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/Concept.lean
|
subset_intentClosure_iff_subset_extentClosure
|
[] |
[
67,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.mem_toAddSubmonoid
|
[] |
[
76,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.cof_zero
|
[] |
[
482,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.Perm.congr_arg
|
[] |
[
138,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
11
] |
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
|
Equiv.Perm.nontrivial_toCycle
|
[
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\n⊢ Cycle.Nontrivial (toCycle f hf)",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nf : Perm α\nhf : IsCycle f\n⊢ Cycle.Nontrivial (toCycle f hf)",
"tactic": "obtain ⟨x, hx, -⟩ := id hf"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\n⊢ Cycle.Nontrivial (toCycle f hf)",
"tactic": "simp [toCycle_eq_toList f hf x hx, hx, Cycle.nontrivial_coe_nodup_iff (nodup_toList _ _)]"
}
] |
[
425,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
infinite_not_isOfFinOrder
|
[
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\n⊢ Set.Infinite {y | ¬IsOfFinOrder y}",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\n⊢ Set.Infinite {y | ¬IsOfFinOrder y}",
"tactic": "let s := { n | 0 < n }.image fun n : ℕ => x ^ n"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ Set.Infinite {y | ¬IsOfFinOrder y}",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\n⊢ Set.Infinite {y | ¬IsOfFinOrder y}",
"tactic": "have hs : s ⊆ { y : G | ¬IsOfFinOrder y } := by\n rintro - ⟨n, hn : 0 < n, rfl⟩ (contra : IsOfFinOrder (x ^ n))\n apply h\n rw [isOfFinOrder_iff_pow_eq_one] at contra⊢\n obtain ⟨m, hm, hm'⟩ := contra\n exact ⟨n * m, mul_pos hn hm, by rwa [pow_mul]⟩"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ Set.Infinite s",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ Set.Infinite {y | ¬IsOfFinOrder y}",
"tactic": "suffices s.Infinite by exact this.mono hs"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\n⊢ IsOfFinOrder x",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\n⊢ Set.Infinite s",
"tactic": "contrapose! h"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nthis : ¬Injective fun n => x ^ n\n⊢ IsOfFinOrder x",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\n⊢ IsOfFinOrder x",
"tactic": "have : ¬Injective fun n : ℕ => x ^ n := by\n have H := Set.not_injOn_infinite_finite_image (Set.Ioi_infinite 0) (Set.not_infinite.mp h)\n contrapose! H rw [not_not] at H exact Set.injOn_of_injective H _"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nthis : ¬Injective fun n => x ^ n\n⊢ IsOfFinOrder x",
"tactic": "rwa [injective_pow_iff_not_isOfFinOrder, Classical.not_not] at this"
},
{
"state_after": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\ncontra : IsOfFinOrder (x ^ n)\n⊢ False",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\n⊢ s ⊆ {y | ¬IsOfFinOrder y}",
"tactic": "rintro - ⟨n, hn : 0 < n, rfl⟩ (contra : IsOfFinOrder (x ^ n))"
},
{
"state_after": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\ncontra : IsOfFinOrder (x ^ n)\n⊢ IsOfFinOrder x",
"state_before": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\ncontra : IsOfFinOrder (x ^ n)\n⊢ False",
"tactic": "apply h"
},
{
"state_after": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\ncontra : ∃ n_1, 0 < n_1 ∧ (x ^ n) ^ n_1 = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"state_before": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\ncontra : IsOfFinOrder (x ^ n)\n⊢ IsOfFinOrder x",
"tactic": "rw [isOfFinOrder_iff_pow_eq_one] at contra⊢"
},
{
"state_after": "case intro.intro.intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m✝ : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\nm : ℕ\nhm : 0 < m\nhm' : (x ^ n) ^ m = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"state_before": "case intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\ncontra : ∃ n_1, 0 < n_1 ∧ (x ^ n) ^ n_1 = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"tactic": "obtain ⟨m, hm, hm'⟩ := contra"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nG : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m✝ : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\nm : ℕ\nhm : 0 < m\nhm' : (x ^ n) ^ m = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"tactic": "exact ⟨n * m, mul_pos hn hm, by rwa [pow_mul]⟩"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn✝ m✝ : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : ℕ\nhn : 0 < n\nm : ℕ\nhm : 0 < m\nhm' : (x ^ n) ^ m = 1\n⊢ x ^ (n * m) = 1",
"tactic": "rwa [pow_mul]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\nh : ¬IsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nthis : Set.Infinite s\n⊢ Set.Infinite {y | ¬IsOfFinOrder y}",
"tactic": "exact this.mono hs"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nH : ¬Set.InjOn (fun n => x ^ n) (Set.Ioi 0)\n⊢ ¬Injective fun n => x ^ n",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\n⊢ ¬Injective fun n => x ^ n",
"tactic": "have H := Set.not_injOn_infinite_finite_image (Set.Ioi_infinite 0) (Set.not_infinite.mp h)"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nH : ¬¬Injective fun n => x ^ n\n⊢ Set.InjOn (fun n => x ^ n) (Set.Ioi 0)",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nH : ¬Set.InjOn (fun n => x ^ n) (Set.Ioi 0)\n⊢ ¬Injective fun n => x ^ n",
"tactic": "contrapose! H"
},
{
"state_after": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nH : Injective fun n => x ^ n\n⊢ Set.InjOn (fun n => x ^ n) (Set.Ioi 0)",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nH : ¬¬Injective fun n => x ^ n\n⊢ Set.InjOn (fun n => x ^ n) (Set.Ioi 0)",
"tactic": "rw [not_not] at H"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝ : LeftCancelMonoid G\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s ⊆ {y | ¬IsOfFinOrder y}\nh : ¬Set.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nH : Injective fun n => x ^ n\n⊢ Set.InjOn (fun n => x ^ n) (Set.Ioi 0)",
"tactic": "exact Set.injOn_of_injective H _"
}
] |
[
529,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.Pairwise.imp₂
|
[] |
[
92,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.limsup_le_limsup_of_le
|
[] |
[
550,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/Data/Set/Pointwise/BigOperators.lean
|
Set.image_finset_prod
|
[] |
[
57,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean
|
Antitone.le_of_tendsto
|
[] |
[
268,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean
|
ContinuousAffineMap.map_vadd
|
[] |
[
97,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Std/Data/BinomialHeap.lean
|
Std.BinomialHeapImp.HeapNode.realSize_toHeap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns✝ : HeapNode α\nn : Nat\nres : Heap α\na : α\nc s : HeapNode α\n⊢ Heap.realSize (toHeap.go (node a c s) n res) = realSize (node a c s) + Heap.realSize res",
"tactic": "simp [toHeap.go, go, Nat.add_assoc, Nat.add_left_comm]"
}
] |
[
234,
77
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
231,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_self
|
[] |
[
404,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
toMul_zero
|
[] |
[
227,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
|
MeasureTheory.Measure.integral_comp_smul
|
[
{
"state_after": "case inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\n⊢ (∫ (x : E), f (0 • x) ∂μ) = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ\n\ncase inr\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nR : ℝ\nhR : R ≠ 0\n⊢ (∫ (x : E), f (R • x) ∂μ) = abs (R ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"state_before": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nR : ℝ\n⊢ (∫ (x : E), f (R • x) ∂μ) = abs (R ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "rcases eq_or_ne R 0 with (rfl | hR)"
},
{
"state_after": "case inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"state_before": "case inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\n⊢ (∫ (x : E), f (0 • x) ∂μ) = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "simp only [zero_smul, integral_const]"
},
{
"state_after": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ\n\ncase inl.inr\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E > 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"state_before": "case inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "rcases Nat.eq_zero_or_pos (finrank ℝ E) with (hE | hE)"
},
{
"state_after": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis : Subsingleton E\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"state_before": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "have : Subsingleton E := finrank_zero_iff.1 hE"
},
{
"state_after": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis✝ : Subsingleton E\nthis : f = fun x => f 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"state_before": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis : Subsingleton E\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim x 0]"
},
{
"state_after": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis✝ : Subsingleton E\nthis : f = fun x => f 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), (fun x => f 0) x ∂μ",
"state_before": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis✝ : Subsingleton E\nthis : f = fun x => f 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "conv_rhs => rw [this]"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis✝ : Subsingleton E\nthis : f = fun x => f 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), (fun x => f 0) x ∂μ",
"tactic": "simp only [hE, pow_zero, inv_one, abs_one, one_smul, integral_const]"
},
{
"state_after": "case h\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis : Subsingleton E\nx : E\n⊢ f x = f 0",
"state_before": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis : Subsingleton E\n⊢ f = fun x => f 0",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E = 0\nthis : Subsingleton E\nx : E\n⊢ f x = f 0",
"tactic": "rw [Subsingleton.elim x 0]"
},
{
"state_after": "case inl.inr\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E > 0\nthis : Nontrivial E\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"state_before": "case inl.inr\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E > 0\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "have : Nontrivial E := finrank_pos_iff.1 hE"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nhE : finrank ℝ E > 0\nthis : Nontrivial E\n⊢ ENNReal.toReal (↑↑μ univ) • f 0 = abs (0 ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "simp only [zero_pow hE, measure_univ_of_isAddLeftInvariant, ENNReal.top_toReal, zero_smul,\n inv_zero, abs_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nR : ℝ\nhR : R ≠ 0\n⊢ (∫ (x : E), f (R • x) ∂μ) = abs (R ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "calc\n (∫ x, f (R • x) ∂μ) = ∫ y, f y ∂Measure.map (fun x => R • x) μ :=\n (integral_map_equiv (Homeomorph.smul (isUnit_iff_ne_zero.2 hR).unit).toMeasurableEquiv\n f).symm\n _ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ := by\n simp only [map_add_haar_smul μ hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nf : E → F\nR : ℝ\nhR : R ≠ 0\n⊢ (∫ (y : E), f y ∂map (fun x => R • x) μ) = abs (R ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ",
"tactic": "simp only [map_add_haar_smul μ hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]"
}
] |
[
85,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
