file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Finset/Lattice.lean
|
Finset.le_max
|
[] |
[
1211,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1210,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
Measurable.comp
|
[] |
[
556,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
11
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
PseudoEpimorphism.comp_apply
|
[] |
[
194,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean
|
CategoryTheory.CostructuredArrow.mk_hom_eq_self
|
[] |
[
312,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
Submonoid.isSubmonoid
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝¹ : Monoid M\ns : Set M\nA : Type ?u.82313\ninst✝ : AddMonoid A\nt : Set A\nS : Submonoid M\n⊢ IsSubmonoid ↑S",
"tactic": "refine' ⟨S.2, S.1.2⟩"
}
] |
[
431,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_assoc
|
[
{
"state_after": "no goals",
"state_before": "m : Nat\nb : Int\nk : Nat\n⊢ ↑m + b + ↑k = ↑m + (b + ↑k)",
"tactic": "rw [Int.add_comm, ← aux1, Int.add_comm k, aux1, Int.add_comm b]"
},
{
"state_after": "no goals",
"state_before": "a : Int\nn k : Nat\n⊢ a + ↑n + ↑k = a + (↑n + ↑k)",
"tactic": "rw [Int.add_comm, Int.add_comm a, ← aux1, Int.add_comm a, Int.add_comm k]"
},
{
"state_after": "no goals",
"state_before": "m n k : Nat\n⊢ -[m+1] + ↑n + -[k+1] = -[m+1] + (↑n + -[k+1])",
"tactic": "rw [Int.add_comm, ← aux2, Int.add_comm n, ← aux2, Int.add_comm -[m+1]]"
},
{
"state_after": "no goals",
"state_before": "m n k : Nat\n⊢ ↑m + -[n+1] + -[k+1] = ↑m + (-[n+1] + -[k+1])",
"tactic": "rw [Int.add_comm, Int.add_comm m, Int.add_comm m, ← aux2, Int.add_comm -[k+1]]"
},
{
"state_after": "no goals",
"state_before": "m n k : Nat\n⊢ -[m+1] + -[n+1] + -[k+1] = -[m+1] + (-[n+1] + -[k+1])",
"tactic": "simp [add_succ, Nat.add_comm, Nat.add_left_comm, neg_ofNat_succ]"
},
{
"state_after": "no goals",
"state_before": "x✝² x✝¹ x✝ : Int\nm n k : Nat\n⊢ ↑m + ↑n + ↑k = ↑m + (↑n + ↑k)",
"tactic": "simp [Nat.add_assoc]"
},
{
"state_after": "no goals",
"state_before": "x✝² x✝¹ x✝ : Int\nm n k : Nat\n⊢ ↑m + ↑n + -[k+1] = ↑m + (↑n + -[k+1])",
"tactic": "simp [subNatNat_add]"
},
{
"state_after": "x✝² x✝¹ x✝ : Int\nm n k : Nat\n⊢ subNatNat k (succ (succ (m + n))) = -[m+1] + subNatNat k (succ n)",
"state_before": "x✝² x✝¹ x✝ : Int\nm n k : Nat\n⊢ -[m+1] + -[n+1] + ↑k = -[m+1] + (-[n+1] + ↑k)",
"tactic": "simp [add_succ]"
},
{
"state_after": "x✝² x✝¹ x✝ : Int\nm n k : Nat\n⊢ subNatNat k (succ (succ (m + n))) = subNatNat k (succ n + succ m)",
"state_before": "x✝² x✝¹ x✝ : Int\nm n k : Nat\n⊢ subNatNat k (succ (succ (m + n))) = -[m+1] + subNatNat k (succ n)",
"tactic": "rw [Int.add_comm, subNatNat_add_negSucc]"
},
{
"state_after": "no goals",
"state_before": "x✝² x✝¹ x✝ : Int\nm n k : Nat\n⊢ subNatNat k (succ (succ (m + n))) = subNatNat k (succ n + succ m)",
"tactic": "simp [add_succ, succ_add, Nat.add_comm]"
}
] |
[
281,
44
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
261,
11
] |
Mathlib/Logic/Small/Basic.lean
|
not_small_type
|
[
{
"state_after": "S : Type u\ne✝ : Type (max u v) ≃ S\na b : Set ((α : S) × ↑e✝.symm α)\ne :\n { fst := ↑e✝ (Set ((α : S) × ↑e✝.symm α)),\n snd := cast (_ : Set ((α : S) × ↑e✝.symm α) = ↑e✝.symm (↑e✝ (Set ((α : S) × ↑e✝.symm α)))) a } =\n { fst := ↑e✝ (Set ((α : S) × ↑e✝.symm α)),\n snd := cast (_ : Set ((α : S) × ↑e✝.symm α) = ↑e✝.symm (↑e✝ (Set ((α : S) × ↑e✝.symm α)))) b }\n⊢ a = b",
"state_before": "S : Type u\ne✝ : Type (max u v) ≃ S\na b : Set ((α : S) × ↑e✝.symm α)\ne :\n (fun a =>\n { fst := Equiv.toFun e✝ (Set ((α : S) × ↑e✝.symm α)),\n snd :=\n cast (_ : Set ((α : S) × ↑e✝.symm α) = Equiv.invFun e✝ (Equiv.toFun e✝ (Set ((α : S) × ↑e✝.symm α)))) a })\n a =\n (fun a =>\n { fst := Equiv.toFun e✝ (Set ((α : S) × ↑e✝.symm α)),\n snd :=\n cast (_ : Set ((α : S) × ↑e✝.symm α) = Equiv.invFun e✝ (Equiv.toFun e✝ (Set ((α : S) × ↑e✝.symm α)))) a })\n b\n⊢ a = b",
"tactic": "dsimp at e"
},
{
"state_after": "S : Type u\ne : Type (max u v) ≃ S\na b : Set ((α : S) × ↑e.symm α)\nh₁ : ↑e (Set ((α : S) × ↑e.symm α)) = ↑e (Set ((α : S) × ↑e.symm α))\nh₂ :\n cast (_ : Set ((α : S) × ↑e.symm α) = ↑e.symm (↑e (Set ((α : S) × ↑e.symm α)))) a =\n cast (_ : Set ((α : S) × ↑e.symm α) = ↑e.symm (↑e (Set ((α : S) × ↑e.symm α)))) b\n⊢ a = b",
"state_before": "S : Type u\ne✝ : Type (max u v) ≃ S\na b : Set ((α : S) × ↑e✝.symm α)\ne :\n { fst := ↑e✝ (Set ((α : S) × ↑e✝.symm α)),\n snd := cast (_ : Set ((α : S) × ↑e✝.symm α) = ↑e✝.symm (↑e✝ (Set ((α : S) × ↑e✝.symm α)))) a } =\n { fst := ↑e✝ (Set ((α : S) × ↑e✝.symm α)),\n snd := cast (_ : Set ((α : S) × ↑e✝.symm α) = ↑e✝.symm (↑e✝ (Set ((α : S) × ↑e✝.symm α)))) b }\n⊢ a = b",
"tactic": "injection e with h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "S : Type u\ne : Type (max u v) ≃ S\na b : Set ((α : S) × ↑e.symm α)\nh₁ : ↑e (Set ((α : S) × ↑e.symm α)) = ↑e (Set ((α : S) × ↑e.symm α))\nh₂ :\n cast (_ : Set ((α : S) × ↑e.symm α) = ↑e.symm (↑e (Set ((α : S) × ↑e.symm α)))) a =\n cast (_ : Set ((α : S) × ↑e.symm α) = ↑e.symm (↑e (Set ((α : S) × ↑e.symm α)))) b\n⊢ a = b",
"tactic": "simpa using h₂"
}
] |
[
158,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.minpoly.degree_le
|
[] |
[
885,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
883,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.natAbs_eq_of_dvd_dvd
|
[] |
[
481,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.orderOf_cycleOf_dvd_orderOf
|
[
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ↑f x = x\n⊢ orderOf (cycleOf f x) ∣ orderOf f\n\ncase neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ orderOf (cycleOf f x) ∣ orderOf f",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\n⊢ orderOf (cycleOf f x) ∣ orderOf f",
"tactic": "by_cases hx : f x = x"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx✝ : ↑f x = x\nhx : cycleOf f x = 1\n⊢ orderOf (cycleOf f x) ∣ orderOf f",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ↑f x = x\n⊢ orderOf (cycleOf f x) ∣ orderOf f",
"tactic": "rw [← cycleOf_eq_one_iff] at hx"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx✝ : ↑f x = x\nhx : cycleOf f x = 1\n⊢ orderOf (cycleOf f x) ∣ orderOf f",
"tactic": "simp [hx]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ orderOf (cycleOf f x) ∈ cycleType f",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ orderOf (cycleOf f x) ∣ orderOf f",
"tactic": "refine dvd_of_mem_cycleType ?_"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ ∃ a, a ∈ (cycleFactorsFinset f).val ∧ (Finset.card ∘ support) a = orderOf (cycleOf f x)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ orderOf (cycleOf f x) ∈ cycleType f",
"tactic": "rw [cycleType, Multiset.mem_map]"
},
{
"state_after": "case neg.refine'_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ cycleOf f x ∈ (cycleFactorsFinset f).val\n\ncase neg.refine'_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ (Finset.card ∘ support) (cycleOf f x) = orderOf (cycleOf f x)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ ∃ a, a ∈ (cycleFactorsFinset f).val ∧ (Finset.card ∘ support) a = orderOf (cycleOf f x)",
"tactic": "refine' ⟨f.cycleOf x, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ cycleOf f x ∈ (cycleFactorsFinset f).val",
"tactic": "rwa [← Finset.mem_def, cycleOf_mem_cycleFactorsFinset_iff, mem_support]"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ (Finset.card ∘ support) (cycleOf f x) = orderOf (cycleOf f x)",
"tactic": "simp [(isCycle_cycleOf _ hx).orderOf]"
}
] |
[
194,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsCofiltered.inf_exists
|
[
{
"state_after": "case empty\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → T mX ≫ f = T mY\n\ncase insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh' : (X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)\nH' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nnmf : ¬h' ∈ H'\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T mX ≫ f = T mY\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ insert h' H' →\n T mX ≫ f = T mY",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H → T mX ≫ f = T mY",
"tactic": "induction' H using Finset.induction with h' H' nmf h''"
},
{
"state_after": "case empty.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nS : C\nf : ∀ {X : C}, X ∈ O → _root_.Nonempty (S ⟶ X)\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → T mX ≫ f = T mY",
"state_before": "case empty\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → T mX ≫ f = T mY",
"tactic": "obtain ⟨S, f⟩ := inf_objs_exists O"
},
{
"state_after": "no goals",
"state_before": "case empty.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nS : C\nf : ∀ {X : C}, X ∈ O → _root_.Nonempty (S ⟶ X)\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → T mX ≫ f = T mY",
"tactic": "refine' ⟨S, fun mX => (f mX).some, by rintro - - - - - ⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nS : C\nf : ∀ {X : C}, X ∈ O → _root_.Nonempty (S ⟶ X)\n⊢ ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f_1 : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f_1 } } } } ∈ ∅ →\n (fun {X} mX => Nonempty.some (_ : _root_.Nonempty (S ⟶ X))) mX ≫ f_1 =\n (fun {X} mX => Nonempty.some (_ : _root_.Nonempty (S ⟶ X))) mY",
"tactic": "rintro - - - - - ⟨⟩"
},
{
"state_after": "case insert.mk.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T mX ≫ f = T mY\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n T mX_1 ≫ f_1 = T mY_1",
"state_before": "case insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh' : (X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)\nH' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nnmf : ¬h' ∈ H'\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T mX ≫ f = T mY\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ insert h' H' →\n T mX ≫ f = T mY",
"tactic": "obtain ⟨X, Y, mX, mY, f⟩ := h'"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n T mX_1 ≫ f_1 = T mY_1",
"state_before": "case insert.mk.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T mX ≫ f = T mY\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n T mX_1 ≫ f_1 = T mY_1",
"tactic": "obtain ⟨S', T', w'⟩ := h''"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\n⊢ ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n (fun {X_2} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mX_1 ≫ f_1 =\n (fun {X_2} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY_1",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n T mX_1 ≫ f_1 = T mY_1",
"tactic": "refine' ⟨eq (T' mX ≫ f) (T' mY), fun mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ, _⟩"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\n⊢ ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n (fun {X_2} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mX_1 ≫ f_1 =\n (fun {X_2} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY_1",
"tactic": "intro X' Y' mX' mY' f' mf'"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "rw [Category.assoc]"
},
{
"state_after": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : X = X' ∧ Y = Y'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'\n\ncase neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "by_cases h : X = X' ∧ Y = Y'"
},
{
"state_after": "case pos.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"state_before": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : X = X' ∧ Y = Y'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "rcases h with ⟨rfl, rfl⟩"
