file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
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stringclasses 4
values | url
stringclasses 4
values | start
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|---|---|---|---|---|---|---|
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
le_mul_of_le_of_one_le_of_nonneg
|
[] |
[
765,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.iUnion_ball_nat
|
[] |
[
468,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.snd_mk
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Subperm.exists_of_length_lt
|
[
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₁ l₂ l : List α\np : l ~ l₁\ns : l <+ l₂\nh : length l₁ < length l₂\n⊢ length l < length l₂ → ∃ a, a :: l <+~ l₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₁ l₂ l : List α\np : l ~ l₁\ns : l <+ l₂\nh : length l₁ < length l₂\n⊢ ∃ a, a :: l₁ <+~ l₂",
"tactic": "suffices length l < length l₂ → ∃ a : α, a :: l <+~ l₂ from\n (this <| p.symm.length_eq ▸ h).imp fun a => (p.cons a).subperm_right.1"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂✝ l₂ l : List α\ns : l <+ l₂\n⊢ length l < length l₂ → ∃ a, a :: l <+~ l₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ l₁ l₂ l : List α\np : l ~ l₁\ns : l <+ l₂\nh : length l₁ < length l₂\n⊢ length l < length l₂ → ∃ a, a :: l <+~ l₂",
"tactic": "clear h p l₁"
},
{
"state_after": "case slnil\nα : Type uu\nβ : Type vv\nl₁ l₂✝ l₂ l : List α\nh : length [] < length []\n⊢ ∃ a, [a] <+~ []\n\ncase cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ l₂✝ l l₁ l₂ : List α\na : α\ns : l₁ <+ l₂\nIH : length l₁ < length l₂ → ∃ a, a :: l₁ <+~ l₂\nh : length l₁ < length (a :: l₂)\n⊢ ∃ a_1, a_1 :: l₁ <+~ a :: l₂\n\ncase cons₂\nα : Type uu\nβ : Type vv\nl₁ l₂✝¹ l₂ l l₁✝ l₂✝ : List α\nb : α\na✝ : l₁✝ <+ l₂✝\nIH : length l₁✝ < length l₂✝ → ∃ a, a :: l₁✝ <+~ l₂✝\nh : length (b :: l₁✝) < length (b :: l₂✝)\n⊢ ∃ a, a :: b :: l₁✝ <+~ b :: l₂✝",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂✝ l₂ l : List α\ns : l <+ l₂\n⊢ length l < length l₂ → ∃ a, a :: l <+~ l₂",
"tactic": "induction' s with l₁ l₂ a s IH _ _ b _ IH <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case slnil\nα : Type uu\nβ : Type vv\nl₁ l₂✝ l₂ l : List α\nh : length [] < length []\n⊢ ∃ a, [a] <+~ []",
"tactic": "cases h"
},
{
"state_after": "case cons.inl\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ l₂✝ l l₁ l₂ : List α\na : α\ns : l₁ <+ l₂\nIH : length l₁ < length l₂ → ∃ a, a :: l₁ <+~ l₂\nh✝ : length l₁ < length (a :: l₂)\nh : length l₁ < length l₂\n⊢ ∃ a_1, a_1 :: l₁ <+~ a :: l₂\n\ncase cons.inr\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ l₂✝ l l₁ l₂ : List α\na : α\ns : l₁ <+ l₂\nIH : length l₁ < length l₂ → ∃ a, a :: l₁ <+~ l₂\nh✝ : length l₁ < length (a :: l₂)\nh : length l₁ = length l₂\n⊢ ∃ a_1, a_1 :: l₁ <+~ a :: l₂",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ l₂✝ l l₁ l₂ : List α\na : α\ns : l₁ <+ l₂\nIH : length l₁ < length l₂ → ∃ a, a :: l₁ <+~ l₂\nh : length l₁ < length (a :: l₂)\n⊢ ∃ a_1, a_1 :: l₁ <+~ a :: l₂",
"tactic": "cases' lt_or_eq_of_le (Nat.le_of_lt_succ h : length l₁ ≤ length l₂) with h h"
},
{
"state_after": "no goals",
"state_before": "case cons.inl\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ l₂✝ l l₁ l₂ : List α\na : α\ns : l₁ <+ l₂\nIH : length l₁ < length l₂ → ∃ a, a :: l₁ <+~ l₂\nh✝ : length l₁ < length (a :: l₂)\nh : length l₁ < length l₂\n⊢ ∃ a_1, a_1 :: l₁ <+~ a :: l₂",
"tactic": "exact (IH h).imp fun a s => s.trans (sublist_cons _ _).subperm"
},
{
"state_after": "no goals",
"state_before": "case cons.inr\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝¹ l₂✝ l l₁ l₂ : List α\na : α\ns : l₁ <+ l₂\nIH : length l₁ < length l₂ → ∃ a, a :: l₁ <+~ l₂\nh✝ : length l₁ < length (a :: l₂)\nh : length l₁ = length l₂\n⊢ ∃ a_1, a_1 :: l₁ <+~ a :: l₂",
"tactic": "exact ⟨a, s.eq_of_length h ▸ Subperm.refl _⟩"
},
{
"state_after": "no goals",
"state_before": "case cons₂\nα : Type uu\nβ : Type vv\nl₁ l₂✝¹ l₂ l l₁✝ l₂✝ : List α\nb : α\na✝ : l₁✝ <+ l₂✝\nIH : length l₁✝ < length l₂✝ → ∃ a, a :: l₁✝ <+~ l₂✝\nh : length (b :: l₁✝) < length (b :: l₂✝)\n⊢ ∃ a, a :: b :: l₁✝ <+~ b :: l₂✝",
"tactic": "exact (IH <| Nat.lt_of_succ_lt_succ h).imp fun a s =>\n (swap _ _ _).subperm_right.1 <| (subperm_cons _).2 s"
}
] |
[
732,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
720,
1
] |
Mathlib/Analysis/ODE/PicardLindelof.lean
|
PicardLindelof.FunSpace.lipschitz
|
[] |
[
181,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
11
] |
Mathlib/Algebra/Divisibility/Basic.lean
|
map_dvd
|
[] |
[
99,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.tuple0_le
|
[] |
[
54,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.EqOnSource.restr
|
[] |
[
980,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
978,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.apply_mem_span_image_of_mem_span
|
[
{
"state_after": "R : Type u_1\nR₂ : Type u_2\nK : Type ?u.203558\nM : Type u_3\nM₂ : Type u_4\nV : Type ?u.203567\nS : Type ?u.203570\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝³ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₂ M₂\ns✝ t : Set M\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\nx : M\ns : Set M\nh : x ∈ span R s\n⊢ ↑f x ∈ map f (span R s)",
"state_before": "R : Type u_1\nR₂ : Type u_2\nK : Type ?u.203558\nM : Type u_3\nM₂ : Type u_4\nV : Type ?u.203567\nS : Type ?u.203570\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝³ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₂ M₂\ns✝ t : Set M\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\nx : M\ns : Set M\nh : x ∈ span R s\n⊢ ↑f x ∈ span R₂ (↑f '' s)",
"tactic": "rw [Submodule.span_image]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₂ : Type u_2\nK : Type ?u.203558\nM : Type u_3\nM₂ : Type u_4\nV : Type ?u.203567\nS : Type ?u.203570\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝³ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R₂ M₂\ns✝ t : Set M\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\nx : M\ns : Set M\nh : x ∈ span R s\n⊢ ↑f x ∈ map f (span R s)",
"tactic": "exact Submodule.mem_map_of_mem h"
}
] |
[
583,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Control/Functor.lean
|
Functor.Comp.run_map
|
[] |
[
203,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
11
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.rOut_pos
|
[] |
[
345,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/RingTheory/Artinian.lean
|
isArtinian_of_fg_of_artinian
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\n⊢ IsArtinian R { x // x ∈ N }",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\n⊢ IsArtinian R { x // x ∈ N }",