Group.rank_range_le
|
[] |
[
419,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/CategoryTheory/Functor/Functorial.lean
|
CategoryTheory.map'_as_map
|
[] |
[
49,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.prod_eq_inf_comap
|
[] |
[
485,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Order/Disjoint.lean
|
Codisjoint.left_le_of_le_inf_right
|
[] |
[
391,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.mem_closedBall'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.187348\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns t : Set α\n⊢ y ∈ closedBall x ε ↔ edist x y ≤ ε",
"tactic": "rw [edist_comm, mem_closedBall]"
}
] |
[
545,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.AEEqFun.Integrable.neg
|
[] |
[
1232,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1231,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.sub_apply
|
[] |
[
1255,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1254,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_mem_eq_prod_of_subset
|
[] |
[
485,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/CategoryTheory/Products/Bifunctor.lean
|
CategoryTheory.Bifunctor.diagonal'
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C × D ⥤ E\nX X' : C\nf : X ⟶ X'\nY Y' : D\ng : Y ⟶ Y'\n⊢ F.map (f, 𝟙 Y) ≫ F.map (𝟙 X', g) = F.map (f, g)",
"tactic": "rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]"
}
] |
[
59,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.comap_comp
|
[] |
[
382,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
11
] |
Mathlib/LinearAlgebra/Determinant.lean
|
is_basis_iff_det
|
[
{
"state_after": "case mp\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤ → IsUnit (↑(Basis.det e) v)\n\ncase mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\n⊢ IsUnit (↑(Basis.det e) v) → LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤ ↔ IsUnit (↑(Basis.det e) v)",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\n⊢ IsUnit (↑(Basis.det e) v)",
"state_before": "case mp\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤ → IsUnit (↑(Basis.det e) v)",
"tactic": "rintro ⟨hli, hspan⟩"
},
{
"state_after": "case mp.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\n⊢ IsUnit (↑(Basis.det e) v)",
"state_before": "case mp.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\n⊢ IsUnit (↑(Basis.det e) v)",
"tactic": "set v' := Basis.mk hli hspan.ge"
},
{
"state_after": "case mp.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\n⊢ IsUnit (det (Basis.toMatrix e v))",
"state_before": "case mp.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\n⊢ IsUnit (↑(Basis.det e) v)",
"tactic": "rw [e.det_apply]"
},
{
"state_after": "case h.e'_3.h.e'_6\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\n⊢ Basis.toMatrix e v = ↑(toMatrix v' e) ↑(LinearEquiv.refl R M)",
"state_before": "case mp.intro\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\n⊢ IsUnit (det (Basis.toMatrix e v))",
"tactic": "convert LinearEquiv.isUnit_det (LinearEquiv.refl R M) v' e using 2"
},
{
"state_after": "case h.e'_3.h.e'_6.a.h\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\ni j : ι\n⊢ Basis.toMatrix e v i j = ↑(toMatrix v' e) (↑(LinearEquiv.refl R M)) i j",
"state_before": "case h.e'_3.h.e'_6\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\n⊢ Basis.toMatrix e v = ↑(toMatrix v' e) ↑(LinearEquiv.refl R M)",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6.a.h\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nhli : LinearIndependent R v\nhspan : span R (Set.range v) = ⊤\nv' : Basis ι R M := Basis.mk hli (_ : ⊤ ≤ span R (Set.range v))\ni j : ι\n⊢ Basis.toMatrix e v i j = ↑(toMatrix v' e) (↑(LinearEquiv.refl R M)) i j",
"tactic": "simp"
},
{
"state_after": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (↑(Basis.det e) v)\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"state_before": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\n⊢ IsUnit (↑(Basis.det e) v) → LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"tactic": "intro h"
},
{
"state_after": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"state_before": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (↑(Basis.det e) v)\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"tactic": "rw [Basis.det_apply, Basis.toMatrix_eq_toMatrix_constr] at h"
},
{
"state_after": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"state_before": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"tactic": "set v' := Basis.map e (LinearEquiv.ofIsUnitDet h) with v'_def"
},
{
"state_after": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\nthis : ↑v' = v\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"state_before": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"tactic": "have : ⇑v' = v := by\n ext i\n rw [v'_def, Basis.map_apply, LinearEquiv.ofIsUnitDet_apply, e.constr_basis]"
},
{
"state_after": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\nthis : ↑v' = v\n⊢ LinearIndependent R ↑v' ∧ span R (Set.range ↑v') = ⊤",
"state_before": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\nthis : ↑v' = v\n⊢ LinearIndependent R v ∧ span R (Set.range v) = ⊤",
"tactic": "rw [← this]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\nthis : ↑v' = v\n⊢ LinearIndependent R ↑v' ∧ span R (Set.range ↑v') = ⊤",
"tactic": "exact ⟨v'.linearIndependent, v'.span_eq⟩"
},
{
"state_after": "case h\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\ni : ι\n⊢ ↑v' i = v i",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\n⊢ ↑v' = v",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type ?u.2551624\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_1\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit (det (↑(toMatrix e e) (↑(Basis.constr e ℕ) v)))\nv' : Basis ι R M := Basis.map e (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = Basis.map e (LinearEquiv.ofIsUnitDet h)\ni : ι\n⊢ ↑v' i = v i",
"tactic": "rw [v'_def, Basis.map_apply, LinearEquiv.ofIsUnitDet_apply, e.constr_basis]"
}
] |
[
567,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Analysis/InnerProductSpace/Positive.lean
|
ContinuousLinearMap.IsPositive.conj_orthogonalProjection
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.262558\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace F\nU : Submodule 𝕜 E\nT : E →L[𝕜] E\nhT : IsPositive T\ninst✝ : CompleteSpace { x // x ∈ U }\nthis :\n IsPositive\n (comp (comp (Submodule.subtypeL U) (orthogonalProjection U))\n (comp T (↑adjoint (comp (Submodule.subtypeL U) (orthogonalProjection U)))))\n⊢ IsPositive\n (comp (Submodule.subtypeL U)\n (comp (orthogonalProjection U) (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U)))))",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.262558\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace F\nU : Submodule 𝕜 E\nT : E →L[𝕜] E\nhT : IsPositive T\ninst✝ : CompleteSpace { x // x ∈ U }\n⊢ IsPositive\n (comp (Submodule.subtypeL U)\n (comp (orthogonalProjection U) (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U)))))",
"tactic": "have := hT.conj_adjoint (U.subtypeL ∘L orthogonalProjection U)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.262558\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace F\nU : Submodule 𝕜 E\nT : E →L[𝕜] E\nhT : IsPositive T\ninst✝ : CompleteSpace { x // x ∈ U }\nthis :\n IsPositive\n (comp (comp (Submodule.subtypeL U) (orthogonalProjection U))\n (comp T (↑adjoint (comp (Submodule.subtypeL U) (orthogonalProjection U)))))\n⊢ IsPositive\n (comp (Submodule.subtypeL U)\n (comp (orthogonalProjection U) (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U)))))",
"tactic": "rwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this"
}
] |
[
112,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Std/Data/List/Lemmas.lean
|
List.reverse_infix
|
[
{
"state_after": "case refine_1\nα✝ : Type u_1\nl₁ l₂ : List α✝\nx✝ : reverse l₁ <:+: reverse l₂\ns t : List α✝\ne : s ++ reverse l₁ ++ t = reverse l₂\n⊢ reverse t ++ l₁ ++ reverse s = l₂\n\ncase refine_2\nα✝ : Type u_1\nl₁ l₂ : List α✝\nx✝ : l₁ <:+: l₂\ns t : List α✝\ne : s ++ l₁ ++ t = l₂\n⊢ reverse t ++ reverse l₁ ++ reverse s = reverse l₂",
"state_before": "α✝ : Type u_1\nl₁ l₂ : List α✝\n⊢ reverse l₁ <:+: reverse l₂ ↔ l₁ <:+: l₂",
"tactic": "refine ⟨fun ⟨s, t, e⟩ => ⟨reverse t, reverse s, ?_⟩, fun ⟨s, t, e⟩ => ⟨reverse t, reverse s, ?_⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα✝ : Type u_1\nl₁ l₂ : List α✝\nx✝ : reverse l₁ <:+: reverse l₂\ns t : List α✝\ne : s ++ reverse l₁ ++ t = reverse l₂\n⊢ reverse t ++ l₁ ++ reverse s = l₂",
"tactic": "rw [← reverse_reverse l₁, append_assoc, ← reverse_append, ← reverse_append, e,\n reverse_reverse]"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα✝ : Type u_1\nl₁ l₂ : List α✝\nx✝ : l₁ <:+: l₂\ns t : List α✝\ne : s ++ l₁ ++ t = l₂\n⊢ reverse t ++ reverse l₁ ++ reverse s = reverse l₂",
"tactic": "rw [append_assoc, ← reverse_append, ← reverse_append, e]"
}
] |
[
1629,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1625,
9
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.HasColimit.isoOfNatIso_hom_desc
|
[] |
[
928,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
924,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.congr'
|
[] |
[
354,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.coe_sort_coe
|
[] |
[
123,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.zmodRepr_lt_p
|
[] |
[
208,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.mem_positive
|
[] |
[
578,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inclusion_mk
|
[] |
[
2782,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2781,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.pmap_res
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\nl : Filter α\ns : Set α\nf : α → β\nt : Set β\n⊢ t ∈ pmap (PFun.res f s) l ↔ t ∈ map f (l ⊓ 𝓟 s)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : Filter α\ns : Set α\nf : α → β\n⊢ pmap (PFun.res f s) l = map f (l ⊓ 𝓟 s)",
"tactic": "ext t"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\nl : Filter α\ns : Set α\nf : α → β\nt : Set β\n⊢ sᶜ ∪ f ⁻¹' t ∈ l ↔ {x | ¬x ∈ s ∨ x ∈ f ⁻¹' t} ∈ l",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\nl : Filter α\ns : Set α\nf : α → β\nt : Set β\n⊢ t ∈ pmap (PFun.res f s) l ↔ t ∈ map f (l ⊓ 𝓟 s)",
"tactic": "simp only [PFun.core_res, mem_pmap, mem_map, mem_inf_principal, imp_iff_not_or]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\nl : Filter α\ns : Set α\nf : α → β\nt : Set β\n⊢ sᶜ ∪ f ⁻¹' t ∈ l ↔ {x | ¬x ∈ s ∨ x ∈ f ⁻¹' t} ∈ l",
"tactic": "rfl"
}
] |
[
243,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.natAbs_eq
|
[] |
[
164,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
162,
1
] |
Mathlib/Data/Dfinsupp/Order.lean
|
Dfinsupp.coeFn_mono
|
[] |
[
83,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.cos_sub
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ cos (x - y) = cos x * cos y + sin x * sin y",
"tactic": "simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]"
}
] |
[
1210,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1209,
1
] |
Mathlib/Combinatorics/Additive/Etransform.lean
|
Finset.mulEtransformRight.card
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Group α\ne : α\nx : Finset α × Finset α\n⊢ Finset.card x.snd + Finset.card (e⁻¹ • x.snd) = 2 * Finset.card x.snd",
"tactic": "rw [card_smul_finset, two_mul]"
}
] |
[
170,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.forall_disjSups_iff
