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Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean
|
AlgebraicTopology.DoldKan.Compatibility.equivalence₁_inverse
|
[] |
[
75,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.isOpen_iff_of_mem
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : ∞ ∈ s\n⊢ IsOpen s ↔ IsClosed ((some ⁻¹' s)ᶜ) ∧ IsCompact ((some ⁻¹' s)ᶜ)",
"tactic": "simp only [isOpen_iff_of_mem' h, isClosed_compl_iff, and_comm]"
}
] |
[
221,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/CategoryTheory/Adhesive.lean
|
CategoryTheory.is_coprod_iff_isPushout
|
[
{
"state_after": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\n⊢ Nonempty (IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)) → IsPushout f (BinaryCofan.inl c) iY fE\n\ncase mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\n⊢ IsPushout f (BinaryCofan.inl c) iY fE → Nonempty (IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY))",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\n⊢ Nonempty (IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)) ↔ IsPushout f (BinaryCofan.inl c) iY fE",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\n⊢ IsPushout f (BinaryCofan.inl c) iY fE",
"state_before": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\n⊢ Nonempty (IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)) → IsPushout f (BinaryCofan.inl c) iY fE",
"tactic": "rintro ⟨h⟩"
},
{
"state_after": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\n⊢ (s : PushoutCocone f (BinaryCofan.inl c)) →\n { l //\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inl s ∧\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inr s ∧\n ∀ {m : (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)).pt ⟶ s.pt},\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m = PushoutCocone.inl s →\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m =\n PushoutCocone.inr s →\n m = l }",
"state_before": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\n⊢ IsPushout f (BinaryCofan.inl c) iY fE",
"tactic": "refine' ⟨H, ⟨Limits.PushoutCocone.isColimitAux' _ _⟩⟩"
},
{
"state_after": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ { l //\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inl s ∧\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inr s ∧\n ∀ {m : (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)).pt ⟶ s.pt},\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m = PushoutCocone.inl s →\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m = PushoutCocone.inr s →\n m = l }",
"state_before": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\n⊢ (s : PushoutCocone f (BinaryCofan.inl c)) →\n { l //\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inl s ∧\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inr s ∧\n ∀ {m : (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)).pt ⟶ s.pt},\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m = PushoutCocone.inl s →\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m =\n PushoutCocone.inr s →\n m = l }",
"tactic": "intro s"
},
{
"state_after": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ { l //\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n l =\n PushoutCocone.inl s ∧\n fE ≫ l = s.ι.app WalkingSpan.right ∧\n ∀ {m : YE ⟶ s.pt},\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s →\n fE ≫ m = s.ι.app WalkingSpan.right → m = l }",
"state_before": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ { l //\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inl s ∧\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ l = PushoutCocone.inr s ∧\n ∀ {m : (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)).pt ⟶ s.pt},\n PushoutCocone.inl (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m = PushoutCocone.inl s →\n PushoutCocone.inr (PushoutCocone.mk iY fE (_ : f ≫ iY = BinaryCofan.inl c ≫ fE)) ≫ m = PushoutCocone.inr s →\n m = l }",
"tactic": "dsimp only [PushoutCocone.inr, PushoutCocone.mk]"
},
{
"state_after": "case mp.intro.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ fE ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n s.ι.app WalkingSpan.right\n\ncase mp.intro.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ ∀ {m : YE ⟶ s.pt},\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s →\n fE ≫ m = s.ι.app WalkingSpan.right →\n m = IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))",
"state_before": "case mp.intro\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ { l //\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n l =\n PushoutCocone.inl s ∧\n fE ≫ l = s.ι.app WalkingSpan.right ∧\n ∀ {m : YE ⟶ s.pt},\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s →\n fE ≫ m = s.ι.app WalkingSpan.right → m = l }",
"tactic": "refine' ⟨h.desc (BinaryCofan.mk (c.inr ≫ s.inr) s.inl), h.fac _ ⟨WalkingPair.right⟩, _, _⟩"
},
{
"state_after": "case mp.intro.refine'_1.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ BinaryCofan.inl c ≫\n fE ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inl c ≫ s.ι.app WalkingSpan.right\n\ncase mp.intro.refine'_1.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ BinaryCofan.inr c ≫\n fE ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inr c ≫ s.ι.app WalkingSpan.right",
"state_before": "case mp.intro.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ fE ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n s.ι.app WalkingSpan.right",
"tactic": "apply BinaryCofan.IsColimit.hom_ext hc"
},
{
"state_after": "case mp.intro.refine'_1.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ f ≫ iY ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inl c ≫ s.ι.app WalkingSpan.right",
"state_before": "case mp.intro.refine'_1.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ BinaryCofan.inl c ≫\n fE ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inl c ≫ s.ι.app WalkingSpan.right",
"tactic": "rw [← H.w_assoc]"
},
{
"state_after": "case mp.intro.refine'_1.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ f ≫\n (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)).ι.app\n { as := WalkingPair.right } =\n BinaryCofan.inl c ≫ s.ι.app WalkingSpan.right",
"state_before": "case mp.intro.refine'_1.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ f ≫ iY ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inl c ≫ s.ι.app WalkingSpan.right",
"tactic": "erw [h.fac _ ⟨WalkingPair.right⟩]"
},
{
"state_after": "no goals",
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"tactic": "exact s.condition"
},
{
"state_after": "case mp.intro.refine'_1.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ (BinaryCofan.inr c ≫ fE) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inr c ≫ s.ι.app WalkingSpan.right",
"state_before": "case mp.intro.refine'_1.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ BinaryCofan.inr c ≫\n fE ≫ IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inr c ≫ s.ι.app WalkingSpan.right",
"tactic": "rw [← Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.refine'_1.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ (BinaryCofan.inr c ≫ fE) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n BinaryCofan.inr c ≫ s.ι.app WalkingSpan.right",
"tactic": "exact h.fac _ ⟨WalkingPair.left⟩"
},
{
"state_after": "case mp.intro.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ m = IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))",
"state_before": "case mp.intro.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\n⊢ ∀ {m : YE ⟶ s.pt},\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s →\n fE ≫ m = s.ι.app WalkingSpan.right →\n m = IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))",
"tactic": "intro m e₁ e₂"
},
{
"state_after": "case mp.intro.refine'_2.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫ m =\n BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))\n\ncase mp.intro.refine'_2.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫ m =\n BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))",
"state_before": "case mp.intro.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ m = IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))",
"tactic": "apply BinaryCofan.IsColimit.hom_ext h"
},
{
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"state_before": "case mp.intro.refine'_2.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫ m =\n BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))",
"tactic": "dsimp only [BinaryCofan.mk, id]"
},
{
"state_after": "case mp.intro.refine'_2.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inl\n { pt := YE,\n ι :=\n NatTrans.mk fun x =>\n WalkingPair.rec (motive := fun t =>\n x.as = t →\n ((pair ((pair X E).obj { as := WalkingPair.right }) Y).obj { as := x.as } ⟶\n ((Functor.const (Discrete WalkingPair)).obj YE).obj { as := x.as }))\n (fun h => (_ : WalkingPair.left = x.as) ▸ BinaryCofan.inr c ≫ fE)\n (fun h => (_ : WalkingPair.right = x.as) ▸ iY) x.as (_ : x.as = x.as) } ≫\n IsColimit.desc h\n { pt := s.pt,\n ι :=\n NatTrans.mk fun x =>\n WalkingPair.rec (motive := fun t =>\n x.as = t →\n ((pair ((pair X E).obj { as := WalkingPair.right }) Y).obj { as := x.as } ⟶\n ((Functor.const (Discrete WalkingPair)).obj s.pt).obj { as := x.as }))\n (fun h => (_ : WalkingPair.left = x.as) ▸ BinaryCofan.inr c ≫ PushoutCocone.inr s)\n (fun h => (_ : WalkingPair.right = x.as) ▸ PushoutCocone.inl s) x.as (_ : x.as = x.as) } =\n BinaryCofan.inr c ≫ s.ι.app WalkingSpan.right",
"state_before": "case mp.intro.refine'_2.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inl\n { pt := YE,\n ι :=\n NatTrans.mk fun x =>\n WalkingPair.rec (motive := fun t =>\n x.as = t →\n ((pair ((pair X E).obj { as := WalkingPair.right }) Y).obj { as := x.as } ⟶\n ((Functor.const (Discrete WalkingPair)).obj YE).obj { as := x.as }))\n (fun h => (_ : WalkingPair.left = x.as) ▸ BinaryCofan.inr c ≫ fE)\n (fun h => (_ : WalkingPair.right = x.as) ▸ iY) x.as (_ : x.as = x.as) } ≫\n m =\n BinaryCofan.inl\n { pt := YE,\n ι :=\n NatTrans.mk fun x =>\n WalkingPair.rec (motive := fun t =>\n x.as = t →\n ((pair ((pair X E).obj { as := WalkingPair.right }) Y).obj { as := x.as } ⟶\n ((Functor.const (Discrete WalkingPair)).obj YE).obj { as := x.as }))\n (fun h => (_ : WalkingPair.left = x.as) ▸ BinaryCofan.inr c ≫ fE)\n (fun h => (_ : WalkingPair.right = x.as) ▸ iY) x.as (_ : x.as = x.as) } ≫\n IsColimit.desc h\n { pt := s.pt,\n ι :=\n NatTrans.mk fun x =>\n WalkingPair.rec (motive := fun t =>\n x.as = t →\n ((pair ((pair X E).obj { as := WalkingPair.right }) Y).obj { as := x.as } ⟶\n ((Functor.const (Discrete WalkingPair)).obj s.pt).obj { as := x.as }))\n (fun h => (_ : WalkingPair.left = x.as) ▸ BinaryCofan.inr c ≫ PushoutCocone.inr s)\n (fun h => (_ : WalkingPair.right = x.as) ▸ PushoutCocone.inl s) x.as (_ : x.as = x.as) }",
"tactic": "rw [Category.assoc, e₂, eq_comm]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.refine'_2.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inl\n { pt := YE,\n ι :=\n NatTrans.mk fun x =>\n WalkingPair.rec (motive := fun t =>\n x.as = t →\n ((pair ((pair X E).obj { as := WalkingPair.right }) Y).obj { as := x.as } ⟶\n ((Functor.const (Discrete WalkingPair)).obj YE).obj { as := x.as }))\n (fun h => (_ : WalkingPair.left = x.as) ▸ BinaryCofan.inr c ≫ fE)\n (fun h => (_ : WalkingPair.right = x.as) ▸ iY) x.as (_ : x.as = x.as) } ≫\n IsColimit.desc h\n { pt := s.pt,\n ι :=\n NatTrans.mk fun x =>\n WalkingPair.rec (motive := fun t =>\n x.as = t →\n ((pair ((pair X E).obj { as := WalkingPair.right }) Y).obj { as := x.as } ⟶\n ((Functor.const (Discrete WalkingPair)).obj s.pt).obj { as := x.as }))\n (fun h => (_ : WalkingPair.left = x.as) ▸ BinaryCofan.inr c ≫ PushoutCocone.inr s)\n (fun h => (_ : WalkingPair.right = x.as) ▸ PushoutCocone.inl s) x.as (_ : x.as = x.as) } =\n BinaryCofan.inr c ≫ s.ι.app WalkingSpan.right",
"tactic": "exact h.fac _ ⟨WalkingPair.left⟩"
},
{
"state_after": "case mp.intro.refine'_2.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n PushoutCocone.inl s",
"state_before": "case mp.intro.refine'_2.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫ m =\n BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s))",
"tactic": "refine' e₁.trans (Eq.symm _)"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.refine'_2.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh✝ : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\nh : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)\ns : PushoutCocone f (BinaryCofan.inl c)\nm : YE ⟶ s.pt\ne₁ :\n PushoutCocone.inl\n { pt := YE, ι := NatTrans.mk fun j => Option.rec (f ≫ iY) (fun val => WalkingPair.rec iY fE val) j } ≫\n m =\n PushoutCocone.inl s\ne₂ : fE ≫ m = s.ι.app WalkingSpan.right\n⊢ BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY) ≫\n IsColimit.desc h (BinaryCofan.mk (BinaryCofan.inr c ≫ PushoutCocone.inr s) (PushoutCocone.inl s)) =\n PushoutCocone.inl s",