"tactic": "let ⟨s, hs⟩ := hN"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis : DecidableEq M\n⊢ IsArtinian R { x // x ∈ N }",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\n⊢ IsArtinian R { x // x ∈ N }",
"tactic": "haveI := Classical.decEq M"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝ : DecidableEq M\nthis : DecidableEq R\n⊢ IsArtinian R { x // x ∈ N }",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis : DecidableEq M\n⊢ IsArtinian R { x // x ∈ N }",
"tactic": "haveI := Classical.decEq R"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ IsArtinian R { x // x ∈ N }",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝ : DecidableEq M\nthis : DecidableEq R\n⊢ IsArtinian R { x // x ∈ N }",
"tactic": "have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx"
},
{
"state_after": "case refine'_1\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ (↑↑s →₀ R) →ₗ[R] { x // x ∈ N }\n\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ Surjective ↑?refine'_1",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ IsArtinian R { x // x ∈ N }",
"tactic": "refine' @isArtinian_of_surjective _ ((↑s : Set M) →₀ R) N _ _ _ _ _ _ _ isArtinian_finsupp"
},
{
"state_after": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ Surjective ↑(Finsupp.total ↑↑s { x // x ∈ N } R fun i => { val := ↑i, property := (_ : ↑i ∈ N) })",
"state_before": "case refine'_1\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ (↑↑s →₀ R) →ₗ[R] { x // x ∈ N }\n\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ Surjective ↑?refine'_1",
"tactic": ". exact Finsupp.total (↑s : Set M) N R (fun i => ⟨i, hs ▸ subset_span i.2⟩)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ Surjective ↑(Finsupp.total ↑↑s { x // x ∈ N } R fun i => { val := ↑i, property := (_ : ↑i ∈ N) })",
"tactic": ". rw [← LinearMap.range_eq_top, eq_top_iff,\n ← map_le_map_iff_of_injective (show Injective (Submodule.subtype N)\n from Subtype.val_injective), Submodule.map_top, range_subtype,\n ← Submodule.map_top, ← Submodule.map_comp, Submodule.map_top]\n subst N\n refine span_le.2 (fun i hi => ?_)\n use Finsupp.single ⟨i, hi⟩ 1\n simp"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ (↑↑s →₀ R) →ₗ[R] { x // x ∈ N }",
"tactic": "exact Finsupp.total (↑s : Set M) N R (fun i => ⟨i, hs ▸ subset_span i.2⟩)"
},
{
"state_after": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ N ≤\n LinearMap.range\n (LinearMap.comp (Submodule.subtype N)\n (Finsupp.total ↑↑s { x // x ∈ N } R fun i => { val := ↑i, property := (_ : ↑i ∈ N) }))",
"state_before": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ Surjective ↑(Finsupp.total ↑↑s { x // x ∈ N } R fun i => { val := ↑i, property := (_ : ↑i ∈ N) })",
"tactic": "rw [← LinearMap.range_eq_top, eq_top_iff,\n ← map_le_map_iff_of_injective (show Injective (Submodule.subtype N)\n from Subtype.val_injective), Submodule.map_top, range_subtype,\n ← Submodule.map_top, ← Submodule.map_comp, Submodule.map_top]"
},
{
"state_after": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinianRing R\ns : Finset M\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nhN : FG (span R ↑s)\nthis : ∀ (x : M), x ∈ s → x ∈ span R ↑s\n⊢ span R ↑s ≤\n LinearMap.range\n (LinearMap.comp (Submodule.subtype (span R ↑s))\n (Finsupp.total ↑↑s { x // x ∈ span R ↑s } R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) }))",
"state_before": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : FG N\ns : Finset M\nhs : span R ↑s = N\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nthis : ∀ (x : M), x ∈ s → x ∈ N\n⊢ N ≤\n LinearMap.range\n (LinearMap.comp (Submodule.subtype N)\n (Finsupp.total ↑↑s { x // x ∈ N } R fun i => { val := ↑i, property := (_ : ↑i ∈ N) }))",
"tactic": "subst N"
},
{
"state_after": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinianRing R\ns : Finset M\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nhN : FG (span R ↑s)\nthis : ∀ (x : M), x ∈ s → x ∈ span R ↑s\ni : M\nhi : i ∈ ↑s\n⊢ i ∈\n ↑(LinearMap.range\n (LinearMap.comp (Submodule.subtype (span R ↑s))\n (Finsupp.total ↑↑s { x // x ∈ span R ↑s } R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) })))",
"state_before": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinianRing R\ns : Finset M\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nhN : FG (span R ↑s)\nthis : ∀ (x : M), x ∈ s → x ∈ span R ↑s\n⊢ span R ↑s ≤\n LinearMap.range\n (LinearMap.comp (Submodule.subtype (span R ↑s))\n (Finsupp.total ↑↑s { x // x ∈ span R ↑s } R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) }))",
"tactic": "refine span_le.2 (fun i hi => ?_)"
},
{
"state_after": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinianRing R\ns : Finset M\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nhN : FG (span R ↑s)\nthis : ∀ (x : M), x ∈ s → x ∈ span R ↑s\ni : M\nhi : i ∈ ↑s\n⊢ ↑(LinearMap.comp (Submodule.subtype (span R ↑s))\n (Finsupp.total ↑↑s { x // x ∈ span R ↑s } R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) }))\n (Finsupp.single { val := i, property := hi } 1) =\n i",
"state_before": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinianRing R\ns : Finset M\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nhN : FG (span R ↑s)\nthis : ∀ (x : M), x ∈ s → x ∈ span R ↑s\ni : M\nhi : i ∈ ↑s\n⊢ i ∈\n ↑(LinearMap.range\n (LinearMap.comp (Submodule.subtype (span R ↑s))\n (Finsupp.total ↑↑s { x // x ∈ span R ↑s } R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) })))",
"tactic": "use Finsupp.single ⟨i, hi⟩ 1"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinianRing R\ns : Finset M\nthis✝¹ : DecidableEq M\nthis✝ : DecidableEq R\nhN : FG (span R ↑s)\nthis : ∀ (x : M), x ∈ s → x ∈ span R ↑s\ni : M\nhi : i ∈ ↑s\n⊢ ↑(LinearMap.comp (Submodule.subtype (span R ↑s))\n (Finsupp.total ↑↑s { x // x ∈ span R ↑s } R fun i => { val := ↑i, property := (_ : ↑i ∈ span R ↑s) }))\n (Finsupp.single { val := i, property := hi } 1) =\n i",
"tactic": "simp"
}
] |
[
355,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Algebra/QuadraticDiscriminant.lean
|
quadratic_eq_zero_iff_discrim_eq_sq
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha : a ≠ 0\nx : R\nh : discrim a b c = (2 * a * x + b) ^ 2\n⊢ a * x * x + b * x + c = 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha : a ≠ 0\nx : R\n⊢ a * x * x + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b) ^ 2",