|
[
{
"state_after": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\np : α → Prop\n⊢ (∀ (c : α), (∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = c) → p c) ↔\n ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → Disjoint a b → p (a ⊔ b)",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\np : α → Prop\n⊢ (∀ (c : α), c ∈ s ○ t → p c) ↔ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → Disjoint a b → p (a ⊔ b)",
"tactic": "simp_rw [mem_disjSups]"
},
{
"state_after": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\np : α → Prop\n⊢ (∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → Disjoint a b → p (a ⊔ b)) →\n ∀ (c : α), (∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = c) → p c",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\np : α → Prop\n⊢ (∀ (c : α), (∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = c) → p c) ↔\n ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → Disjoint a b → p (a ⊔ b)",
"tactic": "refine' ⟨fun h a ha b hb hab => h _ ⟨_, ha, _, hb, hab, rfl⟩, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na✝ b✝ c : α\np : α → Prop\nh : ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → Disjoint a b → p (a ⊔ b)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nhab : Disjoint a b\n⊢ p (a ⊔ b)",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\np : α → Prop\n⊢ (∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → Disjoint a b → p (a ⊔ b)) →\n ∀ (c : α), (∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = c) → p c",
"tactic": "rintro h _ ⟨a, ha, b, hb, hab, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na✝ b✝ c : α\np : α → Prop\nh : ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → Disjoint a b → p (a ⊔ b)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\nhab : Disjoint a b\n⊢ p (a ⊔ b)",
"tactic": "exact h _ ha _ hb hab"
}
] |
[
471,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Products.lean
|
TopCat.prodIsoProd_inv_snd
|
[
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nX Y : TopCat\n⊢ (prodIsoProd X Y).inv ≫ prod.snd = prodSnd",
"tactic": "simp [Iso.inv_comp_eq]"
}
] |
[
237,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/Sites.lean
|
OpenEmbedding.compatiblePreserving
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\n⊢ CompatiblePreserving (Opens.grothendieckTopology ↑Y) (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f)))",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\n⊢ CompatiblePreserving (Opens.grothendieckTopology ↑Y) (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f)))",
"tactic": "haveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.inj"
},
{
"state_after": "case hF\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\n⊢ {c : Opens ↑X} →\n {d : Opens ↑Y} →\n (d ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj c) →\n (c' : Opens ↑X) × ((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj c' ≅ d)",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\n⊢ CompatiblePreserving (Opens.grothendieckTopology ↑Y) (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f)))",
"tactic": "apply compatiblePreservingOfDownwardsClosed"
},
{
"state_after": "case hF\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\nU : Opens ↑X\nV : Opens ↑Y\ni : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj U\n⊢ (c' : Opens ↑X) × ((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj c' ≅ V)",
"state_before": "case hF\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\n⊢ {c : Opens ↑X} →\n {d : Opens ↑Y} →\n (d ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj c) →\n (c' : Opens ↑X) × ((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj c' ≅ d)",
"tactic": "intro U V i"
},
{
"state_after": "case hF\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\nU : Opens ↑X\nV : Opens ↑Y\ni : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj U\nx : ↑Y\nh : x ∈ V.1\n⊢ x ∈ Set.range fun x => (forget TopCat).map f x",
"state_before": "case hF\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\nU : Opens ↑X\nV : Opens ↑Y\ni : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj U\n⊢ (c' : Opens ↑X) × ((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj c' ≅ V)",
"tactic": "refine' ⟨(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h ↦ _⟩"
},
{
"state_after": "case hF.intro.intro\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\nU : Opens ↑X\nV : Opens ↑Y\ni : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj U\nw✝ : (forget TopCat).obj X\nleft✝ : w✝ ∈ ↑U\nh : (forget TopCat).map f w✝ ∈ V.1\n⊢ (forget TopCat).map f w✝ ∈ Set.range fun x => (forget TopCat).map f x",
"state_before": "case hF\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\nU : Opens ↑X\nV : Opens ↑Y\ni : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj U\nx : ↑Y\nh : x ∈ V.1\n⊢ x ∈ Set.range fun x => (forget TopCat).map f x",
"tactic": "obtain ⟨_, _, rfl⟩ := i.le h"
},
{
"state_after": "no goals",
"state_before": "case hF.intro.intro\nC : Type u\ninst✝ : Category C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nhf : OpenEmbedding ((forget TopCat).map f)\nthis : Mono f\nU : Opens ↑X\nV : Opens ↑Y\ni : V ⟶ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map f))).obj U\nw✝ : (forget TopCat).obj X\nleft✝ : w✝ ∈ ↑U\nh : (forget TopCat).map f w✝ ∈ V.1\n⊢ (forget TopCat).map f w✝ ∈ Set.range fun x => (forget TopCat).map f x",
"tactic": "exact ⟨_, rfl⟩"
}
] |
[
171,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Order/Minimal.lean
|
maximals_eq_minimals
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na b : α\ninst✝ : IsSymm α r\n⊢ maximals r s = minimals r s",
"tactic": "rw [minimals_of_symm, maximals_of_symm]"
}
] |
[
118,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.mul_zero
|
[
{
"state_after": "no goals",
"state_before": "a : Rat\n⊢ a * 0 = 0",
"tactic": "simp [mul_def]"
}
] |
[
261,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
261,
19
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
NNReal.le_add_nndist
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.239808\nι : Type ?u.239811\ninst✝ : PseudoMetricSpace α\na b : ℝ≥0\n⊢ ↑a ≤ ↑b + dist a b",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.239808\nι : Type ?u.239811\ninst✝ : PseudoMetricSpace α\na b : ℝ≥0\n⊢ a ≤ b + nndist a b",
"tactic": "suffices (a : ℝ) ≤ (b : ℝ) + dist a b by\n rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]"
},
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.239808\nι : Type ?u.239811\ninst✝ : PseudoMetricSpace α\na b : ℝ≥0\n⊢ ↑a - ↑b ≤ dist a b",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.239808\nι : Type ?u.239811\ninst✝ : PseudoMetricSpace α\na b : ℝ≥0\n⊢ ↑a ≤ ↑b + dist a b",
"tactic": "rw [← sub_le_iff_le_add']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.239808\nι : Type ?u.239811\ninst✝ : PseudoMetricSpace α\na b : ℝ≥0\n⊢ ↑a - ↑b ≤ dist a b",
"tactic": "exact le_of_abs_le (dist_eq a b).ge"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.239808\nι : Type ?u.239811\ninst✝ : PseudoMetricSpace α\na b : ℝ≥0\nthis : ↑a ≤ ↑b + dist a b\n⊢ a ≤ b + nndist a b",
"tactic": "rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]"
}
] |
[
1706,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1702,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.splitAtPred_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl₁ : List α\no : α\nl₂ : List α\nx✝ : ∀ (x : α), x ∈ l₁ → p x = false\nleft✝ : p o = true\nh₃ : [] = l₁ ++ o :: l₂\n⊢ splitAtPred p [] = (l₁, some o, l₂)",
"tactic": "simp at h₃"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nL l₁ : List α\no : Option α\nl₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, o, l₂)",
"state_before": "α : Type u_1\np : α → Bool\na : α\nL l₁ : List α\no : Option α\nl₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o\n⊢ splitAtPred p (a :: L) = (l₁, o, l₂)",
"tactic": "rw [splitAtPred]"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nL l₁ : List α\no : Option α\nl₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, o, l₂)",
"state_before": "α : Type u_1\np : α → Bool\na : α\nL l₁ : List α\no : Option α\nl₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, o, l₂)",
"tactic": "have IH := splitAtPred_eq p L"
},
{
"state_after": "case none\nα : Type u_1\np : α → Bool\na : α\nL l₁ l₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) none\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, none, l₂)\n\ncase some\nα : Type u_1\np : α → Bool\na : α\nL l₁ l₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\no : α\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) (some o)\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, some o, l₂)",
"state_before": "α : Type u_1\np : α → Bool\na : α\nL l₁ : List α\no : Option α\nl₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, o, l₂)",
"tactic": "cases' o with o"
},
{
"state_after": "case none.cons.intro.refl\nα : Type u_1\np : α → Bool\na : α\nL : List α\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₁ : ∀ (x : α), x ∈ a :: L → p x = false\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (a :: L, none, [])",
"state_before": "case none\nα : Type u_1\np : α → Bool\na : α\nL l₁ l₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) none\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, none, l₂)",
"tactic": "cases' l₁ with a' l₁ <;> rcases h₂ with ⟨⟨⟩, rfl⟩"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nL : List α\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₁ : ∀ (x : α), x ∈ a :: L → p x = false\n⊢ ∀ (x : α), x ∈ L → p x = false",
"state_before": "case none.cons.intro.refl\nα : Type u_1\np : α → Bool\na : α\nL : List α\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₁ : ∀ (x : α), x ∈ a :: L → p x = false\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (a :: L, none, [])",
"tactic": "rw [h₁ a (List.Mem.head _), cond, IH L none [] _ ⟨rfl, rfl⟩]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nL : List α\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₁ : ∀ (x : α), x ∈ a :: L → p x = false\n⊢ ∀ (x : α), x ∈ L → p x = false",
"tactic": "exact fun x h => h₁ x (List.Mem.tail _ h)"
},
{
"state_after": "case some.nil.intro.refl\nα : Type u_1\np : α → Bool\na : α\nL : List α\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₁ : ∀ (x : α), x ∈ [] → p x = false\nh₂ : p a = true\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n ([], some a, L)\n\ncase some.cons.intro.refl\nα : Type u_1\np : α → Bool\na : α\nl₂ : List α\no : α\nl₁ : List α\nh₂ : p o = true\nh₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false\nIH :\n ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α),\n (∀ (x : α), x ∈ l₁_1 → p x = false) →\n Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = [])\n (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 →\n splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1)\n⊢ (bif p a then ([], some a, List.append l₁ (o :: l₂))\n else\n match splitAtPred p (List.append l₁ (o :: l₂)) with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (a :: l₁, some o, l₂)",
"state_before": "case some\nα : Type u_1\np : α → Bool\na : α\nL l₁ l₂ : List α\nh₁ : ∀ (x : α), x ∈ l₁ → p x = false\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\no : α\nh₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) (some o)\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (l₁, some o, l₂)",