"tactic": "exact h.fac _ _"
},
{
"state_after": "case mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)",
"state_before": "case mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH : CommSq f (BinaryCofan.inl c) iY fE\n⊢ IsPushout f (BinaryCofan.inl c) iY fE → Nonempty (IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY))",
"tactic": "refine' fun H => ⟨_⟩"
},
{
"state_after": "case mpr.desc\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y) → YE ⟶ s.pt\n\ncase mpr.fac_left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ ∀ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y),\n (BinaryCofan.inr c ≫ fE) ≫ ?mpr.desc s = BinaryCofan.inl s\n\ncase mpr.fac_right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ ∀ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y), iY ≫ ?mpr.desc s = BinaryCofan.inr s\n\ncase mpr.uniq\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ ∀ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y) (m : YE ⟶ s.pt),\n (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s → iY ≫ m = BinaryCofan.inr s → m = ?mpr.desc s",
"state_before": "case mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ fE) iY)",
"tactic": "fapply Limits.BinaryCofan.isColimitMk"
},
{
"state_after": "no goals",
"state_before": "case mpr.desc\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y) → YE ⟶ s.pt",
"tactic": "exact fun s => H.isColimit.desc (PushoutCocone.mk s.inr _ <|\n (hc.fac (BinaryCofan.mk (f ≫ s.inr) s.inl) ⟨WalkingPair.left⟩).symm)"
},
{
"state_after": "case mpr.fac_left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\n⊢ (BinaryCofan.inr c ≫ fE) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n BinaryCofan.inl s",
"state_before": "case mpr.fac_left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ ∀ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y),\n (BinaryCofan.inr c ≫ fE) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n BinaryCofan.inl s",
"tactic": "intro s"
},
{
"state_after": "case mpr.fac_left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\n⊢ (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.right } = BinaryCofan.inl s",
"state_before": "case mpr.fac_left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\n⊢ (BinaryCofan.inr c ≫ fE) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n BinaryCofan.inl s",
"tactic": "erw [Category.assoc, H.isColimit.fac _ WalkingSpan.right, hc.fac]"
},
{
"state_after": "no goals",
"state_before": "case mpr.fac_left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\n⊢ (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.right } = BinaryCofan.inl s",
"tactic": "rfl"
},
{
"state_after": "case mpr.fac_right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\n⊢ iY ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n BinaryCofan.inr s",
"state_before": "case mpr.fac_right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ ∀ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y),\n iY ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n BinaryCofan.inr s",
"tactic": "intro s"
},
{
"state_after": "no goals",
"state_before": "case mpr.fac_right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\n⊢ iY ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n BinaryCofan.inr s",
"tactic": "exact H.isColimit.fac _ WalkingSpan.left"
},
{
"state_after": "case mpr.uniq\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ m =\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s))))",
"state_before": "case mpr.uniq\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\n⊢ ∀ (s : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y) (m : YE ⟶ s.pt),\n (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s →\n iY ≫ m = BinaryCofan.inr s →\n m =\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s))))",
"tactic": "intro s m e₁ e₂"
},
{
"state_after": "case mpr.uniq.h₀\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ PushoutCocone.inl (IsPushout.cocone H) ≫ m =\n PushoutCocone.inl (IsPushout.cocone H) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s))))\n\ncase mpr.uniq.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ PushoutCocone.inr (IsPushout.cocone H) ≫ m =\n PushoutCocone.inr (IsPushout.cocone H) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s))))",
"state_before": "case mpr.uniq\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ m =\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s))))",
"tactic": "apply PushoutCocone.IsColimit.hom_ext H.isColimit"
},
{
"state_after": "case mpr.uniq.h₀\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ PushoutCocone.inl (IsPushout.cocone H) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n PushoutCocone.inl (IsPushout.cocone H) ≫ m",
"state_before": "case mpr.uniq.h₀\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ PushoutCocone.inl (IsPushout.cocone H) ≫ m =\n PushoutCocone.inl (IsPushout.cocone H) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s))))",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "case mpr.uniq.h₀\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ PushoutCocone.inl (IsPushout.cocone H) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))) =\n PushoutCocone.inl (IsPushout.cocone H) ≫ m",
"tactic": "exact (H.isColimit.fac _ WalkingSpan.left).trans e₂.symm"
},
{
"state_after": "case mpr.uniq.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ PushoutCocone.inr (IsPushout.cocone H) ≫ m =\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right",
"state_before": "case mpr.uniq.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ PushoutCocone.inr (IsPushout.cocone H) ≫ m =\n PushoutCocone.inr (IsPushout.cocone H) ≫\n IsColimit.desc (IsPushout.isColimit H)\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s))))",
"tactic": "erw [H.isColimit.fac _ WalkingSpan.right]"
},
{
"state_after": "case mpr.uniq.h₁.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inl c ≫ PushoutCocone.inr (IsPushout.cocone H) ≫ m =\n BinaryCofan.inl c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right\n\ncase mpr.uniq.h₁.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inr c ≫ PushoutCocone.inr (IsPushout.cocone H) ≫ m =\n BinaryCofan.inr c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right",
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"tactic": "apply BinaryCofan.IsColimit.hom_ext hc"
},
{
"state_after": "case mpr.uniq.h₁.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ f ≫ BinaryCofan.inr s = (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left }",
"state_before": "case mpr.uniq.h₁.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inl c ≫ PushoutCocone.inr (IsPushout.cocone H) ≫ m =\n BinaryCofan.inl c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right",
"tactic": "erw [hc.fac, ← H.w_assoc, e₂]"
},
{
"state_after": "no goals",
"state_before": "case mpr.uniq.h₁.h₁\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ f ≫ BinaryCofan.inr s = (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left }",
"tactic": "rfl"
},
{
"state_after": "case mpr.uniq.h₁.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inl s =\n BinaryCofan.inr c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right",
"state_before": "case mpr.uniq.h₁.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inr c ≫ PushoutCocone.inr (IsPushout.cocone H) ≫ m =\n BinaryCofan.inr c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right",
"tactic": "refine' ((Category.assoc _ _ _).symm.trans e₁).trans _"
},
{
"state_after": "case mpr.uniq.h₁.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inr c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right =\n BinaryCofan.inl s",
"state_before": "case mpr.uniq.h₁.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inl s =\n BinaryCofan.inr c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "case mpr.uniq.h₁.h₂\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X✝ Y✝ Z : C\nf✝ : W ⟶ X✝\ng : W ⟶ Y✝\nh : X✝ ⟶ Z\ni : Y✝ ⟶ Z\nX E Y YE : C\nc : BinaryCofan X E\nhc : IsColimit c\nf : X ⟶ Y\niY : Y ⟶ YE\nfE : c.pt ⟶ YE\nH✝ : CommSq f (BinaryCofan.inl c) iY fE\nH : IsPushout f (BinaryCofan.inl c) iY fE\ns : BinaryCofan ((pair X E).obj { as := WalkingPair.right }) Y\nm : YE ⟶ s.pt\ne₁ : (BinaryCofan.inr c ≫ fE) ≫ m = BinaryCofan.inl s\ne₂ : iY ≫ m = BinaryCofan.inr s\n⊢ BinaryCofan.inr c ≫\n (PushoutCocone.mk (BinaryCofan.inr s)\n (IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))\n (_ :\n (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)).ι.app { as := WalkingPair.left } =\n c.ι.app { as := WalkingPair.left } ≫\n IsColimit.desc hc (BinaryCofan.mk (f ≫ BinaryCofan.inr s) (BinaryCofan.inl s)))).ι.app\n WalkingSpan.right =\n BinaryCofan.inl s",
"tactic": "exact hc.fac _ _"
}
] |
[
151,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.toFun_eq_coe
|
[] |
[
102,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean
|
IsOpen.uniqueDiffOn
|
[] |
[
331,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/GroupTheory/Coset.lean
|
QuotientGroup.eq_class_eq_leftCoset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns✝ : Subgroup α\na b : α\ns : Subgroup α\ng z : α\n⊢ z ∈ {x | ↑x = ↑g} ↔ z ∈ g *l ↑s",
"tactic": "rw [mem_leftCoset_iff, Set.mem_setOf_eq, eq_comm, QuotientGroup.eq, SetLike.mem_coe]"
}
] |
[
558,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Surjective.injective_comp_right
|
[] |
[
227,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Algebra/GCDMonoid/Multiset.lean
|
Multiset.extract_gcd'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns t : Multiset α\nhs : ∃ x, x ∈ s ∧ x ≠ 0\nht : s = map ((fun x x_1 => x * x_1) (gcd s)) t\n⊢ gcd s * gcd t = gcd s",
"tactic": "conv_lhs => rw [← normalize_gcd, ← gcd_map_mul, ← ht]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns t : Multiset α\nht : s = map ((fun x x_1 => x * x_1) (gcd s)) t\nhs : gcd s = 0\n⊢ ∀ (x : α), x ∈ s → x = 0",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns t : Multiset α\nhs : ∃ x, x ∈ s ∧ x ≠ 0\nht : s = map ((fun x x_1 => x * x_1) (gcd s)) t\n⊢ ¬gcd s = 0",
"tactic": "contrapose! hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns t : Multiset α\nht : s = map ((fun x x_1 => x * x_1) (gcd s)) t\nhs : gcd s = 0\n⊢ ∀ (x : α), x ∈ s → x = 0",
"tactic": "exact s.gcd_eq_zero_iff.1 hs"
}
] |
[
234,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.card_boundaries_eq_succ_length
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nc : Composition n\n⊢ Finset.card (boundaries c) = length c + 1",
"tactic": "simp [boundaries]"
}
] |
[
281,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/MeasureTheory/Integral/Layercake.lean
|
Measure.countable_meas_le_ne_meas_lt
|
[] |
[
302,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Algebra/Lie/UniversalEnveloping.lean
|
UniversalEnvelopingAlgebra.hom_ext
|
[
{
"state_after": "case h\nR : Type u₁\nL : Type u₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nA : Type u₃\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : L →ₗ⁅R⁆ A\ng₁ g₂ : UniversalEnvelopingAlgebra R L →ₐ[R] A\nh : LieHom.comp (AlgHom.toLieHom g₁) (ι R) = LieHom.comp (AlgHom.toLieHom g₂) (ι R)\nx✝ : L\n⊢ ↑(↑(lift R).symm g₁) x✝ = ↑(↑(lift R).symm g₂) x✝",
"state_before": "R : Type u₁\nL : Type u₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nA : Type u₃\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : L →ₗ⁅R⁆ A\ng₁ g₂ : UniversalEnvelopingAlgebra R L →ₐ[R] A\nh : LieHom.comp (AlgHom.toLieHom g₁) (ι R) = LieHom.comp (AlgHom.toLieHom g₂) (ι R)\n⊢ ↑(lift R).symm g₁ = ↑(lift R).symm g₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u₁\nL : Type u₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nA : Type u₃\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : L →ₗ⁅R⁆ A\ng₁ g₂ : UniversalEnvelopingAlgebra R L →ₐ[R] A\nh : LieHom.comp (AlgHom.toLieHom g₁) (ι R) = LieHom.comp (AlgHom.toLieHom g₂) (ι R)\nx✝ : L\n⊢ ↑(↑(lift R).symm g₁) x✝ = ↑(↑(lift R).symm g₂) x✝",
"tactic": "simp [h]"
}
] |
[
172,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Data/List/Lattice.lean
|
List.forall_mem_inter_of_forall_left
|
[] |
[
188,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_eq_self_of_subset_right
|
[] |
[
981,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
980,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.closure_empty
|
[] |
[
1042,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Computability/Partrec.lean
|
Nat.Partrec.of_eq_tot
|
[] |
[