"tactic": "refine ⟨discrim_eq_sq_of_quadratic_eq_zero, fun h ↦ ?_⟩"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha : a ≠ 0\nx : R\nh : b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2\n⊢ a * x * x + b * x + c = 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha : a ≠ 0\nx : R\nh : discrim a b c = (2 * a * x + b) ^ 2\n⊢ a * x * x + b * x + c = 0",
"tactic": "rw [discrim] at h"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha✝ : a ≠ 0\nx : R\nh : b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2\nha : 2 * 2 * a ≠ 0\n⊢ a * x * x + b * x + c = 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha : a ≠ 0\nx : R\nh : b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2\n⊢ a * x * x + b * x + c = 0",
"tactic": "have ha : 2 * 2 * a ≠ 0 := mul_ne_zero (mul_ne_zero (NeZero.ne _) (NeZero.ne _)) ha"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha✝ : a ≠ 0\nx : R\nh : b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2\nha : 2 * 2 * a ≠ 0\n⊢ 2 * 2 * a * (a * x * x + b * x + c) = 2 * 2 * a * 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha✝ : a ≠ 0\nx : R\nh : b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2\nha : 2 * 2 * a ≠ 0\n⊢ a * x * x + b * x + c = 0",
"tactic": "apply mul_left_cancel₀ ha"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\na b c : R\ninst✝¹ : NeZero 2\ninst✝ : NoZeroDivisors R\nha✝ : a ≠ 0\nx : R\nh : b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2\nha : 2 * 2 * a ≠ 0\n⊢ 2 * 2 * a * (a * x * x + b * x + c) = 2 * 2 * a * 0",
"tactic": "linear_combination -h"
}
] |
[
73,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.normSq_apply
|
[] |
[
584,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Std/Data/List/Lemmas.lean
|
List.forall_mem_map_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nP : β → Prop\n⊢ (∀ (i : β) (x : α), x ∈ l → f x = i → P i) ↔ ∀ (j : α), j ∈ l → P (f j)",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nP : β → Prop\n⊢ (∀ (i : β), i ∈ map f l → P i) ↔ ∀ (j : α), j ∈ l → P (f j)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : List α\nP : β → Prop\n⊢ (∀ (i : β) (x : α), x ∈ l → f x = i → P i) ↔ ∀ (j : α), j ∈ l → P (f j)",
"tactic": "exact ⟨fun H j h => H _ _ h rfl, fun H i x h e => e ▸ H _ h⟩"
}
] |
[
166,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
164,
1
] |
Mathlib/Data/Vector3.lean
|
vectorAllP_iff_forall
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nm n : ℕ\np : α → Prop\nv : Vector3 α n\n⊢ VectorAllP p [] ↔ ∀ (i : Fin2 0), p []\n\ncase refine'_2\nα : Type u_1\nm n : ℕ\np : α → Prop\nv : Vector3 α n\n⊢ ∀ {n : ℕ} (a : α) (w : Vector3 α n),\n (VectorAllP p w ↔ ∀ (i : Fin2 n), p (w i)) → (VectorAllP p (a :: w) ↔ ∀ (i : Fin2 (succ n)), p ((a :: w) i))",
"state_before": "α : Type u_1\nm n : ℕ\np : α → Prop\nv : Vector3 α n\n⊢ VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)",
"tactic": "refine' v.recOn _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nm n : ℕ\np : α → Prop\nv : Vector3 α n\n⊢ VectorAllP p [] ↔ ∀ (i : Fin2 0), p []",
"tactic": "exact ⟨fun _ => Fin2.elim0, fun _ => trivial⟩"
},
{
"state_after": "case refine'_2\nα : Type u_1\nm n : ℕ\np : α → Prop\nv : Vector3 α n\n⊢ ∀ {n : ℕ} (a : α) (w : Vector3 α n),\n (VectorAllP p w ↔ ∀ (i : Fin2 n), p (w i)) → (p a ∧ VectorAllP p w ↔ ∀ (i : Fin2 (succ n)), p ((a :: w) i))",
"state_before": "case refine'_2\nα : Type u_1\nm n : ℕ\np : α → Prop\nv : Vector3 α n\n⊢ ∀ {n : ℕ} (a : α) (w : Vector3 α n),\n (VectorAllP p w ↔ ∀ (i : Fin2 n), p (w i)) → (VectorAllP p (a :: w) ↔ ∀ (i : Fin2 (succ n)), p ((a :: w) i))",
"tactic": "simp"
},
{
"state_after": "case refine'_2.refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\n⊢ p a\n\ncase refine'_2.refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\ni : Fin2 n\n⊢ p (v i)",
"state_before": "case refine'_2\nα : Type u_1\nm n : ℕ\np : α → Prop\nv : Vector3 α n\n⊢ ∀ {n : ℕ} (a : α) (w : Vector3 α n),\n (VectorAllP p w ↔ ∀ (i : Fin2 n), p (w i)) → (p a ∧ VectorAllP p w ↔ ∀ (i : Fin2 (succ n)), p ((a :: w) i))",
"tactic": "refine' @fun n a v IH =>\n (and_congr_right fun _ => IH).trans\n ⟨fun ⟨pa, h⟩ i => by\n refine' i.cases' _ _\n exacts [pa, h], fun h => ⟨_, fun i => _⟩⟩"
},
{
"state_after": "case refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nx✝ : p a ∧ ∀ (i : Fin2 n), p (v i)\ni : Fin2 (succ n)\npa : p a\nh : ∀ (i : Fin2 n), p (v i)\n⊢ p ((a :: v) fz)\n\ncase refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nx✝ : p a ∧ ∀ (i : Fin2 n), p (v i)\ni : Fin2 (succ n)\npa : p a\nh : ∀ (i : Fin2 n), p (v i)\n⊢ ∀ (n_1 : Fin2 n), p ((a :: v) (fs n_1))",
"state_before": "α : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nx✝ : p a ∧ ∀ (i : Fin2 n), p (v i)\ni : Fin2 (succ n)\npa : p a\nh : ∀ (i : Fin2 n), p (v i)\n⊢ p ((a :: v) i)",
"tactic": "refine' i.cases' _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nx✝ : p a ∧ ∀ (i : Fin2 n), p (v i)\ni : Fin2 (succ n)\npa : p a\nh : ∀ (i : Fin2 n), p (v i)\n⊢ p ((a :: v) fz)\n\ncase refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nx✝ : p a ∧ ∀ (i : Fin2 n), p (v i)\ni : Fin2 (succ n)\npa : p a\nh : ∀ (i : Fin2 n), p (v i)\n⊢ ∀ (n_1 : Fin2 n), p ((a :: v) (fs n_1))",
"tactic": "exacts [pa, h]"
},
{
"state_after": "case refine'_2.refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\nh0 : p ((a :: v) fz)\n⊢ p a",
"state_before": "case refine'_2.refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\n⊢ p a",
"tactic": "have h0 := h fz"
},
{
"state_after": "case refine'_2.refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\nh0 : p a\n⊢ p a",
"state_before": "case refine'_2.refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\nh0 : p ((a :: v) fz)\n⊢ p a",
"tactic": "simp at h0"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_1\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\nh0 : p a\n⊢ p a",
"tactic": "exact h0"
},
{
"state_after": "case refine'_2.refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\ni : Fin2 n\nhs : p ((a :: v) (fs i))\n⊢ p (v i)",
"state_before": "case refine'_2.refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\ni : Fin2 n\n⊢ p (v i)",
"tactic": "have hs := h (fs i)"
},
{
"state_after": "case refine'_2.refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\ni : Fin2 n\nhs : p (v i)\n⊢ p (v i)",
"state_before": "case refine'_2.refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\ni : Fin2 n\nhs : p ((a :: v) (fs i))\n⊢ p (v i)",
"tactic": "simp at hs"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_2\nα : Type u_1\nm n✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (succ n)), p ((a :: v) i)\ni : Fin2 n\nhs : p (v i)\n⊢ p (v i)",
"tactic": "exact hs"
}
] |
[
296,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.set_biInter_insert_update
|
[] |
[