"tactic": "cases' l₁ with a' l₁ <;> rcases h₂ with ⟨h₂, ⟨⟩⟩"
},
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl₂ : List α\no : α\nl₁ : List α\nh₂ : p o = true\nh₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false\nIH :\n ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α),\n (∀ (x : α), x ∈ l₁_1 → p x = false) →\n Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = [])\n (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 →\n splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1)\n⊢ ∀ (x : α), x ∈ l₁ → p x = false",
"state_before": "case some.cons.intro.refl\nα : Type u_1\np : α → Bool\na : α\nl₂ : List α\no : α\nl₁ : List α\nh₂ : p o = true\nh₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false\nIH :\n ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α),\n (∀ (x : α), x ∈ l₁_1 → p x = false) →\n Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = [])\n (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 →\n splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1)\n⊢ (bif p a then ([], some a, List.append l₁ (o :: l₂))\n else\n match splitAtPred p (List.append l₁ (o :: l₂)) with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n (a :: l₁, some o, l₂)",
"tactic": "rw [h₁ a (List.Mem.head _), cond, IH l₁ (some o) l₂ _ ⟨h₂, _⟩] <;> try rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl₂ : List α\no : α\nl₁ : List α\nh₂ : p o = true\nh₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false\nIH :\n ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α),\n (∀ (x : α), x ∈ l₁_1 → p x = false) →\n Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = [])\n (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 →\n splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1)\n⊢ ∀ (x : α), x ∈ l₁ → p x = false",
"tactic": "exact fun x h => h₁ x (List.Mem.tail _ h)"
},
{
"state_after": "no goals",
"state_before": "case some.nil.intro.refl\nα : Type u_1\np : α → Bool\na : α\nL : List α\nIH :\n ∀ (l₁ : List α) (o : Option α) (l₂ : List α),\n (∀ (x : α), x ∈ l₁ → p x = false) →\n Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂)\nh₁ : ∀ (x : α), x ∈ [] → p x = false\nh₂ : p a = true\n⊢ (bif p a then ([], some a, L)\n else\n match splitAtPred p L with\n | (l₁, o, l₂) => (a :: l₁, o, l₂)) =\n ([], some a, L)",
"tactic": "rw [h₂, cond]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl₂ : List α\no : α\nl₁ : List α\nh₂ : p o = true\nh₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false\nIH :\n ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α),\n (∀ (x : α), x ∈ l₁_1 → p x = false) →\n Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = [])\n (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 →\n splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1)\n⊢ List.append l₁ (o :: l₂) = l₁ ++ o :: l₂",
"tactic": "rfl"
}
] |
[
1345,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1328,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.prod_insert
|
[
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ insert b t ↔ (x, y) ∈ (fun a => (a, b)) '' s ∪ s ×ˢ t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\n⊢ s ×ˢ insert b t = (fun a => (a, b)) '' s ∪ s ×ˢ t",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ x ∈ s ∧ (y = b ∨ y ∈ t) ↔ (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ insert b t ↔ (x, y) ∈ (fun a => (a, b)) '' s ∪ s ×ˢ t",
"tactic": "simp [image, or_imp]"
},
{
"state_after": "case h.mk.refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nh : x ∈ s ∧ (y = b ∨ y ∈ t)\n⊢ (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t\n\ncase h.mk.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nh : (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t\n⊢ x ∈ s ∧ (y = b ∨ y ∈ t)",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ x ∈ s ∧ (y = b ∨ y ∈ t) ↔ (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t",
"tactic": "refine ⟨fun h => ?_, fun h => ?_⟩"
},
{
"state_after": "case h.mk.refine_1.intro.inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na x : α\ny : β\nhx : x ∈ s\n⊢ (∃ a, a ∈ s ∧ a = x ∧ y = y) ∨ x ∈ s ∧ y ∈ t\n\ncase h.mk.refine_1.intro.inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\n⊢ (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t",
"state_before": "case h.mk.refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nh : x ∈ s ∧ (y = b ∨ y ∈ t)\n⊢ (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t",
"tactic": "obtain ⟨hx, rfl|hy⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case h.mk.refine_1.intro.inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na x : α\ny : β\nhx : x ∈ s\n⊢ (∃ a, a ∈ s ∧ a = x ∧ y = y) ∨ x ∈ s ∧ y ∈ t",
"tactic": "exact Or.inl ⟨x, hx, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk.refine_1.intro.inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\n⊢ (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t",
"tactic": "exact Or.inr ⟨hx, hy⟩"
},
{
"state_after": "case h.mk.refine_2.inl.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\nhx : x ∈ s\n⊢ x ∈ s ∧ (b = b ∨ b ∈ t)\n\ncase h.mk.refine_2.inr.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\n⊢ x ∈ s ∧ (y = b ∨ y ∈ t)",
"state_before": "case h.mk.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nh : (∃ a, a ∈ s ∧ a = x ∧ b = y) ∨ x ∈ s ∧ y ∈ t\n⊢ x ∈ s ∧ (y = b ∨ y ∈ t)",
"tactic": "obtain ⟨x, hx, rfl, rfl⟩|⟨hx, hy⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case h.mk.refine_2.inl.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\nhx : x ∈ s\n⊢ x ∈ s ∧ (b = b ∨ b ∈ t)",
"tactic": "exact ⟨hx, Or.inl rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk.refine_2.inr.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.20280\nδ : Type ?u.20283\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\nhx : x ∈ s\nhy : y ∈ t\n⊢ x ∈ s ∧ (y = b ∨ y ∈ t)",
"tactic": "exact ⟨hx, Or.inr hy⟩"
}
] |
[
196,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lift_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.111454\nβ : Type ?u.111457\nγ : Type ?u.111460\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\n⊢ lift a < lift b ↔ a < b",
"tactic": "simp only [lt_iff_le_not_le, lift_le]"
}
] |
[
778,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
777,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.IsAtomic.liftAt
|
[] |
[
700,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
699,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.mod_eq_self_iff
|
[
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ p % q = p",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\n⊢ p % q = p",
"tactic": "have : ¬degree (q * C (leadingCoeff q)⁻¹) ≤ degree p :=\n not_le_of_gt <| by rwa [degree_mul_leadingCoeff_inv q hq0]"
},
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ (divModByMonicAux p (_ : Monic (q * ↑C (leadingCoeff q)⁻¹))).snd = p",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ p % q = p",
"tactic": "rw [mod_def, modByMonic, dif_pos (monic_mul_leadingCoeff_inv hq0)]"
},
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ (if h : degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹));\n let_fun _wf :=\n (_ :\n degree\n (p -\n ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) *\n (q * ↑C (leadingCoeff q)⁻¹)) <\n degree p);\n let dm := divModByMonicAux (p - z * (q * ↑C (leadingCoeff q)⁻¹)) (_ : Monic (q * ↑C (leadingCoeff q)⁻¹));\n (z + dm.fst, dm.snd)\n else (0, p)).snd =\n p",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ (divModByMonicAux p (_ : Monic (q * ↑C (leadingCoeff q)⁻¹))).snd = p",
"tactic": "unfold divModByMonicAux"
},
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ (if degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) +\n (divModByMonicAux\n (p -\n ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) *\n (q * ↑C (leadingCoeff q)⁻¹))\n (_ : Monic (q * ↑C (leadingCoeff q)⁻¹))).fst,\n (divModByMonicAux\n (p -\n ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) *\n (q * ↑C (leadingCoeff q)⁻¹))\n (_ : Monic (q * ↑C (leadingCoeff q)⁻¹))).snd)\n else (0, p)).snd =\n p",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ (if h : degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹));\n let_fun _wf :=\n (_ :\n degree\n (p -\n ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) *\n (q * ↑C (leadingCoeff q)⁻¹)) <\n degree p);\n let dm := divModByMonicAux (p - z * (q * ↑C (leadingCoeff q)⁻¹)) (_ : Monic (q * ↑C (leadingCoeff q)⁻¹));\n (z + dm.fst, dm.snd)\n else (0, p)).snd =\n p",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\nthis : ¬degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p\n⊢ (if degree (q * ↑C (leadingCoeff q)⁻¹) ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) +\n (divModByMonicAux\n (p -\n ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) *\n (q * ↑C (leadingCoeff q)⁻¹))\n (_ : Monic (q * ↑C (leadingCoeff q)⁻¹))).fst,\n (divModByMonicAux\n (p -\n ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * ↑C (leadingCoeff q)⁻¹)) *\n (q * ↑C (leadingCoeff q)⁻¹))\n (_ : Monic (q * ↑C (leadingCoeff q)⁻¹))).snd)\n else (0, p)).snd =\n p",
"tactic": "simp only [this, false_and_iff, if_false]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : degree p < degree q\n⊢ degree (q * ↑C (leadingCoeff q)⁻¹) > degree p",
"tactic": "rwa [degree_mul_leadingCoeff_inv q hq0]"
}
] |
[
238,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
mul_inv_le_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ b ≤ 1 * a ↔ b ≤ a",
"tactic": "rw [one_mul]"
}
] |
[
271,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Computability/Primrec.lean
|
Nat.Primrec'.comp₁
|
[] |
[
1437,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1435,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.pi_nonempty_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.116518\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\n⊢ Set.Nonempty (pi s t) ↔ ∀ (i : ι), ∃ x, i ∈ s → x ∈ t i",
"tactic": "simp [Classical.skolem, Set.Nonempty]"
}
] |
[
684,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_inf_distrib_right
|
[
{
"state_after": "F : Type ?u.139618\nα : Type u_2\nβ : Type ?u.139624\nγ : Type ?u.139627\nι : Type u_1\nκ : Type ?u.139633\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\ns : Finset ι\nf : ι → α\na : α\n⊢ (sup s fun i => a ⊓ f i) = sup s fun i => f i ⊓ a",
"state_before": "F : Type ?u.139618\nα : Type u_2\nβ : Type ?u.139624\nγ : Type ?u.139627\nι : Type u_1\nκ : Type ?u.139633\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\ns : Finset ι\nf : ι → α\na : α\n⊢ sup s f ⊓ a = sup s fun i => f i ⊓ a",
"tactic": "rw [_root_.inf_comm, s.sup_inf_distrib_left]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.139618\nα : Type u_2\nβ : Type ?u.139624\nγ : Type ?u.139627\nι : Type u_1\nκ : Type ?u.139633\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\ns : Finset ι\nf : ι → α\na : α\n⊢ (sup s fun i => a ⊓ f i) = sup s fun i => f i ⊓ a",
"tactic": "simp_rw [_root_.inf_comm]"
}
] |
[
524,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.neg_def
|
[] |
[
351,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.add_inter_distrib
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.189863\nγ : Type ?u.189866\ninst✝ : DecidableEq α\ns✝ t✝ u✝ : Multiset α\na b : α\ns t u : Multiset α\n⊢ s + t ∩ u = (s + t) ∩ (s + u)",
"tactic": "rw [add_comm, inter_add_distrib, add_comm s, add_comm s]"
}
] |
[
1876,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1875,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_max