},
{
"state_after": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : f = f'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'\n\ncase neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"state_before": "case pos.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "by_cases hf : f = f'"
},
{
"state_after": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"state_before": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : f = f'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "subst hf"
},
{
"state_after": "no goals",
"state_before": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "apply eq_condition"
},
{
"state_after": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "rw [@w' _ _ mX mY f']"
},
{
"state_after": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f ∨ { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "simp only [Finset.mem_insert, PSigma.mk.injEq, heq_eq_eq, true_and] at mf'"
},
{
"state_after": "case neg.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'\n\ncase neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f ∨ { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "rcases mf' with mf' | mf'"
},
{
"state_after": "case neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'\n\ncase neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": ". exfalso\n exact hf mf'.symm"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": ". exact mf'"
},
{
"state_after": "case neg.inl.h\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ False",
"state_before": "case neg.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case neg.inl.h\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ False",
"tactic": "exact hf mf'.symm"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "exact mf'"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ eqHom (T' mX ≫ f) (T' mY) ≫ T' mX' ≫ f' = (fun {X_1} mZ => eqHom (T' mX ≫ f) (T' mY) ≫ T' mZ) mY'",
"tactic": "rw [@w' _ _ mX' mY' f' _]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ≠\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'",
"tactic": "apply Finset.mem_of_mem_insert_of_ne mf'"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } =\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }\n⊢ X = X' ∧ Y = Y'",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ≠\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }",
"tactic": "contrapose! h"
},
{
"state_after": "case refl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ X = X ∧ Y = Y",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } =\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }\n⊢ X = X' ∧ Y = Y'",
"tactic": "obtain ⟨rfl, h⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsCofiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (S' ⟶ X)\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → T' mX ≫ f = T' mY\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ X = X ∧ Y = Y",
"tactic": "trivial"
}
] |
[
669,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
le_csSup_iff
|
[] |
[
476,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
1
] |
Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
|
MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le
|
[
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf : Memℒp f 1\n⊢ ∃ g, snorm (fun x => f x - ↑g x) 1 μ ≤ ε ∧ Memℒp (↑g) 1",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nf : α → E\nhf : Integrable f\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, (∫⁻ (x : α), ↑‖f x - ↑g x‖₊ ∂μ) ≤ ε ∧ Integrable ↑g",
"tactic": "simp only [← memℒp_one_iff_integrable, ← snorm_one_eq_lintegral_nnnorm] at hf ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nf : α → E\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf : Memℒp f 1\n⊢ ∃ g, snorm (fun x => f x - ↑g x) 1 μ ≤ ε ∧ Memℒp (↑g) 1",
"tactic": "exact hf.exists_boundedContinuous_snorm_sub_le ENNReal.one_ne_top hε"
}
] |
[
314,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/LinearAlgebra/Coevaluation.lean
|
contractLeft_assoc_coevaluation'
|
[
{
"state_after": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V))) =\n LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)",
"state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V))) =\n LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)",
"tactic": "letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)"
},
{
"state_after": "case H\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V)))) =\n LinearMap.compr₂ (mk K K V) (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V))",
"state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V))) =\n LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)",
"tactic": "apply TensorProduct.ext"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V)))))\n 1 =\n ↑(LinearMap.compr₂ (mk K K V) (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)))\n 1",
"state_before": "case H\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V)))) =\n LinearMap.compr₂ (mk K K V) (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V))",
"tactic": "apply LinearMap.ext_ring"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ∀ (i : ↑(Basis.ofVectorSpaceIndex K V)),\n ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V))\n (LinearMap.rTensor V (coevaluation K V)))))\n 1)\n (↑(Basis.ofVectorSpace K V) i) =\n ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)))\n 1)\n (↑(Basis.ofVectorSpace K V) i)",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V)))))\n 1 =\n ↑(LinearMap.compr₂ (mk K K V) (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)))\n 1",
"tactic": "apply (Basis.ofVectorSpace K V).ext"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V))\n (LinearMap.rTensor V (coevaluation K V)))))\n 1)\n (↑(Basis.ofVectorSpace K V) j) =\n ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)))\n 1)\n (↑(Basis.ofVectorSpace K V) j)",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ∀ (i : ↑(Basis.ofVectorSpaceIndex K V)),\n ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V))\n (LinearMap.rTensor V (coevaluation K V)))))\n 1)\n (↑(Basis.ofVectorSpace K V) i) =\n ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)))\n 1)\n (↑(Basis.ofVectorSpace K V) i)",
"tactic": "intro j"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V))))\n (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j) =\n ↑(LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V))\n (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j)",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V))\n (LinearMap.rTensor V (coevaluation K V)))))\n 1)\n (↑(Basis.ofVectorSpace K V) j) =\n ↑(↑(LinearMap.compr₂ (mk K K V)\n (LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V)))\n 1)\n (↑(Basis.ofVectorSpace K V) j)",
"tactic": "rw [LinearMap.compr₂_apply, LinearMap.compr₂_apply, TensorProduct.mk_apply]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(TensorProduct.assoc K V (Module.Dual K V) V)\n (↑(LinearMap.rTensor V (coevaluation K V)) (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j))) =\n ↑(LinearEquiv.symm (TensorProduct.rid K V)) (↑(TensorProduct.lid K V) (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j))",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.comp (LinearMap.lTensor V (contractLeft K V))\n (LinearMap.comp (↑(TensorProduct.assoc K V (Module.Dual K V) V)) (LinearMap.rTensor V (coevaluation K V))))\n (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j) =\n ↑(LinearMap.comp ↑(LinearEquiv.symm (TensorProduct.rid K V)) ↑(TensorProduct.lid K V))\n (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j)",
"tactic": "simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(TensorProduct.assoc K V (Module.Dual K V) V)\n (↑(LinearMap.rTensor V (coevaluation K V)) (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j))) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(TensorProduct.assoc K V (Module.Dual K V) V)\n (↑(LinearMap.rTensor V (coevaluation K V)) (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j))) =\n ↑(LinearEquiv.symm (TensorProduct.rid K V)) (↑(TensorProduct.lid K V) (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j))",
"tactic": "rw [lid_tmul, one_smul, rid_symm_apply]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(TensorProduct.assoc K V (Module.Dual K V) V)\n ((∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x) ⊗ₜ[K]\n ↑(Basis.ofVectorSpace K V) j)) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(TensorProduct.assoc K V (Module.Dual K V) V)\n (↑(LinearMap.rTensor V (coevaluation K V)) (1 ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j))) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (∑ i : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(TensorProduct.assoc K V (Module.Dual K V) V)\n ((↑(Basis.ofVectorSpace K V) i ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) i) ⊗ₜ[K]\n ↑(Basis.ofVectorSpace K V) j)) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(TensorProduct.assoc K V (Module.Dual K V) V)\n ((∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x) ⊗ₜ[K]\n ↑(Basis.ofVectorSpace K V) j)) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "rw [TensorProduct.sum_tmul, LinearEquiv.map_sum]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (∑ i : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(TensorProduct.assoc K V (Module.Dual K V) V)\n ((↑(Basis.ofVectorSpace K V) i ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) i) ⊗ₜ[K]\n ↑(Basis.ofVectorSpace K V) j)) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "simp only [assoc_tmul]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ∑ i : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(Basis.ofVectorSpace K V) i ⊗ₜ[K]\n Basis.coord (Basis.ofVectorSpace K V) i ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ↑(LinearMap.lTensor V (contractLeft K V))\n (∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "rw [LinearMap.map_sum]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] ↑(Basis.coord (Basis.ofVectorSpace K V) x) (↑(Basis.ofVectorSpace K V) j) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ∑ i : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(LinearMap.lTensor V (contractLeft K V))\n (↑(Basis.ofVectorSpace K V) i ⊗ₜ[K]\n Basis.coord (Basis.ofVectorSpace K V) i ⊗ₜ[K] ↑(Basis.ofVectorSpace K V) j) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "simp only [LinearMap.lTensor_tmul, contractLeft_apply]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ (∑ x : ↑(Basis.ofVectorSpaceIndex K V), if j = x then ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] 1 else 0) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] ↑(Basis.coord (Basis.ofVectorSpace K V) x) (↑(Basis.ofVectorSpace K V) j) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "simp only [Basis.coord_apply, Basis.repr_self_apply, TensorProduct.tmul_ite]"
},
{
"state_after": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ (if j ∈ Finset.univ then ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1 else 0) = ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ (∑ x : ↑(Basis.ofVectorSpaceIndex K V), if j = x then ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] 1 else 0) =\n ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "rw [Finset.sum_ite_eq]"
},
{
"state_after": "no goals",