2074,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2072,
1
] |
Mathlib/Combinatorics/SimpleGraph/Hasse.lean
|
SimpleGraph.hasseDualIso_symm_apply
|
[] |
[
64,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.up_left_mem
|
[] |
[
99,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.mul_div_cancel
|
[
{
"state_after": "R : Type u\ninst✝ : EuclideanDomain R\na b : R\nb0 : b ≠ 0\n⊢ b * a / b = a",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\na b : R\nb0 : b ≠ 0\n⊢ a * b / b = a",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\na b : R\nb0 : b ≠ 0\n⊢ b * a / b = a",
"tactic": "exact mul_div_cancel_left a b0"
}
] |
[
49,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Data/Finset/Sigma.lean
|
Finset.coe_sigma
|
[] |
[
59,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Std/Data/String/Lemmas.lean
|
String.data_append
|
[] |
[
28,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
28,
9
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.mem_lift_of_splits_of_roots_mem_range
|
[
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : ∀ (a : K), a ∈ roots f → a ∈ RingHom.range (algebraMap R K)\n⊢ Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots f)) ∈ liftsRing (algebraMap R K)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : ∀ (a : K), a ∈ roots f → a ∈ RingHom.range (algebraMap R K)\n⊢ f ∈ lifts (algebraMap R K)",
"tactic": "rw [eq_prod_roots_of_monic_of_splits_id hm hs, lifts_iff_liftsRing]"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : ∀ (a : K), a ∈ roots f → a ∈ RingHom.range (algebraMap R K)\nP : K[X]\nhP : P ∈ Multiset.map (fun a => X - ↑C a) (roots f)\n⊢ P ∈ liftsRing (algebraMap R K)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : ∀ (a : K), a ∈ roots f → a ∈ RingHom.range (algebraMap R K)\n⊢ Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots f)) ∈ liftsRing (algebraMap R K)",
"tactic": "refine' Subring.multiset_prod_mem _ _ fun P hP => _"
},
{
"state_after": "case intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : ∀ (a : K), a ∈ roots f → a ∈ RingHom.range (algebraMap R K)\nb : K\nhb : b ∈ roots f\nhP : X - ↑C b ∈ Multiset.map (fun a => X - ↑C a) (roots f)\n⊢ X - ↑C b ∈ liftsRing (algebraMap R K)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : ∀ (a : K), a ∈ roots f → a ∈ RingHom.range (algebraMap R K)\nP : K[X]\nhP : P ∈ Multiset.map (fun a => X - ↑C a) (roots f)\n⊢ P ∈ liftsRing (algebraMap R K)",
"tactic": "obtain ⟨b, hb, rfl⟩ := Multiset.mem_map.1 hP"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra R K\nf : K[X]\nhs : Splits (RingHom.id K) f\nhm : Monic f\nhr : ∀ (a : K), a ∈ roots f → a ∈ RingHom.range (algebraMap R K)\nb : K\nhb : b ∈ roots f\nhP : X - ↑C b ∈ Multiset.map (fun a => X - ↑C a) (roots f)\n⊢ X - ↑C b ∈ liftsRing (algebraMap R K)",
"tactic": "exact Subring.sub_mem _ (X_mem_lifts _) (C'_mem_lifts (hr _ hb))"
}
] |
[
383,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/MeasureTheory/CardMeasurableSpace.lean
|
MeasurableSpace.self_subset_generateMeasurableRec
|
[
{
"state_after": "α : Type u\ns : Set (Set α)\ni : (Quotient.out (ord (aleph 1))).α\n⊢ s ⊆\n let i := i;\n let S := ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j;\n s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ (n : ℕ), ↑(f n)",
"state_before": "α : Type u\ns : Set (Set α)\ni : (Quotient.out (ord (aleph 1))).α\n⊢ s ⊆ generateMeasurableRec s i",
"tactic": "unfold generateMeasurableRec"
},
{
"state_after": "case h.h.h\nα : Type u\ns : Set (Set α)\ni : (Quotient.out (ord (aleph 1))).α\n⊢ s ⊆ s",
"state_before": "α : Type u\ns : Set (Set α)\ni : (Quotient.out (ord (aleph 1))).α\n⊢ s ⊆\n let i := i;\n let S := ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j;\n s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ (n : ℕ), ↑(f n)",
"tactic": "apply_rules [subset_union_of_subset_left]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u\ns : Set (Set α)\ni : (Quotient.out (ord (aleph 1))).α\n⊢ s ⊆ s",
"tactic": "exact subset_rfl"
}
] |
[
62,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
CompleteLattice.Independent.subtype_ne_bot_le_finrank
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\nι : Type w\np : ι → Submodule K V\nhp : CompleteLattice.Independent p\ninst✝ : Fintype { i // p i ≠ ⊥ }\n⊢ Fintype.card { i // p i ≠ ⊥ } ≤ finrank K V",
"tactic": "simpa using hp.subtype_ne_bot_le_finrank_aux"
}
] |
[
438,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.t1Space
|
[] |
[
290,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
11
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.funCongrLeft_symm
|
[] |
[
2675,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2674,
1
] |
Mathlib/Topology/UrysohnsLemma.lean
|
Urysohns.CU.lim_of_mem_C
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nx : X\nh : x ∈ c.C\n⊢ CU.lim c x = 0",
"tactic": "simp only [CU.lim, approx_of_mem_C, h, ciSup_const]"
}
] |
[
227,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/RingTheory/Coprime/Lemmas.lean
|
IsCoprime.of_prod_left
|
[] |
[
75,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.equivOfEq_rfl
|
[
{
"state_after": "case h.a\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : IntermediateField F E\na✝ : { x // x ∈ S }\n⊢ ↑(↑(equivOfEq (_ : S = S)) a✝) = ↑(↑AlgEquiv.refl a✝)",
"state_before": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : IntermediateField F E\n⊢ equivOfEq (_ : S = S) = AlgEquiv.refl",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.a\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : IntermediateField F E\na✝ : { x // x ∈ S }\n⊢ ↑(↑(equivOfEq (_ : S = S)) a✝) = ↑(↑AlgEquiv.refl a✝)",
"tactic": "rfl"
}
] |
[
202,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Algebra/Polynomial/BigOperators.lean
|
Polynomial.natDegree_multiset_prod_of_monic
|
[
{
"state_after": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n⊢ natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)",
"tactic": "nontriviality R"
},
{
"state_after": "case h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ prod (Multiset.map (fun f => leadingCoeff f) t) ≠ 0",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)",
"tactic": "apply natDegree_multiset_prod'"
},
{
"state_after": "case h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ prod (Multiset.map (fun f => leadingCoeff f) t) = 1",
"state_before": "case h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ prod (Multiset.map (fun f => leadingCoeff f) t) ≠ 0",
"tactic": "suffices (t.map fun f => leadingCoeff f).prod = 1 by\n rw [this]\n simp"
},
{