|
[] |
[
358,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diag_multiset_sum
|
[] |
[
693,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.top_adj
|
[] |
[
381,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.tsupport_normed_eq
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\nX : Type ?u.2093391\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup X\ninst✝⁶ : NormedSpace ℝ X\ninst✝⁵ : HasContDiffBump E\nc : E\nf : ContDiffBump c\nx : E\nn : ℕ∞\ninst✝⁴ : MeasurableSpace E\nμ : MeasureTheory.Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : Measure.IsOpenPosMeasure μ\n⊢ tsupport (ContDiffBump.normed f μ) = closedBall c f.rOut",
"tactic": "rw [tsupport, f.support_normed_eq, closure_ball _ f.rOut_pos.ne']"
}
] |
[
541,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.Lifts.exists_upper_bound
|
[
{
"state_after": "F : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\n⊢ x ≤ upperBound hc",
"state_before": "F : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\n⊢ ∀ (a : Lifts F E K), a ∈ c → a ≤ upperBound hc",
"tactic": "intro x hx"
},
{
"state_after": "case left\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\n⊢ x.fst ≤ (upperBound hc).fst\n\ncase right\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\n⊢ ∀ (s : { x_1 // x_1 ∈ x.fst }) (t : { x // x ∈ (upperBound hc).fst }), ↑s = ↑t → ↑x.snd s = ↑(upperBound hc).snd t",
"state_before": "F : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\n⊢ x ≤ upperBound hc",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case left\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\n⊢ x.fst ≤ (upperBound hc).fst",
"tactic": "exact fun s hs => ⟨x, Set.mem_insert_of_mem ⊥ hx, hs⟩"
},
{
"state_after": "case right\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\n⊢ ↑x.snd s = ↑(upperBound hc).snd t",
"state_before": "case right\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\n⊢ ∀ (s : { x_1 // x_1 ∈ x.fst }) (t : { x // x ∈ (upperBound hc).fst }), ↑s = ↑t → ↑x.snd s = ↑(upperBound hc).snd t",
"tactic": "intro s t hst"
},
{
"state_after": "case right\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\n⊢ ↑x.snd s =\n ↑(Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).snd\n { val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) }",
"state_before": "case right\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\n⊢ ↑x.snd s = ↑(upperBound hc).snd t",
"tactic": "change x.2 s = (Classical.choose t.mem).2 ⟨t, (Classical.choose_spec t.mem).2⟩"
},
{
"state_after": "case right.intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\nz : Lifts F E K\nleft✝ : z ∈ insert ⊥ c\nhxz : x ≤ z\nhyz : Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst) ≤ z\n⊢ ↑x.snd s =\n ↑(Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).snd\n { val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) }",
"state_before": "case right\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\n⊢ ↑x.snd s =\n ↑(Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).snd\n { val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) }",
"tactic": "obtain ⟨z, _, hxz, hyz⟩ :=\n Lifts.exists_max_two hc (Set.mem_insert_of_mem ⊥ hx) (Classical.choose_spec t.mem).1"
},
{
"state_after": "case right.intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\nz : Lifts F E K\nleft✝ : z ∈ insert ⊥ c\nhxz : x ≤ z\nhyz : Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst) ≤ z\n⊢ ↑z.snd { val := ↑s, property := (_ : ↑s ∈ z.fst) } =\n ↑z.snd\n { val := ↑{ val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) },\n property :=\n (_ : ↑{ val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) } ∈ z.fst) }",
"state_before": "case right.intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\nz : Lifts F E K\nleft✝ : z ∈ insert ⊥ c\nhxz : x ≤ z\nhyz : Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst) ≤ z\n⊢ ↑x.snd s =\n ↑(Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).snd\n { val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) }",
"tactic": "rw [Lifts.eq_of_le hxz, Lifts.eq_of_le hyz]"
},
{
"state_after": "no goals",
"state_before": "case right.intro.intro.intro\nF : Type u_3\nE : Type u_2\nK : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nc : Set (Lifts F E K)\nhc : IsChain (fun x x_1 => x ≤ x_1) c\nx : Lifts F E K\nhx : x ∈ c\ns : { x_1 // x_1 ∈ x.fst }\nt : { x // x ∈ (upperBound hc).fst }\nhst : ↑s = ↑t\nz : Lifts F E K\nleft✝ : z ∈ insert ⊥ c\nhxz : x ≤ z\nhyz : Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst) ≤ z\n⊢ ↑z.snd { val := ↑s, property := (_ : ↑s ∈ z.fst) } =\n ↑z.snd\n { val := ↑{ val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) },\n property :=\n (_ : ↑{ val := ↑t, property := (_ : ↑t ∈ (Classical.choose (_ : ↑t ∈ ↑(upperBound hc).fst)).fst) } ∈ z.fst) }",
"tactic": "exact congr_arg z.2 (Subtype.ext hst)"
}
] |
[
1085,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1074,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithTop.add_lt_add_left
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\ninst✝² : Add α\nb c d : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : b < c\na : α\n⊢ ↑a + b < ↑a + c",
"state_before": "α : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nha : a ≠ ⊤\nh : b < c\n⊢ a + b < a + c",
"tactic": "lift a to α using ha"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\ninst✝² : Add α\nc d : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : α\nh' h : ↑b < c\n⊢ ↑a + ↑b < ↑a + c",
"state_before": "case intro\nα : Type u\nβ : Type v\ninst✝² : Add α\nb c d : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : b < c\na : α\n⊢ ↑a + b < ↑a + c",
"tactic": "rcases lt_iff_exists_coe.1 h with ⟨b, rfl, h'⟩"
},
{
"state_after": "case intro.intro.intro.none\nα : Type u\nβ : Type v\ninst✝² : Add α\nd : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : α\nh' h : ↑b < none\n⊢ ↑a + ↑b < ↑a + none\n\ncase intro.intro.intro.some\nα : Type u\nβ : Type v\ninst✝² : Add α\nd : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b val✝ : α\nh' h : ↑b < Option.some val✝\n⊢ ↑a + ↑b < ↑a + Option.some val✝",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\ninst✝² : Add α\nc d : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : α\nh' h : ↑b < c\n⊢ ↑a + ↑b < ↑a + c",
"tactic": "cases c"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.none\nα : Type u\nβ : Type v\ninst✝² : Add α\nd : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : α\nh' h : ↑b < none\n⊢ ↑a + ↑b < ↑a + none",
"tactic": "exact coe_lt_top _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.some\nα : Type u\nβ : Type v\ninst✝² : Add α\nd : WithTop α\nx y : α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b val✝ : α\nh' h : ↑b < Option.some val✝\n⊢ ↑a + ↑b < ↑a + Option.some val✝",
"tactic": "exact coe_lt_coe.2 (add_lt_add_left (coe_lt_coe.1 h) _)"
}
] |
[
248,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
11
] |
Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
|
CategoryTheory.Abelian.isIso_of_isIso_of_isIso_of_isIso_of_isIso
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝⁵ : Category V\ninst✝⁴ : Abelian V\nA B C D A' B' C' D' : V\nf : A ⟶ B\ng : B ⟶ C\nh : C ⟶ D\nf' : A' ⟶ B'\ng' : B' ⟶ C'\nh' : C' ⟶ D'\nα : A ⟶ A'\nβ : B ⟶ B'\nγ : C ⟶ C'\nδ : D ⟶ D'\ncomm₁ : α ≫ f' = f ≫ β\ncomm₂ : β ≫ g' = g ≫ γ\ncomm₃ : γ ≫ h' = h ≫ δ\nE E' : V\ni : D ⟶ E\ni' : D' ⟶ E'\nε : E ⟶ E'\ncomm₄ : δ ≫ i' = i ≫ ε\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst✝³ : IsIso α\ninst✝² : IsIso β\ninst✝¹ : IsIso δ\ninst✝ : IsIso ε\n⊢ Mono γ",
"tactic": "apply mono_of_epi_of_mono_of_mono comm₁ comm₂ comm₃ hfg hgh hf'g' <;> infer_instance"
},
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝⁵ : Category V\ninst✝⁴ : Abelian V\nA B C D A' B' C' D' : V\nf : A ⟶ B\ng : B ⟶ C\nh : C ⟶ D\nf' : A' ⟶ B'\ng' : B' ⟶ C'\nh' : C' ⟶ D'\nα : A ⟶ A'\nβ : B ⟶ B'\nγ : C ⟶ C'\nδ : D ⟶ D'\ncomm₁ : α ≫ f' = f ≫ β\ncomm₂ : β ≫ g' = g ≫ γ\ncomm₃ : γ ≫ h' = h ≫ δ\nE E' : V\ni : D ⟶ E\ni' : D' ⟶ E'\nε : E ⟶ E'\ncomm₄ : δ ≫ i' = i ≫ ε\nhfg : Exact f g\nhgh : Exact g h\nhhi : Exact h i\nhf'g' : Exact f' g'\nhg'h' : Exact g' h'\nhh'i' : Exact h' i'\ninst✝³ : IsIso α\ninst✝² : IsIso β\ninst✝¹ : IsIso δ\ninst✝ : IsIso ε\nthis : Mono γ\n⊢ Epi γ",
"tactic": "apply epi_of_epi_of_epi_of_mono comm₂ comm₃ comm₄ hhi hg'h' hh'i' <;> infer_instance"
}
] |
[
216,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Data/Set/Pairwise/Lattice.lean
|
Set.pairwiseDisjoint_iUnion
|
[] |
[
60,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/Nat/Multiplicity.lean
|
Nat.Prime.multiplicity_pow_self
|
[] |
[
99,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean
|
IsHomeomorphicTrivialFiberBundle.surjective_proj
|
[
{
"state_after": "case intro\nB : Type u_2\nF : Type u_1\nZ : Type u_3\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : TopologicalSpace Z\ninst✝ : Nonempty F\ne : Z ≃ₜ B × F\nh : IsHomeomorphicTrivialFiberBundle F (Prod.fst ∘ ↑e)\n⊢ Function.Surjective (Prod.fst ∘ ↑e)",
"state_before": "B : Type u_2\nF : Type u_1\nZ : Type u_3\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : TopologicalSpace Z\nproj : Z → B\ninst✝ : Nonempty F\nh : IsHomeomorphicTrivialFiberBundle F proj\n⊢ Function.Surjective proj",
"tactic": "obtain ⟨e, rfl⟩ := h.proj_eq"
},
{
"state_after": "no goals",
"state_before": "case intro\nB : Type u_2\nF : Type u_1\nZ : Type u_3\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : TopologicalSpace Z\ninst✝ : Nonempty F\ne : Z ≃ₜ B × F\nh : IsHomeomorphicTrivialFiberBundle F (Prod.fst ∘ ↑e)\n⊢ Function.Surjective (Prod.fst ∘ ↑e)",
"tactic": "exact Prod.fst_surjective.comp e.surjective"
}
] |
[
49,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
11
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castPred_mk
|
[
{
"state_after": "n✝ m n i : ℕ\nh : i < n + 1\nthis : ¬↑castSucc (last n) < { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n⊢ castPred { val := i, isLt := (_ : i < Nat.succ (n + 1)) } = { val := i, isLt := h }",
"state_before": "n✝ m n i : ℕ\nh : i < n + 1\n⊢ castPred { val := i, isLt := (_ : i < Nat.succ (n + 1)) } = { val := i, isLt := h }",
"tactic": "have : ¬castSucc (last n) < ⟨i, lt_succ_of_lt h⟩ := by\n simpa [lt_iff_val_lt_val] using le_of_lt_succ h"
},
{
"state_after": "no goals",
"state_before": "n✝ m n i : ℕ\nh : i < n + 1\nthis : ¬↑castSucc (last n) < { val := i, isLt := (_ : i < Nat.succ (n + 1)) }\n⊢ castPred { val := i, isLt := (_ : i < Nat.succ (n + 1)) } = { val := i, isLt := h }",
"tactic": "simp [castPred, predAbove, this]"
},
{
"state_after": "no goals",
"state_before": "n✝ m n i : ℕ\nh : i < n + 1\n⊢ ¬↑castSucc (last n) < { val := i, isLt := (_ : i < Nat.succ (n + 1)) }",
"tactic": "simpa [lt_iff_val_lt_val] using le_of_lt_succ h"
}
] |
[
2326,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2323,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisCoinsertion.isGLB_of_l_image
|
[] |
[
845,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
843,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformFun.tendsto_iff_tendstoUniformly
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.45029\nι : Type u_3\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\nF : ι → α →ᵤ β\nf : α →ᵤ β\n⊢ (∀ (i : Set (β × β)), i ∈ 𝓤 β → ∀ᶠ (x : ι) in p, F x ∈ {g | (f, g) ∈ UniformFun.gen α β i}) ↔\n ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∀ᶠ (n : ι) in p, ∀ (x : α), (f x, F n x) ∈ u",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.45029\nι : Type u_3\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\nF : ι → α →ᵤ β\nf : α →ᵤ β\n⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly F f p",