"state_before": "case H.h\nK : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nthis : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)\nj : ↑(Basis.ofVectorSpaceIndex K V)\n⊢ (if j ∈ Finset.univ then ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1 else 0) = ↑(Basis.ofVectorSpace K V) j ⊗ₜ[K] 1",
"tactic": "simp only [Finset.mem_univ, if_true]"
}
] |
[
99,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_ofNat_succ
|
[] |
[
40,
81
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
40,
15
] |
Mathlib/Order/BoundedOrder.lean
|
isBot_iff_eq_bot
|
[] |
[
346,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.span_singleton_sup_orthogonal_eq_top
|
[
{
"state_after": "R : Type ?u.260339\nR₁ : Type ?u.260342\nR₂ : Type ?u.260345\nR₃ : Type ?u.260348\nM : Type ?u.260351\nM₁ : Type ?u.260354\nM₂ : Type ?u.260357\nMₗ₁ : Type ?u.260360\nMₗ₁' : Type ?u.260363\nMₗ₂ : Type ?u.260366\nMₗ₂' : Type ?u.260369\nK : Type u_1\nK₁ : Type ?u.260375\nK₂ : Type ?u.260378\nV : Type u_2\nV₁ : Type ?u.260384\nV₂ : Type ?u.260387\nn : Type ?u.260390\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₗ[K] K\nx : V\nhx : ¬IsOrtho B x x\n⊢ Submodule.span K {x} ⊔ ker (↑B x) = ⊤",
"state_before": "R : Type ?u.260339\nR₁ : Type ?u.260342\nR₂ : Type ?u.260345\nR₃ : Type ?u.260348\nM : Type ?u.260351\nM₁ : Type ?u.260354\nM₂ : Type ?u.260357\nMₗ₁ : Type ?u.260360\nMₗ₁' : Type ?u.260363\nMₗ₂ : Type ?u.260366\nMₗ₂' : Type ?u.260369\nK : Type u_1\nK₁ : Type ?u.260375\nK₂ : Type ?u.260378\nV : Type u_2\nV₁ : Type ?u.260384\nV₂ : Type ?u.260387\nn : Type ?u.260390\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₗ[K] K\nx : V\nhx : ¬IsOrtho B x x\n⊢ Submodule.span K {x} ⊔ Submodule.orthogonalBilin (Submodule.span K {x}) B = ⊤",
"tactic": "rw [orthogonal_span_singleton_eq_to_lin_ker]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.260339\nR₁ : Type ?u.260342\nR₂ : Type ?u.260345\nR₃ : Type ?u.260348\nM : Type ?u.260351\nM₁ : Type ?u.260354\nM₂ : Type ?u.260357\nMₗ₁ : Type ?u.260360\nMₗ₁' : Type ?u.260363\nMₗ₂ : Type ?u.260366\nMₗ₂' : Type ?u.260369\nK : Type u_1\nK₁ : Type ?u.260375\nK₂ : Type ?u.260378\nV : Type u_2\nV₁ : Type ?u.260384\nV₂ : Type ?u.260387\nn : Type ?u.260390\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₗ[K] K\nx : V\nhx : ¬IsOrtho B x x\n⊢ Submodule.span K {x} ⊔ ker (↑B x) = ⊤",
"tactic": "exact (B x).span_singleton_sup_ker_eq_top hx"
}
] |
[
393,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_sin_sq
|
[
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, sin x ^ 2) = (sin a * cos a - sin b * cos b + b - a) / 2",
"tactic": "field_simp [integral_sin_pow, add_sub_assoc]"
}
] |
[
663,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/Algebra/Group/Opposite.lean
|
SemiconjBy.unop
|
[] |
[
241,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.mul_lt_mul_right'
|
[] |
[
1015,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1013,
21
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.re_eq_add_conj
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.3487608\ninst✝ : IsROrC K\nz : K\n⊢ ↑(↑re z) = (z + ↑(starRingEnd K) z) / 2",
"tactic": "rw [add_conj, mul_div_cancel_left (re z : K) two_ne_zero]"
}
] |
[
401,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
orthogonalProjection_inner_eq_zero
|
[] |
[
483,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Data/Dfinsupp/WellFounded.lean
|
Dfinsupp.lex_fibration
|
[
{
"state_after": "case mk.mk.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = ↑((fun x => piecewise x.snd.fst x.snd.snd x.fst) (p, x₁, x₂)) j\nhs : s i (↑x i) (↑((fun x => piecewise x.snd.fst x.snd.snd x.fst) (p, x₁, x₂)) i)\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\n⊢ Fibration (InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd) (Dfinsupp.Lex r s) fun x =>\n piecewise x.snd.fst x.snd.snd x.fst",
"tactic": "rintro ⟨p, x₁, x₂⟩ x ⟨i, hr, hs⟩"
},
{
"state_after": "case mk.mk.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhs : s i (↑x i) (if i ∈ p then ↑x₁ i else ↑x₂ i)\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x",
"state_before": "case mk.mk.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = ↑((fun x => piecewise x.snd.fst x.snd.snd x.fst) (p, x₁, x₂)) j\nhs : s i (↑x i) (↑((fun x => piecewise x.snd.fst x.snd.snd x.fst) (p, x₁, x₂)) i)\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x",
"tactic": "simp_rw [piecewise_apply] at hs hr"
},
{
"state_after": "case mk.mk.intro.intro.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x\n\ncase mk.mk.intro.intro.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x",
"state_before": "case mk.mk.intro.intro\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhs : s i (↑x i) (if i ∈ p then ↑x₁ i else ↑x₂ i)\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x",
"tactic": "split_ifs at hs with hp"
},
{
"state_after": "case mk.mk.intro.intro.inl.refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nhj : r j i\n⊢ (if r j i then ↑x₁ j else ↑x j) = ↑x₁ j\n\ncase mk.mk.intro.intro.inl.refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\n⊢ s i (if r i i then ↑x₁ i else ↑x i) (↑x₁ i)\n\ncase mk.mk.intro.intro.inl.refine_3\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\n⊢ piecewise (piecewise x₁ x {j | r j i}) x₂ {j | r j i → j ∈ p} = x",
"state_before": "case mk.mk.intro.intro.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x",
"tactic": "refine ⟨⟨{ j | r j i → j ∈ p }, piecewise x₁ x { j | r j i }, x₂⟩,\n .fst ⟨i, fun j hj ↦ ?_, ?_⟩, ?_⟩ <;> simp only [piecewise_apply, Set.mem_setOf_eq]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inl.refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nhj : r j i\n⊢ (if r j i then ↑x₁ j else ↑x j) = ↑x₁ j",
"tactic": "simp only [if_pos hj]"
},
{
"state_after": "case mk.mk.intro.intro.inl.refine_2.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nhi : r i i\n⊢ s i (↑x₁ i) (↑x₁ i)\n\ncase mk.mk.intro.intro.inl.refine_2.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nhi : ¬r i i\n⊢ s i (↑x i) (↑x₁ i)",
"state_before": "case mk.mk.intro.intro.inl.refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\n⊢ s i (if r i i then ↑x₁ i else ↑x i) (↑x₁ i)",
"tactic": "split_ifs with hi"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inl.refine_2.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nhi : r i i\n⊢ s i (↑x₁ i) (↑x₁ i)",
"tactic": "rwa [hr i hi, if_pos hp] at hs"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inl.refine_2.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nhi : ¬r i i\n⊢ s i (↑x i) (↑x₁ i)",
"tactic": "assumption"
},
{
"state_after": "case mk.mk.intro.intro.inl.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\n⊢ ↑(piecewise (piecewise x₁ x {j | r j i}) x₂ {j | r j i → j ∈ p}) j = ↑x j",
"state_before": "case mk.mk.intro.intro.inl.refine_3\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\n⊢ piecewise (piecewise x₁ x {j | r j i}) x₂ {j | r j i → j ∈ p} = x",
"tactic": "ext1 j"
},
{
"state_after": "case mk.mk.intro.intro.inl.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\n⊢ (if r j i → j ∈ p then if r j i then ↑x₁ j else ↑x j else ↑x₂ j) = ↑x j",
"state_before": "case mk.mk.intro.intro.inl.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\n⊢ ↑(piecewise (piecewise x₁ x {j | r j i}) x₂ {j | r j i → j ∈ p}) j = ↑x j",
"tactic": "simp only [piecewise_apply, Set.mem_setOf_eq]"
},
{
"state_after": "case mk.mk.intro.intro.inl.refine_3.h.inl.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : r j i → j ∈ p\nh₂ : r j i\n⊢ ↑x₁ j = ↑x j\n\ncase mk.mk.intro.intro.inl.refine_3.h.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : ¬(r j i → j ∈ p)\n⊢ ↑x₂ j = ↑x j",
"state_before": "case mk.mk.intro.intro.inl.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\n⊢ (if r j i → j ∈ p then if r j i then ↑x₁ j else ↑x j else ↑x₂ j) = ↑x j",
"tactic": "split_ifs with h₁ h₂ <;> try rfl"
},
{
"state_after": "case mk.mk.intro.intro.inl.refine_3.h.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : ¬(r j i → j ∈ p)\n⊢ ↑x₂ j = ↑x j",
"state_before": "case mk.mk.intro.intro.inl.refine_3.h.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : ¬(r j i → j ∈ p)\n⊢ ↑x₂ j = ↑x j",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inl.refine_3.h.inl.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : r j i → j ∈ p\nh₂ : r j i\n⊢ ↑x₁ j = ↑x j",
"tactic": "rw [hr j h₂, if_pos (h₁ h₂)]"
},
{
"state_after": "case mk.mk.intro.intro.inl.refine_3.h.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : r j i ∧ ¬j ∈ p\n⊢ ↑x₂ j = ↑x j",
"state_before": "case mk.mk.intro.intro.inl.refine_3.h.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : ¬(r j i → j ∈ p)\n⊢ ↑x₂ j = ↑x j",
"tactic": "rw [not_imp] at h₁"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inl.refine_3.h.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : i ∈ p\nhs : s i (↑x i) (↑x₁ i)\nj : ι\nh₁ : r j i ∧ ¬j ∈ p\n⊢ ↑x₂ j = ↑x j",
"tactic": "rw [hr j h₁.1, if_neg h₁.2]"
},
{
"state_after": "case mk.mk.intro.intro.inr.refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nhj : r j i\n⊢ (if r j i then ↑x₂ j else ↑x j) = ↑x₂ j\n\ncase mk.mk.intro.intro.inr.refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\n⊢ s i (if r i i then ↑x₂ i else ↑x i) (↑x₂ i)\n\ncase mk.mk.intro.intro.inr.refine_3\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\n⊢ piecewise x₁ (piecewise x₂ x {j | r j i}) {j | r j i ∧ j ∈ p} = x",
"state_before": "case mk.mk.intro.intro.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\n⊢ ∃ a',\n InvImage (GameAdd (Dfinsupp.Lex r s) (Dfinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => piecewise x.snd.fst x.snd.snd x.fst) a' = x",
"tactic": "refine ⟨⟨{ j | r j i ∧ j ∈ p }, x₁, piecewise x₂ x { j | r j i }⟩,\n .snd ⟨i, fun j hj ↦ ?_, ?_⟩, ?_⟩ <;> simp only [piecewise_apply, Set.mem_setOf_eq]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inr.refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nhj : r j i\n⊢ (if r j i then ↑x₂ j else ↑x j) = ↑x₂ j",
"tactic": "exact if_pos hj"
},
{
"state_after": "case mk.mk.intro.intro.inr.refine_2.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nhi : r i i\n⊢ s i (↑x₂ i) (↑x₂ i)\n\ncase mk.mk.intro.intro.inr.refine_2.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nhi : ¬r i i\n⊢ s i (↑x i) (↑x₂ i)",
"state_before": "case mk.mk.intro.intro.inr.refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\n⊢ s i (if r i i then ↑x₂ i else ↑x i) (↑x₂ i)",
"tactic": "split_ifs with hi"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inr.refine_2.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nhi : r i i\n⊢ s i (↑x₂ i) (↑x₂ i)",
"tactic": "rwa [hr i hi, if_neg hp] at hs"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inr.refine_2.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nhi : ¬r i i\n⊢ s i (↑x i) (↑x₂ i)",
"tactic": "assumption"
},
{
"state_after": "case mk.mk.intro.intro.inr.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\n⊢ ↑(piecewise x₁ (piecewise x₂ x {j | r j i}) {j | r j i ∧ j ∈ p}) j = ↑x j",
"state_before": "case mk.mk.intro.intro.inr.refine_3\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\n⊢ piecewise x₁ (piecewise x₂ x {j | r j i}) {j | r j i ∧ j ∈ p} = x",
"tactic": "ext1 j"
},
{
"state_after": "case mk.mk.intro.intro.inr.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\n⊢ (if r j i ∧ j ∈ p then ↑x₁ j else if r j i then ↑x₂ j else ↑x j) = ↑x j",