"state_after": "case h.e'_2.h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ Multiset.map (fun f => leadingCoeff f) t = replicate (↑Multiset.card t) 1\n\ncase h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ 1 = 1 ^ ↑Multiset.card t",
"state_before": "case h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ prod (Multiset.map (fun f => leadingCoeff f) t) = 1",
"tactic": "convert prod_replicate (Multiset.card t) (1 : R)"
},
{
"state_after": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\nthis : prod (Multiset.map (fun f => leadingCoeff f) t) = 1\n⊢ 1 ≠ 0",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\nthis : prod (Multiset.map (fun f => leadingCoeff f) t) = 1\n⊢ prod (Multiset.map (fun f => leadingCoeff f) t) ≠ 0",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\nthis : prod (Multiset.map (fun f => leadingCoeff f) t) = 1\n⊢ 1 ≠ 0",
"tactic": "simp"
},
{
"state_after": "case h.e'_2.h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ ∀ (b : R), b ∈ Multiset.map (fun f => leadingCoeff f) t → b = 1",
"state_before": "case h.e'_2.h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ Multiset.map (fun f => leadingCoeff f) t = replicate (↑Multiset.card t) 1",
"tactic": "simp only [eq_replicate, Multiset.card_map, eq_self_iff_true, true_and_iff]"
},
{
"state_after": "case h.e'_2.h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\ni : R\nhi : i ∈ Multiset.map (fun f => leadingCoeff f) t\n⊢ i = 1",
"state_before": "case h.e'_2.h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ ∀ (b : R), b ∈ Multiset.map (fun f => leadingCoeff f) t → b = 1",
"tactic": "rintro i hi"
},
{
"state_after": "case h.e'_2.h.e'_3.intro.intro\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\ni : R[X]\nhi✝ : i ∈ t\nhi : leadingCoeff i ∈ Multiset.map (fun f => leadingCoeff f) t\n⊢ leadingCoeff i = 1",
"state_before": "case h.e'_2.h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\ni : R\nhi : i ∈ Multiset.map (fun f => leadingCoeff f) t\n⊢ i = 1",
"tactic": "obtain ⟨i, hi, rfl⟩ := Multiset.mem_map.mp hi"
},
{
"state_after": "case h.e'_2.h.e'_3.intro.intro.a\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\ni : R[X]\nhi✝ : i ∈ t\nhi : leadingCoeff i ∈ Multiset.map (fun f => leadingCoeff f) t\n⊢ i ∈ t",
"state_before": "case h.e'_2.h.e'_3.intro.intro\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\ni : R[X]\nhi✝ : i ∈ t\nhi : leadingCoeff i ∈ Multiset.map (fun f => leadingCoeff f) t\n⊢ leadingCoeff i = 1",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_3.intro.intro.a\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\ni : R[X]\nhi✝ : i ∈ t\nhi : leadingCoeff i ∈ Multiset.map (fun f => leadingCoeff f) t\n⊢ i ∈ t",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\nt : Multiset R[X]\nh : ∀ (f : R[X]), f ∈ t → Monic f\n✝ : Nontrivial R\n⊢ 1 = 1 ^ ↑Multiset.card t",
"tactic": "simp"
}
] |
[
206,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.compl_singleton_eq
|
[] |
[
1700,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1699,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.nth_zero_iterate
|
[] |
[
291,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.toList_singleton
|
[] |
[
488,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
VectorSpace.card_fintype
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1163482\nι' : Type ?u.1163485\nR : Type ?u.1163488\nR₂ : Type ?u.1163491\nK : Type u_1\nM : Type ?u.1163497\nM' : Type ?u.1163500\nM'' : Type ?u.1163503\nV : Type u\nV' : Type ?u.1163508\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module K V\ninst✝² : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\ninst✝¹ : Fintype K\ninst✝ : Fintype V\n⊢ ∃ n, card V = card K ^ n",
"tactic": "classical\nexact ⟨card (Basis.ofVectorSpaceIndex K V), Module.card_fintype (Basis.ofVectorSpace K V)⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1163482\nι' : Type ?u.1163485\nR : Type ?u.1163488\nR₂ : Type ?u.1163491\nK : Type u_1\nM : Type ?u.1163497\nM' : Type ?u.1163500\nM'' : Type ?u.1163503\nV : Type u\nV' : Type ?u.1163508\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module K V\ninst✝² : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\ninst✝¹ : Fintype K\ninst✝ : Fintype V\n⊢ ∃ n, card V = card K ^ n",
"tactic": "exact ⟨card (Basis.ofVectorSpaceIndex K V), Module.card_fintype (Basis.ofVectorSpace K V)⟩"
}
] |
[
1512,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1510,
1
] |
Mathlib/Algebra/Free.lean
|
FreeMagma.toFreeSemigroup_comp_of
|
[] |
[
726,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
726,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.invFun_as_coe
|
[] |
[
206,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.sum_affineCombinationSingleWeights
|
[
{
"state_after": "k : Type u_2\nV : Type ?u.415884\nP : Type ?u.415887\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.416543\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni : ι\nh : i ∈ s\n⊢ ∑ j in s, affineCombinationSingleWeights k i j = affineCombinationSingleWeights k i i",
"state_before": "k : Type u_2\nV : Type ?u.415884\nP : Type ?u.415887\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.416543\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni : ι\nh : i ∈ s\n⊢ ∑ j in s, affineCombinationSingleWeights k i j = 1",
"tactic": "rw [← affineCombinationSingleWeights_apply_self k i]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type ?u.415884\nP : Type ?u.415887\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.416543\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni : ι\nh : i ∈ s\n⊢ ∑ j in s, affineCombinationSingleWeights k i j = affineCombinationSingleWeights k i i",
"tactic": "exact sum_eq_single_of_mem i h fun j _ hj => affineCombinationSingleWeights_apply_of_ne k hj"
}
] |
[
671,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Units.mem_posSubgroup
|
[] |
[
1521,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1519,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
finrank_vectorSpan_range_le
|
[
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p Finset.univ) } ≤ n",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ finrank k { x // x ∈ vectorSpan k (Set.range p) } ≤ n",
"tactic": "rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image]"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Finset.card Finset.univ = n + 1\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p Finset.univ) } ≤ n",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p Finset.univ) } ≤ n",
"tactic": "rw [← Finset.card_univ] at hc"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\nι : Type u_1\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Finset.card Finset.univ = n + 1\n⊢ finrank k { x // x ∈ vectorSpan k ↑(Finset.image p Finset.univ) } ≤ n",