"tactic": "rw [(UniformFun.hasBasis_nhds α β f).tendsto_right_iff, TendstoUniformly]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.45029\nι : Type u_3\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\nF : ι → α →ᵤ β\nf : α →ᵤ β\n⊢ (∀ (i : Set (β × β)), i ∈ 𝓤 β → ∀ᶠ (x : ι) in p, F x ∈ {g | (f, g) ∈ UniformFun.gen α β i}) ↔\n ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∀ᶠ (n : ι) in p, ∀ (x : α), (f x, F n x) ∈ u",
"tactic": "exact Iff.rfl"
}
] |
[
501,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
11
] |
Mathlib/Computability/Partrec.lean
|
Computable.fst
|
[] |
[
305,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.prod_lt_one'
|
[] |
[
527,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/Algebra/Category/GroupCat/Injective.lean
|
AddCommGroupCat.injective_of_injective_as_module
|
[
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\n⊢ ∃ h, f ≫ h = g",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\n⊢ ∃ h, f ≫ h = g",
"tactic": "let G : (⟨X⟩ : ModuleCat ℤ) ⟶ ⟨A⟩ :=\n { g with\n map_smul' := by\n intros\n dsimp\n rw [map_zsmul] }"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\n⊢ ∃ h, f ≫ h = g",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\n⊢ ∃ h, f ≫ h = g",
"tactic": "let F : (⟨X⟩ : ModuleCat ℤ) ⟶ ⟨Y⟩ :=\n { f with\n map_smul' := by\n intros\n dsimp\n rw [map_zsmul] }"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nthis : Mono F\n⊢ ∃ h, f ≫ h = g",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\n⊢ ∃ h, f ≫ h = g",
"tactic": "have : Mono F := by\n refine' ⟨fun {Z} α β eq1 => _⟩\n let α' : AddCommGroupCat.of Z ⟶ X := @LinearMap.toAddMonoidHom _ _ _ _ _ _ _ _ (_) _ _ α\n let β' : AddCommGroupCat.of Z ⟶ X := @LinearMap.toAddMonoidHom _ _ _ _ _ _ _ _ (_) _ _ β\n have eq2 : α' ≫ f = β' ≫ f := by\n ext x\n simp only [CategoryTheory.comp_apply, LinearMap.toAddMonoidHom_coe]\n simpa only [ModuleCat.coe_comp, LinearMap.coe_mk, Function.comp_apply] using\n FunLike.congr_fun eq1 x\n rw [cancel_mono] at eq2\n have : ⇑α' = ⇑β' := congrArg _ eq2\n ext x\n apply congrFun this _"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nthis : Mono F\n⊢ f ≫ LinearMap.toAddMonoidHom (Injective.factorThru G F) = g",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nthis : Mono F\n⊢ ∃ h, f ≫ h = g",
"tactic": "refine' ⟨(Injective.factorThru G F).toAddMonoidHom, _⟩"
},
{
"state_after": "case w\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nthis : Mono F\nx : (forget (Bundled AddCommGroup)).obj X\n⊢ (forget (Bundled AddCommGroup)).map (f ≫ LinearMap.toAddMonoidHom (Injective.factorThru G F)) x =\n (forget (Bundled AddCommGroup)).map g x",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nthis : Mono F\n⊢ f ≫ LinearMap.toAddMonoidHom (Injective.factorThru G F) = g",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case w\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nthis : Mono F\nx : (forget (Bundled AddCommGroup)).obj X\n⊢ (forget (Bundled AddCommGroup)).map (f ≫ LinearMap.toAddMonoidHom (Injective.factorThru G F)) x =\n (forget (Bundled AddCommGroup)).map g x",
"tactic": "convert FunLike.congr_fun (Injective.comp_factorThru G F) x"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ AddHom.toFun\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r✝ • x✝) =\n ↑(RingHom.id ℤ) r✝ •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x✝",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\n⊢ ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x",
"tactic": "intros"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ ↑g (r✝ • x✝) = r✝ • ↑g x✝",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ AddHom.toFun\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r✝ • x✝) =\n ↑(RingHom.id ℤ) r✝ •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x✝",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ ↑g (r✝ • x✝) = r✝ • ↑g x✝",
"tactic": "rw [map_zsmul]"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ AddHom.toFun\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r✝ • x✝) =\n ↑(RingHom.id ℤ) r✝ •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x✝",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\n⊢ ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x",
"tactic": "intros"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ ↑f (r✝ • x✝) = r✝ • ↑f x✝",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ AddHom.toFun\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r✝ • x✝) =\n ↑(RingHom.id ℤ) r✝ •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x✝",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nr✝ : ℤ\nx✝ : ↑(ModuleCat.mk ↑X)\n⊢ ↑f (r✝ • x✝) = r✝ • ↑f x✝",
"tactic": "rw [map_zsmul]"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\n⊢ α = β",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\n⊢ Mono F",
"tactic": "refine' ⟨fun {Z} α β eq1 => _⟩"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\n⊢ α = β",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\n⊢ α = β",
"tactic": "let α' : AddCommGroupCat.of Z ⟶ X := @LinearMap.toAddMonoidHom _ _ _ _ _ _ _ _ (_) _ _ α"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\n⊢ α = β",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\n⊢ α = β",
"tactic": "let β' : AddCommGroupCat.of Z ⟶ X := @LinearMap.toAddMonoidHom _ _ _ _ _ _ _ _ (_) _ _ β"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' ≫ f = β' ≫ f\n⊢ α = β",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\n⊢ α = β",
"tactic": "have eq2 : α' ≫ f = β' ≫ f := by\n ext x\n simp only [CategoryTheory.comp_apply, LinearMap.toAddMonoidHom_coe]\n simpa only [ModuleCat.coe_comp, LinearMap.coe_mk, Function.comp_apply] using\n FunLike.congr_fun eq1 x"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' = β'\n⊢ α = β",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' ≫ f = β' ≫ f\n⊢ α = β",
"tactic": "rw [cancel_mono] at eq2"
},
{
"state_after": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' = β'\nthis : ↑α' = ↑β'\n⊢ α = β",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' = β'\n⊢ α = β",
"tactic": "have : ⇑α' = ⇑β' := congrArg _ eq2"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' = β'\nthis : ↑α' = ↑β'\nx : ↑Z\n⊢ ↑α x = ↑β x",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' = β'\nthis : ↑α' = ↑β'\n⊢ α = β",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\neq2 : α' = β'\nthis : ↑α' = ↑β'\nx : ↑Z\n⊢ ↑α x = ↑β x",
"tactic": "apply congrFun this _"
},
{
"state_after": "case w\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\nx : ↑(of ↑Z)\n⊢ ↑(α' ≫ f) x = ↑(β' ≫ f) x",
"state_before": "A : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\n⊢ α' ≫ f = β' ≫ f",
"tactic": "ext x"
},
{
"state_after": "case w\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\nx : ↑(of ↑Z)\n⊢ ↑(LinearMap.toAddMonoidHom α ≫ f) x = ↑(LinearMap.toAddMonoidHom β ≫ f) x",
"state_before": "case w\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\nx : ↑(of ↑Z)\n⊢ ↑(α' ≫ f) x = ↑(β' ≫ f) x",
"tactic": "simp only [CategoryTheory.comp_apply, LinearMap.toAddMonoidHom_coe]"
},
{
"state_after": "no goals",
"state_before": "case w\nA : Type u\ninst✝¹ : AddCommGroup A\ninst✝ : Injective (ModuleCat.mk A)\nX Y : Bundled AddCommGroup\ng : X ⟶ Bundled.mk A\nf : X ⟶ Y\nm : Mono f\nG : ModuleCat.mk ↑X ⟶ ModuleCat.mk A :=\n {\n toAddHom :=\n { toFun := g.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := g.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑g) (x + y) = ZeroHom.toFun (↑g) x + ZeroHom.toFun (↑g) y) }\n x) }\nF : ModuleCat.mk ↑X ⟶ ModuleCat.mk ↑Y :=\n {\n toAddHom :=\n { toFun := f.toFun,\n map_add' := (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) },\n map_smul' :=\n (_ :\n ∀ (r : ℤ) (x : ↑(ModuleCat.mk ↑X)),\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n (r • x) =\n ↑(RingHom.id ℤ) r •\n AddHom.toFun\n { toFun := f.toFun,\n map_add' :=\n (_ : ∀ (x y : ↑X), ZeroHom.toFun (↑f) (x + y) = ZeroHom.toFun (↑f) x + ZeroHom.toFun (↑f) y) }\n x) }\nZ : ModuleCat ℤ\nα β : Z ⟶ ModuleCat.mk ↑X\neq1 : α ≫ F = β ≫ F\nα' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom α\nβ' : of ↑Z ⟶ X := LinearMap.toAddMonoidHom β\nx : ↑(of ↑Z)\n⊢ ↑(LinearMap.toAddMonoidHom α ≫ f) x = ↑(LinearMap.toAddMonoidHom β ≫ f) x",
"tactic": "simpa only [ModuleCat.coe_comp, LinearMap.coe_mk, Function.comp_apply] using\n FunLike.congr_fun eq1 x"
}
] |
[
71,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean
|
CategoryTheory.CartesianClosed.curry_injective
|
[] |
[
241,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.toCompositionAsSet_blocks
|
[
{
"state_after": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\n⊢ CompositionAsSet.blocks (toCompositionAsSet c) = c.blocks",
"state_before": "n : ℕ\nc : Composition n\n⊢ CompositionAsSet.blocks (toCompositionAsSet c) = c.blocks",
"tactic": "let d := c.toCompositionAsSet"
},
{
"state_after": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\n⊢ CompositionAsSet.blocks d = c.blocks",
"state_before": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\n⊢ CompositionAsSet.blocks (toCompositionAsSet c) = c.blocks",
"tactic": "change d.blocks = c.blocks"
},
{
"state_after": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\n⊢ CompositionAsSet.blocks d = c.blocks",
"state_before": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\n⊢ CompositionAsSet.blocks d = c.blocks",
"tactic": "have length_eq : d.blocks.length = c.blocks.length := by\n convert c.toCompositionAsSet_length\n simp [CompositionAsSet.blocks]"
},
{
"state_after": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\nH :\n ∀ (i : ℕ),\n i ≤ List.length (CompositionAsSet.blocks d) → sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)\n⊢ CompositionAsSet.blocks d = c.blocks\n\ncase H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\n⊢ ∀ (i : ℕ),\n i ≤ List.length (CompositionAsSet.blocks d) → sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\n⊢ CompositionAsSet.blocks d = c.blocks",
"tactic": "suffices H : ∀ i ≤ d.blocks.length, (d.blocks.take i).sum = (c.blocks.take i).sum"
},
{
"state_after": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\n⊢ ∀ (i : ℕ),\n i ≤ List.length (CompositionAsSet.blocks d) → sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\nH :\n ∀ (i : ℕ),\n i ≤ List.length (CompositionAsSet.blocks d) → sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)\n⊢ CompositionAsSet.blocks d = c.blocks\n\ncase H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\n⊢ ∀ (i : ℕ),\n i ≤ List.length (CompositionAsSet.blocks d) → sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "exact eq_of_sum_take_eq length_eq H"
},
{
"state_after": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\n⊢ ∀ (i : ℕ),\n i ≤ List.length (CompositionAsSet.blocks d) → sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "intro i hi"
},
{
"state_after": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "have i_lt : i < d.boundaries.card := by\n simpa [CompositionAsSet.blocks, length_ofFn, Nat.succ_eq_add_one,\n d.card_boundaries_eq_succ_length] using Nat.lt_succ_iff.2 hi"
},
{
"state_after": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "have i_lt' : i < c.boundaries.card := i_lt"
},
{