177,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.coe_monoidHom_injective
|
[] |
[
207,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.map_aeval
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝² : Algebra R S₁\ninst✝¹ : CommSemiring S₂\nf : σ → S₁\nB : Type u_1\ninst✝ : CommSemiring B\ng : σ → S₁\nφ : S₁ →+* B\np : MvPolynomial σ R\n⊢ ↑φ (↑(aeval g) p) = ↑(RingHom.comp φ (eval₂Hom (algebraMap R S₁) fun i => g i)) p",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝² : Algebra R S₁\ninst✝¹ : CommSemiring S₂\nf : σ → S₁\nB : Type u_1\ninst✝ : CommSemiring B\ng : σ → S₁\nφ : S₁ →+* B\np : MvPolynomial σ R\n⊢ ↑φ (↑(aeval g) p) = ↑(eval₂Hom (RingHom.comp φ (algebraMap R S₁)) fun i => ↑φ (g i)) p",
"tactic": "rw [← comp_eval₂Hom]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring S₁\np✝ q : MvPolynomial σ R\ninst✝² : Algebra R S₁\ninst✝¹ : CommSemiring S₂\nf : σ → S₁\nB : Type u_1\ninst✝ : CommSemiring B\ng : σ → S₁\nφ : S₁ →+* B\np : MvPolynomial σ R\n⊢ ↑φ (↑(aeval g) p) = ↑(RingHom.comp φ (eval₂Hom (algebraMap R S₁) fun i => g i)) p",
"tactic": "rfl"
}
] |
[
1476,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1473,
1
] |
Std/Data/Int/DivMod.lean
|
Int.eq_mul_of_div_eq_left
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\nH1 : b ∣ a\nH2 : div a b = c\n⊢ a = c * b",
"tactic": "rw [Int.mul_comm, Int.eq_mul_of_div_eq_right H1 H2]"
}
] |
[
773,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
771,
11
] |
Mathlib/Data/List/Count.lean
|
List.count_join
|
[] |
[
237,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.IsInvertedBy.of_comp
|
[
{
"state_after": "C : Type u\ninst✝⁴ : Category C\nD : Type ?u.46661\ninst✝³ : Category D\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝² : Category C₁\ninst✝¹ : Category C₂\ninst✝ : Category C₃\nW : MorphismProperty C₁\nF : C₁ ⥤ C₂\nhF : IsInvertedBy W F\nG : C₂ ⥤ C₃\nX Y : C₁\nf : X ⟶ Y\nhf : W f\nthis : IsIso (F.map f)\n⊢ IsIso ((F ⋙ G).map f)",
"state_before": "C : Type u\ninst✝⁴ : Category C\nD : Type ?u.46661\ninst✝³ : Category D\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝² : Category C₁\ninst✝¹ : Category C₂\ninst✝ : Category C₃\nW : MorphismProperty C₁\nF : C₁ ⥤ C₂\nhF : IsInvertedBy W F\nG : C₂ ⥤ C₃\nX Y : C₁\nf : X ⟶ Y\nhf : W f\n⊢ IsIso ((F ⋙ G).map f)",
"tactic": "haveI := hF f hf"
},
{
"state_after": "C : Type u\ninst✝⁴ : Category C\nD : Type ?u.46661\ninst✝³ : Category D\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝² : Category C₁\ninst✝¹ : Category C₂\ninst✝ : Category C₃\nW : MorphismProperty C₁\nF : C₁ ⥤ C₂\nhF : IsInvertedBy W F\nG : C₂ ⥤ C₃\nX Y : C₁\nf : X ⟶ Y\nhf : W f\nthis : IsIso (F.map f)\n⊢ IsIso (G.map (F.map f))",
"state_before": "C : Type u\ninst✝⁴ : Category C\nD : Type ?u.46661\ninst✝³ : Category D\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝² : Category C₁\ninst✝¹ : Category C₂\ninst✝ : Category C₃\nW : MorphismProperty C₁\nF : C₁ ⥤ C₂\nhF : IsInvertedBy W F\nG : C₂ ⥤ C₃\nX Y : C₁\nf : X ⟶ Y\nhf : W f\nthis : IsIso (F.map f)\n⊢ IsIso ((F ⋙ G).map f)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\nD : Type ?u.46661\ninst✝³ : Category D\nC₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝² : Category C₁\ninst✝¹ : Category C₂\ninst✝ : Category C₃\nW : MorphismProperty C₁\nF : C₁ ⥤ C₂\nhF : IsInvertedBy W F\nG : C₂ ⥤ C₃\nX Y : C₁\nf : X ⟶ Y\nhf : W f\nthis : IsIso (F.map f)\n⊢ IsIso (G.map (F.map f))",
"tactic": "infer_instance"
}
] |
[
307,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean
|
parallelepiped_eq_convexHull
|
[
{
"state_after": "ι : Type u_2\nι' : Type ?u.96374\nE : Type u_1\nF : Type ?u.96380\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : AddCommGroup F\ninst✝ : Module ℝ F\nv : ι → E\nA : Set E →+ Set E :=\n { toZeroHom := { toFun := ↑(convexHull ℝ).toOrderHom, map_zero' := (_ : ↑(convexHull ℝ).toOrderHom {0} = {0}) },\n map_add' :=\n (_ :\n ∀ (s t : Set E),\n ↑(convexHull ℝ).toOrderHom (s + t) = ↑(convexHull ℝ).toOrderHom s + ↑(convexHull ℝ).toOrderHom t) }\n⊢ parallelepiped v = ↑(convexHull ℝ).toOrderHom (∑ i : ι, {0, v i})",
"state_before": "ι : Type u_2\nι' : Type ?u.96374\nE : Type u_1\nF : Type ?u.96380\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : AddCommGroup F\ninst✝ : Module ℝ F\nv : ι → E\n⊢ parallelepiped v = ↑(convexHull ℝ).toOrderHom (∑ i : ι, {0, v i})",
"tactic": "let A : Set E →+ Set E :=\n { toFun := convexHull ℝ\n map_zero' := convexHull_singleton _\n map_add' := convexHull_add }"
},
{
"state_after": "ι : Type u_2\nι' : Type ?u.96374\nE : Type u_1\nF : Type ?u.96380\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : AddCommGroup F\ninst✝ : Module ℝ F\nv : ι → E\nA : Set E →+ Set E :=\n { toZeroHom := { toFun := ↑(convexHull ℝ).toOrderHom, map_zero' := (_ : ↑(convexHull ℝ).toOrderHom {0} = {0}) },\n map_add' :=\n (_ :\n ∀ (s t : Set E),\n ↑(convexHull ℝ).toOrderHom (s + t) = ↑(convexHull ℝ).toOrderHom s + ↑(convexHull ℝ).toOrderHom t) }\n⊢ ∑ x : ι, ↑(convexHull ℝ).toOrderHom {0, v x} = ↑(convexHull ℝ).toOrderHom (∑ i : ι, {0, v i})",
"state_before": "ι : Type u_2\nι' : Type ?u.96374\nE : Type u_1\nF : Type ?u.96380\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : AddCommGroup F\ninst✝ : Module ℝ F\nv : ι → E\nA : Set E →+ Set E :=\n { toZeroHom := { toFun := ↑(convexHull ℝ).toOrderHom, map_zero' := (_ : ↑(convexHull ℝ).toOrderHom {0} = {0}) },\n map_add' :=\n (_ :\n ∀ (s t : Set E),\n ↑(convexHull ℝ).toOrderHom (s + t) = ↑(convexHull ℝ).toOrderHom s + ↑(convexHull ℝ).toOrderHom t) }\n⊢ parallelepiped v = ↑(convexHull ℝ).toOrderHom (∑ i : ι, {0, v i})",
"tactic": "simp_rw [parallelepiped_eq_sum_segment, ← convexHull_pair]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.96374\nE : Type u_1\nF : Type ?u.96380\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : AddCommGroup F\ninst✝ : Module ℝ F\nv : ι → E\nA : Set E →+ Set E :=\n { toZeroHom := { toFun := ↑(convexHull ℝ).toOrderHom, map_zero' := (_ : ↑(convexHull ℝ).toOrderHom {0} = {0}) },\n map_add' :=\n (_ :\n ∀ (s t : Set E),\n ↑(convexHull ℝ).toOrderHom (s + t) = ↑(convexHull ℝ).toOrderHom s + ↑(convexHull ℝ).toOrderHom t) }\n⊢ ∑ x : ι, ↑(convexHull ℝ).toOrderHom {0, v x} = ↑(convexHull ℝ).toOrderHom (∑ i : ι, {0, v i})",
"tactic": "exact (A.map_sum _ _).symm"
}
] |
[
150,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Infinite.mono
|
[] |
[
1308,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1307,
11
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.algebraMap_matrix_apply
|
[
{
"state_after": "l : Type ?u.519207\nm : Type ?u.519210\nn : Type u_1\no : Type ?u.519216\nm' : o → Type ?u.519221\nn' : o → Type ?u.519226\nR : Type u_2\nS : Type ?u.519232\nα : Type v\nβ : Type w\nγ : Type ?u.519239\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\ni j : n\n⊢ ↑Algebra.toRingHom r * OfNat.ofNat 1 i j = if i = j then ↑Algebra.toRingHom r else 0",
"state_before": "l : Type ?u.519207\nm : Type ?u.519210\nn : Type u_1\no : Type ?u.519216\nm' : o → Type ?u.519221\nn' : o → Type ?u.519226\nR : Type u_2\nS : Type ?u.519232\nα : Type v\nβ : Type w\nγ : Type ?u.519239\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\ni j : n\n⊢ ↑(algebraMap R (Matrix n n α)) r i j = if i = j then ↑(algebraMap R α) r else 0",
"tactic": "dsimp [algebraMap, Algebra.toRingHom, Matrix.scalar]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.519207\nm : Type ?u.519210\nn : Type u_1\no : Type ?u.519216\nm' : o → Type ?u.519221\nn' : o → Type ?u.519226\nR : Type u_2\nS : Type ?u.519232\nα : Type v\nβ : Type w\nγ : Type ?u.519239\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\ni j : n\n⊢ ↑Algebra.toRingHom r * OfNat.ofNat 1 i j = if i = j then ↑Algebra.toRingHom r else 0",
"tactic": "split_ifs with h <;> simp [h, Matrix.one_apply_ne]"
}
] |
[
1315,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1312,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.zero_bot
|
[] |
[
498,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.smul_singleton
|
[] |
[
153,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Topology/Order.lean
|
continuous_le_rng
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.37963\nf : α → β\nι : Sort ?u.37970\nt₁ : TopologicalSpace α\nt₂ t₃ : TopologicalSpace β\nh₁ : t₂ ≤ t₃\nh₂ : coinduced f t₁ ≤ t₂\n⊢ coinduced f t₁ ≤ t₃",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.37963\nf : α → β\nι : Sort ?u.37970\nt₁ : TopologicalSpace α\nt₂ t₃ : TopologicalSpace β\nh₁ : t₂ ≤ t₃\nh₂ : Continuous f\n⊢ Continuous f",
"tactic": "rw [continuous_iff_coinduced_le] at h₂ ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.37963\nf : α → β\nι : Sort ?u.37970\nt₁ : TopologicalSpace α\nt₂ t₃ : TopologicalSpace β\nh₁ : t₂ ≤ t₃\nh₂ : coinduced f t₁ ≤ t₂\n⊢ coinduced f t₁ ≤ t₃",
"tactic": "exact le_trans h₂ h₁"
}
] |
[
733,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.coe_copy
|
[] |
[
139,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/RingTheory/IsTensorProduct.lean
|
IsTensorProduct.inductionOn
|
[
{
"state_after": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C (AddHom.toFun (↑(equiv h)).toAddHom (LinearEquiv.invFun (equiv h) m))",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C m",
"tactic": "rw [← h.equiv.right_inv m]"
},
{
"state_after": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C (AddHom.toFun (↑(equiv h)).toAddHom y)",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C (AddHom.toFun (↑(equiv h)).toAddHom (LinearEquiv.invFun (equiv h) m))",
"tactic": "generalize h.equiv.invFun m = y"
},
{
"state_after": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C (↑(TensorProduct.lift f) y)",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C (AddHom.toFun (↑(equiv h)).toAddHom y)",
"tactic": "change C (TensorProduct.lift f y)"
},
{
"state_after": "case C0\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C (↑(TensorProduct.lift f) 0)\n\ncase C1\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ : M₁\ny✝ : M₂\n⊢ C (↑(TensorProduct.lift f) (x✝ ⊗ₜ[R] y✝))\n\ncase Cp\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ y✝ : M₁ ⊗[R] M₂\na✝¹ : C (↑(TensorProduct.lift f) x✝)\na✝ : C (↑(TensorProduct.lift f) y✝)\n⊢ C (↑(TensorProduct.lift f) (x✝ + y✝))",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\ny : M₁ ⊗[R] M₂\n⊢ C (↑(TensorProduct.lift f) y)",
"tactic": "induction y using TensorProduct.induction_on"
},
{
"state_after": "no goals",
"state_before": "case C0\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\n⊢ C (↑(TensorProduct.lift f) 0)",
"tactic": "rwa [map_zero]"
},
{
"state_after": "case C1\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ : M₁\ny✝ : M₂\n⊢ C (↑(↑f x✝) y✝)",
"state_before": "case C1\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ : M₁\ny✝ : M₂\n⊢ C (↑(TensorProduct.lift f) (x✝ ⊗ₜ[R] y✝))",
"tactic": "rw [TensorProduct.lift.tmul]"
},
{
"state_after": "no goals",
"state_before": "case C1\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ : M₁\ny✝ : M₂\n⊢ C (↑(↑f x✝) y✝)",
"tactic": "apply htmul"
},
{
"state_after": "case Cp\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ y✝ : M₁ ⊗[R] M₂\na✝¹ : C (↑(TensorProduct.lift f) x✝)\na✝ : C (↑(TensorProduct.lift f) y✝)\n⊢ C (↑(TensorProduct.lift f) x✝ + ↑(TensorProduct.lift f) y✝)",
"state_before": "case Cp\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ y✝ : M₁ ⊗[R] M₂\na✝¹ : C (↑(TensorProduct.lift f) x✝)\na✝ : C (↑(TensorProduct.lift f) y✝)\n⊢ C (↑(TensorProduct.lift f) (x✝ + y✝))",
"tactic": "rw [map_add]"
},
{
"state_after": "no goals",
"state_before": "case Cp\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.113149\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.115388\nN₂ : Type ?u.115391\nN : Type ?u.115394\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nC : M → Prop\nm : M\nh0 : C 0\nhtmul : ∀ (x : M₁) (y : M₂), C (↑(↑f x) y)\nhadd : ∀ (x y : M), C x → C y → C (x + y)\nx✝ y✝ : M₁ ⊗[R] M₂\na✝¹ : C (↑(TensorProduct.lift f) x✝)\na✝ : C (↑(TensorProduct.lift f) y✝)\n⊢ C (↑(TensorProduct.lift f) x✝ + ↑(TensorProduct.lift f) y✝)",
"tactic": "apply hadd <;> assumption"
}
] |
[
135,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
iUnion_Ioc_add_int_cast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedRing α\ninst✝ : Archimedean α\na : α\n⊢ (⋃ (n : ℤ), Ioc (a + ↑n) (a + ↑n + 1)) = univ",
"tactic": "simpa only [zsmul_one, Int.cast_add, Int.cast_one, ← add_assoc] using\n iUnion_Ioc_add_zsmul zero_lt_one a"
}
] |
[
1094,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1092,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
Real.ediam_eq
|
[
{
"state_after": "case inl\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\nh : Metric.Bounded ∅\n⊢ EMetric.diam ∅ = ENNReal.ofReal (sSup ∅ - sInf ∅)\n\ncase inr\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ EMetric.diam s = ENNReal.ofReal (sSup s - sInf s)",