"tactic": "exact finrank_vectorSpan_image_finset_le _ _ _ hc"
}
] |
[
163,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean
|
Set.ordConnectedSection_subset
|
[] |
[
153,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
IsGLB.biUnion_Ioi_eq
|
[
{
"state_after": "case refine'_1\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\nh : IsGLB s a\nx : α\nhx : x ∈ s\n⊢ Ioi x ⊆ Ioi a\n\ncase refine'_2\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\nh : IsGLB s a\nx : α\nhx : x ∈ Ioi a\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ioi i",
"state_before": "ι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\nh : IsGLB s a\n⊢ (⋃ (x : α) (_ : x ∈ s), Ioi x) = Ioi a",
"tactic": "refine' (iUnion₂_subset fun x hx => _).antisymm fun x hx => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\nh : IsGLB s a\nx : α\nhx : x ∈ s\n⊢ Ioi x ⊆ Ioi a",
"tactic": "exact Ioi_subset_Ioi (h.1 hx)"
},
{
"state_after": "case refine'_2.intro.intro.intro\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\nh : IsGLB s a\nx : α\nhx : x ∈ Ioi a\ny : α\nhys : y ∈ s\nleft✝ : a ≤ y\nhyx : y < x\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ioi i",
"state_before": "case refine'_2\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\nh : IsGLB s a\nx : α\nhx : x ∈ Ioi a\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ioi i",
"tactic": "rcases h.exists_between hx with ⟨y, hys, _, hyx⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro\nι : Sort u\nα : Type v\nβ : Type w\ninst✝ : LinearOrder α\ns : Set α\na : α\nf : ι → α\nh : IsGLB s a\nx : α\nhx : x ∈ Ioi a\ny : α\nhys : y ∈ s\nleft✝ : a ≤ y\nhyx : y < x\n⊢ x ∈ ⋃ (i : α) (_ : i ∈ s), Ioi i",
"tactic": "exact mem_biUnion hys hyx"
}
] |
[
189,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/CategoryTheory/Bicategory/Strict.lean
|
CategoryTheory.Bicategory.eqToHom_whiskerRight
|
[
{
"state_after": "case refl\nB : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\nh : b ⟶ c\n⊢ eqToHom (_ : f = f) ▷ h = eqToHom (_ : f ≫ h = f ≫ h)",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf g : a ⟶ b\nη : f = g\nh : b ⟶ c\n⊢ eqToHom η ▷ h = eqToHom (_ : f ≫ h = g ≫ h)",
"tactic": "cases η"
},
{
"state_after": "no goals",
"state_before": "case refl\nB : Type u\ninst✝ : Bicategory B\na b c : B\nf : a ⟶ b\nh : b ⟶ c\n⊢ eqToHom (_ : f = f) ▷ h = eqToHom (_ : f ≫ h = f ≫ h)",
"tactic": "simp only [id_whiskerRight, eqToHom_refl]"
}
] |
[
92,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_iff
|
[] |
[
192,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Algebra/Lie/Semisimple.lean
|
LieAlgebra.abelian_radical_of_semisimple
|
[
{
"state_after": "R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\n⊢ IsLieAbelian { x // x ∈ ↑⊥ }",
"state_before": "R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\n⊢ IsLieAbelian { x // x ∈ ↑(radical R L) }",
"tactic": "rw [IsSemisimple.semisimple]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\n⊢ IsLieAbelian { x // x ∈ ↑⊥ }",
"tactic": "exact isLieAbelian_bot R L"
}
] |
[
104,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.restrict_withDensity
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1771367\nγ : Type ?u.1771370\nδ : Type ?u.1771373\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(Measure.restrict (withDensity μ f) s) t = ↑↑(withDensity (Measure.restrict μ s) f) t",
"state_before": "α : Type u_1\nβ : Type ?u.1771367\nγ : Type ?u.1771370\nδ : Type ?u.1771373\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\n⊢ Measure.restrict (withDensity μ f) s = withDensity (Measure.restrict μ s) f",
"tactic": "ext1 t ht"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1771367\nγ : Type ?u.1771370\nδ : Type ?u.1771373\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(Measure.restrict (withDensity μ f) s) t = ↑↑(withDensity (Measure.restrict μ s) f) t",
"tactic": "rw [restrict_apply ht, withDensity_apply _ ht, withDensity_apply _ (ht.inter hs),\n restrict_restrict ht]"
}
] |
[
1666,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1662,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
sup_eq_prod_inf_factors
|
[
{
"state_after": "R : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\n⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)",
"state_before": "R : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\n⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)",
"tactic": "have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =\n normalizedFactors I ∩ normalizedFactors J := by\n apply normalizedFactors_prod_of_prime\n intro p hp\n rw [mem_inter] at hp\n exact prime_of_normalized_factor p hp.left"
},
{
"state_after": "R : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)",
"state_before": "R : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\n⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)",
"tactic": "have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>\n prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1"
},
{
"state_after": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ I ⊔ J ≤ Multiset.prod (normalizedFactors I ∩ normalizedFactors J)\n\ncase a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≤ I ⊔ J",
"state_before": "R : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case h\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\n⊢ ∀ (p : Ideal T), p ∈ normalizedFactors I ∩ normalizedFactors J → Prime p",
"state_before": "R : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\n⊢ normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J",
"tactic": "apply normalizedFactors_prod_of_prime"
},
{
"state_after": "case h\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\np : Ideal T\nhp : p ∈ normalizedFactors I ∩ normalizedFactors J\n⊢ Prime p",
"state_before": "case h\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\n⊢ ∀ (p : Ideal T), p ∈ normalizedFactors I ∩ normalizedFactors J → Prime p",
"tactic": "intro p hp"
},
{
"state_after": "case h\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\np : Ideal T\nhp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J\n⊢ Prime p",
"state_before": "case h\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\np : Ideal T\nhp : p ∈ normalizedFactors I ∩ normalizedFactors J\n⊢ Prime p",
"tactic": "rw [mem_inter] at hp"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\np : Ideal T\nhp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J\n⊢ Prime p",
"tactic": "exact prime_of_normalized_factor p hp.left"
},
{
