind tail✝ f ++ List.bind ys f\n⊢ List.bind (head✝ :: tail✝ ++ ys) f = List.bind (head✝ :: tail✝) f ++ List.bind ys f",
"state_before": "case nil\nα : Type u_1\nβ : Type u_2\nys : List α\nf : α → List β\n⊢ List.bind ([] ++ ys) f = List.bind [] f ++ List.bind ys f\n\ncase cons\nα : Type u_1\nβ : Type u_2\nys : List α\nf : α → List β\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : List.bind (tail✝ ++ ys) f = List.bind tail✝ f ++ List.bind ys f\n⊢ List.bind (head✝ :: tail✝ ++ ys) f = List.bind (head✝ :: tail✝) f ++ List.bind ys f",
"tactic": "{rfl}"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type u_2\nys : List α\nf : α → List β\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : List.bind (tail✝ ++ ys) f = List.bind tail✝ f ++ List.bind ys f\n⊢ List.bind (head✝ :: tail✝ ++ ys) f = List.bind (head✝ :: tail✝) f ++ List.bind ys f",
"tactic": "simp_all [cons_bind, append_assoc]"
}
] | [
107,
58
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
105,
9
] |
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