"state_before": "α : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\n⊢ EMetric.diam s = ENNReal.ofReal (sSup s - sInf s)",
"tactic": "rcases eq_empty_or_nonempty s with (rfl | hne)"
},
{
"state_after": "case inr.refine'_1\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\nx : ℝ\nhx : x ∈ s\ny : ℝ\nhy : y ∈ s\n⊢ dist x y ≤ sSup s - sInf s\n\ncase inr.refine'_2\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ ENNReal.ofReal (sSup s - sInf s) ≤ EMetric.diam s",
"state_before": "case inr\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ EMetric.diam s = ENNReal.ofReal (sSup s - sInf s)",
"tactic": "refine' le_antisymm (Metric.ediam_le_of_forall_dist_le fun x hx y hy => _) _"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\nh : Metric.Bounded ∅\n⊢ EMetric.diam ∅ = ENNReal.ofReal (sSup ∅ - sInf ∅)",
"tactic": "simp"
},
{
"state_after": "case inr.refine'_1\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\nx : ℝ\nhx : x ∈ s\ny : ℝ\nhy : y ∈ s\nthis : s ⊆ Icc (sInf s) (sSup s)\n⊢ dist x y ≤ sSup s - sInf s",
"state_before": "case inr.refine'_1\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\nx : ℝ\nhx : x ∈ s\ny : ℝ\nhy : y ∈ s\n⊢ dist x y ≤ sSup s - sInf s",
"tactic": "have := Real.subset_Icc_sInf_sSup_of_bounded h"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_1\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\nx : ℝ\nhx : x ∈ s\ny : ℝ\nhy : y ∈ s\nthis : s ⊆ Icc (sInf s) (sSup s)\n⊢ dist x y ≤ sSup s - sInf s",
"tactic": "exact Real.dist_le_of_mem_Icc (this hx) (this hy)"
},
{
"state_after": "case inr.refine'_2.h\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ sSup s - sInf s ≤ ENNReal.toReal (EMetric.diam s)",
"state_before": "case inr.refine'_2\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ ENNReal.ofReal (sSup s - sInf s) ≤ EMetric.diam s",
"tactic": "apply ENNReal.ofReal_le_of_le_toReal"
},
{
"state_after": "case inr.refine'_2.h\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ sSup s - sInf s ≤ Metric.diam (closure s)",
"state_before": "case inr.refine'_2.h\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ sSup s - sInf s ≤ ENNReal.toReal (EMetric.diam s)",
"tactic": "rw [← Metric.diam, ← Metric.diam_closure]"
},
{
"state_after": "case inr.refine'_2.h\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\nh' : BddBelow s ∧ BddAbove s\n⊢ sSup s - sInf s ≤ Metric.diam (closure s)",
"state_before": "case inr.refine'_2.h\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\n⊢ sSup s - sInf s ≤ Metric.diam (closure s)",
"tactic": "have h' := Real.bounded_iff_bddBelow_bddAbove.1 h"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_2.h\nα : Type ?u.434159\nβ : Type ?u.434162\nγ : Type ?u.434165\ninst✝ : PseudoEMetricSpace α\ns : Set ℝ\nh : Metric.Bounded s\nhne : Set.Nonempty s\nh' : BddBelow s ∧ BddAbove s\n⊢ sSup s - sInf s ≤ Metric.diam (closure s)",
"tactic": "calc sSup s - sInf s ≤ dist (sSup s) (sInf s) := le_abs_self _\n_ ≤ Metric.diam (closure s) := dist_le_diam_of_mem h.closure (csSup_mem_closure hne h'.2)\n (csInf_mem_closure hne h'.1)"
}
] |
[
1519,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1507,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
EReal.measurable_of_measurable_real
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3039973\nγ : Type ?u.3039976\nγ₂ : Type ?u.3039979\nδ : Type ?u.3039982\nι : Sort y\ns t u : Set α\ninst✝ : MeasurableSpace α\nf : EReal → α\nh : Measurable fun p => f ↑p\n⊢ Set.Finite {⊥, ⊤}",
"tactic": "simp"
}
] |
[
1990,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1987,
1
] |
Mathlib/ModelTheory/Encoding.lean
|
FirstOrder.Language.BoundedFormula.card_le
|
[
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (#List BoundedFormula.encoding.Γ) ≤ lift (max ℵ₀ (lift (#α) + lift (card L)))",
"state_before": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ (#(n : ℕ) × BoundedFormula L α n) ≤ max ℵ₀ (lift (#α) + lift (card L))",
"tactic": "refine' lift_le.1 (BoundedFormula.encoding.card_le_card_list.trans _)"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (#(k : ℕ) × Term L (α ⊕ Fin k) ⊕ (n : ℕ) × Relations L n ⊕ ℕ) ≤ max ℵ₀ (lift (lift (#α) + lift (card L))) ∧\n ℵ₀ ≤ max ℵ₀ (lift (lift (#α) + lift (card L)))",
"state_before": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (#List BoundedFormula.encoding.Γ) ≤ lift (max ℵ₀ (lift (#α) + lift (card L)))",
"tactic": "rw [encoding_Γ, mk_list_eq_max_mk_aleph0, lift_max, lift_aleph0, lift_max, lift_aleph0,\n max_le_iff]"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (#(k : ℕ) × Term L (α ⊕ Fin k) ⊕ (n : ℕ) × Relations L n ⊕ ℕ) ≤ max ℵ₀ (lift (lift (#α) + lift (card L)))",
"state_before": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (#(k : ℕ) × Term L (α ⊕ Fin k) ⊕ (n : ℕ) × Relations L n ⊕ ℕ) ≤ max ℵ₀ (lift (lift (#α) + lift (card L))) ∧\n ℵ₀ ≤ max ℵ₀ (lift (lift (#α) + lift (card L)))",
"tactic": "refine' ⟨_, le_max_left _ _⟩"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (lift (ℵ₀ + (lift (#α) + lift (#(i : ℕ) × Functions L i))) + lift (lift (#(n : ℕ) × Relations L n) + lift ℵ₀)) ≤\n max ℵ₀ (lift (lift (#α) + lift (card L)))",
"state_before": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (#(k : ℕ) × Term L (α ⊕ Fin k) ⊕ (n : ℕ) × Relations L n ⊕ ℕ) ≤ max ℵ₀ (lift (lift (#α) + lift (card L)))",
"tactic": "rw [mk_sum, Term.card_sigma, mk_sum, ← add_eq_max le_rfl, mk_sum, mk_nat]"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ ℵ₀ + (lift (#α) + lift (#(i : ℕ) × Functions L i)) + (lift (#(n : ℕ) × Relations L n) + ℵ₀) ≤\n max ℵ₀ (lift (#α) + lift (card L))",
"state_before": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ lift (lift (ℵ₀ + (lift (#α) + lift (#(i : ℕ) × Functions L i))) + lift (lift (#(n : ℕ) × Relations L n) + lift ℵ₀)) ≤\n max ℵ₀ (lift (lift (#α) + lift (card L)))",
"tactic": "simp only [lift_add, lift_lift, lift_aleph0]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\nN : Type ?u.158067\nP : Type ?u.158070\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\n⊢ ℵ₀ + (lift (#α) + lift (#(i : ℕ) × Functions L i)) + (lift (#(n : ℕ) × Relations L n) + ℵ₀) ≤\n max ℵ₀ (lift (#α) + lift (card L))",
"tactic": "rw [← add_assoc, add_comm, ← add_assoc, ← add_assoc, aleph0_add_aleph0, add_assoc,\n add_eq_max le_rfl, add_assoc, card, Symbols, mk_sum, lift_add, lift_lift, lift_lift]"
}
] |
[
323,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.rowLens_length_ofRowLens
|
[
{
"state_after": "w : List ℕ\nhw : List.Sorted (fun x x_1 => x ≥ x_1) w\nhpos : ∀ (x : ℕ), x ∈ w → 0 < x\n⊢ True ∧ ∀ (n : ℕ), n < List.length w → ∃ h, 0 < List.get w { val := n, isLt := (_ : (n, 0).fst < List.length w) }",
"state_before": "w : List ℕ\nhw : List.Sorted (fun x x_1 => x ≥ x_1) w\nhpos : ∀ (x : ℕ), x ∈ w → 0 < x\n⊢ List.length (rowLens (ofRowLens w hw)) = List.length w",
"tactic": "simp only [length_rowLens, colLen, Nat.find_eq_iff, mem_cells, mem_ofRowLens,\n lt_self_iff_false, IsEmpty.exists_iff, Classical.not_not]"
},
{
"state_after": "no goals",
"state_before": "w : List ℕ\nhw : List.Sorted (fun x x_1 => x ≥ x_1) w\nhpos : ∀ (x : ℕ), x ∈ w → 0 < x\n⊢ True ∧ ∀ (n : ℕ), n < List.length w → ∃ h, 0 < List.get w { val := n, isLt := (_ : (n, 0).fst < List.length w) }",
"tactic": "refine' ⟨True.intro, fun n hn => ⟨hn, hpos _ (List.get_mem _ _ hn)⟩⟩"
}
] |
[
505,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
501,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.restrict_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.474375\nm inst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\nv : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(restrict v Set.univ) i = ↑v i",
"tactic": "rw [restrict_apply v MeasurableSet.univ hi, Set.inter_univ]"
}
] |
[
718,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
717,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_inj
|
[
{
"state_after": "no goals",
"state_before": "a b : Int\nh : -a = -b\n⊢ a = b",
"tactic": "rw [← Int.neg_neg a, ← Int.neg_neg b, h]"
}
] |
[
88,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
87,
11
] |
Mathlib/Dynamics/Ergodic/Conservative.lean
|
MeasureTheory.Conservative.exists_gt_measure_inter_ne_zero
|
[] |
[
116,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.of_rat_div
|
[] |
[
411,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Topology/MetricSpace/Equicontinuity.lean
|
Metric.equicontinuous_of_continuity_modulus
|
[] |
[
125,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.Tendsto.atTop_div_const
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.221463\nι' : Type ?u.221466\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.221475\ninst✝ : LinearOrderedSemifield α\nl : Filter β\nf : β → α\nr c : α\nn : ℕ\nhr : 0 < r\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x => f x / r) l atTop",
"tactic": "simpa only [div_eq_mul_inv] using hf.atTop_mul_const (inv_pos.2 hr)"
}
] |
[
1052,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aevalTower_algebraMap
|
[] |
[
416,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
fderiv_mul'
|
[] |
[
379,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.mk_cardinal
|
[
{
"state_after": "no goals",
"state_before": "⊢ (#Cardinal) = univ",
"tactic": "simpa only [card_type, card_univ] using congr_arg card type_cardinal"
}
] |
[
151,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Order/SuccPred/IntervalSucc.lean
|
Monotone.biUnion_Ico_Ioc_map_succ
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhnm : n ≤ m\n⊢ (⋃ (i : α) (_ : i ∈ Ico m n), Ioc (f i) (f (succ i))) = Ioc (f m) (f n)\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\n⊢ (⋃ (i : α) (_ : i ∈ Ico m n), Ioc (f i) (f (succ i))) = Ioc (f m) (f n)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\n⊢ (⋃ (i : α) (_ : i ∈ Ico m n), Ioc (f i) (f (succ i))) = Ioc (f m) (f n)",
"tactic": "cases' le_total n m with hnm hmn"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhnm : n ≤ m\n⊢ (⋃ (i : α) (_ : i ∈ Ico m n), Ioc (f i) (f (succ i))) = Ioc (f m) (f n)",
"tactic": "rw [Ico_eq_empty_of_le hnm, Ioc_eq_empty_of_le (hf hnm), biUnion_empty]"
},
{
"state_after": "case inr.refine'_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\n⊢ (⋃ (i : α) (_ : i ∈ Ico m m), Ioc (f i) (f (succ i))) = Ioc (f m) (f m)\n\ncase inr.refine'_2\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\n⊢ ∀ (n : α),\n m ≤ n →\n (⋃ (i : α) (_ : i ∈ Ico m n), Ioc (f i) (f (succ i))) = Ioc (f m) (f n) →\n (⋃ (i : α) (_ : i ∈ Ico m (succ n)), Ioc (f i) (f (succ i))) = Ioc (f m) (f (succ n))",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\n⊢ (⋃ (i : α) (_ : i ∈ Ico m n), Ioc (f i) (f (succ i))) = Ioc (f m) (f n)",
"tactic": "refine' Succ.rec _ _ hmn"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_1\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\n⊢ (⋃ (i : α) (_ : i ∈ Ico m m), Ioc (f i) (f (succ i))) = Ioc (f m) (f m)",
"tactic": "simp only [Ioc_self, Ico_self, biUnion_empty]"
},
{
"state_after": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) = Ioc (f m) (f (succ k))",
"state_before": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\n⊢ ∀ (n : α),\n m ≤ n →\n (⋃ (i : α) (_ : i ∈ Ico m n), Ioc (f i) (f (succ i))) = Ioc (f m) (f n) →\n (⋃ (i : α) (_ : i ∈ Ico m (succ n)), Ioc (f i) (f (succ i))) = Ioc (f m) (f (succ n))",
"tactic": "intro k hmk ihk"
},
{
"state_after": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) =\n Ioc (f k) (f (succ k)) ∪ ⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))",
"state_before": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) = Ioc (f m) (f (succ k))",
"tactic": "rw [← Ioc_union_Ioc_eq_Ioc (hf hmk) (hf <| le_succ _), union_comm, ← ihk]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\nhk : IsMax k\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) =\n Ioc (f k) (f (succ k)) ∪ ⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))\n\ncase neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\nhk : ¬IsMax k\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) =\n Ioc (f k) (f (succ k)) ∪ ⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))",
"state_before": "case inr.refine'_2\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) =\n Ioc (f k) (f (succ k)) ∪ ⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))",
"tactic": "by_cases hk : IsMax k"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\nhk : IsMax k\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) =\n Ioc (f k) (f (succ k)) ∪ ⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))",