"state_after": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ I ∧\n Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ J",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ I ⊔ J ≤ Multiset.prod (normalizedFactors I ∩ normalizedFactors J)",
"tactic": "rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]"
},
{
"state_after": "case a.left\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ I\n\ncase a.right\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ J",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ I ∧\n Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ J",
"tactic": "constructor"
},
{
"state_after": "case a.left\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ normalizedFactors I ∩ normalizedFactors J ≤ normalizedFactors I",
"state_before": "case a.left\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ I",
"tactic": "rw [dvd_iff_normalizedFactors_le_normalizedFactors this hI, H]"
},
{
"state_after": "no goals",
"state_before": "case a.left\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ normalizedFactors I ∩ normalizedFactors J ≤ normalizedFactors I",
"tactic": "exact inf_le_left"
},
{
"state_after": "case a.right\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ normalizedFactors I ∩ normalizedFactors J ≤ normalizedFactors J",
"state_before": "case a.right\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ J",
"tactic": "rw [dvd_iff_normalizedFactors_le_normalizedFactors this hJ, H]"
},
{
"state_after": "no goals",
"state_before": "case a.right\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ normalizedFactors I ∩ normalizedFactors J ≤ normalizedFactors J",
"tactic": "exact inf_le_right"
},
{
"state_after": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ ∀ (a : Ideal T), count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)\n\ncase a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ ∀ (p : Ideal T), p ∈ normalizedFactors I ∩ normalizedFactors J → Prime p\n\ncase a.hx\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ I ⊔ J ≠ 0\n\ncase a.hy\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≤ I ⊔ J",
"tactic": "rw [← dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors,\n normalizedFactors_prod_of_prime, le_iff_count]"
},
{
"state_after": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\na : Ideal T\n⊢ count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ ∀ (a : Ideal T), count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)",
"tactic": "intro a"
},
{
"state_after": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\na : Ideal T\n⊢ count a (normalizedFactors (I ⊔ J)) ≤ min (count a (normalizedFactors I)) (count a (normalizedFactors J))",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\na : Ideal T\n⊢ count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)",
"tactic": "rw [Multiset.count_inter]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\na : Ideal T\n⊢ count a (normalizedFactors (I ⊔ J)) ≤ min (count a (normalizedFactors I)) (count a (normalizedFactors J))",
"tactic": "exact le_min (count_le_of_ideal_ge le_sup_left hI a) (count_le_of_ideal_ge le_sup_right hJ a)"
},
{
"state_after": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\np : Ideal T\nhp : p ∈ normalizedFactors I ∩ normalizedFactors J\n⊢ Prime p",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ ∀ (p : Ideal T), p ∈ normalizedFactors I ∩ normalizedFactors J → Prime p",
"tactic": "intro p hp"
},
{
"state_after": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\np : Ideal T\nhp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J\n⊢ Prime p",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\np : Ideal T\nhp : p ∈ normalizedFactors I ∩ normalizedFactors J\n⊢ Prime p",
"tactic": "rw [mem_inter] at hp"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\np : Ideal T\nhp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J\n⊢ Prime p",
"tactic": "exact prime_of_normalized_factor p hp.left"
},
{
"state_after": "no goals",
"state_before": "case a.hx\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ I ⊔ J ≠ 0",
"tactic": "exact ne_bot_of_le_ne_bot hI le_sup_left"
},
{
"state_after": "no goals",
"state_before": "case a.hy\nR : Type ?u.906300\nA : Type ?u.906303\nK : Type ?u.906306\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\nT : Type u_1\ninst✝² : CommRing T\ninst✝¹ : IsDomain T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH :\n normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =\n normalizedFactors I ∩ normalizedFactors J\nthis : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0\n⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0",
"tactic": "exact this"
}
] |
[
937,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
exists_square_le
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\n⊢ ∃ b, b * b ≤ a\n\ncase neg\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : ¬a < 1\n⊢ ∃ b, b * b ≤ a",
"state_before": "α : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\n⊢ ∃ b, b * b ≤ a",
"tactic": "by_cases h:a < 1"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\n⊢ a * a ≤ a",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\n⊢ ∃ b, b * b ≤ a",
"tactic": "use a"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\nthis : a * a < a * 1\n⊢ a * a ≤ a",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\n⊢ a * a ≤ a",
"tactic": "have : a * a < a * 1 := mul_lt_mul_left' h a"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\nthis : a * a < a\n⊢ a * a ≤ a",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\nthis : a * a < a * 1\n⊢ a * a ≤ a",
"tactic": "rw [mul_one] at this"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : a < 1\nthis : a * a < a\n⊢ a * a ≤ a",
"tactic": "exact le_of_lt this"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : ¬a < 1\n⊢ 1 * 1 ≤ a",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : ¬a < 1\n⊢ ∃ b, b * b ≤ a",
"tactic": "use 1"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : 1 ≤ a\n⊢ 1 * 1 ≤ a",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : ¬a < 1\n⊢ 1 * 1 ≤ a",
"tactic": "push_neg at h"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.59597\ninst✝² : MulOneClass α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na : α\nh : 1 ≤ a\n⊢ 1 * 1 ≤ a",
"tactic": "rwa [mul_one]"
}
] |
[
1200,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1192,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean
|
Nat.bodd_two
|
[] |
[
65,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/GroupTheory/SpecificGroups/Quaternion.lean
|
QuaternionGroup.orderOf_a
|
[
{
