file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/SuccPred/Limit.lean
|
Order.isPredLimit_iff_lt_pred
|
[] |
[
368,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_sub
|
[] |
[
1387,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1386,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.direction_bot
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_1\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\n⊢ direction ⊥ = ⊥",
"tactic": "rw [direction_eq_vectorSpan, bot_coe, vectorSpan_def, vsub_empty, Submodule.span_empty]"
}
] |
[
830,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.BinaryBicone.binary_cofan_inr_toCocone
|
[] |
[
1045,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1044,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.LiftRelRec.lem
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\n⊢ ∀ (cb : Computation β), C ca cb → LiftRel R ca cb",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\ncb : Computation β\nHc : C ca cb\na : α\nha : a ∈ ca\n⊢ LiftRel R ca cb",
"tactic": "revert cb"
},
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\ncb : Computation β\nHc : C (pure a) cb\nh : LiftRelAux R C (destruct (pure a)) (destruct cb)\n⊢ LiftRel R (pure a) cb\n\ncase refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\nh : LiftRelAux R C (destruct (think ca')) (destruct cb)\n⊢ LiftRel R (think ca') cb",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\n⊢ ∀ (cb : Computation β), C ca cb → LiftRel R ca cb",
"tactic": "refine' memRecOn (C := (λ ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb))\n ha _ (fun ca' IH => _) <;> intro cb Hc <;> have h := H Hc"
},
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\ncb : Computation β\nHc : C (pure a) cb\nh : ∃ b, b ∈ cb ∧ R a b\n⊢ LiftRel R (pure a) cb",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\ncb : Computation β\nHc : C (pure a) cb\nh : LiftRelAux R C (destruct (pure a)) (destruct cb)\n⊢ LiftRel R (pure a) cb",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\ncb : Computation β\nHc : C (pure a) cb\nh : ∃ b, b ∈ cb ∧ R a b\n⊢ LiftRel R (pure a) cb",
"tactic": "simp [h]"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\nh : LiftRelAux R C (destruct (think ca')) (destruct cb)\n⊢ LiftRel R ca' cb",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\nh : LiftRelAux R C (destruct (think ca')) (destruct cb)\n⊢ LiftRel R (think ca') cb",
"tactic": "simp"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\n⊢ LiftRelAux R C (destruct (think ca')) (destruct cb) → LiftRel R ca' cb",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\nh : LiftRelAux R C (destruct (think ca')) (destruct cb)\n⊢ LiftRel R ca' cb",
"tactic": "revert h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\ncb'✝ : Computation β\nh : C ca' cb'✝\n⊢ LiftRel R ca' cb'✝",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\n⊢ LiftRelAux R C (destruct (think ca')) (destruct cb) → LiftRel R ca' cb",
"tactic": "apply cb.recOn (fun b => _) fun cb' => _ <;> intros _ h <;> simp at h <;> simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nH : ∀ {ca : Computation α} {cb : Computation β}, C ca cb → LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation α\na : α\nha : a ∈ ca\nca' : Computation α\nIH : (fun ca => ∀ (cb : Computation β), C ca cb → LiftRel R ca cb) ca'\ncb : Computation β\nHc : C (think ca') cb\ncb'✝ : Computation β\nh : C ca' cb'✝\n⊢ LiftRel R ca' cb'✝",
"tactic": "exact IH _ h"
}
] |
[
1296,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1285,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.mem_inv_smul_finset_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.832295\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.832304\ninst✝² : DecidableEq β\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns t : Finset β\na : α\nb : β\n⊢ b ∈ a⁻¹ • s ↔ a • b ∈ s",
"tactic": "rw [← smul_mem_smul_finset_iff a, smul_inv_smul]"
}
] |
[
1964,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1963,
1
] |
Mathlib/Topology/Separation.lean
|
SeparatedNhds.disjoint_closure_right
|
[] |
[
147,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Algebra/Ring/Commute.lean
|
Commute.add_right
|
[] |
[
38,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Data/Vector3.lean
|
vectorAllP_cons
|
[] |
[
278,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.encode_iff
|
[] |
[
415,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
SubgroupClass.inclusion_self
|
[
{
"state_after": "case mk\nG : Type u_1\nG' : Type ?u.59883\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.59892\ninst✝⁴ : AddGroup A\nM : Type ?u.59898\nS : Type u_2\ninst✝³ : DivInvMonoid M\ninst✝² : SetLike S M\nhSM : SubgroupClass S M\nH K : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nval✝ : G\nproperty✝ : val✝ ∈ H\n⊢ ↑(inclusion (_ : H ≤ H)) { val := val✝, property := property✝ } = { val := val✝, property := property✝ }",
"state_before": "G : Type u_1\nG' : Type ?u.59883\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.59892\ninst✝⁴ : AddGroup A\nM : Type ?u.59898\nS : Type u_2\ninst✝³ : DivInvMonoid M\ninst✝² : SetLike S M\nhSM : SubgroupClass S M\nH K : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nx : { x // x ∈ H }\n⊢ ↑(inclusion (_ : H ≤ H)) x = x",
"tactic": "cases x"
},
{
"state_after": "no goals",
"state_before": "case mk\nG : Type u_1\nG' : Type ?u.59883\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.59892\ninst✝⁴ : AddGroup A\nM : Type ?u.59898\nS : Type u_2\ninst✝³ : DivInvMonoid M\ninst✝² : SetLike S M\nhSM : SubgroupClass S M\nH K : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nval✝ : G\nproperty✝ : val✝ ∈ H\n⊢ ↑(inclusion (_ : H ≤ H)) { val := val✝, property := property✝ } = { val := val✝, property := property✝ }",
"tactic": "rfl"
}
] |
[
320,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
Balanced.mem_smul_iff
|
[
{
"state_after": "case inl\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖0‖\n⊢ a • x ∈ s ↔ 0 • x ∈ s\n\ncase inr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\n⊢ a • x ∈ s ↔ b • x ∈ s",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\n⊢ a • x ∈ s ↔ b • x ∈ s",
"tactic": "obtain rfl | hb := eq_or_ne b 0"
},
{
"state_after": "case inr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\nha : a ≠ 0\n⊢ a • x ∈ s ↔ b • x ∈ s",
"state_before": "case inr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\n⊢ a • x ∈ s ↔ b • x ∈ s",
"tactic": "have ha : a ≠ 0 := norm_ne_zero_iff.1 (ne_of_eq_of_ne h <| norm_ne_zero_iff.2 hb)"
},
{
"state_after": "case inl\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na : 𝕜\nhs : Balanced 𝕜 s\nh : a = 0\n⊢ a • x ∈ s ↔ 0 • x ∈ s",
"state_before": "case inl\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖0‖\n⊢ a • x ∈ s ↔ 0 • x ∈ s",
"tactic": "rw [norm_zero, norm_eq_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na : 𝕜\nhs : Balanced 𝕜 s\nh : a = 0\n⊢ a • x ∈ s ↔ 0 • x ∈ s",
"tactic": "rw [h]"
},
{
"state_after": "case inr.mpr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\nha : a ≠ 0\nh' : b • x ∈ s\n⊢ (a * b⁻¹) • b • x ∈ s",
"state_before": "case inr.mpr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\nha : a ≠ 0\nh' : b • x ∈ s\n⊢ (a * b⁻¹ * b) • x ∈ s",
"tactic": "rw [← smul_eq_mul, smul_assoc]"
},
{
"state_after": "case inr.mpr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\nha : a ≠ 0\nh' : b • x ∈ s\n⊢ ‖a * b⁻¹‖ ≤ 1",
"state_before": "case inr.mpr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\nha : a ≠ 0\nh' : b • x ∈ s\n⊢ (a * b⁻¹) • b • x ∈ s",
"tactic": "refine' hs.smul_mem _ h'"
},
{
"state_after": "no goals",
"state_before": "case inr.mpr\n𝕜 : Type u_1\n𝕝 : Type ?u.103382\nE : Type u_2\nι : Sort ?u.103388\nκ : ι → Sort ?u.103393\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Balanced 𝕜 s\nh : ‖a‖ = ‖b‖\nhb : b ≠ 0\nha : a ≠ 0\nh' : b • x ∈ s\n⊢ ‖a * b⁻¹‖ ≤ 1",
"tactic": "simp [← h, ha]"
}
] |
[
310,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Computability/Partrec.lean
|
Nat.rfind_min
|
[] |
[
87,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Algebra/Homology/Exact.lean
|
CategoryTheory.Functor.exact_of_exact_map
|
[] |
[
367,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.comp_symm
|
[
{
"state_after": "case H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne✝ e : A₁ ≃ₐ[R] A₂\nx✝ : A₂\n⊢ ↑(AlgHom.comp ↑e ↑(symm e)) x✝ = ↑(AlgHom.id R A₂) x✝",
"state_before": "R : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne✝ e : A₁ ≃ₐ[R] A₂\n⊢ AlgHom.comp ↑e ↑(symm e) = AlgHom.id R A₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A₁\ninst✝⁴ : Semiring A₂\ninst✝³ : Semiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne✝ e : A₁ ≃ₐ[R] A₂\nx✝ : A₂\n⊢ ↑(AlgHom.comp ↑e ↑(symm e)) x✝ = ↑(AlgHom.id R A₂) x✝",
"tactic": "simp"
}
] |
[
423,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean
|
CategoryTheory.IsKernelPair.of_isIso_of_mono
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ IsPullback a a f f",
"state_before": "C : Type u\ninst✝² : Category C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ IsKernelPair f a a",
"tactic": "change IsPullback _ _ _ _"
},
{
"state_after": "case h.e'_8\nC : Type u\ninst✝² : Category C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ a = a ≫ 𝟙 X\n\ncase h.e'_9\nC : Type u\ninst✝² : Category C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ f = 𝟙 X ≫ f",
"state_before": "C : Type u\ninst✝² : Category C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ IsPullback a a f f",
"tactic": "convert (IsPullback.of_horiz_isIso ⟨(rfl : a ≫ 𝟙 X = _ )⟩).paste_vert (IsKernelPair.id_of_mono f)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_8\nC : Type u\ninst✝² : Category C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ a = a ≫ 𝟙 X\n\ncase h.e'_9\nC : Type u\ninst✝² : Category C\nR X Y Z : C\nf : X ⟶ Y\na b : R ⟶ X\ninst✝¹ : IsIso a\ninst✝ : Mono f\n⊢ f = 𝟙 X ≫ f",
"tactic": "all_goals { simp }"
}
] |
[
220,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.opNorm_nonneg
|
[] |
[
238,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.nat_lt_limit
|
[] |
[
2443,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2441,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitePos_abs_iff_infinite_abs
|
[] |
[
520,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.isCircuit_copy
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' : V\np : Walk G u' u'\n⊢ IsCircuit (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCircuit p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu u' : V\np : Walk G u u\nhu : u = u'\n⊢ IsCircuit (Walk.copy p hu hu) ↔ IsCircuit p",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' : V\np : Walk G u' u'\n⊢ IsCircuit (Walk.copy p (_ : u' = u') (_ : u' = u')) ↔ IsCircuit p",
"tactic": "rfl"
}
] |
[
902,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Subsingleton.pi_mulSingle_eq
|
[
{
"state_after": "no goals",
"state_before": "I : Type u\nα✝ : Type ?u.35145\nβ : Type ?u.35148\nγ : Type ?u.35151\nf : I → Type v₁\ng : I → Type v₂\nh : I → Type v₃\nx✝ y : (i : I) → f i\ni✝ : I\nα : Type u_1\ninst✝² : DecidableEq I\ninst✝¹ : Subsingleton I\ninst✝ : One α\ni : I\nx : α\nj : I\n⊢ Pi.mulSingle i x j = x",
"tactic": "rw [Subsingleton.elim j i, Pi.mulSingle_eq_same]"