"state_before": "case mk.mk.intro.intro.inr.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\n⊢ ↑(piecewise x₁ (piecewise x₂ x {j | r j i}) {j | r j i ∧ j ∈ p}) j = ↑x j",
"tactic": "simp only [piecewise_apply, Set.mem_setOf_eq]"
},
{
"state_after": "case mk.mk.intro.intro.inr.refine_3.h.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nh₁ : r j i ∧ j ∈ p\n⊢ ↑x₁ j = ↑x j\n\ncase mk.mk.intro.intro.inr.refine_3.h.inr.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nh₁ : ¬(r j i ∧ j ∈ p)\nh₂ : r j i\n⊢ ↑x₂ j = ↑x j",
"state_before": "case mk.mk.intro.intro.inr.refine_3.h\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\n⊢ (if r j i ∧ j ∈ p then ↑x₁ j else if r j i then ↑x₂ j else ↑x j) = ↑x j",
"tactic": "split_ifs with h₁ h₂ <;> try rfl"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inr.refine_3.h.inr.inr\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nh₁ : ¬(r j i ∧ j ∈ p)\nh₂ : ¬r j i\n⊢ ↑x j = ↑x j",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inr.refine_3.h.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nh₁ : r j i ∧ j ∈ p\n⊢ ↑x₁ j = ↑x j",
"tactic": "rw [hr j h₁.1, if_pos h₁.2]"
},
{
"state_after": "case mk.mk.intro.intro.inr.refine_3.h.inr.inl.hnc\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nh₁ : ¬(r j i ∧ j ∈ p)\nh₂ : r j i\n⊢ ¬j ∈ p",
"state_before": "case mk.mk.intro.intro.inr.refine_3.h.inr.inl\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nh₁ : ¬(r j i ∧ j ∈ p)\nh₂ : r j i\n⊢ ↑x₂ j = ↑x j",
"tactic": "rw [hr j h₂, if_neg]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.inr.refine_3.h.inr.inl.hnc\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → ↑x j = if j ∈ p then ↑x₁ j else ↑x₂ j\nhp : ¬i ∈ p\nhs : s i (↑x i) (↑x₂ i)\nj : ι\nh₁ : ¬(r j i ∧ j ∈ p)\nh₂ : r j i\n⊢ ¬j ∈ p",
"tactic": "simpa [h₂] using h₁"
}
] |
[
100,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.coeff_natDegree_eq_zero_of_degree_lt
|
[] |
[
582,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.decode_unit_succ
|
[] |
[
157,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
|
CategoryTheory.Pretriangulated.rot_of_dist_triangle
|
[] |
[
110,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.id_apply
|
[] |
[
440,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
439,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.map_right_cancel
|
[
{
"state_after": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.626659\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x ∈ S }\nh : ↑(toMap f) ↑c * ↑(toMap f) x = ↑(toMap f) ↑c * ↑(toMap f) y\n⊢ ↑(toMap f) x = ↑(toMap f) y",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.626659\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x ∈ S }\nh : ↑(toMap f) (↑c * x) = ↑(toMap f) (↑c * y)\n⊢ ↑(toMap f) x = ↑(toMap f) y",
"tactic": "rw [f.toMap.map_mul, f.toMap.map_mul] at h"
},
{
"state_after": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.626659\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x ∈ S }\nh : ↑(toMap f) ↑c * ↑(toMap f) x = ↑(toMap f) ↑c * ↑(toMap f) y\nu : ((fun x => N) ↑c)ˣ\nhu : ↑u = ↑(toMap f) ↑c\n⊢ ↑(toMap f) x = ↑(toMap f) y",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.626659\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x ∈ S }\nh : ↑(toMap f) ↑c * ↑(toMap f) x = ↑(toMap f) ↑c * ↑(toMap f) y\n⊢ ↑(toMap f) x = ↑(toMap f) y",
"tactic": "cases' f.map_units c with u hu"
},
{
"state_after": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.626659\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x ∈ S }\nu : ((fun x => N) ↑c)ˣ\nh : ↑u * ↑(toMap f) x = ↑u * ↑(toMap f) y\nhu : ↑u = ↑(toMap f) ↑c\n⊢ ↑(toMap f) x = ↑(toMap f) y",
"state_before": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.626659\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x ∈ S }\nh : ↑(toMap f) ↑c * ↑(toMap f) x = ↑(toMap f) ↑c * ↑(toMap f) y\nu : ((fun x => N) ↑c)ˣ\nhu : ↑u = ↑(toMap f) ↑c\n⊢ ↑(toMap f) x = ↑(toMap f) y",
"tactic": "rw [← hu] at h"
},
{
"state_after": "no goals",
"state_before": "case intro\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.626659\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx y : M\nc : { x // x ∈ S }\nu : ((fun x => N) ↑c)ˣ\nh : ↑u * ↑(toMap f) x = ↑u * ↑(toMap f) y\nhu : ↑u = ↑(toMap f) ↑c\n⊢ ↑(toMap f) x = ↑(toMap f) y",
"tactic": "exact (Units.mul_right_inj u).1 h"
}
] |
[
686,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.volume_preserving_piFinTwo
|
[] |
[
804,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.Eventually.atTop_of_arithmetic
|
[
{
"state_after": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nhp : ∀ (k : ℕ), k < n → ∃ a, ∀ (b : ℕ), b ≥ a → p (n * b + k)\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → p b",
"state_before": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nhp : ∀ (k : ℕ), k < n → ∀ᶠ (a : ℕ) in atTop, p (n * a + k)\n⊢ ∀ᶠ (a : ℕ) in atTop, p a",
"tactic": "simp only [eventually_atTop] at hp ⊢"
},
{
"state_after": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → p b",
"state_before": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nhp : ∀ (k : ℕ), k < n → ∃ a, ∀ (b : ℕ), b ≥ a → p (n * b + k)\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → p b",
"tactic": "choose! N hN using hp"
},
{
"state_after": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\n⊢ p (n * (b / n) + b % n)",
"state_before": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\n⊢ p b",
"tactic": "rw [← Nat.div_add_mod b n]"
},
{
"state_after": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n⊢ p (n * (b / n) + b % n)",
"state_before": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\n⊢ p (n * (b / n) + b % n)",
"tactic": "have hlt := Nat.mod_lt b hn.bot_lt"
},
{
"state_after": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n⊢ b / n ≥ N (b % n)",
"state_before": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n⊢ p (n * (b / n) + b % n)",
"tactic": "refine hN _ hlt _ ?_"
},
{
"state_after": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n⊢ n * N (b % n) ≤ b",
"state_before": "ι : Type ?u.63705\nι' : Type ?u.63708\nα : Type ?u.63711\nβ : Type ?u.63714\nγ : Type ?u.63717\np : ℕ → Prop\nn : ℕ\nhn : n ≠ 0\nN : ℕ → ℕ\nhN : ∀ (k : ℕ), k < n → ∀ (b : ℕ), b ≥ N k → p (n * b + k)\nb : ℕ\nhb : b ≥ Finset.sup (Finset.range n) fun x => n * N x\nhlt : b % n < n\n⊢ b / n ≥ N (b % n)",
"tactic": "rw [ge_iff_le, Nat.le_div_iff_mul_le hn.bot_lt, mul_comm]"
}
] |
[
510,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
501,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.cosh_sub_sinh
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ cosh x - sinh x = exp (-x)",
"tactic": "rw [← mul_right_inj' (two_ne_zero' ℂ), mul_sub, two_cosh, two_sinh, add_sub_sub_cancel, two_mul]"
}
] |
[
747,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.imageMonoIsoSource_hom_self
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (imageMonoIsoSource f).hom ≫ (imageMonoIsoSource f).inv ≫ image.ι f = image.ι f",
"state_before": "C : Type u\ninst✝¹ : Category C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (imageMonoIsoSource f).hom ≫ f = image.ι f",
"tactic": "simp only [← imageMonoIsoSource_inv_ι f]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (imageMonoIsoSource f).hom ≫ (imageMonoIsoSource f).inv ≫ image.ι f = image.ι f",
"tactic": "rw [← Category.assoc, Iso.hom_inv_id, Category.id_comp]"
}
] |
[
432,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.SimpleFunc.norm_eq_sum_mul
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑‖↑(toSimpleFunc f) x‖₊ ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\n⊢ ‖f‖ = ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "rw [norm_toSimpleFunc, snorm_one_eq_lintegral_nnnorm]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑‖↑(toSimpleFunc f) x‖₊ ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑‖↑(toSimpleFunc f) x‖₊ ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "have h_eq := SimpleFunc.map_apply (fun x => (‖x‖₊ : ℝ≥0∞)) (toSimpleFunc f)"
},
{
"state_after": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑‖↑(toSimpleFunc f) x‖₊ ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑‖↑(toSimpleFunc f) x‖₊ ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "dsimp only at h_eq"
},
{
"state_after": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) x ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑‖↑(toSimpleFunc f) x‖₊ ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "simp_rw [← h_eq]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ∑ a in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑‖a‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {a})) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖\n\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ∀ (a : G), a ∈ SimpleFunc.range (toSimpleFunc f) → ↑‖a‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {a}) ≠ ⊤",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ENNReal.toReal (∫⁻ (x : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) x ∂μ) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "rw [SimpleFunc.lintegral_eq_lintegral, SimpleFunc.map_lintegral, ENNReal.toReal_sum]"
},
{
"state_after": "case e_f\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ (fun a => ENNReal.toReal (↑‖a‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {a}))) = fun x =>\n ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ∑ a in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑‖a‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {a})) =\n ∑ x in SimpleFunc.range (toSimpleFunc f), ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "congr"
},
{
"state_after": "case e_f.h\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\n⊢ ENNReal.toReal (↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) = ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"state_before": "case e_f\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ (fun a => ENNReal.toReal (↑‖a‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {a}))) = fun x =>\n ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case e_f.h\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\n⊢ ENNReal.toReal (↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) = ENNReal.toReal (↑↑μ (↑(toSimpleFunc f) ⁻¹' {x})) * ‖x‖",
"tactic": "rw [ENNReal.toReal_mul, mul_comm, ← ofReal_norm_eq_coe_nnnorm,\n ENNReal.toReal_ofReal (norm_nonneg _)]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\n⊢ ↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x}) ≠ ⊤",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\n⊢ ∀ (a : G), a ∈ SimpleFunc.range (toSimpleFunc f) → ↑‖a‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {a}) ≠ ⊤",
"tactic": "intro x _"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n⊢ ↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x}) ≠ ⊤\n\ncase neg\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\nhx0 : ¬x = 0\n⊢ ↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x}) ≠ ⊤",
"state_before": "α : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\n⊢ ↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x}) ≠ ⊤",
"tactic": "by_cases hx0 : x = 0"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n⊢ ↑‖0‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {0}) ≠ ⊤",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n⊢ ↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x}) ≠ ⊤",