"state_after": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\ni_lt'' : i < length c + 1\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "have i_lt'' : i < c.length + 1 := by rwa [c.card_boundaries_eq_succ_length] at i_lt'"
},
{
"state_after": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\ni_lt'' : i < length c + 1\nA :\n ↑(Finset.orderEmbOfFin d.boundaries (_ : Finset.card d.boundaries = Finset.card d.boundaries))\n { val := i, isLt := i_lt } =\n ↑(Finset.orderEmbOfFin (boundaries c) (_ : Finset.card (boundaries c) = length c + 1)) { val := i, isLt := i_lt'' }\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\ni_lt'' : i < length c + 1\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "have A :\n d.boundaries.orderEmbOfFin rfl ⟨i, i_lt⟩ =\n c.boundaries.orderEmbOfFin c.card_boundaries_eq_succ_length ⟨i, i_lt''⟩ :=\n rfl"
},
{
"state_after": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\ni_lt'' : i < length c + 1\nA :\n ↑(Finset.orderEmbOfFin d.boundaries (_ : Finset.card d.boundaries = Finset.card d.boundaries))\n { val := i, isLt := i_lt } =\n ↑(Finset.orderEmbOfFin (boundaries c) (_ : Finset.card (boundaries c) = length c + 1)) { val := i, isLt := i_lt'' }\nB : sizeUpTo c i = ↑(↑(boundary c) { val := i, isLt := i_lt'' })\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"state_before": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\ni_lt'' : i < length c + 1\nA :\n ↑(Finset.orderEmbOfFin d.boundaries (_ : Finset.card d.boundaries = Finset.card d.boundaries))\n { val := i, isLt := i_lt } =\n ↑(Finset.orderEmbOfFin (boundaries c) (_ : Finset.card (boundaries c) = length c + 1)) { val := i, isLt := i_lt'' }\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "have B : c.sizeUpTo i = c.boundary ⟨i, i_lt''⟩ := rfl"
},
{
"state_after": "no goals",
"state_before": "case H\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\ni_lt'' : i < length c + 1\nA :\n ↑(Finset.orderEmbOfFin d.boundaries (_ : Finset.card d.boundaries = Finset.card d.boundaries))\n { val := i, isLt := i_lt } =\n ↑(Finset.orderEmbOfFin (boundaries c) (_ : Finset.card (boundaries c) = length c + 1)) { val := i, isLt := i_lt'' }\nB : sizeUpTo c i = ↑(↑(boundary c) { val := i, isLt := i_lt'' })\n⊢ sum (take i (CompositionAsSet.blocks d)) = sum (take i c.blocks)",
"tactic": "rw [d.blocks_partial_sum i_lt, CompositionAsSet.boundary, ← Composition.sizeUpTo, B, A,\n c.orderEmbOfFin_boundaries]"
},
{
"state_after": "case h.e'_2\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\n⊢ List.length (CompositionAsSet.blocks d) = CompositionAsSet.length (toCompositionAsSet c)",
"state_before": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\n⊢ List.length (CompositionAsSet.blocks d) = List.length c.blocks",
"tactic": "convert c.toCompositionAsSet_length"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nn : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\n⊢ List.length (CompositionAsSet.blocks d) = CompositionAsSet.length (toCompositionAsSet c)",
"tactic": "simp [CompositionAsSet.blocks]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\n⊢ i < Finset.card d.boundaries",
"tactic": "simpa [CompositionAsSet.blocks, length_ofFn, Nat.succ_eq_add_one,\n d.card_boundaries_eq_succ_length] using Nat.lt_succ_iff.2 hi"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nc : Composition n\nd : CompositionAsSet n := toCompositionAsSet c\nlength_eq : List.length (CompositionAsSet.blocks d) = List.length c.blocks\ni : ℕ\nhi : i ≤ List.length (CompositionAsSet.blocks d)\ni_lt : i < Finset.card d.boundaries\ni_lt' : i < Finset.card (boundaries c)\n⊢ i < length c + 1",
"tactic": "rwa [c.card_boundaries_eq_succ_length] at i_lt'"
}
] |
[
1037,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1015,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.tendsto_prod_self_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.47951\nι : Sort ?u.47954\nf✝ : Filter α\nf : α × α → β\nx : Filter α\ny : Filter β\n⊢ Tendsto f (x ×ˢ x) y ↔ ∀ (W : Set β), W ∈ y → ∃ U, U ∈ x ∧ ∀ (x x' : α), x ∈ U → x' ∈ U → f (x, x') ∈ W",
"tactic": "simp only [tendsto_def, mem_prod_same_iff, prod_sub_preimage_iff, exists_prop, iff_self_iff]"
}
] |
[
438,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
HasDerivWithinAt.smul_const
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nc : 𝕜 → 𝕜'\nc' : 𝕜'\nhc : HasDerivWithinAt c c' s x\nf : F\nthis : HasDerivWithinAt (fun y => c y • f) (c x • 0 + c' • f) s x\n⊢ HasDerivWithinAt (fun y => c y • f) (c' • f) s x",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nc : 𝕜 → 𝕜'\nc' : 𝕜'\nhc : HasDerivWithinAt c c' s x\nf : F\n⊢ HasDerivWithinAt (fun y => c y • f) (c' • f) s x",
"tactic": "have := hc.smul (hasDerivWithinAt_const x s f)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nc : 𝕜 → 𝕜'\nc' : 𝕜'\nhc : HasDerivWithinAt c c' s x\nf : F\nthis : HasDerivWithinAt (fun y => c y • f) (c x • 0 + c' • f) s x\n⊢ HasDerivWithinAt (fun y => c y • f) (c' • f) s x",
"tactic": "rwa [smul_zero, zero_add] at this"
}
] |
[
99,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.imClm_apply
|
[] |
[
292,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.continuous_proj
|
[] |
[
769,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
8
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.transnumAuxSeq_dist_lt
|
[
{
"state_after": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < 2 ^ (n + 1)\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) < 1 / 2 / 2 ^ n",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) < 1 / 2 / 2 ^ n",
"tactic": "have : 0 < (2 ^ (n + 1) : ℝ) := pow_pos zero_lt_two _"
},
{
"state_after": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < 2 ^ (n + 1)\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) < 1 / abs (2 ^ (n + 1))",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < 2 ^ (n + 1)\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) < 1 / 2 / 2 ^ n",
"tactic": "rw [div_div, ← pow_succ, ← abs_of_pos this]"
},
{
"state_after": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < abs (2 ^ (n + 1))\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) < 1 / abs (2 ^ (n + 1))",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < 2 ^ (n + 1)\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) < 1 / abs (2 ^ (n + 1))",
"tactic": "replace := abs_pos.2 (ne_of_gt this)"
},
{
"state_after": "case h.e'_3\nf g : CircleDeg1Lift\nn : ℕ\nthis : 0 < abs (2 ^ (n + 1))\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) =\n dist (↑(f ^ 2 ^ n) 0 + ↑(f ^ 2 ^ n) 0) (↑(f ^ 2 ^ n) (↑(f ^ 2 ^ n) 0)) / abs (2 ^ (n + 1))",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\nthis : 0 < abs (2 ^ (n + 1))\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) < 1 / abs (2 ^ (n + 1))",
"tactic": "convert (div_lt_div_right this).2 ((f ^ 2 ^ n).dist_map_map_zero_lt (f ^ 2 ^ n)) using 1"
},
{
"state_after": "case h.e'_3\nf g : CircleDeg1Lift\nn : ℕ\nthis : 0 < abs (2 ^ (n + 1))\n⊢ abs (↑(f ^ 2 ^ n) 0 / 2 ^ n - ↑(f ^ 2 ^ (n + 1)) 0 / 2 ^ (n + 1)) =\n abs (↑(f ^ 2 ^ n) 0 + ↑(f ^ 2 ^ n) 0 - ↑(f ^ 2 ^ n) (↑(f ^ 2 ^ n) 0)) / abs (2 ^ (n + 1))",
"state_before": "case h.e'_3\nf g : CircleDeg1Lift\nn : ℕ\nthis : 0 < abs (2 ^ (n + 1))\n⊢ dist (transnumAuxSeq f n) (transnumAuxSeq f (n + 1)) =\n dist (↑(f ^ 2 ^ n) 0 + ↑(f ^ 2 ^ n) 0) (↑(f ^ 2 ^ n) (↑(f ^ 2 ^ n) 0)) / abs (2 ^ (n + 1))",
"tactic": "simp_rw [transnumAuxSeq, Real.dist_eq]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nf g : CircleDeg1Lift\nn : ℕ\nthis : 0 < abs (2 ^ (n + 1))\n⊢ abs (↑(f ^ 2 ^ n) 0 / 2 ^ n - ↑(f ^ 2 ^ (n + 1)) 0 / 2 ^ (n + 1)) =\n abs (↑(f ^ 2 ^ n) 0 + ↑(f ^ 2 ^ n) 0 - ↑(f ^ 2 ^ n) (↑(f ^ 2 ^ n) 0)) / abs (2 ^ (n + 1))",
"tactic": "rw [← abs_div, sub_div, pow_succ', pow_succ, ← two_mul, mul_div_mul_left _ _ (two_ne_zero' ℝ),\n pow_mul, sq, mul_apply]"
}
] |
[
679,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
671,
1
] |
Mathlib/RingTheory/Adjoin/Basic.lean
|
Algebra.adjoin_insert_adjoin
|
[] |
[
230,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.ImageMap.map_uniq_aux
|
[
{
"state_after": "C : Type u\ninst✝⁴ : Category C\nf✝ g✝ : Arrow C\ninst✝³ : HasImage f✝.hom\ninst✝² : HasImage g✝.hom\nsq✝ : f✝ ⟶ g✝\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nmap : image f.hom ⟶ image g.hom\nmap_ι : autoParam (map ≫ image.ι g.hom = image.ι f.hom ≫ sq.right) _auto✝\nmap' : image f.hom ⟶ image g.hom\nmap_ι' : map' ≫ image.ι g.hom = image.ι f.hom ≫ sq.right\nthis : map ≫ image.ι g.hom = map' ≫ image.ι g.hom\n⊢ map = map'",
"state_before": "C : Type u\ninst✝⁴ : Category C\nf✝ g✝ : Arrow C\ninst✝³ : HasImage f✝.hom\ninst✝² : HasImage g✝.hom\nsq✝ : f✝ ⟶ g✝\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nmap : image f.hom ⟶ image g.hom\nmap_ι : autoParam (map ≫ image.ι g.hom = image.ι f.hom ≫ sq.right) _auto✝\nmap' : image f.hom ⟶ image g.hom\nmap_ι' : map' ≫ image.ι g.hom = image.ι f.hom ≫ sq.right\n⊢ map = map'",
"tactic": "have : map ≫ image.ι g.hom = map' ≫ image.ι g.hom := by rw [map_ι,map_ι']"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\nf✝ g✝ : Arrow C\ninst✝³ : HasImage f✝.hom\ninst✝² : HasImage g✝.hom\nsq✝ : f✝ ⟶ g✝\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nmap : image f.hom ⟶ image g.hom\nmap_ι : autoParam (map ≫ image.ι g.hom = image.ι f.hom ≫ sq.right) _auto✝\nmap' : image f.hom ⟶ image g.hom\nmap_ι' : map' ≫ image.ι g.hom = image.ι f.hom ≫ sq.right\nthis : map ≫ image.ι g.hom = map' ≫ image.ι g.hom\n⊢ map = map'",
"tactic": "apply (cancel_mono (image.ι g.hom)).1 this"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\nf✝ g✝ : Arrow C\ninst✝³ : HasImage f✝.hom\ninst✝² : HasImage g✝.hom\nsq✝ : f✝ ⟶ g✝\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nmap : image f.hom ⟶ image g.hom\nmap_ι : autoParam (map ≫ image.ι g.hom = image.ι f.hom ≫ sq.right) _auto✝\nmap' : image f.hom ⟶ image g.hom\nmap_ι' : map' ≫ image.ι g.hom = image.ι f.hom ≫ sq.right\n⊢ map ≫ image.ι g.hom = map' ≫ image.ι g.hom",
"tactic": "rw [map_ι,map_ι']"
}
] |
[
770,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Logic/Basic.lean
|
or_not_of_imp
|
[] |
[
369,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Data/Int/Dvd/Basic.lean
|
Int.coe_nat_dvd
|
[
{
"state_after": "m n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nm0 : m = 0\nae : n = 0\n⊢ m ∣ n",
"state_before": "m n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nae : ↑n = ↑m * a\nm0 : m = 0\n⊢ m ∣ n",
"tactic": "simp [m0] at ae"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nm0 : m = 0\nae : n = 0\n⊢ m ∣ n",
"tactic": "simp [ae, m0]"
},
{
"state_after": "case intro\nm n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nae : ↑n = ↑m * a\nm0l : m > 0\nk : ℕ\ne : a = ↑k\n⊢ m ∣ n",
"state_before": "m n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nae : ↑n = ↑m * a\nm0l : m > 0\n⊢ m ∣ n",
"tactic": "cases'\n eq_ofNat_of_zero_le\n (@nonneg_of_mul_nonneg_right ℤ _ m a (by simp [ae.symm]) (by simpa using m0l)) with\n k e"
},
{
"state_after": "case intro\nm n : ℕ\nx✝ : ↑m ∣ ↑n\nm0l : m > 0\nk : ℕ\nae : ↑n = ↑m * ↑k\n⊢ m ∣ n",
"state_before": "case intro\nm n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nae : ↑n = ↑m * a\nm0l : m > 0\nk : ℕ\ne : a = ↑k\n⊢ m ∣ n",
"tactic": "subst a"
},
{
"state_after": "no goals",
"state_before": "case intro\nm n : ℕ\nx✝ : ↑m ∣ ↑n\nm0l : m > 0\nk : ℕ\nae : ↑n = ↑m * ↑k\n⊢ m ∣ n",
"tactic": "exact ⟨k, Int.ofNat.inj ae⟩"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nae : ↑n = ↑m * a\nm0l : m > 0\n⊢ 0 ≤ ↑m * a",