}
] |
[
439,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.All.append
|
[
{
"state_after": "α✝ : Type u_1\nl r : RBNode α✝\np : α✝ → Prop\nhl : All p l\nhr : All p r\n⊢ All p\n (match l, r with\n | nil, x => x\n | x, nil => x\n | node red a x b, node red c y d =>\n match append b c with\n | node red b' z c' => node red (node red a x b') z (node red c' y d)\n | bc => node red a x (node red bc y d)\n | node black a x b, node black c y d =>\n match append b c with\n | node red b' z c' => node red (node black a x b') z (node black c' y d)\n | bc => balLeft a x (node black bc y d)\n | a@h:(node black l v r), node red b x c => node red (append a b) x c\n | node red a x b, c@h:(node black l v r) => node red a x (append b c))",
"state_before": "α✝ : Type u_1\nl r : RBNode α✝\np : α✝ → Prop\nhl : All p l\nhr : All p r\n⊢ All p (append l r)",
"tactic": "unfold append"
},
{
"state_after": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))\n\ncase h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n\ncase h_5\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ l✝ : RBNode α✝\nv✝ : α✝\nr✝ b✝ : RBNode α✝\nx✝ : α✝\nc✝ : RBNode α✝\nhl : All p (node black l✝ v✝ r✝)\nhr : All p (node red b✝ x✝ c✝)\n⊢ p x✝ ∧ All p (append (node black l✝ v✝ r✝) b✝) ∧ All p c✝\n\ncase h_6\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ l✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node black l✝ v✝ r✝)\n⊢ p x✝ ∧ All p a✝ ∧ All p (append b✝ (node black l✝ v✝ r✝))",
"state_before": "α✝ : Type u_1\nl r : RBNode α✝\np : α✝ → Prop\nhl : All p l\nhr : All p r\n⊢ All p\n (match l, r with\n | nil, x => x\n | x, nil => x\n | node red a x b, node red c y d =>\n match append b c with\n | node red b' z c' => node red (node red a x b') z (node red c' y d)\n | bc => node red a x (node red bc y d)\n | node black a x b, node black c y d =>\n match append b c with\n | node red b' z c' => node red (node black a x b') z (node black c' y d)\n | bc => balLeft a x (node black bc y d)\n | a@h:(node black l v r), node red b x c => node red (append a b) x c\n | node red a x b, c@h:(node black l v r) => node red a x (append b c))",
"tactic": "split <;> simp [*]"
},
{
"state_after": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))",
"state_before": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))",
"tactic": "have ⟨hx, ha, hb⟩ := hl"
},
{
"state_after": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))",
"state_before": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))",
"tactic": "have ⟨hy, hc, hd⟩ := hr"
},
{
"state_after": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\nthis : All p (append b✝ c✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))",
"state_before": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))",
"tactic": "have := hb.append hc"
},
{
"state_after": "no goals",
"state_before": "case h_3\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node red c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\nthis : All p (append b✝ c✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))",
"tactic": "split <;> simp_all"
},
{
"state_after": "case h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))",
"state_before": "case h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))",
"tactic": "have ⟨hx, ha, hb⟩ := hl"
},
{
"state_after": "case h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))",
"state_before": "case h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))",
"tactic": "have ⟨hy, hc, hd⟩ := hr"
},
{
"state_after": "case h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\nthis : All p (append b✝ c✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))",
"state_before": "case h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))",
"tactic": "have := hb.append hc"
},
{
"state_after": "no goals",
"state_before": "case h_4\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ c✝ : RBNode α✝\ny✝ : α✝\nd✝ : RBNode α✝\nhl : All p (node black a✝ x✝ b✝)\nhr : All p (node black c✝ y✝ d✝)\nhx : p x✝\nha : All p a✝\nhb : All p b✝\nhy : p y✝\nhc : All p c✝\nhd : All p d✝\nthis : All p (append b✝ c✝)\n⊢ All p\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))",
"tactic": "split <;> simp_all [All.balLeft]"
},
{
"state_after": "no goals",
"state_before": "case h_5\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ l✝ : RBNode α✝\nv✝ : α✝\nr✝ b✝ : RBNode α✝\nx✝ : α✝\nc✝ : RBNode α✝\nhl : All p (node black l✝ v✝ r✝)\nhr : All p (node red b✝ x✝ c✝)\n⊢ p x✝ ∧ All p (append (node black l✝ v✝ r✝) b✝) ∧ All p c✝",
"tactic": "simp_all [hl.append hr.2.1]"
},
{
"state_after": "no goals",
"state_before": "case h_6\nα✝ : Type u_1\np : α✝ → Prop\nx✝² x✝¹ a✝ : RBNode α✝\nx✝ : α✝\nb✝ l✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nhl : All p (node red a✝ x✝ b✝)\nhr : All p (node black l✝ v✝ r✝)\n⊢ p x✝ ∧ All p a✝ ∧ All p (append b✝ (node black l✝ v✝ r✝))",
"tactic": "simp_all [hl.2.2.append hr]"
}
] |
[
311,
36
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
303,
11
] |
Mathlib/Algebra/Star/StarAlgHom.lean
|
StarAlgHom.ext
|
[] |
[
380,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.centroid_def
|
[] |
[
846,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
845,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.inf_eq_bot_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.86298\nι : Sort x\nf✝ g✝ : Filter α\ns t : Set α\nf g : Filter α\n⊢ f ⊓ g = ⊥ ↔ ∃ U, U ∈ f ∧ ∃ V, V ∈ g ∧ U ∩ V = ∅",
"tactic": "simp only [← disjoint_iff, Filter.disjoint_iff, Set.disjoint_iff_inter_eq_empty]"
}
] |
[
726,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
725,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.nat_cast_eq_zero
|
[] |
[
2321,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2320,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.equiv'_toAlgHom
|
[] |
[
686,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
684,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.mem_cons
|
[] |
[
191,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoLocallyUniformly.continuous
|
[] |
[
902,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
11
] |
Mathlib/RingTheory/Nilpotent.lean
|
Commute.isNilpotent_mul_left
|
[
{
"state_after": "case intro\nR S : Type u\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nn : ℕ\nhn : x ^ n = 0\n⊢ IsNilpotent (x * y)",
"state_before": "R S : Type u\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nh : IsNilpotent x\n⊢ IsNilpotent (x * y)",
"tactic": "obtain ⟨n, hn⟩ := h"
},
{
"state_after": "case intro\nR S : Type u\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nn : ℕ\nhn : x ^ n = 0\n⊢ (x * y) ^ n = 0",
"state_before": "case intro\nR S : Type u\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nn : ℕ\nhn : x ^ n = 0\n⊢ IsNilpotent (x * y)",
"tactic": "use n"
},
{
"state_after": "no goals",
"state_before": "case intro\nR S : Type u\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nn : ℕ\nhn : x ^ n = 0\n⊢ (x * y) ^ n = 0",
"tactic": "rw [h_comm.mul_pow, hn, MulZeroClass.zero_mul]"
}
] |
[
161,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.IsBasis.filter_eq_generate
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.9602\nγ : Type ?u.9605\nι : Sort u_2\nι' : Sort ?u.9611\np : ι → Prop\ns : ι → Set α\nh : IsBasis p s\n⊢ IsBasis.filter h = FilterBasis.filter (IsBasis.filterBasis h)",
"state_before": "α : Type u_1\nβ : Type ?u.9602\nγ : Type ?u.9605\nι : Sort u_2\nι' : Sort ?u.9611\np : ι → Prop\ns : ι → Set α\nh : IsBasis p s\n⊢ IsBasis.filter h = generate {U | ∃ i, p i ∧ s i = U}",
"tactic": "erw [h.filterBasis.generate]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9602\nγ : Type ?u.9605\nι : Sort u_2\nι' : Sort ?u.9611\np : ι → Prop\ns : ι → Set α\nh : IsBasis p s\n⊢ IsBasis.filter h = FilterBasis.filter (IsBasis.filterBasis h)",
"tactic": "rfl"
}
] |
[
225,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.length_le
|
[] |
[
401,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
Real.differentiable_rpow_const
|
[] |
[
359,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseEmbedding.dense_image
|
[] |
[
296,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.pi_div_two_eq_arcsin
|
[] |
[
250,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.monomial_zero_one
|
[] |
[
201,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean
|
CategoryTheory.Limits.hasCoproducts_of_opposite
|
[] |
[
375,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum_mul_neg
|
[
{
"state_after": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\nthis : -((∑ i in range n, x ^ i) * (x - 1)) = -(x ^ n - 1)\n⊢ (∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n",
"state_before": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\n⊢ (∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n",
"tactic": "have := congr_arg Neg.neg (geom_sum_mul x n)"
},
{
"state_after": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\nthis : (∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n\n⊢ (∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n",
"state_before": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\nthis : -((∑ i in range n, x ^ i) * (x - 1)) = -(x ^ n - 1)\n⊢ (∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n",
"tactic": "rw [neg_sub, ← mul_neg, neg_sub] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\nthis : (∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n\n⊢ (∑ i in range n, x ^ i) * (1 - x) = 1 - x ^ n",
"tactic": "exact this"
}
] |
[
233,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Order/Hom/Basic.lean
|
RelEmbedding.orderEmbeddingOfLTEmbedding_apply
|
[] |
[
633,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
630,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_gt_iff
|
[] |
[
82,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isCoseparator_iff_faithful_yoneda_obj
|
[] |
[
521,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval_eq_sum_range
|
[
{
"state_after": "case h\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r : R[X]\nx✝ : R\np : R[X]\nx : R\n⊢ ∀ (n : ℕ), 0 * x ^ n = 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r : R[X]\nx✝ : R\np : R[X]\nx : R\n⊢ eval x p = ∑ i in range (natDegree p + 1), coeff p i * x ^ i",
"tactic": "rw [eval_eq_sum, sum_over_range]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r : R[X]\nx✝ : R\np : R[X]\nx : R\n⊢ ∀ (n : ℕ), 0 * x ^ n = 0",
"tactic": "simp"
}
] |
[
323,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.get?_push_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : Array α\nx : α\ni : Nat\nh : i < size a\n⊢ (push a x)[i]? = some a[i]",
"tactic": "rw [getElem?_pos, get_push_lt]"
}
] |
[
80,
33
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
78,
1
] |
Mathlib/Combinatorics/Colex.lean
|
Colex.lt_trans
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\n⊢ a < c",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\n⊢ a < b → b < c → a < c",
"tactic": "rintro ⟨k₁, k₁z, notinA, inB⟩ ⟨k₂, k₂z, notinB, inC⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.inl\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ a < c\n\ncase intro.intro.intro.intro.intro.intro.inr\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ > k₂\n⊢ a < c",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\n⊢ a < c",