"tactic": "rw [hx0]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\nhx0 : x = 0\n⊢ ↑‖0‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {0}) ≠ ⊤",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.465938\nF : Type ?u.465941\nF' : Type ?u.465944\nG : Type u_2\n𝕜 : Type ?u.465950\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nf : { x // x ∈ simpleFunc G 1 μ }\nh_eq : ∀ (a : α), ↑(SimpleFunc.map (fun x => ↑‖x‖₊) (toSimpleFunc f)) a = ↑‖↑(toSimpleFunc f) a‖₊\nx : G\na✝ : x ∈ SimpleFunc.range (toSimpleFunc f)\nhx0 : ¬x = 0\n⊢ ↑‖x‖₊ * ↑↑μ (↑(toSimpleFunc f) ⁻¹' {x}) ≠ ⊤",
"tactic": "exact\n ENNReal.mul_ne_top ENNReal.coe_ne_top\n (SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0).ne"
}
] |
[
677,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
661,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
NormedSpace.isVonNBounded_iff'
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\n𝕜' : Type ?u.254043\nE : Type u_1\nE' : Type ?u.254049\nF : Type ?u.254052\nι : Type ?u.254055\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ Bornology.IsVonNBounded 𝕜 s ↔ ∃ r, ∀ (x : E), x ∈ s → ‖x‖ ≤ r",
"tactic": "rw [NormedSpace.isVonNBounded_iff, ← Metric.bounded_iff_isBounded, bounded_iff_forall_norm_le]"
}
] |
[
302,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.image_const_sub_Ioo
|
[
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ioo b c = Ioo (a - c) (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a - x) '' Ioo b c = Ioo (a - c) (a - b)",
"tactic": "have := image_comp (fun x => a + x) fun x => -x"
},
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ioo b c = Ioo (a - c) (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ioo b c = Ioo (a - c) (a - b)",
"tactic": "dsimp [Function.comp] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ioo b c = Ioo (a - c) (a - b)",
"tactic": "simp [sub_eq_add_neg, this, add_comm]"
}
] |
[
365,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.continuous_ofReal
|
[] |
[
108,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.bind_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.31281\na : α\ns t : Multiset α\nf✝ g : α → Multiset β\nm : Multiset α\nn : β → Multiset γ\nf : α → β\n⊢ bind (map f 0) n = bind 0 fun a => n (f a)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.31281\na : α\ns t : Multiset α\nf✝ g : α → Multiset β\nm : Multiset α\nn : β → Multiset γ\nf : α → β\n⊢ ∀ ⦃a : α⦄ {s : Multiset α},\n (bind (map f s) n = bind s fun a => n (f a)) → bind (map f (a ::ₘ s)) n = bind (a ::ₘ s) fun a => n (f a)",
"tactic": "simp (config := { contextual := true })"
}
] |
[
169,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/CategoryTheory/Preadditive/Generator.lean
|
CategoryTheory.Preadditive.isCoseparator_iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\nhG : IsCoseparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ G), f ≫ h = 0\n⊢ ∀ (h : Y ⟶ G), f ≫ h = 0 ≫ h",
"tactic": "simpa only [Limits.zero_comp] using hf"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\nhG : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ ∀ (h : Y ⟶ G), (f - g) ≫ h = 0",
"tactic": "simpa only [Preadditive.sub_comp, sub_eq_zero] using hfg"
}
] |
[
54,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_le_factorization_mul_right
|
[
{
"state_after": "a b : ℕ\nha : a ≠ 0\n⊢ factorization b ≤ factorization (b * a)",
"state_before": "a b : ℕ\nha : a ≠ 0\n⊢ factorization b ≤ factorization (a * b)",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nha : a ≠ 0\n⊢ factorization b ≤ factorization (b * a)",
"tactic": "apply factorization_le_factorization_mul_left ha"
}
] |
[
487,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/CategoryTheory/Category/Grpd.lean
|
CategoryTheory.Grpd.piIsoPi_hom_π
|
[
{
"state_after": "J : Type u\nf : J → Grpd\nj : J\n⊢ (piLimitFan f).π.app { as := j } = Pi.eval (fun i => ↑(f i)) j",
"state_before": "J : Type u\nf : J → Grpd\nj : J\n⊢ (piIsoPi J f).hom ≫ Limits.Pi.π f j = Pi.eval (fun i => ↑(f i)) j",
"tactic": "simp [piIsoPi]"
},
{
"state_after": "no goals",
"state_before": "J : Type u\nf : J → Grpd\nj : J\n⊢ (piLimitFan f).π.app { as := j } = Pi.eval (fun i => ↑(f i)) j",
"tactic": "rfl"
}
] |
[
158,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integrable_prod_iff'
|
[
{
"state_after": "case h.e'_1.a\nα : Type u_1\nα' : Type ?u.2403525\nβ : Type u_2\nβ' : Type ?u.2403531\nγ : Type ?u.2403534\nE : Type u_3\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nf : α × β → E\nh1f : AEStronglyMeasurable f (Measure.prod μ ν)\n⊢ Integrable f ↔ Integrable fun z => f (Prod.swap z)",
"state_before": "α : Type u_1\nα' : Type ?u.2403525\nβ : Type u_2\nβ' : Type ?u.2403531\nγ : Type ?u.2403534\nE : Type u_3\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nf : α × β → E\nh1f : AEStronglyMeasurable f (Measure.prod μ ν)\n⊢ Integrable f ↔ (∀ᵐ (y : β) ∂ν, Integrable fun x => f (x, y)) ∧ Integrable fun y => ∫ (x : α), ‖f (x, y)‖ ∂μ",
"tactic": "convert integrable_prod_iff h1f.prod_swap using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1.a\nα : Type u_1\nα' : Type ?u.2403525\nβ : Type u_2\nβ' : Type ?u.2403531\nγ : Type ?u.2403534\nE : Type u_3\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nf : α × β → E\nh1f : AEStronglyMeasurable f (Measure.prod μ ν)\n⊢ Integrable f ↔ Integrable fun z => f (Prod.swap z)",
"tactic": "rw [funext fun _ => Function.comp_apply.symm, integrable_swap_iff]"
}
] |
[
290,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_left_surjective₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.29573\nM₀ : Type ?u.29576\nG₀ : Type u_1\nM₀' : Type ?u.29582\nG₀' : Type ?u.29585\nF : Type ?u.29588\nF' : Type ?u.29591\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : a ≠ 0\ng : G₀\n⊢ (fun g => a * g) (a⁻¹ * g) = g",
"tactic": "simp [← mul_assoc, mul_inv_cancel h]"
}
] |
[
425,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.eq_interpolate_iff
|
[
{
"state_after": "case mp\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\nh : degree f < ↑(card s) ∧ ∀ (i : ι), i ∈ s → eval (v i) f = r i\n⊢ f = ↑(interpolate s v) r\n\ncase mpr\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\nh : f = ↑(interpolate s v) r\n⊢ degree f < ↑(card s) ∧ ∀ (i : ι), i ∈ s → eval (v i) f = r i",
"state_before": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\n⊢ (degree f < ↑(card s) ∧ ∀ (i : ι), i ∈ s → eval (v i) f = r i) ↔ f = ↑(interpolate s v) r",
"tactic": "constructor <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\nh : degree f < ↑(card s) ∧ ∀ (i : ι), i ∈ s → eval (v i) f = r i\n⊢ f = ↑(interpolate s v) r",
"tactic": "exact eq_interpolate_of_eval_eq _ hvs h.1 h.2"
},
{
"state_after": "case mpr\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\nh : f = ↑(interpolate s v) r\n⊢ degree (↑(interpolate s v) r) < ↑(card s) ∧ ∀ (i : ι), i ∈ s → eval (v i) (↑(interpolate s v) r) = r i",
"state_before": "case mpr\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\nh : f = ↑(interpolate s v) r\n⊢ degree f < ↑(card s) ∧ ∀ (i : ι), i ∈ s → eval (v i) f = r i",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nf : F[X]\nhvs : Set.InjOn v ↑s\nh : f = ↑(interpolate s v) r\n⊢ degree (↑(interpolate s v) r) < ↑(card s) ∧ ∀ (i : ι), i ∈ s → eval (v i) (↑(interpolate s v) r) = r i",
"tactic": "exact ⟨degree_interpolate_lt _ hvs, fun _ hi => eval_interpolate_at_node _ hvs hi⟩"
}
] |
[
394,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
Monotone.mem_lowerBounds_image
|
[] |
[
1269,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1268,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean
|
Matrix.IsSymm.ext_iff
|
[] |
[
45,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.ker_eq_bot_of_cancel
|
[
{
"state_after": "R : Type u_1\nR₁ : Type ?u.1573042\nR₂ : Type u_2\nR₃ : Type ?u.1573048\nR₄ : Type ?u.1573051\nS : Type ?u.1573054\nK : Type ?u.1573057\nK₂ : Type ?u.1573060\nM : Type u_3\nM' : Type ?u.1573066\nM₁ : Type ?u.1573069\nM₂ : Type u_4\nM₃ : Type ?u.1573075\nM₄ : Type ?u.1573078\nN : Type ?u.1573081\nN₂ : Type ?u.1573084\nι : Type ?u.1573087\nV : Type ?u.1573090\nV₂ : Type ?u.1573093\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : Semiring R₃\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : AddCommMonoid M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\nh : ∀ (u v : { x // x ∈ ker f } →ₗ[R] M), comp f u = comp f v → u = v\nh₁ : comp f 0 = 0\n⊢ ker f = ⊥",
"state_before": "R : Type u_1\nR₁ : Type ?u.1573042\nR₂ : Type u_2\nR₃ : Type ?u.1573048\nR₄ : Type ?u.1573051\nS : Type ?u.1573054\nK : Type ?u.1573057\nK₂ : Type ?u.1573060\nM : Type u_3\nM' : Type ?u.1573066\nM₁ : Type ?u.1573069\nM₂ : Type u_4\nM₃ : Type ?u.1573075\nM₄ : Type ?u.1573078\nN : Type ?u.1573081\nN₂ : Type ?u.1573084\nι : Type ?u.1573087\nV : Type ?u.1573090\nV₂ : Type ?u.1573093\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : Semiring R₃\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : AddCommMonoid M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\nh : ∀ (u v : { x // x ∈ ker f } →ₗ[R] M), comp f u = comp f v → u = v\n⊢ ker f = ⊥",
"tactic": "have h₁ : f.comp (0 : ker f →ₗ[R] M) = 0 := comp_zero _"
},
{
"state_after": "R : Type u_1\nR₁ : Type ?u.1573042\nR₂ : Type u_2\nR₃ : Type ?u.1573048\nR₄ : Type ?u.1573051\nS : Type ?u.1573054\nK : Type ?u.1573057\nK₂ : Type ?u.1573060\nM : Type u_3\nM' : Type ?u.1573066\nM₁ : Type ?u.1573069\nM₂ : Type u_4\nM₃ : Type ?u.1573075\nM₄ : Type ?u.1573078\nN : Type ?u.1573081\nN₂ : Type ?u.1573084\nι : Type ?u.1573087\nV : Type ?u.1573090\nV₂ : Type ?u.1573093\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : Semiring R₃\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : AddCommMonoid M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\nh : ∀ (u v : { x // x ∈ ker f } →ₗ[R] M), comp f u = comp f v → u = v\nh₁ : comp f 0 = 0\n⊢ range 0 = ⊥",
"state_before": "R : Type u_1\nR₁ : Type ?u.1573042\nR₂ : Type u_2\nR₃ : Type ?u.1573048\nR₄ : Type ?u.1573051\nS : Type ?u.1573054\nK : Type ?u.1573057\nK₂ : Type ?u.1573060\nM : Type u_3\nM' : Type ?u.1573066\nM₁ : Type ?u.1573069\nM₂ : Type u_4\nM₃ : Type ?u.1573075\nM₄ : Type ?u.1573078\nN : Type ?u.1573081\nN₂ : Type ?u.1573084\nι : Type ?u.1573087\nV : Type ?u.1573090\nV₂ : Type ?u.1573093\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : Semiring R₃\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : AddCommMonoid M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\nh : ∀ (u v : { x // x ∈ ker f } →ₗ[R] M), comp f u = comp f v → u = v\nh₁ : comp f 0 = 0\n⊢ ker f = ⊥",
"tactic": "rw [← Submodule.range_subtype (ker f),\n ← h 0 f.ker.subtype (Eq.trans h₁ (comp_ker_subtype f).symm)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1573042\nR₂ : Type u_2\nR₃ : Type ?u.1573048\nR₄ : Type ?u.1573051\nS : Type ?u.1573054\nK : Type ?u.1573057\nK₂ : Type ?u.1573060\nM : Type u_3\nM' : Type ?u.1573066\nM₁ : Type ?u.1573069\nM₂ : Type u_4\nM₃ : Type ?u.1573075\nM₄ : Type ?u.1573078\nN : Type ?u.1573081\nN₂ : Type ?u.1573084\nι : Type ?u.1573087\nV : Type ?u.1573090\nV₂ : Type ?u.1573093\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : Semiring R₃\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : AddCommMonoid M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nf : M →ₛₗ[τ₁₂] M₂\nh : ∀ (u v : { x // x ∈ ker f } →ₗ[R] M), comp f u = comp f v → u = v\nh₁ : comp f 0 = 0\n⊢ range 0 = ⊥",