"tactic": "rw [hk.succ_eq, Ioc_self, empty_union]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : (⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))) = Ioc (f m) (f k)\nhk : ¬IsMax k\n⊢ (⋃ (i : α) (_ : i ∈ Ico m (succ k)), Ioc (f i) (f (succ i))) =\n Ioc (f k) (f (succ k)) ∪ ⋃ (i : α) (_ : i ∈ Ico m k), Ioc (f i) (f (succ i))",
"tactic": "rw [Ico_succ_right_eq_insert_of_not_isMax hmk hk, biUnion_insert]"
}
] |
[
50,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Data/Multiset/NatAntidiagonal.lean
|
Multiset.Nat.antidiagonal_zero
|
[] |
[
52,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/RingTheory/Localization/AtPrime.lean
|
IsLocalization.AtPrime.Nontrivial
|
[
{
"state_after": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : ↑(algebraMap R S) 0 = ↑(algebraMap R S) 1\n⊢ False",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : 0 = 1\n⊢ False",
"tactic": "rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : ↑(algebraMap R S) 0 = ↑(algebraMap R S) 1\nt : { x // x ∈ Ideal.primeCompl I }\nht : ↑t * 0 = ↑t * 1\n⊢ False",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : ↑(algebraMap R S) 0 = ↑(algebraMap R S) 1\n⊢ False",
"tactic": "obtain ⟨t, ht⟩ := (eq_iff_exists I.primeCompl S).1 hze"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : ↑(algebraMap R S) 0 = ↑(algebraMap R S) 1\nt : { x // x ∈ Ideal.primeCompl I }\nht : ↑t * 0 = ↑t * 1\nhtz : ↑t = 0\n⊢ False",
"state_before": "case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : ↑(algebraMap R S) 0 = ↑(algebraMap R S) 1\nt : { x // x ∈ Ideal.primeCompl I }\nht : ↑t * 0 = ↑t * 1\n⊢ False",
"tactic": "have htz : (t : R) = 0 := by simpa using ht.symm"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : ↑(algebraMap R S) 0 = ↑(algebraMap R S) 1\nt : { x // x ∈ Ideal.primeCompl I }\nht : ↑t * 0 = ↑t * 1\nhtz : ↑t = 0\n⊢ False",
"tactic": "exact t.2 (htz.symm ▸ I.zero_mem : ↑t ∈ I)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.10429\ninst✝¹ : CommSemiring P\nI : Ideal R\nhp : Ideal.IsPrime I\ninst✝ : IsLocalization.AtPrime S I\nhze : ↑(algebraMap R S) 0 = ↑(algebraMap R S) 1\nt : { x // x ∈ Ideal.primeCompl I }\nht : ↑t * 0 = ↑t * 1\n⊢ ↑t = 0",
"tactic": "simpa using ht.symm"
}
] |
[
80,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.sub_tmul
|
[] |
[
1256,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1255,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.even_mul_self_pred
|
[] |
[
219,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.support_map_subset
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ x ∈ support (↑(map f) p) → x ∈ support p",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\n⊢ support (↑(map f) p) ⊆ support p",
"tactic": "intro x"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x (↑(map f) p) ≠ 0 → coeff x p ≠ 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ x ∈ support (↑(map f) p) → x ∈ support p",
"tactic": "simp only [mem_support_iff]"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x p = 0 → coeff x (↑(map f) p) = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x (↑(map f) p) ≠ 0 → coeff x p ≠ 0",
"tactic": "contrapose!"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x p = 0 → coeff x (↑(map f) p) = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x p = 0 → coeff x (↑(map f) p) = 0",
"tactic": "change p.coeff x = 0 → (map f p).coeff x = 0"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x p = 0 → ↑f (coeff x p) = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x p = 0 → coeff x (↑(map f) p) = 0",
"tactic": "rw [coeff_map]"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\nhx : coeff x p = 0\n⊢ ↑f (coeff x p) = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\n⊢ coeff x p = 0 → ↑f (coeff x p) = 0",
"tactic": "intro hx"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\nhx : coeff x p = 0\n⊢ ↑f 0 = 0",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\nhx : coeff x p = 0\n⊢ ↑f (coeff x p) = 0",
"tactic": "rw [hx]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\np : MvPolynomial σ R\nx : σ →₀ ℕ\nhx : coeff x p = 0\n⊢ ↑f 0 = 0",
"tactic": "exact RingHom.map_zero f"
}
] |
[
1356,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1348,
1
] |
Mathlib/Topology/Instances/Int.lean
|
Int.closedBall_eq_Icc
|
[
{
"state_after": "no goals",
"state_before": "x : ℤ\nr : ℝ\n⊢ closedBall x r = Icc ⌈↑x - r⌉ ⌊↑x + r⌋",
"tactic": "rw [← preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc]"
}
] |
[
67,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
mem_uniformity_edist
|
[] |
[
183,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_nonneg_of_nonpos
|
[] |
[
697,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
696,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
SameRay.exists_nonneg_left
|
[
{
"state_after": "case inl\nR : Type u_1\ninst✝² : LinearOrderedField R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx v₁ v₂ : M\nhx : x ≠ 0\nh : SameRay R x 0\n⊢ ∃ r, 0 ≤ r ∧ r • x = 0\n\ncase inr\nR : Type u_1\ninst✝² : LinearOrderedField R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y v₁ v₂ : M\nh : SameRay R x y\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ∃ r, 0 ≤ r ∧ r • x = y",
"state_before": "R : Type u_1\ninst✝² : LinearOrderedField R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y v₁ v₂ : M\nh : SameRay R x y\nhx : x ≠ 0\n⊢ ∃ r, 0 ≤ r ∧ r • x = y",
"tactic": "obtain rfl | hy := eq_or_ne y 0"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝² : LinearOrderedField R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx v₁ v₂ : M\nhx : x ≠ 0\nh : SameRay R x 0\n⊢ ∃ r, 0 ≤ r ∧ r • x = 0",
"tactic": "exact ⟨0, le_rfl, zero_smul _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u_1\ninst✝² : LinearOrderedField R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y v₁ v₂ : M\nh : SameRay R x y\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ∃ r, 0 ≤ r ∧ r • x = y",
"tactic": "exact (h.exists_pos_left hx hy).imp fun _ => And.imp_left le_of_lt"
}
] |
[
670,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Submodule.starConvex
|
[] |
[
604,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
11
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.ball_finset_sup'
|
[
{
"state_after": "case h₀\nR : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\ne : E\nr : ℝ\na : ι\n⊢ ball (Finset.sup' {a} (_ : Finset.Nonempty {a}) p) e r =\n Finset.inf' {a} (_ : Finset.Nonempty {a}) fun i => ball (p i) e r\n\ncase h₁\nR : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns✝ : Finset ι\ne : E\nr : ℝ\na : ι\ns : Finset ι\nha : ¬a ∈ s\nhs : Finset.Nonempty s\nih : ball (Finset.sup' s hs p) e r = Finset.inf' s hs fun i => ball (p i) e r\n⊢ ball (Finset.sup' (Finset.cons a s ha) (_ : Finset.Nonempty (Finset.cons a s ha)) p) e r =\n Finset.inf' (Finset.cons a s ha) (_ : Finset.Nonempty (Finset.cons a s ha)) fun i => ball (p i) e r",
"state_before": "R : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nH : Finset.Nonempty s\ne : E\nr : ℝ\n⊢ ball (Finset.sup' s H p) e r = Finset.inf' s H fun i => ball (p i) e r",
"tactic": "induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih"
},
{
"state_after": "no goals",
"state_before": "case h₀\nR : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\ne : E\nr : ℝ\na : ι\n⊢ ball (Finset.sup' {a} (_ : Finset.Nonempty {a}) p) e r =\n Finset.inf' {a} (_ : Finset.Nonempty {a}) fun i => ball (p i) e r",
"tactic": "classical simp"
},
{
"state_after": "no goals",
"state_before": "case h₀\nR : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\ne : E\nr : ℝ\na : ι\n⊢ ball (Finset.sup' {a} (_ : Finset.Nonempty {a}) p) e r =\n Finset.inf' {a} (_ : Finset.Nonempty {a}) fun i => ball (p i) e r",
"tactic": "simp"
},
{
"state_after": "case h₁\nR : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns✝ : Finset ι\ne : E\nr : ℝ\na : ι\ns : Finset ι\nha : ¬a ∈ s\nhs : Finset.Nonempty s\nih : ball (Finset.sup' s hs p) e r = Finset.inf' s hs fun i => ball (p i) e r\n⊢ ball (p a) e r ∩ ball (Finset.sup' s hs p) e r = ball (p a) e r ⊓ Finset.inf' s hs fun i => ball (p i) e r",
"state_before": "case h₁\nR : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns✝ : Finset ι\ne : E\nr : ℝ\na : ι\ns : Finset ι\nha : ¬a ∈ s\nhs : Finset.Nonempty s\nih : ball (Finset.sup' s hs p) e r = Finset.inf' s hs fun i => ball (p i) e r\n⊢ ball (Finset.sup' (Finset.cons a s ha) (_ : Finset.Nonempty (Finset.cons a s ha)) p) e r =\n Finset.inf' (Finset.cons a s ha) (_ : Finset.Nonempty (Finset.cons a s ha)) fun i => ball (p i) e r",
"tactic": "rw [Finset.sup'_cons hs, Finset.inf'_cons hs, ball_sup]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nR : Type ?u.865125\nR' : Type ?u.865128\n𝕜 : Type u_1\n𝕜₂ : Type ?u.865134\n𝕜₃ : Type ?u.865137\n𝕝 : Type ?u.865140\nE : Type u_2\nE₂ : Type ?u.865146\nE₃ : Type ?u.865149\nF : Type ?u.865152\nG : Type ?u.865155\nι : Type u_3\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y : E\nr✝ : ℝ\np : ι → Seminorm 𝕜 E\ns✝ : Finset ι\ne : E\nr : ℝ\na : ι\ns : Finset ι\nha : ¬a ∈ s\nhs : Finset.Nonempty s\nih : ball (Finset.sup' s hs p) e r = Finset.inf' s hs fun i => ball (p i) e r\n⊢ ball (p a) e r ∩ ball (Finset.sup' s hs p) e r = ball (p a) e r ⊓ Finset.inf' s hs fun i => ball (p i) e r",
"tactic": "simp only [inf_eq_inter, ih]"
}
] |
[
725,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
719,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_restrictScalars'
|
[] |
[
1685,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1684,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.setToFun_indicator_const
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1503019\nG : Type ?u.1503022\n𝕜 : Type ?u.1503025\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ setToFun μ T hT ↑↑(indicatorConstLp 1 hs hμs x) = ↑(T s) x",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1503019\nG : Type ?u.1503022\n𝕜 : Type ?u.1503025\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ setToFun μ T hT (indicator s fun x_1 => x) = ↑(T s) x",
"tactic": "rw [setToFun_congr_ae hT (@indicatorConstLp_coeFn _ _ _ 1 _ _ _ hs hμs x).symm]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1503019\nG : Type ?u.1503022\n𝕜 : Type ?u.1503025\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ ↑(L1.setToL1 hT) (indicatorConstLp 1 hs hμs x) = ↑(T s) x",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1503019\nG : Type ?u.1503022\n𝕜 : Type ?u.1503025\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ setToFun μ T hT ↑↑(indicatorConstLp 1 hs hμs x) = ↑(T s) x",
"tactic": "rw [L1.setToFun_eq_setToL1 hT]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1503019\nG : Type ?u.1503022\n𝕜 : Type ?u.1503025\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ ↑(L1.setToL1 hT) (indicatorConstLp 1 hs hμs x) = ↑(T s) x",
"tactic": "exact L1.setToL1_indicatorConstLp hT hs hμs x"
}
] |
[
1454,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1449,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.image2_eq_iUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.256231\nι' : Sort ?u.256234\nι₂ : Sort ?u.256237\nκ : ι → Sort ?u.256242\nκ₁ : ι → Sort ?u.256247\nκ₂ : ι → Sort ?u.256252\nκ' : ι' → Sort ?u.256257\nf : α → β → γ\ns✝ : Set α\nt✝ : Set β\ns : Set α\nt : Set β\n⊢ image2 f s t = ⋃ (i : α) (_ : i ∈ s) (j : β) (_ : j ∈ t), {f i j}",
"tactic": "simp_rw [← image_eq_iUnion, iUnion_image_left]"
}
] |
[
1927,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1926,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
cauchySeq_prod_of_eventually_eq
|
[
{
"state_after": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\n⊢ CauchySeq fun n => ∏ k in range (n + 1), u k",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\n⊢ CauchySeq fun n => ∏ k in range (n + 1), u k",
"tactic": "let d : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k"
},
{
"state_after": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\n⊢ CauchySeq (d * fun n => ∏ k in range (n + 1), v k)",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\n⊢ CauchySeq fun n => ∏ k in range (n + 1), u k",
"tactic": "rw [show (fun n => ∏ k in range (n + 1), u k) = d * fun n => ∏ k in range (n + 1), v k\n by ext n; simp]"
},
{
"state_after": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\n⊢ ∀ (n : ℕ), n ≥ N → d n = d N",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\n⊢ CauchySeq (d * fun n => ∏ k in range (n + 1), v k)",
"tactic": "suffices ∀ n ≥ N, d n = d N by exact (tendsto_atTop_of_eventually_const this).cauchySeq.mul hv"
},
{
"state_after": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\n⊢ d n = d N",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\n⊢ ∀ (n : ℕ), n ≥ N → d n = d N",
"tactic": "intro n hn"
},
{