"tactic": "cases' lt_or_gt_of_ne (ne_of_mem_of_not_mem inB notinB) with h h"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\nx : α\nhx : k₂ < x\n⊢ x ∈ a ↔ x ∈ c\n\ncase intro.intro.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ ¬k₂ ∈ a",
"state_before": "case intro.intro.intro.intro.intro.intro.inl\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ a < c",
"tactic": "refine' ⟨k₂, @fun x hx => _, _, inC⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\nx : α\nhx : k₂ < x\n⊢ x ∈ a ↔ x ∈ b\n\ncase intro.intro.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ ¬k₂ ∈ a",
"state_before": "case intro.intro.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\nx : α\nhx : k₂ < x\n⊢ x ∈ a ↔ x ∈ c\n\ncase intro.intro.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ ¬k₂ ∈ a",
"tactic": "rw [← k₂z hx]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ ¬k₂ ∈ a",
"state_before": "case intro.intro.intro.intro.intro.intro.inl.refine'_1\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\nx : α\nhx : k₂ < x\n⊢ x ∈ a ↔ x ∈ b\n\ncase intro.intro.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ ¬k₂ ∈ a",
"tactic": "apply k₁z (Trans.trans h hx)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.inl.refine'_2\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ < k₂\n⊢ ¬k₂ ∈ a",
"tactic": "rwa [k₁z h]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.inr\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ > k₂\nx : α\nhx : k₁ < x\n⊢ x ∈ a ↔ x ∈ c",
"state_before": "case intro.intro.intro.intro.intro.intro.inr\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ > k₂\n⊢ a < c",
"tactic": "refine' ⟨k₁, @fun x hx => _, notinA, by rwa [← k₂z h]⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.inr\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ > k₂\nx : α\nhx : k₁ < x\n⊢ x ∈ b ↔ x ∈ c",
"state_before": "case intro.intro.intro.intro.intro.intro.inr\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ > k₂\nx : α\nhx : k₁ < x\n⊢ x ∈ a ↔ x ∈ c",
"tactic": "rw [k₁z hx]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.inr\nα : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ > k₂\nx : α\nhx : k₁ < x\n⊢ x ∈ b ↔ x ∈ c",
"tactic": "apply k₂z (Trans.trans h hx)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\na b c : Colex α\nk₁ : α\nk₁z : ∀ {x : α}, k₁ < x → (x ∈ a ↔ x ∈ b)\nnotinA : ¬k₁ ∈ a\ninB : k₁ ∈ b\nk₂ : α\nk₂z : ∀ {x : α}, k₂ < x → (x ∈ b ↔ x ∈ c)\nnotinB : ¬k₂ ∈ b\ninC : k₂ ∈ c\nh : k₁ > k₂\n⊢ k₁ ∈ c",
"tactic": "rwa [← k₂z h]"
}
] |
[
160,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
extChartAt_preimage_inter_eq
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nM : Type u_2\nH : Type u_4\nE' : Type ?u.225289\nM' : Type ?u.225292\nH' : Type ?u.225295\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\n⊢ ↑(LocalEquiv.symm (extChartAt I x)) ⁻¹' (s ∩ t) ∩ range ↑I =\n ↑(LocalEquiv.symm (extChartAt I x)) ⁻¹' s ∩ range ↑I ∩ ↑(LocalEquiv.symm (extChartAt I x)) ⁻¹' t",
"tactic": "mfld_set_tac"
}
] |
[
1236,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1233,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.mapFun.zero
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_2\nS : Type u_1\nT : Type ?u.30320\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.30335\nβ : Type ?u.30338\nf : R →+* S\nx y : 𝕎 R\n⊢ mapFun (↑f) 0 = 0",
"tactic": "map_fun_tac"
}
] |
[
109,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Data/Nat/Sqrt.lean
|
Nat.sqrt_succ_le_succ_sqrt
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ sqrt n * sqrt n + sqrt n ≤ Nat.mul (succ (succ (sqrt n))) (succ (sqrt n))",
"tactic": "refine' add_le_add (Nat.mul_le_mul_right _ _) _ <;> exact Nat.le_add_right _ 2"
}
] |
[
171,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean
|
Nat.testBit_lxor'
|
[] |
[
520,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Order/Heyting/Regular.lean
|
Heyting.IsRegular.disjoint_compl_right_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : HeytingAlgebra α\na b : α\nhb : IsRegular b\n⊢ Disjoint a (bᶜ) ↔ a ≤ b",
"tactic": "rw [← le_compl_iff_disjoint_right, hb.eq]"
}
] |
[
86,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
11
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂_eq_eval_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\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g p = ↑(eval₂Hom (RingHom.id S₁) g) (↑(eval₂Hom (RingHom.comp C f) X) 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\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g p = ↑(eval g) (↑(map f) p)",
"tactic": "unfold map eval"
},
{
"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\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g p = eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X 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\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g p = ↑(eval₂Hom (RingHom.id S₁) g) (↑(eval₂Hom (RingHom.comp C f) X) p)",
"tactic": "simp only [coe_eval₂Hom]"
},
{
"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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n ↑(eval₂Hom (RingHom.id S₁) g) (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (↑(eval₂Hom (RingHom.id S₁) g) ∘ X) p\n⊢ eval₂ f g p = eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X 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\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ f g p = eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p)",
"tactic": "have h := eval₂_comp_left (eval₂Hom (RingHom.id S₁) g) (C.comp f) X p"
},
{
"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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ eval₂ f g p = eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n ↑(eval₂Hom (RingHom.id S₁) g) (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (↑(eval₂Hom (RingHom.id S₁) g) ∘ X) p\n⊢ eval₂ f g p = eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p)",
"tactic": "dsimp [-eval₂_id] at h"
},
{
"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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ eval₂ f g p = eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ eval₂ f g p = eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p)",
"tactic": "rw [h]"
},
{
"state_after": "case e_f\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ f = RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)\n\ncase e_g\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ g = eval₂ (RingHom.id S₁) g ∘ X",
"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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ eval₂ f g p = eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p",
"tactic": "congr"
},
{
"state_after": "case e_f.a\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\na : R\n⊢ ↑f a = ↑(RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) a",
"state_before": "case e_f\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ f = RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)",
"tactic": "ext1 a"
},
{
"state_after": "no goals",
"state_before": "case e_f.a\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\na : R\n⊢ ↑f a = ↑(RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) a",
"tactic": "simp only [coe_eval₂Hom, RingHom.id_apply, comp_apply, eval₂_C, RingHom.coe_comp]"
},
{
"state_after": "case e_g.h\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\nn : σ\n⊢ g n = (eval₂ (RingHom.id S₁) g ∘ X) n",
"state_before": "case e_g\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\n⊢ g = eval₂ (RingHom.id S₁) g ∘ X",
"tactic": "ext1 n"
},
{
"state_after": "no goals",
"state_before": "case e_g.h\nR : 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\ng : σ → S₁\np : MvPolynomial σ R\nh :\n eval₂ (RingHom.id S₁) g (eval₂ (RingHom.comp C f) X p) =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id S₁) g) (RingHom.comp C f)) (eval₂ (RingHom.id S₁) g ∘ X) p\nn : σ\n⊢ g n = (eval₂ (RingHom.id S₁) g ∘ X) n",
"tactic": "simp only [comp_apply, eval₂_X]"
}
] |
[
1242,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1230,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Nat.cast_prod
|
[] |
[
2218,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2216,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.LiftRel.trans
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t u✝ : WSeq α\nh1 : LiftRel R s✝ t\nh2 : LiftRel R t u✝\ns u : WSeq α\nh : (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) s u\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns t u : WSeq α\nh1 : LiftRel R s t\nh2 : LiftRel R t u\n⊢ LiftRel R s u",
"tactic": "refine' ⟨fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u, ⟨t, h1, h2⟩, fun {s u} h => _⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1 : LiftRel R s t\nh2 : LiftRel R t u\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t u✝ : WSeq α\nh1 : LiftRel R s✝ t\nh2 : LiftRel R t u✝\ns u : WSeq α\nh : (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) s u\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"tactic": "rcases h with ⟨t, h1, h2⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2 : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1 : LiftRel R s t\nh2 : LiftRel R t u\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"tactic": "have h1 := liftRel_destruct h1"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2 : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"tactic": "have h2 := liftRel_destruct h2"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\na c : Option (α × WSeq α)\nha : a ∈ destruct s\nhc : c ∈ destruct u\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) a c",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\n⊢ Computation.LiftRel (LiftRelO R fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (destruct s) (destruct u)",
"tactic": "refine'\n Computation.liftRel_def.2\n ⟨(Computation.terminates_of_LiftRel h1).trans (Computation.terminates_of_LiftRel h2),\n fun {a c} ha hc => _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\na c : Option (α × WSeq α)\nha : a ∈ destruct s\nhc : c ∈ destruct u\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\nt1 : LiftRelO R (LiftRel R) a b\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) a c",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\na c : Option (α × WSeq α)\nha : a ∈ destruct s\nhc : c ∈ destruct u\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) a c",
"tactic": "rcases h1.left ha with ⟨b, hb, t1⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\na c : Option (α × WSeq α)\nha : a ∈ destruct s\nhc : c ∈ destruct u\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\nt1 : LiftRelO R (LiftRel R) a b\nt2 : LiftRelO R (LiftRel R) b c\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) a c",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\na c : Option (α × WSeq α)\nha : a ∈ destruct s\nhc : c ∈ destruct u\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\nt1 : LiftRelO R (LiftRel R) a b\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) a c",
"tactic": "have t2 := Computation.rel_of_LiftRel h2 hb hc"
},
{