"tactic": "exact range_zero"
}
] |
[
1713,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1708,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
ContinuousLinearMap.smul_compLpL
|
[
{
"state_after": "case h\nα : Type u_5\nE : Type u_4\nF : Type u_2\nG : Type ?u.7594842\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ng : E → F\nc✝ : ℝ≥0\n𝕜 : Type u_3\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : Fact (1 ≤ p)\n𝕜' : Type u_1\ninst✝³ : NormedRing 𝕜'\ninst✝² : Module 𝕜' F\ninst✝¹ : BoundedSMul 𝕜' F\ninst✝ : SMulCommClass 𝕜 𝕜' F\nc : 𝕜'\nL : E →L[𝕜] F\nf : { x // x ∈ Lp E p }\n⊢ ↑(compLpL p μ (c • L)) f = ↑(c • compLpL p μ L) f",
"state_before": "α : Type u_5\nE : Type u_4\nF : Type u_2\nG : Type ?u.7594842\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ng : E → F\nc✝ : ℝ≥0\n𝕜 : Type u_3\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : Fact (1 ≤ p)\n𝕜' : Type u_1\ninst✝³ : NormedRing 𝕜'\ninst✝² : Module 𝕜' F\ninst✝¹ : BoundedSMul 𝕜' F\ninst✝ : SMulCommClass 𝕜 𝕜' F\nc : 𝕜'\nL : E →L[𝕜] F\n⊢ compLpL p μ (c • L) = c • compLpL p μ L",
"tactic": "ext1 f"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_5\nE : Type u_4\nF : Type u_2\nG : Type ?u.7594842\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ng : E → F\nc✝ : ℝ≥0\n𝕜 : Type u_3\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : Fact (1 ≤ p)\n𝕜' : Type u_1\ninst✝³ : NormedRing 𝕜'\ninst✝² : Module 𝕜' F\ninst✝¹ : BoundedSMul 𝕜' F\ninst✝ : SMulCommClass 𝕜 𝕜' F\nc : 𝕜'\nL : E →L[𝕜] F\nf : { x // x ∈ Lp E p }\n⊢ ↑(compLpL p μ (c • L)) f = ↑(c • compLpL p μ L) f",
"tactic": "exact smul_compLp c L f"
}
] |
[
1057,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.disjoint_cycleFactorsFinset
|
[
{
"state_after": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\n⊢ _root_.Disjoint (cycleFactorsFinset f) (cycleFactorsFinset g)",
"state_before": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh : Disjoint f g\n⊢ _root_.Disjoint (cycleFactorsFinset f) (cycleFactorsFinset g)",
"tactic": "rw [disjoint_iff_disjoint_support] at h"
},
{
"state_after": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\n⊢ ∀ ⦃a : Perm α⦄, a ∈ cycleFactorsFinset f → ¬a ∈ cycleFactorsFinset g",
"state_before": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\n⊢ _root_.Disjoint (cycleFactorsFinset f) (cycleFactorsFinset g)",
"tactic": "rw [Finset.disjoint_left]"
},
{
"state_after": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhx : x ∈ cycleFactorsFinset f\nhy : x ∈ cycleFactorsFinset g\n⊢ False",
"state_before": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\n⊢ ∀ ⦃a : Perm α⦄, a ∈ cycleFactorsFinset f → ¬a ∈ cycleFactorsFinset g",
"tactic": "intro x hx hy"
},
{
"state_after": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhx : IsCycle x ∧ ∀ (a : α), ↑x a ≠ a → ↑x a = ↑f a\nhy : IsCycle x ∧ ∀ (a : α), ↑x a ≠ a → ↑x a = ↑g a\n⊢ False",
"state_before": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhx : x ∈ cycleFactorsFinset f\nhy : x ∈ cycleFactorsFinset g\n⊢ False",
"tactic": "simp only [mem_cycleFactorsFinset_iff, mem_support] at hx hy"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhf : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑f a\na : α\nha : ↑x a ≠ a\nhg : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑g a\n⊢ False",
"state_before": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhx : IsCycle x ∧ ∀ (a : α), ↑x a ≠ a → ↑x a = ↑f a\nhy : IsCycle x ∧ ∀ (a : α), ↑x a ≠ a → ↑x a = ↑g a\n⊢ False",
"tactic": "obtain ⟨⟨⟨a, ha, -⟩, hf⟩, -, hg⟩ := hx, hy"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhf : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑f a\na : α\nha : ↑x a ≠ a\nhg : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑g a\nthis : a ∈ ⊥\n⊢ False",
"state_before": "case intro.intro.intro.intro\nι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhf : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑f a\na : α\nha : ↑x a ≠ a\nhg : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑g a\n⊢ False",
"tactic": "have := h.le_bot (by simp [ha, ← hf a ha, ← hg a ha] : a ∈ f.support ∩ g.support)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhf : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑f a\na : α\nha : ↑x a ≠ a\nhg : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑g a\nthis : a ∈ ⊥\n⊢ False",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2757665\nα : Type u_1\nβ : Type ?u.2757671\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f g : Perm α\nh✝ : Disjoint f g\nh : _root_.Disjoint (support f) (support g)\nx : Perm α\nhf : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑f a\na : α\nha : ↑x a ≠ a\nhg : ∀ (a : α), ↑x a ≠ a → ↑x a = ↑g a\n⊢ a ∈ support f ∩ support g",
"tactic": "simp [ha, ← hf a ha, ← hg a ha]"
}
] |
[
1485,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1477,
1
] |
Mathlib/Init/Data/Nat/Lemmas.lean
|
Nat.one_lt_bit0
|
[
{
"state_after": "no goals",
"state_before": "h : 0 ≠ 0\n⊢ 1 < bit0 0",
"tactic": "contradiction"
},
{
"state_after": "n : ℕ\nx✝ : succ n ≠ 0\n⊢ 1 < succ (succ (bit0 n))",
"state_before": "n : ℕ\nx✝ : succ n ≠ 0\n⊢ 1 < bit0 (succ n)",
"tactic": "rw [Nat.bit0_succ_eq]"
},
{
"state_after": "case a\nn : ℕ\nx✝ : succ n ≠ 0\n⊢ 0 < bit0 n + 1",
"state_before": "n : ℕ\nx✝ : succ n ≠ 0\n⊢ 1 < succ (succ (bit0 n))",
"tactic": "apply succ_lt_succ"
},
{
"state_after": "no goals",
"state_before": "case a\nn : ℕ\nx✝ : succ n ≠ 0\n⊢ 0 < bit0 n + 1",
"tactic": "apply zero_lt_succ"
}
] |
[
174,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
11
] |
Mathlib/Topology/Basic.lean
|
all_mem_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nx : α\nP : Set α → Prop\nhP : ∀ (s t : Set α), s ⊆ t → P s → P t\n⊢ (∀ (i : Set α), x ∈ i ∧ IsOpen i → P i) ↔ ∀ (s : Set α), IsOpen s → x ∈ s → P s",
"tactic": "simp only [@and_comm (x ∈ _), and_imp]"
}
] |
[
1017,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1015,
1
] |
Mathlib/Deprecated/Group.lean
|
IsMonoidHom.inv
|
[] |
[
185,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/LinearAlgebra/Multilinear/TensorProduct.lean
|
MultilinearMap.domCoprod'_apply
|
[] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.monomial_def
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\n⊢ LinearMap.stdBasis R (fun x => R) n = LinearMap.stdBasis R (fun x => R) n",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\n⊢ monomial R n = LinearMap.stdBasis R (fun x => R) n",
"tactic": "rw [monomial]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\n⊢ LinearMap.stdBasis R (fun x => R) n = LinearMap.stdBasis R (fun x => R) n",
"tactic": "convert rfl"
}
] |
[
150,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.succ_eq_succ_iff_of_not_isMax
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : SuccOrder α\na b : α\nha : ¬IsMax a\nhb : ¬IsMax b\n⊢ succ a = succ b ↔ a = b",
"tactic": "rw [eq_iff_le_not_lt, eq_iff_le_not_lt, succ_le_succ_iff_of_not_isMax ha hb,\n succ_lt_succ_iff_of_not_isMax ha hb]"
}
] |
[
414,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_le_left
|
[] |
[
155,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Algebra/RingQuot.lean
|
RingQuot.add_quot
|
[
{
"state_after": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\na b : R\n⊢ RingQuot.add r { toQuot := Quot.mk (Rel r) a } { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a + b) }",
"state_before": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\na b : R\n⊢ { toQuot := Quot.mk (Rel r) a } + { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a + b) }",
"tactic": "show add r _ _ = _"
},
{
"state_after": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\na b : R\n⊢ (match { toQuot := Quot.mk (Rel r) a }, { toQuot := Quot.mk (Rel r) b } with\n | { toQuot := a }, { toQuot := b } =>\n {\n toQuot :=\n Quot.map₂ (fun x x_1 => x + x_1) (_ : ∀ ⦃a b c : R⦄, Rel r b c → Rel r (a + b) (a + c))\n (_ : ∀ ⦃a b c : R⦄, Rel r a b → Rel r (a + c) (b + c)) a b }) =\n { toQuot := Quot.mk (Rel r) (a + b) }",
"state_before": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\na b : R\n⊢ RingQuot.add r { toQuot := Quot.mk (Rel r) a } { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a + b) }",
"tactic": "rw [add_def]"
},
{
"state_after": "no goals",
"state_before": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\na b : R\n⊢ (match { toQuot := Quot.mk (Rel r) a }, { toQuot := Quot.mk (Rel r) b } with\n | { toQuot := a }, { toQuot := b } =>\n {\n toQuot :=\n Quot.map₂ (fun x x_1 => x + x_1) (_ : ∀ ⦃a b c : R⦄, Rel r b c → Rel r (a + b) (a + c))\n (_ : ∀ ⦃a b c : R⦄, Rel r a b → Rel r (a + c) (b + c)) a b }) =\n { toQuot := Quot.mk (Rel r) (a + b) }",
"tactic": "rfl"
}
] |
[
239,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.sub_apply
|
[] |
[
338,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean
|
SimplexCategory.len_le_of_epi
|
[
{
"state_after": "x y : SimplexCategory\nf : x ⟶ y\nhyp_f_epi : Epi f\n⊢ len y ≤ len x",
"state_before": "x y : SimplexCategory\nf : x ⟶ y\n⊢ Epi f → len y ≤ len x",
"tactic": "intro hyp_f_epi"
},
{
"state_after": "x y : SimplexCategory\nf : x ⟶ y\nhyp_f_epi : Epi f\nf_surj : Function.Surjective ↑(Hom.toOrderHom f)\n⊢ len y ≤ len x",
"state_before": "x y : SimplexCategory\nf : x ⟶ y\nhyp_f_epi : Epi f\n⊢ len y ≤ len x",
"tactic": "have f_surj : Function.Surjective f.toOrderHom.toFun := epi_iff_surjective.1 hyp_f_epi"
},
{
"state_after": "no goals",
"state_before": "x y : SimplexCategory\nf : x ⟶ y\nhyp_f_epi : Epi f\nf_surj : Function.Surjective ↑(Hom.toOrderHom f)\n⊢ len y ≤ len x",
"tactic": "simpa using Fintype.card_le_of_surjective f.toOrderHom.toFun f_surj"
}
] |
[
512,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.equiv_def₃
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abv (↑(f - g) j) < ε / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\nj : ℕ\nij : j ≥ i\nk : ℕ\njk : k ≥ j\nh₁ : abv (↑(f - g) j) < ε / 2\nh₂ : ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\n⊢ abv (↑f k - ↑g j) < ε",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abv (↑(f - g) j) < ε / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\nj : ℕ\nij : j ≥ i\nk : ℕ\njk : k ≥ j\n⊢ abv (↑f k - ↑g j) < ε",
"tactic": "let ⟨h₁, h₂⟩ := H _ ij"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abv (↑(f - g) j) < ε / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\nj : ℕ\nij : j ≥ i\nk : ℕ\njk : k ≥ j\nh₁ : abv (↑(f - g) j) < ε / 2\nh₂ : ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\nthis : abv (↑f j - ↑g j + (↑f k - ↑f j)) < ε / 2 + ε / 2\n⊢ abv (↑f k - ↑g j) < ε",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abv (↑(f - g) j) < ε / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\nj : ℕ\nij : j ≥ i\nk : ℕ\njk : k ≥ j\nh₁ : abv (↑(f - g) j) < ε / 2\nh₂ : ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\n⊢ abv (↑f k - ↑g j) < ε",
"tactic": "have := lt_of_le_of_lt (abv_add abv (f j - g j) _) (add_lt_add h₁ (h₂ _ jk))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abv (↑(f - g) j) < ε / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\nj : ℕ\nij : j ≥ i\nk : ℕ\njk : k ≥ j\nh₁ : abv (↑(f - g) j) < ε / 2\nh₂ : ∀ (k : ℕ), k ≥ j → abv (↑f k - ↑f j) < ε / 2\nthis : abv (↑f j - ↑g j + (↑f k - ↑f j)) < ε / 2 + ε / 2\n⊢ abv (↑f k - ↑g j) < ε",
"tactic": "rwa [sub_add_sub_cancel', add_halves] at this"
}
] |
[