"state_after": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\n⊢ ∏ k in range (n + 1), u k / v k = ∏ k in range (N + 1), u k / v k",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\n⊢ d n = d N",
"tactic": "dsimp"
},
{
"state_after": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\n⊢ ∀ (n : ℕ), n ≥ N → u n / v n = 1",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\n⊢ ∏ k in range (n + 1), u k / v k = ∏ k in range (N + 1), u k / v k",
"tactic": "rw [eventually_constant_prod _ hn]"
},
{
"state_after": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\nm : ℕ\nhm : m ≥ N\n⊢ u m / v m = 1",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\n⊢ ∀ (n : ℕ), n ≥ N → u n / v n = 1",
"tactic": "intro m hm"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\nm : ℕ\nhm : m ≥ N\n⊢ u m / v m = 1",
"tactic": "simp [huv m hm]"
},
{
"state_after": "case h\n𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\n⊢ ∏ k in range (n + 1), u k = (d * fun n => ∏ k in range (n + 1), v k) n",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\n⊢ (fun n => ∏ k in range (n + 1), u k) = d * fun n => ∏ k in range (n + 1), v k",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\n𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nn : ℕ\n⊢ ∏ k in range (n + 1), u k = (d * fun n => ∏ k in range (n + 1), v k) n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.1218103\n𝕜 : Type ?u.1218106\nα : Type ?u.1218109\nι : Type ?u.1218112\nκ : Type ?u.1218115\nE : Type u_1\nF : Type ?u.1218121\nG : Type ?u.1218124\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : ℕ → E\nN : ℕ\nhuv : ∀ (n : ℕ), n ≥ N → u n = v n\nhv : CauchySeq fun n => ∏ k in range (n + 1), v k\nd : ℕ → E := fun n => ∏ k in range (n + 1), u k / v k\nthis : ∀ (n : ℕ), n ≥ N → d n = d N\n⊢ CauchySeq (d * fun n => ∏ k in range (n + 1), v k)",
"tactic": "exact (tendsto_atTop_of_eventually_const this).cauchySeq.mul hv"
}
] |
[
1954,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1943,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_multiset_sum
|
[] |
[
320,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean
|
LieSubmodule.comap_map_eq
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nhf : LieModuleHom.ker f = ⊥\n⊢ ↑(comap f (map f N)) = ↑N",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nhf : LieModuleHom.ker f = ⊥\n⊢ comap f (map f N) = N",
"tactic": "rw [SetLike.ext'_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nf : M →ₗ⁅R,L⁆ M₂\nhf : LieModuleHom.ker f = ⊥\n⊢ ↑(comap f (map f N)) = ↑N",
"tactic": "exact (N : Set M).preimage_image_eq (f.ker_eq_bot.mp hf)"
}
] |
[
239,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.iInf_mul_left'
|
[
{
"state_after": "case pos\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : a = ⊤ ∧ (⨅ (i : ι), f i) = 0\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i\n\ncase neg\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬(a = ⊤ ∧ (⨅ (i : ι), f i) = 0)\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"state_before": "α : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"tactic": "by_cases H : a = ⊤ ∧ (⨅ i, f i) = 0"
},
{
"state_after": "case pos.intro\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : a = ⊤ ∧ (⨅ (i : ι), f i) = 0\ni : ι\nhi : f i = 0\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"state_before": "case pos\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : a = ⊤ ∧ (⨅ (i : ι), f i) = 0\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"tactic": "rcases h H.1 H.2 with ⟨i, hi⟩"
},
{
"state_after": "case pos.intro\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : a = ⊤ ∧ (⨅ (i : ι), f i) = 0\ni : ι\nhi : f i = 0\n⊢ ∀ (b : ℝ≥0∞), b > ⊥ → ∃ i, a * f i < b",
"state_before": "case pos.intro\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : a = ⊤ ∧ (⨅ (i : ι), f i) = 0\ni : ι\nhi : f i = 0\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"tactic": "rw [H.2, mul_zero, ← bot_eq_zero, iInf_eq_bot]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : a = ⊤ ∧ (⨅ (i : ι), f i) = 0\ni : ι\nhi : f i = 0\n⊢ ∀ (b : ℝ≥0∞), b > ⊥ → ∃ i, a * f i < b",
"tactic": "exact fun b hb => ⟨i, by rwa [hi, mul_zero, ← bot_eq_zero]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : a = ⊤ ∧ (⨅ (i : ι), f i) = 0\ni : ι\nhi : f i = 0\nb : ℝ≥0∞\nhb : b > ⊥\n⊢ a * f i < b",
"tactic": "rwa [hi, mul_zero, ← bot_eq_zero]"
},
{
"state_after": "case neg\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"state_before": "case neg\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬(a = ⊤ ∧ (⨅ (i : ι), f i) = 0)\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"tactic": "rw [not_and_or] at H"
},
{
"state_after": "case neg.inl\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\nh✝ : IsEmpty ι\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i\n\ncase neg.inr\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\nh✝ : Nonempty ι\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"state_before": "case neg\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"tactic": "cases isEmpty_or_nonempty ι"
},
{
"state_after": "case neg.inl\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\nh✝ : IsEmpty ι\n⊢ a ≠ 0",
"state_before": "case neg.inl\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\nh✝ : IsEmpty ι\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"tactic": "rw [iInf_of_empty, iInf_of_empty, mul_top]"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\nh✝ : IsEmpty ι\n⊢ a ≠ 0",
"tactic": "exact mt h0 (not_nonempty_iff.2 ‹_›)"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nα : Type ?u.115719\nβ : Type ?u.115722\nγ : Type ?u.115725\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nh : a = ⊤ → (⨅ (i : ι), f i) = 0 → ∃ i, f i = 0\nh0 : a = 0 → Nonempty ι\nH : ¬a = ⊤ ∨ ¬(⨅ (i : ι), f i) = 0\nh✝ : Nonempty ι\n⊢ (⨅ (i : ι), a * f i) = a * ⨅ (i : ι), f i",
"tactic": "exact (ENNReal.mul_left_mono.map_iInf_of_continuousAt'\n (ENNReal.continuousAt_const_mul H)).symm"
}
] |
[
497,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/GroupTheory/SemidirectProduct.lean
|
SemidirectProduct.rightHom_comp_inr
|
[
{
"state_after": "case h\nN : Type u_2\nG : Type u_1\nH : Type ?u.109697\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nx✝ : G\n⊢ ↑(MonoidHom.comp rightHom inr) x✝ = ↑(MonoidHom.id G) x✝",
"state_before": "N : Type u_2\nG : Type u_1\nH : Type ?u.109697\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\n⊢ MonoidHom.comp rightHom inr = MonoidHom.id G",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nN : Type u_2\nG : Type u_1\nH : Type ?u.109697\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nx✝ : G\n⊢ ↑(MonoidHom.comp rightHom inr) x✝ = ↑(MonoidHom.id G) x✝",
"tactic": "simp [rightHom]"
}
] |
[
194,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/List/Zip.lean
|
List.unzip_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type ?u.84917\nδ : Type ?u.84920\nε : Type ?u.84923\nl : List (α × β)\n⊢ (unzip l).fst = map Prod.fst l",
"tactic": "simp only [unzip_eq_map]"
}
] |
[
217,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
Setoid.isPartition_classes
|
[] |
[
220,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
tendsto_iff_of_dist
|
[] |
[
1479,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1477,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
finrank_euclideanSpace
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.75574\n𝕜 : Type u_1\ninst✝⁹ : IsROrC 𝕜\nE : Type ?u.75583\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace 𝕜 E\nE' : Type ?u.75603\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : InnerProductSpace 𝕜 E'\nF : Type ?u.75621\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nF' : Type ?u.75641\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : Fintype ι\n⊢ FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 ι) = Fintype.card ι",
"tactic": "simp [EuclideanSpace, PiLp]"
}
] |
[
164,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.294672\nE : Type ?u.294675\nF : Type u_3\nG : Type ?u.294681\nE' : Type u_2\nF' : Type ?u.294687\nG' : Type ?u.294690\nE'' : Type ?u.294693\nF'' : Type ?u.294696\nG'' : Type ?u.294699\nR : Type ?u.294702\nR' : Type ?u.294705\n𝕜 : Type ?u.294708\n𝕜' : Type ?u.294711\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝¹ : One F\ninst✝ : NormOneClass F\n⊢ (f' =o[l] fun _x => 1) ↔ Tendsto f' l (𝓝 0)",
"tactic": "simp only [isLittleO_iff, norm_one, mul_one, Metric.nhds_basis_closedBall.tendsto_right_iff,\n Metric.mem_closedBall, dist_zero_right]"
}
] |
[
1325,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1323,
1
] |
Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
|
Filter.Tendsto.cesaro
|
[] |
[
169,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.pure_mul
|
[] |
[
354,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Algebra/AlgebraicCard.lean
|
Algebraic.cardinal_mk_of_countable_of_charZero
|
[] |
[
86,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.trans_refl_restr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.48193\nδ : Type ?u.48196\ne : LocalEquiv α β\ne' : LocalEquiv β γ\ns : Set β\n⊢ (LocalEquiv.trans e (LocalEquiv.restr (LocalEquiv.refl β) s)).source = (LocalEquiv.restr e (↑e ⁻¹' s)).source",
"tactic": "simp [trans_source]"
}
] |
[
761,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
760,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.le_ωSup_of_le
|
[] |
[
201,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
Filter.coprod_cocompact
|
[
{
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},
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},
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{
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"tactic": "refine' ⟨⟨(Prod.fst '' t)ᶜ, _, _⟩, ⟨(Prod.snd '' t)ᶜ, _, _⟩⟩"
},
{
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"tactic": "exact ⟨Prod.fst '' t, ht.image continuous_fst, Subset.rfl⟩"
},
{
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"tactic": "rw [compl_subset_comm] at htS⊢"
},
{
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"tactic": "exact htS.trans (subset_preimage_image Prod.fst _)"
},
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},
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"tactic": "rw [preimage_compl]"
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"tactic": "rw [compl_subset_comm] at htS⊢"
},
{
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"tactic": "exact htS.trans (subset_preimage_image Prod.snd _)"
}
] |
[
1001,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
980,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.iterate_id
|
[
{
"state_after": "α : Type u\nβ✝ : Type v\nδ : Type w\na : α\nβ : Type u\nfr : Stream' α → β\nch : fr (iterate id a) = fr (const a)\n⊢ fr (iterate id (id a)) = fr (const a)",
"state_before": "α : Type u\nβ✝ : Type v\nδ : Type w\na : α\nβ : Type u\nfr : Stream' α → β\nch : fr (iterate id a) = fr (const a)\n⊢ fr (tail (iterate id a)) = fr (tail (const a))",
"tactic": "rw [tail_iterate, tail_const]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ✝ : Type v\nδ : Type w\na : α\nβ : Type u\nfr : Stream' α → β\nch : fr (iterate id a) = fr (const a)\n⊢ fr (iterate id (id a)) = fr (const a)",
"tactic": "exact ch"
}
] |
[
358,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.inf_relindex_left
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\n⊢ relindex (H ⊓ K) H = relindex K H",
"tactic": "rw [inf_comm, inf_relindex_right]"
}
] |
[
143,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag
|
[
{
"state_after": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrderTop ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : CommMonoid α\nf : ι → ι → α\n⊢ (∏ x : ι, ∏ x_1 in Ioi x, f x_1 x) * ∏ x : ι, ∏ x_1 in Ioi x, f x x_1 =\n (∏ x : ι, ∏ x_1 in Ioi x, f x_1 x) * ∏ x : ι, ∏ x_1 in Iio x, f x_1 x",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrderTop ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : CommMonoid α\nf : ι → ι → α\n⊢ ∏ i : ι, ∏ j in Ioi i, f j i * f i j = ∏ i : ι, ∏ j in {i}ᶜ, f j i",
"tactic": "simp_rw [← Ioi_disjUnion_Iio, prod_disjUnion, prod_mul_distrib]"
},
{
"state_after": "case e_a\nι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrderTop ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : CommMonoid α\nf : ι → ι → α\n⊢ ∏ x : ι, ∏ x_1 in Ioi x, f x x_1 = ∏ x : ι, ∏ x_1 in Iio x, f x_1 x",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrderTop ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : CommMonoid α\nf : ι → ι → α\n⊢ (∏ x : ι, ∏ x_1 in Ioi x, f x_1 x) * ∏ x : ι, ∏ x_1 in Ioi x, f x x_1 =\n (∏ x : ι, ∏ x_1 in Ioi x, f x_1 x) * ∏ x : ι, ∏ x_1 in Iio x, f x_1 x",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrderTop ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : CommMonoid α\nf : ι → ι → α\n⊢ ∏ x in Finset.sigma univ fun x => Ioi x, f x.fst x.snd = ∏ x in Finset.sigma univ fun x => Iio x, f x.snd x.fst",