"state_after": "case intro.intro.intro.intro.none.none\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\nha : none ∈ destruct s\nt1 : LiftRelO R (LiftRel R) none b\nhc : none ∈ destruct u\nt2 : LiftRelO R (LiftRel R) b none\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) none none\n\ncase intro.intro.intro.intro.none.some\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\nha : none ∈ destruct s\nt1 : LiftRelO R (LiftRel R) none b\nc : α × WSeq α\nhc : some c ∈ destruct u\nt2 : LiftRelO R (LiftRel R) b (some c)\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) none (some c)\n\ncase intro.intro.intro.intro.some.none\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\na : α × WSeq α\nha : some a ∈ destruct s\nt1 : LiftRelO R (LiftRel R) (some a) b\nhc : none ∈ destruct u\nt2 : LiftRelO R (LiftRel R) b none\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some a) none\n\ncase intro.intro.intro.intro.some.some\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\na : α × WSeq α\nha : some a ∈ destruct s\nt1 : LiftRelO R (LiftRel R) (some a) b\nc : α × WSeq α\nhc : some c ∈ destruct u\nt2 : LiftRelO R (LiftRel R) b (some c)\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some a) (some c)",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\na c : Option (α × WSeq α)\nha : a ∈ destruct s\nhc : c ∈ destruct u\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\nt1 : LiftRelO R (LiftRel R) a b\nt2 : LiftRelO R (LiftRel R) b c\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) a c",
"tactic": "cases' a with a <;> cases' c with c"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.none.none\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nb : Option (α × WSeq α)\nhb : b ∈ destruct t\nha : none ∈ destruct s\nt1 : LiftRelO R (LiftRel R) none b\nhc : none ∈ destruct u\nt2 : LiftRelO R (LiftRel R) b none\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) none none",
"tactic": "trivial"
},
{
"state_after": "case intro.intro.intro.intro.none.some.none\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nha : none ∈ destruct s\nc : α × WSeq α\nhc : some c ∈ destruct u\nhb : none ∈ destruct t\nt1 : LiftRelO R (LiftRel R) none none\nt2 : LiftRelO R (LiftRel R) none (some c)\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) none (some c)\n\ncase intro.intro.intro.intro.none.some.some\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns u t : WSeq α\nh1✝ : LiftRel R s t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nha : none ∈ destruct s\nc : α × WSeq α\nhc : some c ∈ destruct u\nval✝ : α × WSeq α\nhb : some val✝ ∈ destruct t\nt1 : LiftRelO R (LiftRel R) none (some val✝)\nt2 : LiftRelO R (LiftRel R) (some val✝) (some c)\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) none (some c)",
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"tactic": "cases' b with b"
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"tactic": "cases' b with b t"
},
{
"state_after": "case intro.intro.intro.intro.some.some.mk.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝¹ u✝¹ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝¹\nh2✝¹ : LiftRel R t✝¹ u✝¹\ns✝ u✝ t✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t✝) (destruct u✝)\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nb : α\nt : WSeq α\nhb : some (b, t) ∈ destruct t✝\nt1 : LiftRelO R (LiftRel R) (some (a, s)) (some (b, t))\nc : α\nu : WSeq α\nhc : some (c, u) ∈ destruct u✝\nt2 : LiftRelO R (LiftRel R) (some (b, t)) (some (c, u))\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some (c, u))",
"state_before": "case intro.intro.intro.intro.some.some.mk.some.mk\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝¹ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝¹\nh2✝¹ : LiftRel R t✝¹ u✝\ns✝ u t✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t✝) (destruct u)\nc : α × WSeq α\nhc : some c ∈ destruct u\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nb : α\nt : WSeq α\nhb : some (b, t) ∈ destruct t✝\nt2 : LiftRelO R (LiftRel R) (some (b, t)) (some c)\nt1 : LiftRelO R (LiftRel R) (some (a, s)) (some (b, t))\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some c)",
"tactic": "cases' c with c u"
},
{
"state_after": "case intro.intro.intro.intro.some.some.mk.some.mk.mk.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝¹ u✝¹ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝¹\nh2✝¹ : LiftRel R t✝¹ u✝¹\ns✝ u✝ t✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t✝) (destruct u✝)\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nb : α\nt : WSeq α\nhb : some (b, t) ∈ destruct t✝\nc : α\nu : WSeq α\nhc : some (c, u) ∈ destruct u✝\nt2 : LiftRelO R (LiftRel R) (some (b, t)) (some (c, u))\nab : R a b\nst : LiftRel R s t\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some (c, u))",
"state_before": "case intro.intro.intro.intro.some.some.mk.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝¹ u✝¹ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝¹\nh2✝¹ : LiftRel R t✝¹ u✝¹\ns✝ u✝ t✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t✝) (destruct u✝)\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nb : α\nt : WSeq α\nhb : some (b, t) ∈ destruct t✝\nt1 : LiftRelO R (LiftRel R) (some (a, s)) (some (b, t))\nc : α\nu : WSeq α\nhc : some (c, u) ∈ destruct u✝\nt2 : LiftRelO R (LiftRel R) (some (b, t)) (some (c, u))\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some (c, u))",
"tactic": "cases' t1 with ab st"
},
{
"state_after": "case intro.intro.intro.intro.some.some.mk.some.mk.mk.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝¹ u✝¹ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝¹\nh2✝¹ : LiftRel R t✝¹ u✝¹\ns✝ u✝ t✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t✝) (destruct u✝)\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nb : α\nt : WSeq α\nhb : some (b, t) ∈ destruct t✝\nc : α\nu : WSeq α\nhc : some (c, u) ∈ destruct u✝\nab : R a b\nst : LiftRel R s t\nbc : R b c\ntu : LiftRel R t u\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some (c, u))",
"state_before": "case intro.intro.intro.intro.some.some.mk.some.mk.mk.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝¹ u✝¹ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝¹\nh2✝¹ : LiftRel R t✝¹ u✝¹\ns✝ u✝ t✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t✝) (destruct u✝)\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nb : α\nt : WSeq α\nhb : some (b, t) ∈ destruct t✝\nc : α\nu : WSeq α\nhc : some (c, u) ∈ destruct u✝\nt2 : LiftRelO R (LiftRel R) (some (b, t)) (some (c, u))\nab : R a b\nst : LiftRel R s t\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some (c, u))",
"tactic": "cases' t2 with bc tu"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.some.some.mk.some.mk.mk.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝¹ u✝¹ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝¹\nh2✝¹ : LiftRel R t✝¹ u✝¹\ns✝ u✝ t✝ : WSeq α\nh1✝ : LiftRel R s✝ t✝\nh2✝ : LiftRel R t✝ u✝\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t✝) (destruct u✝)\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nb : α\nt : WSeq α\nhb : some (b, t) ∈ destruct t✝\nc : α\nu : WSeq α\nhc : some (c, u) ∈ destruct u✝\nab : R a b\nst : LiftRel R s t\nbc : R b c\ntu : LiftRel R t u\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some (c, u))",
"tactic": "exact ⟨H ab bc, t, st, tu⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.some.some.mk.none\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nH : Transitive R\ns✝¹ t✝ u✝ : WSeq α\nh1✝¹ : LiftRel R s✝¹ t✝\nh2✝¹ : LiftRel R t✝ u✝\ns✝ u t : WSeq α\nh1✝ : LiftRel R s✝ t\nh2✝ : LiftRel R t u\nh1 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t)\nh2 : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct u)\nc : α × WSeq α\nhc : some c ∈ destruct u\na : α\ns : WSeq α\nha : some (a, s) ∈ destruct s✝\nhb : none ∈ destruct t\nt2 : LiftRelO R (LiftRel R) none (some c)\nt1 : LiftRelO R (LiftRel R) (some (a, s)) none\n⊢ LiftRelO R (fun s u => ∃ t, LiftRel R s t ∧ LiftRel R t u) (some (a, s)) (some c)",
"tactic": "cases t1"
}
] |
[
599,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
|
SimpleGraph.end_hom_mk_of_mk
|
[
{
"state_after": "V : Type u\nG : SimpleGraph V\nK✝ L✝ L' M : Set V\ns : (j : (Finset V)ᵒᵖ) → (componentComplFunctor G).obj j\nsec : s ∈ SimpleGraph.end G\nK L : (Finset V)ᵒᵖ\nh : L ⟶ K\nv : V\nvnL : ¬v ∈ L.unop\nhs : s L = componentComplMk G vnL\n⊢ (componentComplFunctor G).map h (componentComplMk G vnL) = componentComplMk G (_ : ¬v ∈ fun a => a ∈ K.unop.val)",
"state_before": "V : Type u\nG : SimpleGraph V\nK✝ L✝ L' M : Set V\ns : (j : (Finset V)ᵒᵖ) → (componentComplFunctor G).obj j\nsec : s ∈ SimpleGraph.end G\nK L : (Finset V)ᵒᵖ\nh : L ⟶ K\nv : V\nvnL : ¬v ∈ L.unop\nhs : s L = componentComplMk G vnL\n⊢ s K = componentComplMk G (_ : ¬v ∈ fun a => a ∈ K.unop.val)",
"tactic": "rw [← sec h, hs]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nG : SimpleGraph V\nK✝ L✝ L' M : Set V\ns : (j : (Finset V)ᵒᵖ) → (componentComplFunctor G).obj j\nsec : s ∈ SimpleGraph.end G\nK L : (Finset V)ᵒᵖ\nh : L ⟶ K\nv : V\nvnL : ¬v ∈ L.unop\nhs : s L = componentComplMk G vnL\n⊢ (componentComplFunctor G).map h (componentComplMk G vnL) = componentComplMk G (_ : ¬v ∈ fun a => a ∈ K.unop.val)",
"tactic": "apply ComponentCompl.hom_mk _ (le_of_op_hom h : _ ⊆ _)"
}
] |
[
276,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Data/Set/Intervals/Infinite.lean
|
Set.Ioc_infinite
|
[] |
[
60,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.lt_max'_of_mem_erase_max'
|
[] |
[
1489,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1487,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.div_two_mul_two_add_one_of_odd
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nh : Odd n\n⊢ n / 2 * 2 + 1 = n",
"tactic": "rw [← odd_iff.mp h, div_add_mod']"
}
] |
[
235,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.of_leftInverse
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\nG : Type u_4\nα : Type ?u.38455\nM : Type u_2\nN : Type u_3\ninst✝³ : Monoid M\ninst✝² : Monoid N\ninst✝¹ : MonoidHomClass F M N\ninst✝ : MonoidHomClass G N M\nf : F\nx : M\ng : G\nhfg : LeftInverse ↑g ↑f\nh : IsUnit (↑f x)\n⊢ IsUnit x",
"tactic": "simpa only [hfg x] using h.map g"
}
] |
[
231,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.iInter_iUnion_of_monotone
|
[] |
[
1540,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1537,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean
|
nndist_eq_nnnorm_vsub
|
[] |
[
81,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.zero_testAgainstNN_apply
|
[
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝⁵ : MeasurableSpace Ω\nR : Type ?u.53064\ninst✝⁴ : SMul R ℝ≥0\ninst✝³ : SMul R ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : TopologicalSpace Ω\nf : Ω →ᵇ ℝ≥0\n⊢ testAgainstNN 0 f = 0",
"tactic": "simp only [testAgainstNN, toMeasure_zero, lintegral_zero_measure, ENNReal.zero_toNNReal]"
}
] |
[
366,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasDerivWithinAt.sinh
|
[] |
[
896,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_univ
|
[] |
[
702,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/Data/List/Cycle.lean
|
List.nextOr_mem
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\n⊢ d ∈ xs → nextOr xs x d ∈ xs",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nhd : d ∈ xs\n⊢ nextOr xs x d ∈ xs",