"tactic": "simp [ae.symm]"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nx✝ : ↑m ∣ ↑n\na : ℤ\nae : ↑n = ↑m * a\nm0l : m > 0\n⊢ 0 < ↑m",
"tactic": "simpa using m0l"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nx✝ : m ∣ n\nk : ℕ\ne : n = m * k\n⊢ ↑m * ↑k = ↑n",
"tactic": "rw [e, Int.ofNat_mul]"
}
] |
[
34,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
25,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_sub_atTop_nat
|
[] |
[
1654,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1653,
1
] |
Mathlib/Data/Nat/Hyperoperation.lean
|
hyperoperation_ge_three_eq_one
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ hyperoperation (n + 3) m 0 = 1",
"tactic": "rw [hyperoperation]"
}
] |
[
55,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
ContDiffOn.fderivWithin
|
[
{
"state_after": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : none + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s\n\ncase some\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nm : ℕ\nhmn : some m + 1 ≤ n\n⊢ ContDiffOn 𝕜 (some m) (fun y => fderivWithin 𝕜 f s y) s",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : m + 1 ≤ n\n⊢ ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s",
"tactic": "cases' m with m"
},
{
"state_after": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : ⊤ + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s",
"state_before": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : none + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s",
"tactic": "change ∞ + 1 ≤ n at hmn"
},
{
"state_after": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s",
"state_before": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : ⊤ + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s",
"tactic": "have : n = ∞ := by simpa using hmn"
},
{
"state_after": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 ⊤ f s\nhs : UniqueDiffOn 𝕜 s\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s",
"state_before": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s",
"tactic": "rw [this] at hf"
},
{
"state_after": "no goals",
"state_before": "case none\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 ⊤ f s\nhs : UniqueDiffOn 𝕜 s\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (fun y => fderivWithin 𝕜 f s y) s",
"tactic": "exact ((contDiffOn_top_iff_fderivWithin hs).1 hf).2"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nhmn : ⊤ + 1 ≤ n\n⊢ n = ⊤",
"tactic": "simpa using hmn"
},
{
"state_after": "case some\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nm : ℕ\nhmn : ↑(Nat.succ m) ≤ n\n⊢ ContDiffOn 𝕜 (some m) (fun y => fderivWithin 𝕜 f s y) s",
"state_before": "case some\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nm : ℕ\nhmn : some m + 1 ≤ n\n⊢ ContDiffOn 𝕜 (some m) (fun y => fderivWithin 𝕜 f s y) s",
"tactic": "change (m.succ : ℕ∞) ≤ n at hmn"
},
{
"state_after": "no goals",
"state_before": "case some\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nhf : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nm : ℕ\nhmn : ↑(Nat.succ m) ≤ n\n⊢ ContDiffOn 𝕜 (some m) (fun y => fderivWithin 𝕜 f s y) s",
"tactic": "exact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2"
}
] |
[
1190,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1182,
11
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.le_sup_of_le
|
[] |
[
125,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/CategoryTheory/Bicategory/Free.lean
|
CategoryTheory.FreeBicategory.mk_id
|
[] |
[
275,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
Function.update_exp
|
[
{
"state_after": "case h\n𝕂 : Type u_3\n𝔸✝ : Type ?u.766093\n𝔹 : Type ?u.766096\ninst✝¹⁰ : IsROrC 𝕂\ninst✝⁹ : NormedRing 𝔸✝\ninst✝⁸ : NormedAlgebra 𝕂 𝔸✝\ninst✝⁷ : NormedRing 𝔹\ninst✝⁶ : NormedAlgebra 𝕂 𝔹\ninst✝⁵ : CompleteSpace 𝔸✝\nι : Type u_1\n𝔸 : ι → Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → NormedRing (𝔸 i)\ninst✝¹ : (i : ι) → NormedAlgebra 𝕂 (𝔸 i)\ninst✝ : ∀ (i : ι), CompleteSpace (𝔸 i)\nx : (i : ι) → 𝔸 i\nj : ι\nxj : 𝔸 j\ni : ι\n⊢ update (exp 𝕂 x) j (exp 𝕂 xj) i = exp 𝕂 (update x j xj) i",
"state_before": "𝕂 : Type u_3\n𝔸✝ : Type ?u.766093\n𝔹 : Type ?u.766096\ninst✝¹⁰ : IsROrC 𝕂\ninst✝⁹ : NormedRing 𝔸✝\ninst✝⁸ : NormedAlgebra 𝕂 𝔸✝\ninst✝⁷ : NormedRing 𝔹\ninst✝⁶ : NormedAlgebra 𝕂 𝔹\ninst✝⁵ : CompleteSpace 𝔸✝\nι : Type u_1\n𝔸 : ι → Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → NormedRing (𝔸 i)\ninst✝¹ : (i : ι) → NormedAlgebra 𝕂 (𝔸 i)\ninst✝ : ∀ (i : ι), CompleteSpace (𝔸 i)\nx : (i : ι) → 𝔸 i\nj : ι\nxj : 𝔸 j\n⊢ update (exp 𝕂 x) j (exp 𝕂 xj) = exp 𝕂 (update x j xj)",
"tactic": "ext i"
},
{
"state_after": "case h\n𝕂 : Type u_3\n𝔸✝ : Type ?u.766093\n𝔹 : Type ?u.766096\ninst✝¹⁰ : IsROrC 𝕂\ninst✝⁹ : NormedRing 𝔸✝\ninst✝⁸ : NormedAlgebra 𝕂 𝔸✝\ninst✝⁷ : NormedRing 𝔹\ninst✝⁶ : NormedAlgebra 𝕂 𝔹\ninst✝⁵ : CompleteSpace 𝔸✝\nι : Type u_1\n𝔸 : ι → Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → NormedRing (𝔸 i)\ninst✝¹ : (i : ι) → NormedAlgebra 𝕂 (𝔸 i)\ninst✝ : ∀ (i : ι), CompleteSpace (𝔸 i)\nx : (i : ι) → 𝔸 i\nj : ι\nxj : 𝔸 j\ni : ι\n⊢ update (fun i => exp 𝕂 (x i)) j (exp 𝕂 xj) i = exp 𝕂 (update x j xj i)",
"state_before": "case h\n𝕂 : Type u_3\n𝔸✝ : Type ?u.766093\n𝔹 : Type ?u.766096\ninst✝¹⁰ : IsROrC 𝕂\ninst✝⁹ : NormedRing 𝔸✝\ninst✝⁸ : NormedAlgebra 𝕂 𝔸✝\ninst✝⁷ : NormedRing 𝔹\ninst✝⁶ : NormedAlgebra 𝕂 𝔹\ninst✝⁵ : CompleteSpace 𝔸✝\nι : Type u_1\n𝔸 : ι → Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → NormedRing (𝔸 i)\ninst✝¹ : (i : ι) → NormedAlgebra 𝕂 (𝔸 i)\ninst✝ : ∀ (i : ι), CompleteSpace (𝔸 i)\nx : (i : ι) → 𝔸 i\nj : ι\nxj : 𝔸 j\ni : ι\n⊢ update (exp 𝕂 x) j (exp 𝕂 xj) i = exp 𝕂 (update x j xj) i",
"tactic": "simp_rw [Pi.exp_def]"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕂 : Type u_3\n𝔸✝ : Type ?u.766093\n𝔹 : Type ?u.766096\ninst✝¹⁰ : IsROrC 𝕂\ninst✝⁹ : NormedRing 𝔸✝\ninst✝⁸ : NormedAlgebra 𝕂 𝔸✝\ninst✝⁷ : NormedRing 𝔹\ninst✝⁶ : NormedAlgebra 𝕂 𝔹\ninst✝⁵ : CompleteSpace 𝔸✝\nι : Type u_1\n𝔸 : ι → Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → NormedRing (𝔸 i)\ninst✝¹ : (i : ι) → NormedAlgebra 𝕂 (𝔸 i)\ninst✝ : ∀ (i : ι), CompleteSpace (𝔸 i)\nx : (i : ι) → 𝔸 i\nj : ι\nxj : 𝔸 j\ni : ι\n⊢ update (fun i => exp 𝕂 (x i)) j (exp 𝕂 xj) i = exp 𝕂 (update x j xj i)",
"tactic": "exact (Function.apply_update (fun i => exp 𝕂) x j xj i).symm"
}
] |
[
569,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.spanSingleton_inv
|
[] |
[
153,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.35154\nβ : Type u_1\nβ' : Type ?u.35160\nγ : Type ?u.35163\nγ' : Type ?u.35166\nδ : Type ?u.35169\nδ' : Type ?u.35172\nε : Type ?u.35175\nε' : Type ?u.35178\nζ : Type ?u.35181\nζ' : Type ?u.35184\nν : Type ?u.35187\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nh : Set.Nonempty t\n⊢ image2 (fun x x_1 => x) s t = s",
"tactic": "simp [nonempty_def.mp h, ext_iff]"
}
] |
[
302,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/CategoryTheory/Monoidal/End.lean
|
CategoryTheory.ε_hom_inv_app
|
[] |
[
110,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.haar.is_left_invariant_prehaar
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nhU : Set.Nonempty (interior U)\ng : G\nK : Compacts G\n⊢ prehaar (↑K₀) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (↑K₀) U K",
"tactic": "simp only [prehaar, Compacts.coe_map, is_left_invariant_index K.isCompact _ hU]"
}
] |
[
358,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Computability/Reduce.lean
|
OneOneReducible.disjoin_right
|
[] |
[
308,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
StrictConvexOn.comp
|
[] |
[
160,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Data/Set/UnionLift.lean
|
Set.iUnionLift_const
|
[
{
"state_after": "α : Type u_1\nι : Sort u_3\nβ : Sort u_2\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf :\n ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set α\nhT : T ⊆ iUnion S\nhT' : T = iUnion S\nc : ↑T\nci : (i : ι) → ↑(S i)\nhci : ∀ (i : ι), ↑(ci i) = ↑c\ncβ : β\nh : ∀ (i : ι), f i (ci i) = cβ\ni : ι\nhi : ↑c ∈ S i\n⊢ iUnionLift S f hf T hT c = cβ",
"state_before": "α : Type u_1\nι : Sort u_3\nβ : Sort u_2\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf :\n ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set α\nhT : T ⊆ iUnion S\nhT' : T = iUnion S\nc : ↑T\nci : (i : ι) → ↑(S i)\nhci : ∀ (i : ι), ↑(ci i) = ↑c\ncβ : β\nh : ∀ (i : ι), f i (ci i) = cβ\n⊢ iUnionLift S f hf T hT c = cβ",
"tactic": "let ⟨i, hi⟩ := Set.mem_iUnion.1 (hT c.prop)"
},
{
"state_after": "α : Type u_1\nι : Sort u_3\nβ : Sort u_2\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf :\n ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set α\nhT : T ⊆ iUnion S\nhT' : T = iUnion S\nc : ↑T\nci : (i : ι) → ↑(S i)\nhci : ∀ (i : ι), ↑(ci i) = ↑c\ncβ : β\nh : ∀ (i : ι), f i (ci i) = cβ\ni : ι\nhi : ↑c ∈ S i\nthis : ci i = { val := ↑c, property := hi }\n⊢ iUnionLift S f hf T hT c = cβ",
"state_before": "α : Type u_1\nι : Sort u_3\nβ : Sort u_2\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf :\n ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set α\nhT : T ⊆ iUnion S\nhT' : T = iUnion S\nc : ↑T\nci : (i : ι) → ↑(S i)\nhci : ∀ (i : ι), ↑(ci i) = ↑c\ncβ : β\nh : ∀ (i : ι), f i (ci i) = cβ\ni : ι\nhi : ↑c ∈ S i\n⊢ iUnionLift S f hf T hT c = cβ",
"tactic": "have : ci i = ⟨c, hi⟩ := Subtype.ext (hci i)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Sort u_3\nβ : Sort u_2\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf :\n ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j),\n f i { val := x, property := hxi } = f j { val := x, property := hxj }\nT : Set α\nhT : T ⊆ iUnion S\nhT' : T = iUnion S\nc : ↑T\nci : (i : ι) → ↑(S i)\nhci : ∀ (i : ι), ↑(ci i) = ↑c\ncβ : β\nh : ∀ (i : ι), f i (ci i) = cβ\ni : ι\nhi : ↑c ∈ S i\nthis : ci i = { val := ↑c, property := hi }\n⊢ iUnionLift S f hf T hT c = cβ",
"tactic": "rw [iUnionLift_of_mem _ hi, ← this, h]"
}
] |
[
103,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isNat_one
|
[] |
[
35,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.singleton_mul_subgroup
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.166670\nG : Type u_1\nA : Type ?u.166676\nS : Type ?u.166679\ninst✝³ : Group G\ninst✝² : AddGroup A\ns : Set G\ninst✝¹ : Group α\ninst✝ : MulDistribMulAction α G\nH : Subgroup G\nh : G\nhh : h ∈ H\nthis : {x | x ∈ H} = ↑H\n⊢ {h} * ↑H = ↑H",
"tactic": "simpa [preimage, mul_mem_cancel_left (inv_mem hh)]"
}
] |
[
383,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_compl_iff
|
[] |
[
1641,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1640,
1
] |
Mathlib/Order/Hom/Basic.lean
|
Disjoint.map_orderIso
|
[
{
"state_after": "F : Type ?u.113997\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.114006\nδ : Type ?u.114009\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ ↑f (a ⊓ b) ≤ ↑f ⊥",
"state_before": "F : Type ?u.113997\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.114006\nδ : Type ?u.114009\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ Disjoint (↑f a) (↑f b)",
"tactic": "rw [disjoint_iff_inf_le, ← f.map_inf, ← f.map_bot]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.113997\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.114006\nδ : Type ?u.114009\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ ↑f (a ⊓ b) ≤ ↑f ⊥",
"tactic": "exact f.monotone ha.le_bot"
}
] |
[
1212,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1209,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.