491,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
DirectSum.decompose_mul_add_right
|
[] |
[
163,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/CovariantAndContravariant.lean
|
Antitone.covariant_of_const
|
[] |
[
275,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Algebra/Order/Nonneg/Ring.lean
|
Nonneg.mk_eq_zero
|
[] |
[
122,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Topology/Separation.lean
|
exists_mem_nhds_isClosed_subset
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns✝ : Set α\na : α\ns : Set α\nh : s ∈ 𝓝 a\nh' : RegularSpace α ↔ ∀ (a : α) (s : Set α), s ∈ 𝓝 a → ∃ t, t ∈ 𝓝 a ∧ IsClosed t ∧ t ⊆ s\n⊢ ∃ t, t ∈ 𝓝 a ∧ IsClosed t ∧ t ⊆ s",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns✝ : Set α\na : α\ns : Set α\nh : s ∈ 𝓝 a\n⊢ ∃ t, t ∈ 𝓝 a ∧ IsClosed t ∧ t ⊆ s",
"tactic": "have h' := (regularSpace_TFAE α).out 0 3"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns✝ : Set α\na : α\ns : Set α\nh : s ∈ 𝓝 a\nh' : RegularSpace α ↔ ∀ (a : α) (s : Set α), s ∈ 𝓝 a → ∃ t, t ∈ 𝓝 a ∧ IsClosed t ∧ t ⊆ s\n⊢ ∃ t, t ∈ 𝓝 a ∧ IsClosed t ∧ t ⊆ s",
"tactic": "exact h'.mp ‹_› _ _ h"
}
] |
[
1534,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1531,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
|
[] |
[
247,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.of_apply
|
[] |
[
644,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.card_support_eq_two
|
[
{
"state_after": "case mp\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\n⊢ IsSwap f\n\ncase mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : IsSwap f\n⊢ card (support f) = 2",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\n⊢ card (support f) = 2 ↔ IsSwap f",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx : α\nt : Finset α\nhmem : ¬x ∈ t\nhins : insert x t = support f\nht : card t = 1\n⊢ IsSwap f",
"state_before": "case mp\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\n⊢ IsSwap f",
"tactic": "obtain ⟨x, t, hmem, hins, ht⟩ := card_eq_succ.1 h"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x ∈ {y}\nhins : {x, y} = support f\nht : card {y} = 1\n⊢ IsSwap f",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx : α\nt : Finset α\nhmem : ¬x ∈ t\nhins : insert x t = support f\nht : card t = 1\n⊢ IsSwap f",
"tactic": "obtain ⟨y, rfl⟩ := card_eq_one.1 ht"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\n⊢ IsSwap f",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x ∈ {y}\nhins : {x, y} = support f\nht : card {y} = 1\n⊢ IsSwap f",
"tactic": "rw [mem_singleton] at hmem"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\n⊢ f = swap x y",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\n⊢ IsSwap f",
"tactic": "refine' ⟨x, y, hmem, _⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.H\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\n⊢ ↑f a = ↑(swap x y) a",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\n⊢ f = swap x y",
"tactic": "ext a"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.H\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\n⊢ ↑f a = ↑(swap x y) a",
"state_before": "case mp.intro.intro.intro.intro.intro.H\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\n⊢ ↑f a = ↑(swap x y) a",
"tactic": "have key : ∀ b, f b ≠ b ↔ _ := fun b => by rw [← mem_support, ← hins, mem_insert, mem_singleton]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ↑f a = a\n⊢ ↑f a = ↑(swap x y) a\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ¬↑f a = a\n⊢ ↑f a = ↑(swap x y) a",
"state_before": "case mp.intro.intro.intro.intro.intro.H\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\n⊢ ↑f a = ↑(swap x y) a",
"tactic": "by_cases ha : f a = a"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na b : α\n⊢ ↑f b ≠ b ↔ ?m.185118 b",
"tactic": "rw [← mem_support, ← hins, mem_insert, mem_singleton]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ↑f a = a\nha' : ¬a = x ∧ ¬a = y\n⊢ ↑f a = ↑(swap x y) a",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ↑f a = a\n⊢ ↑f a = ↑(swap x y) a",
"tactic": "have ha' := not_or.mp (mt (key a).mpr (not_not.mpr ha))"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ↑f a = a\nha' : ¬a = x ∧ ¬a = y\n⊢ ↑f a = ↑(swap x y) a",
"tactic": "rw [ha, swap_apply_of_ne_of_ne ha'.1 ha'.2]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ¬↑f a = a\nha' : ↑f a = x ∨ ↑f a = y\n⊢ ↑f a = ↑(swap x y) a",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ¬↑f a = a\n⊢ ↑f a = ↑(swap x y) a",
"tactic": "have ha' := (key (f a)).mp (mt f.apply_eq_iff_eq.mp ha)"
},
{
"state_after": "case neg.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\ny : α\nht : card {y} = 1\na : α\nha : ¬↑f a = a\nhmem : ¬a = y\nhins : {a, y} = support f\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = a ∨ b = y\nha' : ↑f a = a ∨ ↑f a = y\n⊢ ↑f a = ↑(swap a y) a\n\ncase neg.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx a : α\nha : ¬↑f a = a\nhmem : ¬x = a\nhins : {x, a} = support f\nht : card {a} = 1\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = a\nha' : ↑f a = x ∨ ↑f a = a\n⊢ ↑f a = ↑(swap x a) a",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx y : α\nhmem : ¬x = y\nhins : {x, y} = support f\nht : card {y} = 1\na : α\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = y\nha : ¬↑f a = a\nha' : ↑f a = x ∨ ↑f a = y\n⊢ ↑f a = ↑(swap x y) a",
"tactic": "obtain rfl | rfl := (key a).mp ha"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\ny : α\nht : card {y} = 1\na : α\nha : ¬↑f a = a\nhmem : ¬a = y\nhins : {a, y} = support f\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = a ∨ b = y\nha' : ↑f a = a ∨ ↑f a = y\n⊢ ↑f a = ↑(swap a y) a",
"tactic": "rw [Or.resolve_left ha' ha, swap_apply_left]"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : card (support f) = 2\nx a : α\nha : ¬↑f a = a\nhmem : ¬x = a\nhins : {x, a} = support f\nht : card {a} = 1\nkey : ∀ (b : α), ↑f b ≠ b ↔ b = x ∨ b = a\nha' : ↑f a = x ∨ ↑f a = a\n⊢ ↑f a = ↑(swap x a) a",
"tactic": "rw [Or.resolve_right ha' ha, swap_apply_right]"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nx y : α\nhxy : x ≠ y\n⊢ card (support (swap x y)) = 2",
"state_before": "case mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nh : IsSwap f\n⊢ card (support f) = 2",
"tactic": "obtain ⟨x, y, hxy, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nx y : α\nhxy : x ≠ y\n⊢ card (support (swap x y)) = 2",
"tactic": "exact card_support_swap hxy"
}
] |
[
637,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.set_integral_eq_subtype
|
[
{
"state_after": "α✝ : Type ?u.1476210\nE : Type u_2\nF : Type ?u.1476216\n𝕜 : Type ?u.1476219\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α✝ → E\nm : MeasurableSpace α✝\nμ : Measure α✝\nX : Type ?u.1478910\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nν : Measure α✝\nα : Type u_1\ninst✝ : MeasureSpace α\ns : Set α\nhs : MeasurableSet s\nf : α → E\n⊢ (∫ (x : α), f x ∂Measure.map Subtype.val (Measure.comap Subtype.val volume)) = ∫ (x : ↑s), f ↑x",
"state_before": "α✝ : Type ?u.1476210\nE : Type u_2\nF : Type ?u.1476216\n𝕜 : Type ?u.1476219\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α✝ → E\nm : MeasurableSpace α✝\nμ : Measure α✝\nX : Type ?u.1478910\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nν : Measure α✝\nα : Type u_1\ninst✝ : MeasureSpace α\ns : Set α\nhs : MeasurableSet s\nf : α → E\n⊢ (∫ (x : α) in s, f x) = ∫ (x : ↑s), f ↑x",
"tactic": "rw [← map_comap_subtype_coe hs]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1476210\nE : Type u_2\nF : Type ?u.1476216\n𝕜 : Type ?u.1476219\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α✝ → E\nm : MeasurableSpace α✝\nμ : Measure α✝\nX : Type ?u.1478910\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nν : Measure α✝\nα : Type u_1\ninst✝ : MeasureSpace α\ns : Set α\nhs : MeasurableSet s\nf : α → E\n⊢ (∫ (x : α), f x ∂Measure.map Subtype.val (Measure.comap Subtype.val volume)) = ∫ (x : ↑s), f ↑x",
"tactic": "exact (MeasurableEmbedding.subtype_coe hs).integral_map _"
}
] |
[
1581,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1577,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.conj_bit1
|
[] |
[
369,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finsupp.prod_embDomain
|
[
{
"state_after": "α : Type u_3\nι : Type ?u.531151\nγ : Type ?u.531154\nA : Type ?u.531157\nB : Type ?u.531160\nC : Type ?u.531163\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_4\nM : Type u_1\nM' : Type ?u.534299\nN : Type u_2\nP : Type ?u.534305\nG : Type ?u.534308\nH : Type ?u.534311\nR : Type ?u.534314\nS : Type ?u.534317\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nv : α →₀ M\nf : α ↪ β\ng : β → M → N\n⊢ ∏ x in v.support, g (↑f x) (↑(embDomain f v) (↑f x)) = ∏ a in v.support, g (↑f a) (↑v a)",
"state_before": "α : Type u_3\nι : Type ?u.531151\nγ : Type ?u.531154\nA : Type ?u.531157\nB : Type ?u.531160\nC : Type ?u.531163\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_4\nM : Type u_1\nM' : Type ?u.534299\nN : Type u_2\nP : Type ?u.534305\nG : Type ?u.534308\nH : Type ?u.534311\nR : Type ?u.534314\nS : Type ?u.534317\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nv : α →₀ M\nf : α ↪ β\ng : β → M → N\n⊢ prod (embDomain f v) g = prod v fun a b => g (↑f a) b",
"tactic": "rw [prod, prod, support_embDomain, Finset.prod_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nι : Type ?u.531151\nγ : Type ?u.531154\nA : Type ?u.531157\nB : Type ?u.531160\nC : Type ?u.531163\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_4\nM : Type u_1\nM' : Type ?u.534299\nN : Type u_2\nP : Type ?u.534305\nG : Type ?u.534308\nH : Type ?u.534311\nR : Type ?u.534314\nS : Type ?u.534317\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nv : α →₀ M\nf : α ↪ β\ng : β → M → N\n⊢ ∏ x in v.support, g (↑f x) (↑(embDomain f v) (↑f x)) = ∏ a in v.support, g (↑f a) (↑v a)",
"tactic": "simp_rw [embDomain_apply]"
}
] |
[
517,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/Data/Analysis/Topology.lean
|
Ctop.Realizer.ofEquiv_σ
|
[] |
[
203,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.dim_pos
|
[] |
[
127,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
src/lean/Init/SimpLemmas.lean
|
Bool.true_or
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ (true || b) = true",
"tactic": "cases b <;> rfl"
}
] |
[
105,
83
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
105,
9
] |
Mathlib/Combinatorics/Colex.lean
|
Colex.sdiff_le_sdiff_iff_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\nA B : Finset α\n⊢ toColex (A \\ B) ≤ toColex (B \\ A) ↔ toColex A ≤ toColex B",
"tactic": "rw [le_iff_le_iff_lt_iff_lt, sdiff_lt_sdiff_iff_lt]"
}
] |
[
334,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_smul_real_left
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1383018\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nr : ℝ\n⊢ ↑r * inner x y = ↑(algebraMap ℝ 𝕜) r * inner x y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1383018\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nr : ℝ\n⊢ inner (↑r • x) y = r • inner x y",
"tactic": "rw [inner_smul_left, conj_ofReal, Algebra.smul_def]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.1383018\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\nr : ℝ\n⊢ ↑r * inner x y = ↑(algebraMap ℝ 𝕜) r * inner x y",
"tactic": "rfl"
}
] |
[
485,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Data/List/Func.lean
|
List.Func.get_neg