"tactic": "revert hd"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\n⊢ ∀ (xs' : List α), (∀ (x : α), x ∈ xs → x ∈ xs') → d ∈ xs' → nextOr xs x d ∈ xs'",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\n⊢ d ∈ xs → nextOr xs x d ∈ xs",
"tactic": "suffices ∀ (xs' : List α) (_ : ∀ x ∈ xs, x ∈ xs') (_ : d ∈ xs'), nextOr xs x d ∈ xs' by\n exact this xs fun _ => id"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs' : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\n⊢ nextOr xs x d ∈ xs'",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\n⊢ ∀ (xs' : List α), (∀ (x : α), x ∈ xs → x ∈ xs') → d ∈ xs' → nextOr xs x d ∈ xs'",
"tactic": "intro xs' hxs' hd"
},
{
"state_after": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\nhxs' : ∀ (x : α), x ∈ [] → x ∈ xs'\n⊢ nextOr [] x d ∈ xs'\n\ncase cons\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny : α\nys : List α\nih : (∀ (x : α), x ∈ ys → x ∈ xs') → nextOr ys x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: ys → x ∈ xs'\n⊢ nextOr (y :: ys) x d ∈ xs'",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs' : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\n⊢ nextOr xs x d ∈ xs'",
"tactic": "induction' xs with y ys ih"
},
{
"state_after": "case cons.nil\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny : α\nih : (∀ (x : α), x ∈ [] → x ∈ xs') → nextOr [] x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ [y] → x ∈ xs'\n⊢ nextOr [y] x d ∈ xs'\n\ncase cons.cons\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\n⊢ nextOr (y :: z :: zs) x d ∈ xs'",
"state_before": "case cons\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny : α\nys : List α\nih : (∀ (x : α), x ∈ ys → x ∈ xs') → nextOr ys x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: ys → x ∈ xs'\n⊢ nextOr (y :: ys) x d ∈ xs'",
"tactic": "cases' ys with z zs"
},
{
"state_after": "case cons.cons\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\n⊢ (if x = y then z else nextOr (z :: zs) x d) ∈ xs'",
"state_before": "case cons.cons\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\n⊢ nextOr (y :: z :: zs) x d ∈ xs'",
"tactic": "rw [nextOr]"
},
{
"state_after": "case cons.cons.inl\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\nh : x = y\n⊢ z ∈ xs'\n\ncase cons.cons.inr\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\nh : ¬x = y\n⊢ nextOr (z :: zs) x d ∈ xs'",
"state_before": "case cons.cons\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\n⊢ (if x = y then z else nextOr (z :: zs) x d) ∈ xs'",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nthis : ∀ (xs' : List α), (∀ (x : α), x ∈ xs → x ∈ xs') → d ∈ xs' → nextOr xs x d ∈ xs'\n⊢ d ∈ xs → nextOr xs x d ∈ xs",
"tactic": "exact this xs fun _ => id"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\nhxs' : ∀ (x : α), x ∈ [] → x ∈ xs'\n⊢ nextOr [] x d ∈ xs'",
"tactic": "exact hd"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny : α\nih : (∀ (x : α), x ∈ [] → x ∈ xs') → nextOr [] x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ [y] → x ∈ xs'\n⊢ nextOr [y] x d ∈ xs'",
"tactic": "exact hd"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.inl\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\nh : x = y\n⊢ z ∈ xs'",
"tactic": "exact hxs' _ (mem_cons_of_mem _ (mem_cons_self _ _))"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.inr\nα : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\nxs' : List α\nhxs'✝ : ∀ (x : α), x ∈ xs → x ∈ xs'\nhd : d ∈ xs'\ny z : α\nzs : List α\nih : (∀ (x : α), x ∈ z :: zs → x ∈ xs') → nextOr (z :: zs) x d ∈ xs'\nhxs' : ∀ (x : α), x ∈ y :: z :: zs → x ∈ xs'\nh : ¬x = y\n⊢ nextOr (z :: zs) x d ∈ xs'",
"tactic": "exact ih fun _ h => hxs' _ (mem_cons_of_mem _ h)"
}
] |
[
110,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.eventually_hasSum_sub
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.458834\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\n⊢ ∀ᶠ (y : E) in 𝓝 x, HasSum (fun n => ↑(p n) fun x_1 => y - x) (f y)",
"tactic": "filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub"
}
] |
[
497,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Algebra/BigOperators/Option.lean
|
Finset.prod_eraseNone
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in ↑eraseNone s, f x = ∏ x in s, Option.elim' 1 f x",
"tactic": "classical calc\n (∏ x in eraseNone s, f x) = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n _ = ∏ x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n _ = ∏ x in s, Option.elim' 1 f x := prod_erase _ rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in ↑eraseNone s, f x = ∏ x in s, Option.elim' 1 f x",
"tactic": "calc\n(∏ x in eraseNone s, f x) = ∏ x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n(prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n_ = ∏ x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n_ = ∏ x in s, Option.elim' 1 f x := prod_erase _ rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝ : CommMonoid M\nf : α → M\ns : Finset (Option α)\n⊢ ∏ x in map Embedding.some (↑eraseNone s), Option.elim' 1 f x = ∏ x in erase s none, Option.elim' 1 f x",
"tactic": "rw [map_some_eraseNone]"
}
] |
[
42,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
IsSubmonoid.preimage
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\n⊢ f 1 ∈ s",
"tactic": "(rw [IsMonoidHom.map_one hf]; exact hs.one_mem)"
},
{
"state_after": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\n⊢ 1 ∈ s",
"state_before": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\n⊢ f 1 ∈ s",
"tactic": "rw [IsMonoidHom.map_one hf]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\n⊢ 1 ∈ s",
"tactic": "exact hs.one_mem"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\na b : M\nha : f a ∈ s\nhb : f b ∈ s\n⊢ f (a * b) ∈ s",
"tactic": "(rw [IsMonoidHom.map_mul' hf]; exact hs.mul_mem ha hb)"
},
{
"state_after": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\na b : M\nha : f a ∈ s\nhb : f b ∈ s\n⊢ f a * f b ∈ s",
"state_before": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\na b : M\nha : f a ∈ s\nhb : f b ∈ s\n⊢ f (a * b) ∈ s",
"tactic": "rw [IsMonoidHom.map_mul' hf]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝² : Monoid M\ns✝ : Set M\nA : Type ?u.24553\ninst✝¹ : AddMonoid A\nt : Set A\nN : Type u_1\ninst✝ : Monoid N\nf : M → N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\na b : M\nha : f a ∈ s\nhb : f b ∈ s\n⊢ f a * f b ∈ s",
"tactic": "exact hs.mul_mem ha hb"
}
] |
[
183,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Data/List/Sort.lean
|
List.Sorted.rel_of_mem_take_of_mem_drop
|
[
{
"state_after": "case intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk : ℕ\nx : α\nhx : x ∈ take k l\niy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\n⊢ r x (get (drop k l) { val := iy, isLt := hiy })",
"state_before": "α : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk : ℕ\nx y : α\nhx : x ∈ take k l\nhy : y ∈ drop k l\n⊢ r x y",
"tactic": "obtain ⟨⟨iy, hiy⟩, rfl⟩ := get_of_mem hy"
},
{
"state_after": "case intro.mk.intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk iy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\nix : ℕ\nhix : ix < length (take k l)\nhx : get (take k l) { val := ix, isLt := hix } ∈ take k l\n⊢ r (get (take k l) { val := ix, isLt := hix }) (get (drop k l) { val := iy, isLt := hiy })",
"state_before": "case intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk : ℕ\nx : α\nhx : x ∈ take k l\niy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\n⊢ r x (get (drop k l) { val := iy, isLt := hiy })",
"tactic": "obtain ⟨⟨ix, hix⟩, rfl⟩ := get_of_mem hx"
},
{
"state_after": "case intro.mk.intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk iy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\nix : ℕ\nhix : ix < length (take k l)\nhx : get (take k l) { val := ix, isLt := hix } ∈ take k l\n⊢ r (get l { val := ↑{ val := ix, isLt := hix }, isLt := (_ : ↑{ val := ix, isLt := hix } < length l) })\n (get l { val := k + ↑{ val := iy, isLt := hiy }, isLt := (_ : k + ↑{ val := iy, isLt := hiy } < length l) })",
"state_before": "case intro.mk.intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk iy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\nix : ℕ\nhix : ix < length (take k l)\nhx : get (take k l) { val := ix, isLt := hix } ∈ take k l\n⊢ r (get (take k l) { val := ix, isLt := hix }) (get (drop k l) { val := iy, isLt := hiy })",
"tactic": "rw [get_take', get_drop']"
},
{
"state_after": "case intro.mk.intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk iy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\nix : ℕ\nhix✝ : ix < length (take k l)\nhix : ix < min k (length l)\nhx : get (take k l) { val := ix, isLt := hix✝ } ∈ take k l\n⊢ r (get l { val := ↑{ val := ix, isLt := hix✝ }, isLt := (_ : ↑{ val := ix, isLt := hix✝ } < length l) })\n (get l { val := k + ↑{ val := iy, isLt := hiy }, isLt := (_ : k + ↑{ val := iy, isLt := hiy } < length l) })",
"state_before": "case intro.mk.intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk iy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\nix : ℕ\nhix : ix < length (take k l)\nhx : get (take k l) { val := ix, isLt := hix } ∈ take k l\n⊢ r (get l { val := ↑{ val := ix, isLt := hix }, isLt := (_ : ↑{ val := ix, isLt := hix } < length l) })\n (get l { val := k + ↑{ val := iy, isLt := hiy }, isLt := (_ : k + ↑{ val := iy, isLt := hiy } < length l) })",
"tactic": "rw [length_take] at hix"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.mk\nα : Type uu\nr : α → α → Prop\na : α\nl✝ l : List α\nh : Sorted r l\nk iy : ℕ\nhiy : iy < length (drop k l)\nhy : get (drop k l) { val := iy, isLt := hiy } ∈ drop k l\nix : ℕ\nhix✝ : ix < length (take k l)\nhix : ix < min k (length l)\nhx : get (take k l) { val := ix, isLt := hix✝ } ∈ take k l\n⊢ r (get l { val := ↑{ val := ix, isLt := hix✝ }, isLt := (_ : ↑{ val := ix, isLt := hix✝ } < length l) })\n (get l { val := k + ↑{ val := iy, isLt := hiy }, isLt := (_ : k + ↑{ val := iy, isLt := hiy } < length l) })",
"tactic": "exact h.rel_nthLe_of_lt _ _ (ix.lt_add_right _ _ (lt_min_iff.mp hix).left)"
}
] |
[
138,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Order/Basic.lean
|
le_of_forall_le_of_dense
|
[] |
[
1310,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1306,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.withDensity_mul
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1814114\nγ : Type ?u.1814117\nδ : Type ?u.1814120\nm m0 : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(Measure.withDensity μ (f * g)) s = ↑↑(Measure.withDensity (Measure.withDensity μ f) g) s",
"state_before": "α : Type u_1\nβ : Type ?u.1814114\nγ : Type ?u.1814117\nδ : Type ?u.1814120\nm m0 : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ Measure.withDensity μ (f * g) = Measure.withDensity (Measure.withDensity μ f) g",
"tactic": "ext1 s hs"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1814114\nγ : Type ?u.1814117\nδ : Type ?u.1814120\nm m0 : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(Measure.withDensity μ (f * g)) s = ↑↑(Measure.withDensity (Measure.withDensity μ f) g) s",
"tactic": "simp [withDensity_apply _ hs, restrict_withDensity hs,\n lintegral_withDensity_eq_lintegral_mul _ hf hg]"
}
] |
[