"state_before": "case e_a\nι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrderTop ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : CommMonoid α\nf : ι → ι → α\n⊢ ∏ x : ι, ∏ x_1 in Ioi x, f x x_1 = ∏ x : ι, ∏ x_1 in Iio x, f x_1 x",
"tactic": "rw [prod_sigma', prod_sigma']"
},
{
"state_after": "no goals",
"state_before": "case e_a\nι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrderTop ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : CommMonoid α\nf : ι → ι → α\n⊢ ∏ x in Finset.sigma univ fun x => Ioi x, f x.fst x.snd = ∏ x in Finset.sigma univ fun x => Iio x, f x.snd x.fst",
"tactic": "refine' prod_bij' (fun i _ => ⟨i.2, i.1⟩) _ _ (fun i _ => ⟨i.2, i.1⟩) _ _ _ <;> simp"
}
] |
[
1152,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1146,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.coe_inv_le
|
[
{
"state_after": "α : Type ?u.217125\nβ : Type ?u.217128\na b✝ c d : ℝ≥0∞\nr p q b : ℝ≥0\nhb : 1 ≤ ↑r * ↑b\n⊢ r⁻¹ ≤ b",
"state_before": "α : Type ?u.217125\nβ : Type ?u.217128\na b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\nb : ℝ≥0∞\nhb : 1 ≤ ↑r * b\n⊢ ∀ (p : ℝ≥0), b = ↑p → r⁻¹ ≤ p",
"tactic": "rintro b rfl"
},
{
"state_after": "case h\nα : Type ?u.217125\nβ : Type ?u.217128\na b✝ c d : ℝ≥0∞\nr p q b : ℝ≥0\nhb : 1 ≤ ↑r * ↑b\n⊢ 1 ≤ r * b",
"state_before": "α : Type ?u.217125\nβ : Type ?u.217128\na b✝ c d : ℝ≥0∞\nr p q b : ℝ≥0\nhb : 1 ≤ ↑r * ↑b\n⊢ r⁻¹ ≤ b",
"tactic": "apply NNReal.inv_le_of_le_mul"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.217125\nβ : Type ?u.217128\na b✝ c d : ℝ≥0∞\nr p q b : ℝ≥0\nhb : 1 ≤ ↑r * ↑b\n⊢ 1 ≤ r * b",
"tactic": "rwa [← coe_mul, ← coe_one, coe_le_coe] at hb"
}
] |
[
1346,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1341,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.transnumAuxSeq_def
|
[] |
[
648,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
647,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
iInf_eq_iInf_finset
|
[] |
[
1834,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1833,
1
] |
Mathlib/Logic/Equiv/Option.lean
|
Equiv.removeNone_aux_none
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.2701\ne : Option α ≃ Option β\nx : α\nh : ↑e (some x) = none\n⊢ some (removeNone_aux e x) = ↑e none",
"tactic": "simp [removeNone_aux, Option.not_isSome_iff_eq_none.mpr h]"
}
] |
[
98,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Surjective.forall₃
|
[] |
[
205,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
11
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_toReal
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (f a)) ∂μ) = ENNReal.toReal (∫⁻ (a : α), f a ∂μ)\n\nα : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ 0 ≤ᵐ[μ] fun x => ENNReal.toReal (f x)",
"state_before": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ (∫ (a : α), ENNReal.toReal (f a) ∂μ) = ENNReal.toReal (∫⁻ (a : α), f a ∂μ)",
"tactic": "rw [integral_eq_lintegral_of_nonneg_ae _ hfm.ennreal_toReal.aestronglyMeasurable]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) =ᵐ[μ] fun a => f a",
"state_before": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (f a)) ∂μ) = ENNReal.toReal (∫⁻ (a : α), f a ∂μ)",
"tactic": "rw [lintegral_congr_ae]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ ∀ (x : α), f x < ⊤ → (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) x = (fun a => f a) x",
"state_before": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) =ᵐ[μ] fun a => f a",
"tactic": "refine' hf.mp (eventually_of_forall _)"
},
{
"state_after": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\nx : α\nhx : f x < ⊤\n⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) x = (fun a => f a) x",
"state_before": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ ∀ (x : α), f x < ⊤ → (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) x = (fun a => f a) x",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\nx : α\nhx : f x ≠ ⊤\n⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) x = (fun a => f a) x",
"state_before": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\nx : α\nhx : f x < ⊤\n⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) x = (fun a => f a) x",
"tactic": "rw [lt_top_iff_ne_top] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\nx : α\nhx : f x ≠ ⊤\n⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (f a))) x = (fun a => f a) x",
"tactic": "simp [hx]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1083893\nF : Type ?u.1083896\n𝕜 : Type ?u.1083899\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1086590\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : ∀ᵐ (x : α) ∂μ, f x < ⊤\n⊢ 0 ≤ᵐ[μ] fun x => ENNReal.toReal (f x)",
"tactic": "exact eventually_of_forall fun x => ENNReal.toReal_nonneg"
}
] |
[
1192,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1187,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
|
Matrix.SpecialLinearGroup.fin_two_exists_eq_mk_of_apply_zero_one_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type ?u.848548\ninst✝ : Field R\ng : SL(2, R)\nhg : ↑g 1 0 = 0\na b : R\nh : a ≠ 0\n⊢ det (↑of ![![a, b], ![0, a⁻¹]]) = 1",
"tactic": "simp [h]"
},
{
"state_after": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : ↑{ val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } 1 0 = 0\n⊢ ∃ a_1 b_1 h,\n { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a_1, b_1], ![0, a_1⁻¹]], property := (_ : det (↑of ![![a_1, b_1], ![0, a_1⁻¹]]) = 1) }",
"state_before": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg : ↑g 1 0 = 0\n⊢ ∃ a b h, g = { val := ↑of ![![a, b], ![0, a⁻¹]], property := (_ : det (↑of ![![a, b], ![0, a⁻¹]]) = 1) }",
"tactic": "induction' g using Matrix.SpecialLinearGroup.fin_two_induction with a b c d h_det"
},
{
"state_after": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n⊢ ∃ a_1 b_1 h,\n { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a_1, b_1], ![0, a_1⁻¹]], property := (_ : det (↑of ![![a_1, b_1], ![0, a_1⁻¹]]) = 1) }",
"state_before": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : ↑{ val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } 1 0 = 0\n⊢ ∃ a_1 b_1 h,\n { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a_1, b_1], ![0, a_1⁻¹]], property := (_ : det (↑of ![![a_1, b_1], ![0, a_1⁻¹]]) = 1) }",
"tactic": "replace hg : c = 0 := by simpa using hg"
},
{
"state_after": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ∃ a_1 b_1 h,\n { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a_1, b_1], ![0, a_1⁻¹]], property := (_ : det (↑of ![![a_1, b_1], ![0, a_1⁻¹]]) = 1) }",
"state_before": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n⊢ ∃ a_1 b_1 h,\n { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a_1, b_1], ![0, a_1⁻¹]], property := (_ : det (↑of ![![a_1, b_1], ![0, a_1⁻¹]]) = 1) }",
"tactic": "have had : a * d = 1 := by rwa [hg, MulZeroClass.mul_zero, sub_zero] at h_det"
},
{
"state_after": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a, b], ![0, a⁻¹]], property := (_ : det (↑of ![![a, b], ![0, a⁻¹]]) = 1) }",
"state_before": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ∃ a_1 b_1 h,\n { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a_1, b_1], ![0, a_1⁻¹]], property := (_ : det (↑of ![![a_1, b_1], ![0, a_1⁻¹]]) = 1) }",
"tactic": "refine' ⟨a, b, left_ne_zero_of_mul_eq_one had, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ { val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } =\n { val := ↑of ![![a, b], ![0, a⁻¹]], property := (_ : det (↑of ![![a, b], ![0, a⁻¹]]) = 1) }",
"tactic": "simp_rw [eq_inv_of_mul_eq_one_right had, hg]"
},
{
"state_after": "no goals",
"state_before": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : ↑{ val := ↑of ![![a, b], ![c, d]], property := (_ : det (↑of ![![a, b], ![c, d]]) = 1) } 1 0 = 0\n⊢ c = 0",
"tactic": "simpa using hg"
},
{
"state_after": "no goals",
"state_before": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type ?u.848542\ninst✝¹ : CommRing S\nR : Type u_1\ninst✝ : Field R\ng : SL(2, R)\nhg✝ : ↑g 1 0 = 0\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n⊢ a * d = 1",
"tactic": "rwa [hg, MulZeroClass.mul_zero, sub_zero] at h_det"
}
] |
[
322,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.divInt_div_divInt_cancel_right
|
[
{
"state_after": "no goals",
"state_before": "a b c : ℚ\nx : ℤ\nhx : x ≠ 0\nn d : ℤ\n⊢ x /. n / (x /. d) = d /. n",
"tactic": "rw [div_eq_mul_inv, inv_def', mul_comm, divInt_mul_divInt_cancel hx]"
}
] |
[
500,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.measurable_of_measurable_coe
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → Measure α\nh : ∀ (s : Set α), MeasurableSet s → Measurable fun b => ↑↑(f b) s\ns : Set α\nhs : MeasurableSet s\n⊢ borel ℝ≥0∞ ≤ MeasurableSpace.map ((fun μ => ↑↑μ s) ∘ f) inst✝",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → Measure α\nh : ∀ (s : Set α), MeasurableSet s → Measurable fun b => ↑↑(f b) s\ns : Set α\nhs : MeasurableSet s\n⊢ borel ℝ≥0∞ ≤ MeasurableSpace.map (fun μ => ↑↑μ s) (MeasurableSpace.map f inst✝)",
"tactic": "rw [MeasurableSpace.map_comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → Measure α\nh : ∀ (s : Set α), MeasurableSet s → Measurable fun b => ↑↑(f b) s\ns : Set α\nhs : MeasurableSet s\n⊢ borel ℝ≥0∞ ≤ MeasurableSpace.map ((fun μ => ↑↑μ s) ∘ f) inst✝",
"tactic": "exact h s hs"
}
] |
[
63,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Polynomial/Reverse.lean
|
Polynomial.reflect_neg
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Ring R\nf : R[X]\nN : ℕ\n⊢ reflect N (-f) = -reflect N f",
"tactic": "rw [neg_eq_neg_one_mul, ← C_1, ← C_neg, reflect_C_mul, C_neg, C_1, ← neg_eq_neg_one_mul]"
}
] |
[
373,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.ne_zero_of_trailingDegree_lt
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.130658\nn : ℕ∞\nh : trailingDegree p < n\nh₀ : p = 0\n⊢ n ≤ trailingDegree p",
"tactic": "simp [h₀]"
}
] |
[
513,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/CategoryTheory/Monoidal/End.lean
|
CategoryTheory.ε_app_obj
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ F.ε.app ((F.obj n).obj X) =\n 𝟙 ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X)) ≫ (F.map (ρ_ n).inv).app X ≫ (MonoidalFunctor.μIso F n (𝟙_ M)).inv.app X",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ F.ε.app ((F.obj n).obj X) = (F.map (ρ_ n).inv).app X ≫ (MonoidalFunctor.μIso F n (𝟙_ M)).inv.app X",
"tactic": "refine' Eq.trans _ (Category.id_comp _)"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ F.ε.app ((F.obj n).obj X) ≫ inv ((MonoidalFunctor.μIso F n (𝟙_ M)).inv.app X) ≫ inv ((F.map (ρ_ n).inv).app X) =\n 𝟙 ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X))",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ F.ε.app ((F.obj n).obj X) =\n 𝟙 ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X)) ≫ (F.map (ρ_ n).inv).app X ≫ (MonoidalFunctor.μIso F n (𝟙_ M)).inv.app X",
"tactic": "rw [← Category.assoc, ← IsIso.comp_inv_eq, ← IsIso.comp_inv_eq, Category.assoc]"
},
{
"state_after": "case h.e'_2.h\nC : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\ne_1✝ :\n ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X) ⟶ (𝟙_ (C ⥤ C)).obj ((F.obj n).obj X)) =\n ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X) ⟶ (F.obj n).obj X)\n⊢ F.ε.app ((F.obj n).obj X) ≫ inv ((MonoidalFunctor.μIso F n (𝟙_ M)).inv.app X) ≫ inv ((F.map (ρ_ n).inv).app X) =\n F.ε.app ((F.obj n).obj X) ≫ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ F.ε.app ((F.obj n).obj X) ≫ inv ((MonoidalFunctor.μIso F n (𝟙_ M)).inv.app X) ≫ inv ((F.map (ρ_ n).inv).app X) =\n 𝟙 ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X))",
"tactic": "convert right_unitality_app F n X using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h\nC : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\ne_1✝ :\n ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X) ⟶ (𝟙_ (C ⥤ C)).obj ((F.obj n).obj X)) =\n ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X) ⟶ (F.obj n).obj X)\n⊢ F.ε.app ((F.obj n).obj X) ≫ inv ((MonoidalFunctor.μIso F n (𝟙_ M)).inv.app X) ≫ inv ((F.map (ρ_ n).inv).app X) =\n F.ε.app ((F.obj n).obj X) ≫ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X",
"tactic": "simp"
}
] |
[
227,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Algebra/Module/Zlattice.lean
|
Zspan.isAddFundamentalDomain
|
[
{