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝ : AddGroup α\nk : ℕ\nas : List α\n⊢ get k (map (fun a => -a) as) = -get k as",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝ : AddGroup α\nk : ℕ\nas : List α\n⊢ get k (neg as) = -get k as",
"tactic": "unfold neg"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ninst✝ : AddGroup α\nk : ℕ\nas : List α\n⊢ -default = default",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝ : AddGroup α\nk : ℕ\nas : List α\n⊢ get k (map (fun a => -a) as) = -get k as",
"tactic": "rw [@get_map' α α ⟨0⟩ ⟨0⟩]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ninst✝ : AddGroup α\nk : ℕ\nas : List α\n⊢ -default = default",
"tactic": "apply neg_zero"
}
] |
[
266,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.add_equiv_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 + g1 ≈ f2 + g2",
"tactic": "simpa only [← add_sub_add_comm] using add_limZero hf hg"
}
] |
[
475,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.head_iterate
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_bot
|
[] |
[
2165,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2165,
9
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_pi_def
|
[] |
[
1992,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1991,
1
] |
Mathlib/Order/RelClasses.lean
|
IsOrderConnected.neg_trans
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nr✝ : α → α → Prop\ns : β → β → Prop\nr : α → α → Prop\ninst✝ : IsOrderConnected α r\na b c : α\nh₁ : ¬r a b\nh₂ : ¬r b c\n⊢ ¬(r a b ∨ r b c)",
"tactic": "simp [h₁, h₂]"
}
] |
[
249,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Topology/MetricSpace/Completion.lean
|
UniformSpace.Completion.dist_triangle
|
[
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\n⊢ IsClosed {x | dist x.fst x.snd.snd ≤ dist x.fst x.snd.fst + dist x.snd.fst x.snd.snd}\n\ncase refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\n⊢ ∀ (a b c : α), dist (↑α a) (↑α c) ≤ dist (↑α a) (↑α b) + dist (↑α b) (↑α c)",
"state_before": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\n⊢ dist x z ≤ dist x y + dist y z",
"tactic": "refine' induction_on₃ x y z _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\n⊢ IsClosed {x | dist x.fst x.snd.snd ≤ dist x.fst x.snd.fst + dist x.snd.fst x.snd.snd}",
"tactic": "refine' isClosed_le _ (Continuous.add _ _) <;>\n apply_rules [Completion.continuous_dist, Continuous.fst, Continuous.snd, continuous_id]"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\na b c : α\n⊢ dist (↑α a) (↑α c) ≤ dist (↑α a) (↑α b) + dist (↑α b) (↑α c)",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\n⊢ ∀ (a b c : α), dist (↑α a) (↑α c) ≤ dist (↑α a) (↑α b) + dist (↑α b) (↑α c)",
"tactic": "intro a b c"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\na b c : α\n⊢ dist a c ≤ dist a b + dist b c",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\na b c : α\n⊢ dist (↑α a) (↑α c) ≤ dist (↑α a) (↑α b) + dist (↑α b) (↑α c)",
"tactic": "rw [Completion.dist_eq, Completion.dist_eq, Completion.dist_eq]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y z : Completion α\na b c : α\n⊢ dist a c ≤ dist a b + dist b c",
"tactic": "exact dist_triangle a b c"
}
] |
[
84,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
11
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.forall_mem_nonempty_iff_neBot
|
[] |
[
768,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
DistribMulActionHom.ext_ring_iff
|
[] |
[
405,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Algebra/Algebra/Tower.lean
|
IsScalarTower.coe_toAlgHom
|
[] |
[
130,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/Nat/Bits.lean
|
Nat.bit_add'
|
[] |
[
99,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/CategoryTheory/Elements.lean
|
CategoryTheory.CategoryOfElements.to_comma_map_right
|
[] |
[
153,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/List/Perm.lean
|
List.perm_singleton
|
[] |
[
210,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Analysis/NormedSpace/MazurUlam.lean
|
IsometryEquiv.toRealLinearIsometryEquiv_symm_apply
|
[] |
[
145,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
monovary_toDual_left
|
[] |
[
249,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Data/PNat/Basic.lean
|
PNat.div_add_mod'
|
[
{
"state_after": "m k : ℕ+\n⊢ ↑k * div m k + ↑(mod m k) = ↑m",
"state_before": "m k : ℕ+\n⊢ div m k * ↑k + ↑(mod m k) = ↑m",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "m k : ℕ+\n⊢ ↑k * div m k + ↑(mod m k) = ↑m",
"tactic": "exact div_add_mod _ _"
}
] |
[
376,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Order/MinMax.lean
|
max_cases
|
[] |
[
170,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
RingEquiv.ofLeftInverseS_apply
|
[] |
[
1295,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1293,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.mk_smul
|
[
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ns : Finset ι\nc : γ\nx : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (if H : i ∈ s then (c • x) { val := i, property := H } else 0) =\n c • if H : i ∈ s then x { val := i, property := H } else 0",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ns : Finset ι\nc : γ\nx : (i : ↑↑s) → β ↑i\ni : ι\n⊢ ↑(mk s (c • x)) i = ↑(c • mk s x) i",
"tactic": "simp only [smul_apply, mk_apply]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : Monoid γ\ninst✝¹ : (i : ι) → AddMonoid (β i)\ninst✝ : (i : ι) → DistribMulAction γ (β i)\ns : Finset ι\nc : γ\nx : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (if H : i ∈ s then (c • x) { val := i, property := H } else 0) =\n c • if H : i ∈ s then x { val := i, property := H } else 0",
"tactic": "split_ifs <;> [rfl; rw [smul_zero]]"
}
] |
[
1074,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1072,
1
] |
Mathlib/RingTheory/Nakayama.lean
|
Submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"tactic": "have hNN' : N ⊔ N' = N ⊔ I • N' :=\n le_antisymm hNN (sup_le_sup_left (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _) _)"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"tactic": "have h_comap := Submodule.comap_injective_of_surjective (LinearMap.range_eq_top.1 N.range_mkQ)"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\nthis : map (mkQ N) (I • N') = map (mkQ N) N'\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"tactic": "have : (I • N').map N.mkQ = N'.map N.mkQ := by\n rw [← h_comap.eq_iff]\n simpa [comap_map_eq, sup_comm, eq_comm] using hNN'"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\nthis✝ : map (mkQ N) (I • N') = map (mkQ N) N'\nthis : map (mkQ N) N' = J • map (mkQ N) N'\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\nthis : map (mkQ N) (I • N') = map (mkQ N) N'\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"tactic": "have :=\n @Submodule.eq_smul_of_le_smul_of_le_jacobson _ _ _ _ _ I J (N'.map N.mkQ) (hN'.map _)\n (by rw [← map_smul'', this]) hIJ"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\nthis✝ : map (mkQ N) (I • N') = map (mkQ N) N'\nthis : N ⊔ I • N' = J • N' ⊔ N\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\nthis✝ : map (mkQ N) (I • N') = map (mkQ N) N'\nthis : map (mkQ N) N' = J • map (mkQ N) N'\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"tactic": "rw [← map_smul'', ← h_comap.eq_iff, comap_map_eq, comap_map_eq, Submodule.ker_mkQ, sup_comm,\n hNN'] at this"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\nthis✝ : map (mkQ N) (I • N') = map (mkQ N) N'\nthis : N ⊔ I • N' = J • N' ⊔ N\n⊢ N ⊔ I • N' = N ⊔ J • N'",
"tactic": "rw [this, sup_comm]"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\n⊢ comap (mkQ N) (map (mkQ N) (I • N')) = comap (mkQ N) (map (mkQ N) N')",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\n⊢ map (mkQ N) (I • N') = map (mkQ N) N'",
"tactic": "rw [← h_comap.eq_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\n⊢ comap (mkQ N) (map (mkQ N) (I • N')) = comap (mkQ N) (map (mkQ N) N')",
"tactic": "simpa [comap_map_eq, sup_comm, eq_comm] using hNN'"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI J : Ideal R\nN N' : Submodule R M\nhN' : FG N'\nhIJ : I ≤ jacobson J\nhNN : N ⊔ N' ≤ N ⊔ I • N'\nhNN' : N ⊔ N' = N ⊔ I • N'\nh_comap : Function.Injective (comap (mkQ N))\nthis : map (mkQ N) (I • N') = map (mkQ N) N'\n⊢ map (mkQ N) N' ≤ I • map (mkQ N) N'",
"tactic": "rw [← map_smul'', this]"
}
] |
[
92,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_mono
|
[] |
[
1322,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1321,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.inlCokernelCofork_π
|
[] |
[
1715,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1714,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.HasColimit.isoOfNatIso_ι_hom
|
[] |
[
914,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
912,
1
] |
Mathlib/Combinatorics/Young/SemistandardTableau.lean
|
Ssyt.col_weak
|
[
{
"state_after": "case inl\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 = i2\n⊢ ↑T i1 j ≤ ↑T i2 j\n\ncase inr\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 < i2\n⊢ ↑T i1 j ≤ ↑T i2 j",
"state_before": "μ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\n⊢ ↑T i1 j ≤ ↑T i2 j",
"tactic": "cases' eq_or_lt_of_le hi with h h"
},
{
"state_after": "case inr\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 < i2\n⊢ ↑T i1 j ≤ ↑T i2 j",
"state_before": "case inl\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 = i2\n⊢ ↑T i1 j ≤ ↑T i2 j\n\ncase inr\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 < i2\n⊢ ↑T i1 j ≤ ↑T i2 j",
"tactic": ". rw [h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 < i2\n⊢ ↑T i1 j ≤ ↑T i2 j",
"tactic": ". exact le_of_lt (T.col_strict h cell)"
},
{
"state_after": "no goals",
"state_before": "case inl\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 = i2\n⊢ ↑T i1 j ≤ ↑T i2 j",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nμ : YoungDiagram\nT : Ssyt μ\ni1 i2 j : ℕ\nhi : i1 ≤ i2\ncell : (i2, j) ∈ μ\nh : i1 < i2\n⊢ ↑T i1 j ≤ ↑T i2 j",
"tactic": "exact le_of_lt (T.col_strict h cell)"
}
] |
[
139,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Data/Nat/Squarefree.lean
|
Nat.sq_mul_squarefree
|
[
{
"state_after": "case zero\n\n⊢ ∃ a b, b ^ 2 * a = zero ∧ Squarefree a\n\ncase succ\nn : ℕ\n⊢ ∃ a b, b ^ 2 * a = succ n ∧ Squarefree a",
"state_before": "n : ℕ\n⊢ ∃ a b, b ^ 2 * a = n ∧ Squarefree a",
"tactic": "cases' n with n"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ ∃ a b, b ^ 2 * a = zero ∧ Squarefree a",
"tactic": "exact ⟨1, 0, by simp, squarefree_one⟩"
},
{
"state_after": "no goals",
"state_before": "⊢ 0 ^ 2 * 1 = zero",
"tactic": "simp"
},
{
"state_after": "case succ.intro.intro.intro.intro.intro\nn a b : ℕ\nh₁ : b ^ 2 * a = succ n\nh₂ : Squarefree a\n⊢ ∃ a b, b ^ 2 * a = succ n ∧ Squarefree a",
"state_before": "case succ\nn : ℕ\n⊢ ∃ a b, b ^ 2 * a = succ n ∧ Squarefree a",
"tactic": "obtain ⟨a, b, -, -, h₁, h₂⟩ := sq_mul_squarefree_of_pos (succ_pos n)"
},
{
"state_after": "no goals",
"state_before": "case succ.intro.intro.intro.intro.intro\nn a b : ℕ\nh₁ : b ^ 2 * a = succ n\nh₂ : Squarefree a\n⊢ ∃ a b, b ^ 2 * a = succ n ∧ Squarefree a",
"tactic": "exact ⟨a, b, h₁, h₂⟩"
}
] |
[
365,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Data/Real/Cardinality.lean
|
Cardinal.mk_Icc_real
|
[] |
[
285,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
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