1898,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1894,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.support_symmDiff_support_subset_support_add
|
[
{
"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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support p \\ support q ∪ support q \\ support p ⊆ support (p + q)",
"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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support p ∆ support q ⊆ support (p + q)",
"tactic": "rw [symmDiff_def, Finset.sup_eq_union]"
},
{
"state_after": "case hs\nR : 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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support p \\ support q ⊆ support (p + q)\n\ncase a\nR : 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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support q \\ support p ⊆ support (p + q)",
"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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support p \\ support q ∪ support q \\ support p ⊆ support (p + q)",
"tactic": "apply Finset.union_subset"
},
{
"state_after": "no goals",
"state_before": "case hs\nR : 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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support p \\ support q ⊆ support (p + q)",
"tactic": "exact support_sdiff_support_subset_support_add p q"
},
{
"state_after": "case a\nR : 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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support q \\ support p ⊆ support (q + p)",
"state_before": "case a\nR : 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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support q \\ support p ⊆ support (p + q)",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : 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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support q \\ support p ⊆ support (q + p)",
"tactic": "exact support_sdiff_support_subset_support_add q p"
}
] |
[
760,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.xgcdAux_fst
|
[
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y x✝ s✝ t✝ s'✝ t'✝ : R\n⊢ (xgcdAux 0 s✝ t✝ x✝ s'✝ t'✝).fst = gcd 0 x✝",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\n⊢ ∀ (x s t s' t' : R), (xgcdAux 0 s t x s' t').fst = gcd 0 x",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y x✝ s✝ t✝ s'✝ t'✝ : R\n⊢ (xgcdAux 0 s✝ t✝ x✝ s'✝ t'✝).fst = gcd 0 x✝",
"tactic": "rw [xgcd_zero_left, gcd_zero_left]"
},
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx✝ y✝ x y : R\nh : x ≠ 0\nIH : ∀ (s t s' t' : R), (xgcdAux (y % x) s t x s' t').fst = gcd (y % x) x\ns t s' t' : R\n⊢ gcd (y % x) x = gcd x y",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx✝ y✝ x y : R\nh : x ≠ 0\nIH : ∀ (s t s' t' : R), (xgcdAux (y % x) s t x s' t').fst = gcd (y % x) x\ns t s' t' : R\n⊢ (xgcdAux x s t y s' t').fst = gcd x y",
"tactic": "simp only [xgcdAux_rec h, if_neg h, IH]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx✝ y✝ x y : R\nh : x ≠ 0\nIH : ∀ (s t s' t' : R), (xgcdAux (y % x) s t x s' t').fst = gcd (y % x) x\ns t s' t' : R\n⊢ gcd (y % x) x = gcd x y",
"tactic": "rw [← gcd_val]"
}
] |
[
203,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.equiv_refl
|
[] |
[
758,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleHom.coe_sub
|
[] |
[
888,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
887,
1
] |
Mathlib/GroupTheory/Coset.lean
|
rightCosetEquivalence_rel
|
[] |
[
116,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
|
CategoryTheory.GrothendieckTopology.sheafificationWhiskerLeftIso_hom_app
|
[
{
"state_after": "C : Type u\ninst✝¹⁰ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category D\nE : Type w₂\ninst✝⁸ : Category E\nF✝ : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ P : Cᵒᵖ ⥤ D\nF : D ⥤ E\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom ≫ 𝟙 (plusObj J (plusObj J (P ⋙ F))) =\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom",
"state_before": "C : Type u\ninst✝¹⁰ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category D\nE : Type w₂\ninst✝⁸ : Category E\nF✝ : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ P : Cᵒᵖ ⥤ D\nF : D ⥤ E\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (sheafificationWhiskerLeftIso J P).hom.app F = (sheafifyCompIso J F P).hom",
"tactic": "dsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹⁰ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁹ : Category D\nE : Type w₂\ninst✝⁸ : Category E\nF✝ : D ⥤ E\ninst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst✝⁵ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝⁴ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E\ninst✝³ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F✝\ninst✝² : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F✝\nP✝ P : Cᵒᵖ ⥤ D\nF : D ⥤ E\ninst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F\ninst✝ :\n (F : D ⥤ E) →\n (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\n⊢ (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom ≫ 𝟙 (plusObj J (plusObj J (P ⋙ F))) =\n (plusCompIso J F (plusObj J P)).hom ≫ plusMap J (plusCompIso J F P).hom",
"tactic": "rw [Category.comp_id]"
}
] |
[
88,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bsup_eq_of_brange_eq
|
[] |
[
1574,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1572,
1
] |
Mathlib/Topology/PathConnected.lean
|
Joined.trans
|
[] |
[
789,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_sub_le_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\n⊢ abs (a - b) ≤ c ↔ a - b ≤ c ∧ b - a ≤ c",
"tactic": "rw [abs_le, neg_le_sub_iff_le_add, sub_le_iff_le_add', and_comm, sub_le_iff_le_add']"
}
] |
[
285,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Algebra/MonoidAlgebra/Support.lean
|
MonoidAlgebra.mem_span_support
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\ninst✝¹ : Semiring k\ninst✝ : MulOneClass G\nf : MonoidAlgebra k G\n⊢ f ∈ Submodule.span k (↑(of k G) '' ↑f.support)",
"tactic": "erw [of, MonoidHom.coe_mk, ← supported_eq_span_single, Finsupp.mem_supported]"
}
] |
[
110,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
div_neg_eq_neg_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.27142\nβ : Type ?u.27145\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b✝ a b : K\n⊢ b / -a = b * (1 / -a)",
"tactic": "rw [← inv_eq_one_div, division_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.27142\nβ : Type ?u.27145\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b✝ a b : K\n⊢ b * (1 / -a) = b * -(1 / a)",
"tactic": "rw [one_div_neg_eq_neg_one_div]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.27142\nβ : Type ?u.27145\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b✝ a b : K\n⊢ b * -(1 / a) = -(b * (1 / a))",
"tactic": "rw [neg_mul_eq_mul_neg]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.27142\nβ : Type ?u.27145\nK : Type u_1\ninst✝¹ : DivisionMonoid K\ninst✝ : HasDistribNeg K\na✝ b✝ a b : K\n⊢ -(b * (1 / a)) = -(b / a)",
"tactic": "rw [mul_one_div]"
}
] |
[
113,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.normalizedFactors_prod_eq
|
[
{
"state_after": "case empty\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns : Multiset α\nhs✝ : ∀ (a : α), a ∈ s → Irreducible a\nhs : ∀ (a : α), a ∈ 0 → Irreducible a\n⊢ normalizedFactors (Multiset.prod 0) = Multiset.map (↑normalize) 0\n\ncase cons\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns : Multiset α\nhs : ∀ (a : α), a ∈ s → Irreducible a\n⊢ normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s",
"tactic": "induction' s using Multiset.induction with a s ih"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns : Multiset α\nhs✝ : ∀ (a : α), a ∈ s → Irreducible a\nhs : ∀ (a : α), a ∈ 0 → Irreducible a\n⊢ normalizedFactors (Multiset.prod 0) = Multiset.map (↑normalize) 0",
"tactic": "rw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero]"
},
{
"state_after": "case cons\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"state_before": "case cons\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"tactic": "have ia := hs a (Multiset.mem_cons_self a _)"
},
{
"state_after": "case cons\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\nib : ∀ (b : α), b ∈ s → Irreducible b\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"state_before": "case cons\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"tactic": "have ib := fun b h => hs b (Multiset.mem_cons_of_mem h)"
},
{
"state_after": "case cons.inl\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns : Multiset α\nhs✝ : ∀ (a : α), a ∈ s → Irreducible a\na : α\nia : Irreducible a\nih : (∀ (a : α), a ∈ 0 → Irreducible a) → normalizedFactors (Multiset.prod 0) = Multiset.map (↑normalize) 0\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ 0 → Irreducible a_1\nib : ∀ (b : α), b ∈ 0 → Irreducible b\n⊢ normalizedFactors (Multiset.prod (a ::ₘ 0)) = Multiset.map (↑normalize) (a ::ₘ 0)\n\ncase cons.inr.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\nib : ∀ (b : α), b ∈ s → Irreducible b\nb : α\nhb : b ∈ s\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"state_before": "case cons\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\nib : ∀ (b : α), b ∈ s → Irreducible b\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"tactic": "obtain rfl | ⟨b, hb⟩ := s.empty_or_exists_mem"
},
{
"state_after": "case cons.inr.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\nib : ∀ (b : α), b ∈ s → Irreducible b\nb : α\nhb : b ∈ s\nthis : Nontrivial α\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"state_before": "case cons.inr.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\nib : ∀ (b : α), b ∈ s → Irreducible b\nb : α\nhb : b ∈ s\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"tactic": "haveI := nontrivial_of_ne b 0 (ib b hb).ne_zero"
},
{
"state_after": "no goals",
"state_before": "case cons.inr.intro\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns✝ : Multiset α\nhs✝ : ∀ (a : α), a ∈ s✝ → Irreducible a\na : α\ns : Multiset α\nih : (∀ (a : α), a ∈ s → Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (↑normalize) s\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Irreducible a_1\nia : Irreducible a\nib : ∀ (b : α), b ∈ s → Irreducible b\nb : α\nhb : b ∈ s\nthis : Nontrivial α\n⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (↑normalize) (a ::ₘ s)",
"tactic": "rw [Multiset.prod_cons, Multiset.map_cons,\n normalizedFactors_mul ia.ne_zero (Multiset.prod_ne_zero fun h => (ib 0 h).ne_zero rfl),\n normalizedFactors_irreducible ia, ih ib, Multiset.singleton_add]"
},
{
"state_after": "no goals",
"state_before": "case cons.inl\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\ns : Multiset α\nhs✝ : ∀ (a : α), a ∈ s → Irreducible a\na : α\nia : Irreducible a\nih : (∀ (a : α), a ∈ 0 → Irreducible a) → normalizedFactors (Multiset.prod 0) = Multiset.map (↑normalize) 0\nhs : ∀ (a_1 : α), a_1 ∈ a ::ₘ 0 → Irreducible a_1\nib : ∀ (b : α), b ∈ 0 → Irreducible b\n⊢ normalizedFactors (Multiset.prod (a ::ₘ 0)) = Multiset.map (↑normalize) (a ::ₘ 0)",
"tactic": "rw [Multiset.cons_zero, Multiset.prod_singleton, Multiset.map_singleton,\n normalizedFactors_irreducible ia]"
}
] |
[
718,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/CategoryTheory/Adjunction/Mates.lean
|