"state_after": "case intro\nE : Type u_2\nι : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝² : Finite ι\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nμ : MeasureTheory.Measure E\nval✝ : Fintype ι\n⊢ IsAddFundamentalDomain { x // x ∈ toAddSubgroup (span ℤ (Set.range ↑b)) } (fundamentalDomain b)",
"state_before": "E : Type u_2\nι : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝² : Finite ι\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nμ : MeasureTheory.Measure E\n⊢ IsAddFundamentalDomain { x // x ∈ toAddSubgroup (span ℤ (Set.range ↑b)) } (fundamentalDomain b)",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "no goals",
"state_before": "case intro\nE : Type u_2\nι : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝² : Finite ι\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nμ : MeasureTheory.Measure E\nval✝ : Fintype ι\n⊢ IsAddFundamentalDomain { x // x ∈ toAddSubgroup (span ℤ (Set.range ↑b)) } (fundamentalDomain b)",
"tactic": "exact IsAddFundamentalDomain.mk' (nullMeasurableSet (fundamentalDomain_measurableSet b))\n fun x => exist_unique_vadd_mem_fundamentalDomain b x"
}
] |
[
258,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
11
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.mul_div_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.62857\nG : Type ?u.62860\nα : Type u_1\nM : Type ?u.62866\nN : Type ?u.62869\ninst✝ : DivisionCommMonoid α\na b✝ c d : α\nh : IsUnit a\nb : α\n⊢ a * b / a = b",
"tactic": "rw [mul_comm, h.mul_div_cancel]"
}
] |
[
472,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
11
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Sized.node'
|
[] |
[
113,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_mul_const_Ioc_of_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a b c : α\nh : c < 0\n⊢ (fun x => x * c) ⁻¹' Ioc a b = Ico (b / c) (a / c)",
"tactic": "simp [← Ioi_inter_Iic, ← Ici_inter_Iio, h, inter_comm]"
}
] |
[
586,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.ring_inverse_eq_map_inverse
|
[
{
"state_after": "case h.h\nR : Type u_2\nM : Type u_1\nM₂ : Type ?u.2031615\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : TopologicalSpace M₂\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : TopologicalAddGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\nx✝¹ : M →L[R] M\nx✝ : M\n⊢ ↑(Ring.inverse x✝¹) x✝ = ↑(inverse x✝¹) x✝",
"state_before": "R : Type u_2\nM : Type u_1\nM₂ : Type ?u.2031615\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : TopologicalSpace M₂\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : TopologicalAddGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\n⊢ Ring.inverse = inverse",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u_2\nM : Type u_1\nM₂ : Type ?u.2031615\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : TopologicalSpace M₂\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : TopologicalAddGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\nx✝¹ : M →L[R] M\nx✝ : M\n⊢ ↑(Ring.inverse x✝¹) x✝ = ↑(inverse x✝¹) x✝",
"tactic": "simp [to_ring_inverse (ContinuousLinearEquiv.refl R M)]"
}
] |
[
2550,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2548,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.not_lt_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.81518\nβ : Type ?u.81521\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ℝ≥0∞\n⊢ ¬x < ⊤ ↔ x = ⊤",
"tactic": "rw [lt_top_iff_ne_top, Classical.not_not]"
}
] |
[
543,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
Infinite.orderOf_eq_zero_of_forall_mem_zpowers
|
[
{
"state_after": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\n⊢ ∀ (n : ℕ), 0 < n → g ^ n ≠ 1",
"state_before": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\n⊢ orderOf g = 0",
"tactic": "rw [orderOf_eq_zero_iff']"
},
{
"state_after": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\n⊢ False",
"state_before": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\n⊢ ∀ (n : ℕ), 0 < n → g ^ n ≠ 1",
"tactic": "refine' fun n hn hgn => _"
},
{
"state_after": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\n⊢ False",
"state_before": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\n⊢ False",
"tactic": "have ho := orderOf_pos' ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)"
},
{
"state_after": "case intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : ?m.100048\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\n⊢ False",
"state_before": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\n⊢ False",
"tactic": "obtain ⟨x, hx⟩ :=\n Infinite.exists_not_mem_finset\n (Finset.image (fun x => g ^ x) <| Finset.range <| orderOf g)"
},
{
"state_after": "case intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\n⊢ x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))",
"state_before": "case intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : ?m.100048\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\n⊢ False",
"tactic": "apply hx"
},
{
"state_after": "case intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\n⊢ ∃ n, g ^ n = x",
"state_before": "case intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\n⊢ x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))",
"tactic": "rw [← mem_powers_iff_mem_range_order_of' (x:=g) (y:=x) ho, Submonoid.mem_powers_iff]"
},
{
"state_after": "case intro.intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℤ\nhk : (fun x x_1 => x ^ x_1) g k = x\n⊢ ∃ n, g ^ n = x",
"state_before": "case intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\n⊢ ∃ n, g ^ n = x",
"tactic": "obtain ⟨k, hk⟩ := h x"
},
{
"state_after": "case intro.intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℤ\nhk : g ^ k = x\n⊢ ∃ n, g ^ n = x",
"state_before": "case intro.intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℤ\nhk : (fun x x_1 => x ^ x_1) g k = x\n⊢ ∃ n, g ^ n = x",
"tactic": "dsimp at hk"
},
{
"state_after": "case intro.intro.intro.inl\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ ↑k = x\n⊢ ∃ n, g ^ n = x\n\ncase intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k) = x\n⊢ ∃ n, g ^ n = x",
"state_before": "case intro.intro\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℤ\nhk : g ^ k = x\n⊢ ∃ n, g ^ n = x",
"tactic": "obtain ⟨k, rfl | rfl⟩ := k.eq_nat_or_neg"
},
{
"state_after": "case intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k % ↑(orderOf g)) = x\n⊢ ∃ n, g ^ n = x",
"state_before": "case intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k) = x\n⊢ ∃ n, g ^ n = x",
"tactic": "rw [zpow_eq_mod_orderOf] at hk"
},
{
"state_after": "case intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k % ↑(orderOf g)) = x\nthis : 0 ≤ -↑k % ↑(orderOf g)\n⊢ ∃ n, g ^ n = x",
"state_before": "case intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k % ↑(orderOf g)) = x\n⊢ ∃ n, g ^ n = x",
"tactic": "have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')"
},
{
"state_after": "case intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k % ↑(orderOf g)) = x\nthis : 0 ≤ -↑k % ↑(orderOf g)\n⊢ g ^ Int.toNat (-↑k % ↑(orderOf g)) = x",
"state_before": "case intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k % ↑(orderOf g)) = x\nthis : 0 ≤ -↑k % ↑(orderOf g)\n⊢ ∃ n, g ^ n = x",
"tactic": "refine' ⟨(-k % orderOf g : ℤ).toNat, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inr\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k % ↑(orderOf g)) = x\nthis : 0 ≤ -↑k % ↑(orderOf g)\n⊢ g ^ Int.toNat (-↑k % ↑(orderOf g)) = x",
"tactic": "rwa [← zpow_ofNat, Int.toNat_of_nonneg this]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inl\nα : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ ↑k = x\n⊢ ∃ n, g ^ n = x",
"tactic": "exact ⟨k, by exact_mod_cast hk⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ ↑k = x\n⊢ g ^ k = x",
"tactic": "exact_mod_cast hk"
},
{
"state_after": "no goals",
"state_before": "α : Type u\na : α\ninst✝¹ : Group α\ninst✝ : Infinite α\ng : α\nh : ∀ (x : α), x ∈ zpowers g\nn : ℕ\nhn : 0 < n\nhgn : g ^ n = 1\nho : 0 < orderOf g\nx : α\nhx : ¬x ∈ Finset.image (fun x => g ^ x) (Finset.range (orderOf g))\nk : ℕ\nhk : g ^ (-↑k % ↑(orderOf g)) = x\n⊢ ↑(orderOf g) ≠ 0",
"tactic": "exact_mod_cast ho.ne'"
}
] |
[
177,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.cos_eq_neg_one_iff_angle_eq_pi
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ cos (angle x y) = cos π ↔ angle x y = π",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ cos (angle x y) = -1 ↔ angle x y = π",
"tactic": "rw [← cos_pi]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ cos (angle x y) = cos π ↔ angle x y = π",
"tactic": "exact injOn_cos.eq_iff ⟨angle_nonneg x y, angle_le_pi x y⟩ (right_mem_Icc.2 pi_pos.le)"
}
] |
[
361,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
EuclideanGeometry.tan_angle_of_angle_eq_pi_div_two
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\n⊢ Real.tan (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₃ p₂",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\n⊢ Real.tan (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₃ p₂",
"tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n inner_neg_left, neg_vsub_eq_vsub_rev] at h"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\n⊢ Real.tan (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₃ p₂",
"tactic": "rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub' V p₃ p₂, ← vsub_add_vsub_cancel p₁ p₂ p₃,\n add_comm, tan_angle_add_of_inner_eq_zero h]"
}
] |
[
458,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/Analysis/Complex/Liouville.lean
|
Complex.deriv_eq_smul_circleIntegral
|
[
{
"state_after": "case intro\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nf : ℂ → F\nR : ℝ≥0\nhR : 0 < ↑R\nhf : DiffContOnCl ℂ f (ball c ↑R)\n⊢ deriv f c = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - c) ^ (-2) • f z",
"state_before": "E : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nR : ℝ\nc : ℂ\nf : ℂ → F\nhR : 0 < R\nhf : DiffContOnCl ℂ f (ball c R)\n⊢ deriv f c = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c) ^ (-2) • f z",
"tactic": "lift R to ℝ≥0 using hR.le"
},
{
"state_after": "case intro\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nf : ℂ → F\nR : ℝ≥0\nhR : 0 < ↑R\nhf : DiffContOnCl ℂ f (ball c ↑R)\n⊢ (↑(cauchyPowerSeries f c (↑R) 1) fun x => 1) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - c) ^ (-2) • f z",
"state_before": "case intro\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nf : ℂ → F\nR : ℝ≥0\nhR : 0 < ↑R\nhf : DiffContOnCl ℂ f (ball c ↑R)\n⊢ deriv f c = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - c) ^ (-2) • f z",
"tactic": "refine' (hf.hasFPowerSeriesOnBall hR).hasFPowerSeriesAt.deriv.trans _"
},
{
"state_after": "no goals",
"state_before": "case intro\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nf : ℂ → F\nR : ℝ≥0\nhR : 0 < ↑R\nhf : DiffContOnCl ℂ f (ball c ↑R)\n⊢ (↑(cauchyPowerSeries f c (↑R) 1) fun x => 1) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - c) ^ (-2) • f z",
"tactic": "simp only [cauchyPowerSeries_apply, one_div, zpow_neg, pow_one, smul_smul, zpow_two, mul_inv]"
}
] |
[
56,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
expNegInvGlue.zero
|
[] |
[
71,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
11
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.norm_id_of_nontrivial_seminorm
|
[
{
"state_after": "V : Type u_1\nV₁ : Type ?u.320008\nV₂ : Type ?u.320011\nV₃ : Type ?u.320014\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g : NormedAddGroupHom V₁ V₂\nh : ∃ x, ‖x‖ ≠ 0\nx : V\nhx : ‖x‖ ≠ 0\n⊢ 1 ≤ ‖id V‖",
"state_before": "V : Type u_1\nV₁ : Type ?u.320008\nV₂ : Type ?u.320011\nV₃ : Type ?u.320014\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g : NormedAddGroupHom V₁ V₂\nh : ∃ x, ‖x‖ ≠ 0\n⊢ 1 ≤ ‖id V‖",
"tactic": "let ⟨x, hx⟩ := h"
},
{
"state_after": "V : Type u_1\nV₁ : Type ?u.320008\nV₂ : Type ?u.320011\nV₃ : Type ?u.320014\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g : NormedAddGroupHom V₁ V₂\nh : ∃ x, ‖x‖ ≠ 0\nx : V\nhx : ‖x‖ ≠ 0\nthis : ‖↑(id V) x‖ / ‖x‖ ≤ ‖id V‖\n⊢ 1 ≤ ‖id V‖",
"state_before": "V : Type u_1\nV₁ : Type ?u.320008\nV₂ : Type ?u.320011\nV₃ : Type ?u.320014\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g : NormedAddGroupHom V₁ V₂\nh : ∃ x, ‖x‖ ≠ 0\nx : V\nhx : ‖x‖ ≠ 0\n⊢ 1 ≤ ‖id V‖",
"tactic": "have := (id V).ratio_le_opNorm x"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nV₁ : Type ?u.320008\nV₂ : Type ?u.320011\nV₃ : Type ?u.320014\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g : NormedAddGroupHom V₁ V₂\nh : ∃ x, ‖x‖ ≠ 0\nx : V\nhx : ‖x‖ ≠ 0\nthis : ‖↑(id V) x‖ / ‖x‖ ≤ ‖id V‖\n⊢ 1 ≤ ‖id V‖",
"tactic": "rwa [id_apply, div_self hx] at this"
}
] |
[
437,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
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