CategoryTheory.transferNatTransSelf_comp
|
[
{
"state_after": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : L₂ ⟶ L₁\ng : L₃ ⟶ L₂\nx✝ : D\n⊢ (↑(transferNatTransSelf adj₁ adj₂) f ≫ ↑(transferNatTransSelf adj₂ adj₃) g).app x✝ =\n (↑(transferNatTransSelf adj₁ adj₃) (g ≫ f)).app x✝",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : L₂ ⟶ L₁\ng : L₃ ⟶ L₂\n⊢ ↑(transferNatTransSelf adj₁ adj₂) f ≫ ↑(transferNatTransSelf adj₂ adj₃) g = ↑(transferNatTransSelf adj₁ adj₃) (g ≫ f)",
"tactic": "ext"
},
{
"state_after": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : L₂ ⟶ L₁\ng : L₃ ⟶ L₂\nx✝ : D\n⊢ (𝟙 (R₁.obj x✝) ≫\n (adj₂.unit.app (R₁.obj x✝) ≫\n R₂.map ((𝟙 (L₂.obj (R₁.obj x✝)) ≫ f.app (R₁.obj x✝) ≫ 𝟙 (L₁.obj (R₁.obj x✝))) ≫ adj₁.counit.app x✝)) ≫\n 𝟙 (R₂.obj x✝)) ≫\n 𝟙 (R₂.obj x✝) ≫\n (adj₃.unit.app (R₂.obj x✝) ≫\n R₃.map ((𝟙 (L₃.obj (R₂.obj x✝)) ≫ g.app (R₂.obj x✝) ≫ 𝟙 (L₂.obj (R₂.obj x✝))) ≫ adj₂.counit.app x✝)) ≫\n 𝟙 (R₃.obj x✝) =\n 𝟙 (R₁.obj x✝) ≫\n (adj₃.unit.app (R₁.obj x✝) ≫\n R₃.map\n ((𝟙 (L₃.obj (R₁.obj x✝)) ≫ (g.app (R₁.obj x✝) ≫ f.app (R₁.obj x✝)) ≫ 𝟙 (L₁.obj (R₁.obj x✝))) ≫\n adj₁.counit.app x✝)) ≫\n 𝟙 (R₃.obj x✝)",
"state_before": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : L₂ ⟶ L₁\ng : L₃ ⟶ L₂\nx✝ : D\n⊢ (↑(transferNatTransSelf adj₁ adj₂) f ≫ ↑(transferNatTransSelf adj₂ adj₃) g).app x✝ =\n (↑(transferNatTransSelf adj₁ adj₃) (g ≫ f)).app x✝",
"tactic": "dsimp [transferNatTransSelf, transferNatTrans]"
},
{
"state_after": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : L₂ ⟶ L₁\ng : L₃ ⟶ L₂\nx✝ : D\n⊢ (adj₂.unit.app (R₁.obj x✝) ≫ R₂.map (f.app (R₁.obj x✝) ≫ adj₁.counit.app x✝)) ≫\n adj₃.unit.app (R₂.obj x✝) ≫ R₃.map (g.app (R₂.obj x✝) ≫ adj₂.counit.app x✝) =\n adj₃.unit.app (R₁.obj x✝) ≫ R₃.map ((g.app (R₁.obj x✝) ≫ f.app (R₁.obj x✝)) ≫ adj₁.counit.app x✝)",
"state_before": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : L₂ ⟶ L₁\ng : L₃ ⟶ L₂\nx✝ : D\n⊢ (𝟙 (R₁.obj x✝) ≫\n (adj₂.unit.app (R₁.obj x✝) ≫\n R₂.map ((𝟙 (L₂.obj (R₁.obj x✝)) ≫ f.app (R₁.obj x✝) ≫ 𝟙 (L₁.obj (R₁.obj x✝))) ≫ adj₁.counit.app x✝)) ≫\n 𝟙 (R₂.obj x✝)) ≫\n 𝟙 (R₂.obj x✝) ≫\n (adj₃.unit.app (R₂.obj x✝) ≫\n R₃.map ((𝟙 (L₃.obj (R₂.obj x✝)) ≫ g.app (R₂.obj x✝) ≫ 𝟙 (L₂.obj (R₂.obj x✝))) ≫ adj₂.counit.app x✝)) ≫\n 𝟙 (R₃.obj x✝) =\n 𝟙 (R₁.obj x✝) ≫\n (adj₃.unit.app (R₁.obj x✝) ≫\n R₃.map\n ((𝟙 (L₃.obj (R₁.obj x✝)) ≫ (g.app (R₁.obj x✝) ≫ f.app (R₁.obj x✝)) ≫ 𝟙 (L₁.obj (R₁.obj x✝))) ≫\n adj₁.counit.app x✝)) ≫\n 𝟙 (R₃.obj x✝)",
"tactic": "simp only [id_comp, comp_id]"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nf : L₂ ⟶ L₁\ng : L₃ ⟶ L₂\nx✝ : D\n⊢ (adj₂.unit.app (R₁.obj x✝) ≫ R₂.map (f.app (R₁.obj x✝) ≫ adj₁.counit.app x✝)) ≫\n adj₃.unit.app (R₂.obj x✝) ≫ R₃.map (g.app (R₂.obj x✝) ≫ adj₂.counit.app x✝) =\n adj₃.unit.app (R₁.obj x✝) ≫ R₃.map ((g.app (R₁.obj x✝) ≫ f.app (R₁.obj x✝)) ≫ adj₁.counit.app x✝)",
"tactic": "rw [← adj₃.unit_naturality_assoc, ← R₃.map_comp, g.naturality_assoc, L₂.map_comp, assoc,\n adj₂.counit_naturality, adj₂.left_triangle_components_assoc, assoc]"
}
] |
[
205,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.relindex_bot_left_eq_card
|
[] |
[
262,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
normalize_apply
|
[] |
[
136,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.single_algebraMap_eq_algebraMap_mul_of
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.1010908\nA : Type u_1\ninst✝³ : CommSemiring k\ninst✝² : Semiring A\ninst✝¹ : Algebra k A\ninst✝ : Monoid G\na : G\nb : k\n⊢ single a (↑(algebraMap k A) b) = ↑(algebraMap k (MonoidAlgebra A G)) b * ↑(of A G) a",
"tactic": "simp"
}
] |
[
834,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/CategoryTheory/Preadditive/Mat.lean
|
CategoryTheory.Mat_.id_def
|
[] |
[
128,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
|
Profinite.exists_locallyConstant_finite_aux
|
[
{
"state_after": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "J : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "cases nonempty_fintype α"
},
{
"state_after": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1"
},
{
"state_after": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let ff := (f.map ι).flip"
},
{
"state_after": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nhff : ∀ (a : α), ∃ j g, ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "have hff := fun a : α => exists_locallyConstant_fin_two _ hC (ff a)"
},
{
"state_after": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nhff : ∀ (a : α), ∃ j g, ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "choose j g h using hff"
},
{
"state_after": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let G : Finset J := Finset.univ.image j"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists G"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "have hj : ∀ a, j a ∈ (Finset.univ.image j : Finset J) := by\n intro a\n simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply]"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let fs : ∀ a : α, j0 ⟶ j a := fun a => (hj0 (hj a)).some"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let gg : α → LocallyConstant (F.obj j0) (Fin 2) := fun a => (g a).comap (F.map (fs _))"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let ggg := LocallyConstant.unflip gg"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ ∃ j g,\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "refine' ⟨j0, ggg, _⟩"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f))\n⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"tactic": "have : f.map ι = LocallyConstant.unflip (f.map ι).flip := by simp"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f))\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f))\n⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"tactic": "rw [this]"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis : LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f))\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"tactic": "clear this"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis :\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"tactic": "have :\n LocallyConstant.comap (C.π.app j0) ggg =\n LocallyConstant.unflip (LocallyConstant.comap (C.π.app j0) ggg).flip :=\n by simp"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis :\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis :\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg",
"tactic": "rw [this]"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\nthis :\n LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))",
"tactic": "clear this"
},
{
"state_after": "case intro.intro.e_f\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.flip (LocallyConstant.map ι f) =\n LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg)",
"state_before": "case intro.intro\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f)) =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))",
"tactic": "congr 1"
},
{
"state_after": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ LocallyConstant.flip (LocallyConstant.map ι f) a =\n LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg) a",
"state_before": "case intro.intro.e_f\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.flip (LocallyConstant.map ι f) =\n LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg)",
"tactic": "ext1 a"
},
{
"state_after": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ ff a = LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg) a",
"state_before": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ LocallyConstant.flip (LocallyConstant.map ι f) a =\n LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg) a",
"tactic": "change ff a = _"
},
{
"state_after": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a) =\n LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg) a",
"state_before": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ ff a = LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg) a",
"tactic": "rw [h]"
},
{
"state_after": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a) =\n LocallyConstant.flip\n (LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n (LocallyConstant.unflip fun a =>\n LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)))\n a",
"state_before": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a) =\n LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg) a",
"tactic": "dsimp"
},
{
"state_after": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)) x =\n ↑(LocallyConstant.flip\n (LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n (LocallyConstant.unflip fun a =>\n LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)))\n a)\n x",
"state_before": "case intro.intro.e_f.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\n⊢ LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a) =\n LocallyConstant.flip\n (LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n (LocallyConstant.unflip fun a =>\n LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)))\n a",
"tactic": "ext1 x"
},
{
"state_after": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ (↑(g a) ∘ (forget Profinite).map (C.π.1 (j a))) x =\n ↑(LocallyConstant.flip\n (LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n (LocallyConstant.unflip fun a =>\n LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)))\n a)\n x",
"state_before": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)) x =\n ↑(LocallyConstant.flip\n (LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n (LocallyConstant.unflip fun a =>\n LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)))\n a)\n x",
"tactic": "rw [LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app (j a)).continuous]"
},
{
"state_after": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n ↑(LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n {\n toFun := fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))\n x,\n isLocallyConstant :=\n (_ :\n IsLocallyConstant fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))\n (g a))\n x) })\n x a",
"state_before": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ (↑(g a) ∘ (forget Profinite).map (C.π.1 (j a))) x =\n ↑(LocallyConstant.flip\n (LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n (LocallyConstant.unflip fun a =>\n LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a)))\n a)\n x",
"tactic": "dsimp [LocallyConstant.flip, LocallyConstant.unflip]"
},
{
"state_after": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n (↑{\n toFun := fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))\n x,\n isLocallyConstant :=\n (_ :\n IsLocallyConstant fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))\n (g a))\n x) } ∘\n (forget Profinite).map (C.π.1 j0))\n x a",
"state_before": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n ↑(LocallyConstant.comap ((forget Profinite).map (C.π.app j0))\n {\n toFun := fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))\n x,\n isLocallyConstant :=\n (_ :\n IsLocallyConstant fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))\n (g a))\n x) })\n x a",
"tactic": "rw [LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app j0).continuous]"
},
{
"state_after": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))\n ((forget Profinite).map (C.π.1 j0) x)",
"state_before": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n (↑{\n toFun := fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))\n x,\n isLocallyConstant :=\n (_ :\n IsLocallyConstant fun x a =>\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))\n (g a))\n x) } ∘\n (forget Profinite).map (C.π.1 j0))\n x a",
"tactic": "dsimp"
},
{
"state_after": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n (↑(g a) ∘ (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))\n ((forget Profinite).map (C.π.1 j0) x)\n\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ Continuous ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))",
"state_before": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n ↑(LocallyConstant.comap ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a))))) (g a))\n ((forget Profinite).map (C.π.1 j0) x)",
"tactic": "rw [LocallyConstant.coe_comap ((forget Profinite).map _) _ _]"
},
{
"state_after": "J : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\na : α\n⊢ j a ∈ Finset.image j Finset.univ",
"state_before": "J : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\n⊢ ∀ (a : α), j a ∈ Finset.image j Finset.univ",
"tactic": "intro a"
},
{
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"state_before": "J : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\na : α\n⊢ j a ∈ Finset.image j Finset.univ",
"tactic": "simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply]"
},
{
"state_after": "no goals",
"state_before": "J : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.map ι f = LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.map ι f))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "J : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\n⊢ LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg =\n LocallyConstant.unflip (LocallyConstant.flip (LocallyConstant.comap ((forget Profinite).map (C.π.app j0)) ggg))",
"tactic": "simp"
},
{
"state_after": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n ↑(g a)\n ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) ((forget Profinite).map (C.π.1 j0) x))",
"state_before": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n (↑(g a) ∘ (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))\n ((forget Profinite).map (C.π.1 j0) x)",
"tactic": "dsimp"
},
{
"state_after": "case intro.intro.e_f.h.h.h.e'_6\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ (forget Profinite).map (C.π.1 (j a)) x =\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) ((forget Profinite).map (C.π.1 j0) x)",
"state_before": "case intro.intro.e_f.h.h\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ ↑(g a) ((forget Profinite).map (C.π.1 (j a)) x) =\n ↑(g a)\n ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) ((forget Profinite).map (C.π.1 j0) x))",
"tactic": "congr! 1"
},
{
"state_after": "case intro.intro.e_f.h.h.h.e'_6\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ (forget Profinite).map (C.π.1 (j a)) x = (forget Profinite).map (C.π.app j0 ≫ F.map (fs a)) x",
"state_before": "case intro.intro.e_f.h.h.h.e'_6\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ (forget Profinite).map (C.π.1 (j a)) x =\n (forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))) ((forget Profinite).map (C.π.1 j0) x)",
"tactic": "change _ = (C.π.app j0 ≫ F.map (fs a)) x"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.e_f.h.h.h.e'_6\nJ : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ (forget Profinite).map (C.π.1 (j a)) x = (forget Profinite).map (C.π.app j0 ≫ F.map (fs a)) x",
"tactic": "rw [C.w]"
},
{
"state_after": "no goals",
"state_before": "J : Type u\ninst✝² : SmallCategory J\ninst✝¹ : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝ : Finite α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\nval✝ : Fintype α\nι : α → α → Fin 2 := fun x y => if x = y then 0 else 1\nff : α → LocallyConstant (↑C.pt.toCompHaus.toTop) (Fin 2) := LocallyConstant.flip (LocallyConstant.map ι f)\nj : α → J\ng : (a : α) → LocallyConstant (↑(F.obj (j a)).toCompHaus.toTop) (Fin 2)\nh : ∀ (a : α), ff a = LocallyConstant.comap ((forget Profinite).map (C.π.app (j a))) (g a)\nG : Finset J := Finset.image j Finset.univ\nj0 : J\nhj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)\nhj : ∀ (a : α), j a ∈ Finset.image j Finset.univ\nfs : (a : α) → j0 ⟶ j a := fun a => Nonempty.some (_ : Nonempty (j0 ⟶ j a))\ngg : α → LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (Fin 2) :=\n fun a => LocallyConstant.comap ((forget Profinite).map (F.map (fs a))) (g a)\nggg : LocallyConstant (↑(F.obj j0).toCompHaus.toTop) (α → Fin 2) := LocallyConstant.unflip gg\na : α\nx : (forget Profinite).obj C.pt\n⊢ Continuous ((forget Profinite).map (F.map (Nonempty.some (_ : Nonempty (j0 ⟶ j a)))))",
"tactic": "exact (F.map _).continuous"
}
] |
[
182,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.noZeroSMulDivisors_of_prime_of_degree_ne_zero
|
[] |
[
381,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Data/Int/Parity.lean
|
Int.emod_two_ne_one
|
[
{
"state_after": "no goals",
"state_before": "m n : ℤ\n⊢ ¬n % 2 = 1 ↔ n % 2 = 0",
"tactic": "cases' emod_two_eq_zero_or_one n with h h <;> simp [h]"
}
] |
[
30,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
29,
1
] |
Mathlib/CategoryTheory/Limits/Presheaf.lean
|
CategoryTheory.ColimitAdj.restrictYonedaHomEquiv_natural
|
[
{
"state_after": "case w.h.h\nC : Type u₁\ninst✝¹ : SmallCategory C\nℰ : Type u₂\ninst✝ : Category ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type u₁\nE₁ E₂ : ℰ\ng : E₁ ⟶ E₂\nc : Cocone ((CategoryOfElements.π P).leftOp ⋙ A)\nt : IsColimit c\nk : c.pt ⟶ E₁\nx : Cᵒᵖ\nX : P.obj x\n⊢ (↑(restrictYonedaHomEquiv A P E₂ t) (k ≫ g)).app x X =\n (↑(restrictYonedaHomEquiv A P E₁ t) k ≫ (restrictedYoneda A).map g).app x X",
"state_before": "C : Type u₁\ninst✝¹ : SmallCategory C\nℰ : Type u₂\ninst✝ : Category ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type u₁\nE₁ E₂ : ℰ\ng : E₁ ⟶ E₂\nc : Cocone ((CategoryOfElements.π P).leftOp ⋙ A)\nt : IsColimit c\nk : c.pt ⟶ E₁\n⊢ ↑(restrictYonedaHomEquiv A P E₂ t) (k ≫ g) = ↑(restrictYonedaHomEquiv A P E₁ t) k ≫ (restrictedYoneda A).map g",
"tactic": "ext x X"
},
{
"state_after": "no goals",
"state_before": "case w.h.h\nC : Type u₁\ninst✝¹ : SmallCategory C\nℰ : Type u₂\ninst✝ : Category ℰ\nA : C ⥤ ℰ\nP : Cᵒᵖ ⥤ Type u₁\nE₁ E₂ : ℰ\ng : E₁ ⟶ E₂\nc : Cocone ((CategoryOfElements.π P).leftOp ⋙ A)\nt : IsColimit c\nk : c.pt ⟶ E₁\nx : Cᵒᵖ\nX : P.obj x\n⊢ (↑(restrictYonedaHomEquiv A P E₂ t) (k ≫ g)).app x X =\n (↑(restrictYonedaHomEquiv A P E₁ t) k ≫ (restrictedYoneda A).map g).app x X",
"tactic": "apply (assoc _ _ _).symm"
}
] |
[
127,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
convex_Iio
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • x + b • y ∈ Iio r",
"state_before": "𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr : β\n⊢ Convex 𝕜 (Iio r)",
"tactic": "intro x hx y hy a b ha hb hab"
},
{
"state_after": "case inl\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\nb : 𝕜\nhb : 0 ≤ b\nha : 0 ≤ 0\nhab : 0 + b = 1\n⊢ 0 • x + b • y ∈ Iio r\n\ncase inr\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nha' : 0 < a\n⊢ a • x + b • y ∈ Iio r",
"state_before": "𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • x + b • y ∈ Iio r",
"tactic": "obtain rfl | ha' := ha.eq_or_lt"
},
{
"state_after": "case inr\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x < r\ny : β\nhy : y < r\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nha' : 0 < a\n⊢ a • x + b • y ∈ Iio r",
"state_before": "case inr\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nha' : 0 < a\n⊢ a • x + b • y ∈ Iio r",
"tactic": "rw [mem_Iio] at hx hy"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x < r\ny : β\nhy : y < r\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nha' : 0 < a\n⊢ a • x + b • y ∈ Iio r",
"tactic": "calc\n a • x + b • y < a • r + b • r :=\n add_lt_add_of_lt_of_le (smul_lt_smul_of_pos hx ha') (smul_le_smul_of_nonneg hy.le hb)\n _ = r := Convex.combo_self hab _"
},
{
"state_after": "case inl\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\nb : 𝕜\nhb : 0 ≤ b\nha : 0 ≤ 0\nhab : b = 1\n⊢ 0 • x + b • y ∈ Iio r",
"state_before": "case inl\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\nb : 𝕜\nhb : 0 ≤ b\nha : 0 ≤ 0\nhab : 0 + b = 1\n⊢ 0 • x + b • y ∈ Iio r",
"tactic": "rw [zero_add] at hab"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\nE : Type ?u.86645\nF : Type ?u.86648\nβ : Type u_2\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ns : Set E\nx✝ : E\ninst✝² : OrderedCancelAddCommMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\nb : 𝕜\nhb : 0 ≤ b\nha : 0 ≤ 0\nhab : b = 1\n⊢ 0 • x + b • y ∈ Iio r",
"tactic": "rwa [zero_smul, zero_add, hab, one_smul]"
}
] |
[
287,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Data/List/Sort.lean
|
List.split_cons_of_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nr : α → α → Prop\ninst✝ : DecidableRel r\na : α\nl l₁ l₂ : List α\nh : split l = (l₁, l₂)\n⊢ split (a :: l) = (a :: l₂, l₁)",
"tactic": "rw [split, h]"
}
] |
[
315,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.trailingDegree_eq_top
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p = 0\n⊢ trailingDegree p = ⊤",
"tactic": "simp [h]"
}
] |
[
102,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.one_le_blocksFun
|
[] |
[
198,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.aleph0_le_continuum
|
[] |
[
83,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.241077\nF : Type u_2\nF' : Type u_3\nG : Type ?u.241086\n𝕜 : Type ?u.241089\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nT : Set α → F →L[ℝ] F'\nc : ℝ\nf : α →ₛ F\n⊢ setToSimpleFunc (fun s => c • T s) f = c • setToSimpleFunc T f",
"tactic": "simp_rw [setToSimpleFunc, ContinuousLinearMap.smul_apply, smul_sum]"
}
] |
[
433,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean
|
AddMonoidAlgebra.toDirectSum_toAddMonoidAlgebra
|
[] |
[
103,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.opow_mul_add_lt_opow_succ
|
[
{
"state_after": "case h.e'_4\nb u v w : Ordinal\nhvb : v < b\nhw : w < b ^ u\n⊢ b ^ succ u = b ^ u * b",
"state_before": "b u v w : Ordinal\nhvb : v < b\nhw : w < b ^ u\n⊢ b ^ u * v + w < b ^ succ u",
"tactic": "convert (opow_mul_add_lt_opow_mul_succ v hw).trans_le (mul_le_mul_left' (succ_le_of_lt hvb) _)\n using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nb u v w : Ordinal\nhvb : v < b\nhw : w < b ^ u\n⊢ b ^ succ u = b ^ u * b",
"tactic": "exact opow_succ b u"
}
] |
[
410,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
ULower.down_up
|
[] |
[
525,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/GroupTheory/Torsion.lean
|
Monoid.IsTorsion.torsion_eq_top
|
[
{
"state_after": "case h\nG : Type u_1\nH : Type ?u.193982\ninst✝ : CommMonoid G\ntG : IsTorsion G\nx✝ : G\n⊢ x✝ ∈ torsion G ↔ x✝ ∈ ⊤",
"state_before": "G : Type u_1\nH : Type ?u.193982\ninst✝ : CommMonoid G\ntG : IsTorsion G\n⊢ torsion G = ⊤",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nG : Type u_1\nH : Type ?u.193982\ninst✝ : CommMonoid G\ntG : IsTorsion G\nx✝ : G\n⊢ x✝ ∈ torsion G ↔ x✝ ∈ ⊤",
"tactic": "tauto"
}
] |
[
270,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
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