"state_after": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\n⊢ orderOf (a ↑(ZMod.val i)) = 2 * n / Nat.gcd (2 * n) (ZMod.val i)",
"state_before": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\n⊢ orderOf (a i) = 2 * n / Nat.gcd (2 * n) (ZMod.val i)",
"tactic": "conv_lhs => rw [← ZMod.nat_cast_zmod_val i]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\n⊢ orderOf (a ↑(ZMod.val i)) = 2 * n / Nat.gcd (2 * n) (ZMod.val i)",
"tactic": "rw [← a_one_pow, orderOf_pow, orderOf_a_one]"
}
] |
[
263,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Nat.cast_multiset_prod
|
[] |
[
2206,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2205,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.mul_bot
|
[] |
[
318,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Data/Fintype/BigOperators.lean
|
Fintype.prod_eq_single
|
[] |
[
83,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Antiperiodic.neg_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.157720\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh : Antiperiodic f c\n⊢ f (-c) = -f 0",
"tactic": "simpa only [zero_add] using h.neg 0"
}
] |
[
420,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.div_zero
|
[
{
"state_after": "n : Nat\n⊢ (if 0 < 0 ∧ 0 ≤ n then (n - 0) / 0 + 1 else 0) = 0",
"state_before": "n : Nat\n⊢ n / 0 = 0",
"tactic": "rw [div_eq]"
},
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ (if 0 < 0 ∧ 0 ≤ n then (n - 0) / 0 + 1 else 0) = 0",
"tactic": "simp [Nat.lt_irrefl]"
}
] |
[
262,
36
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
261,
19
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.nhds_coe_coe
|
[] |
[
104,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
Convex.mem_Icc
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.291676\nF : Type ?u.291679\nG : Type ?u.291682\nι : Type ?u.291685\nπ : ι → Type ?u.291690\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ [x-[𝕜]y] ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z",
"state_before": "𝕜 : Type u_1\nE : Type ?u.291676\nF : Type ?u.291679\nG : Type ?u.291682\nι : Type ?u.291685\nπ : ι → Type ?u.291690\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z",
"tactic": "rw [← segment_eq_Icc h]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.291676\nF : Type ?u.291679\nG : Type ?u.291682\nι : Type ?u.291685\nπ : ι → Type ?u.291690\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ [x-[𝕜]y] ↔ ∃ a b _h _h _h, a * x + b * y = z",
"state_before": "𝕜 : Type u_1\nE : Type ?u.291676\nF : Type ?u.291679\nG : Type ?u.291682\nι : Type ?u.291685\nπ : ι → Type ?u.291690\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ [x-[𝕜]y] ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z",
"tactic": "simp_rw [← exists_prop]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.291676\nF : Type ?u.291679\nG : Type ?u.291682\nι : Type ?u.291685\nπ : ι → Type ?u.291690\ninst✝ : LinearOrderedField 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ [x-[𝕜]y] ↔ ∃ a b _h _h _h, a * x + b * y = z",
"tactic": "rfl"
}
] |
[
538,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.IsTopologicalBasis.prod
|
[
{
"state_after": "case refine'_1\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\n⊢ ∀ (u : Set (α × β)), u ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ → IsOpen u\n\ncase refine'_2\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\n⊢ ∀ (a : α × β) (u : Set (α × β)), a ∈ u → IsOpen u → ∃ v, v ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ ∧ a ∈ v ∧ v ⊆ u",
"state_before": "α : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\n⊢ IsTopologicalBasis (image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂)",
"tactic": "refine' isTopologicalBasis_of_open_of_nhds _ _"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\nu₁ : Set α\nu₂ : Set β\nhu₁ : u₁ ∈ B₁\nhu₂ : u₂ ∈ B₂\n⊢ IsOpen ((fun x x_1 => x ×ˢ x_1) u₁ u₂)",
"state_before": "case refine'_1\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\n⊢ ∀ (u : Set (α × β)), u ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ → IsOpen u",
"tactic": "rintro _ ⟨u₁, u₂, hu₁, hu₂, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\nu₁ : Set α\nu₂ : Set β\nhu₁ : u₁ ∈ B₁\nhu₂ : u₂ ∈ B₂\n⊢ IsOpen ((fun x x_1 => x ×ˢ x_1) u₁ u₂)",
"tactic": "exact (h₁.isOpen hu₁).prod (h₂.isOpen hu₂)"
},
{
"state_after": "case refine'_2.mk\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\na : α\nb : β\nu : Set (α × β)\nhu : (a, b) ∈ u\nuo : IsOpen u\n⊢ ∃ v, v ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ ∧ (a, b) ∈ v ∧ v ⊆ u",
"state_before": "case refine'_2\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\n⊢ ∀ (a : α × β) (u : Set (α × β)), a ∈ u → IsOpen u → ∃ v, v ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ ∧ a ∈ v ∧ v ⊆ u",
"tactic": "rintro ⟨a, b⟩ u hu uo"
},
{
"state_after": "case refine'_2.mk.intro.mk.intro.intro.intro.intro\nα : Type u\nt✝ : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\na : α\nb : β\nu : Set (α × β)\nhu✝ : (a, b) ∈ u\nuo : IsOpen u\ns : Set α\nt : Set β\nhu : (s, t).fst ×ˢ (s, t).snd ⊆ u\nhs : (s, t).fst ∈ B₁\nha : a ∈ (s, t).fst\nht : (s, t).snd ∈ B₂\nhb : b ∈ (s, t).snd\n⊢ ∃ v, v ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ ∧ (a, b) ∈ v ∧ v ⊆ u",
"state_before": "case refine'_2.mk\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\na : α\nb : β\nu : Set (α × β)\nhu : (a, b) ∈ u\nuo : IsOpen u\n⊢ ∃ v, v ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ ∧ (a, b) ∈ v ∧ v ⊆ u",
"tactic": "rcases(h₁.nhds_hasBasis.prod_nhds h₂.nhds_hasBasis).mem_iff.1 (IsOpen.mem_nhds uo hu) with\n ⟨⟨s, t⟩, ⟨⟨hs, ha⟩, ht, hb⟩, hu⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk.intro.mk.intro.intro.intro.intro\nα : Type u\nt✝ : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\na : α\nb : β\nu : Set (α × β)\nhu✝ : (a, b) ∈ u\nuo : IsOpen u\ns : Set α\nt : Set β\nhu : (s, t).fst ×ˢ (s, t).snd ⊆ u\nhs : (s, t).fst ∈ B₁\nha : a ∈ (s, t).fst\nht : (s, t).snd ∈ B₂\nhb : b ∈ (s, t).snd\n⊢ ∃ v, v ∈ image2 (fun x x_1 => x ×ˢ x_1) B₁ B₂ ∧ (a, b) ∈ v ∧ v ⊆ u",
"tactic": "exact ⟨s ×ˢ t, mem_image2_of_mem hs ht, ⟨ha, hb⟩, hu⟩"
}
] |
[
254,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
11
] |
Mathlib/Order/Hom/Lattice.lean
|
SupBotHom.id_apply
|
[] |
[
784,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
783,
1
] |
Mathlib/Data/ZMod/Quotient.lean
|
IsOfFinOrder.finite_zpowers
|
[
{
"state_after": "n : ℕ\nA : Type ?u.324794\nR : Type ?u.324797\ninst✝² : AddGroup A\ninst✝¹ : Ring R\nα : Type u_1\ninst✝ : Group α\na : α\nh : 0 < Nat.card { x // x ∈ zpowers a }\n⊢ Finite { x // x ∈ zpowers a }",
"state_before": "n : ℕ\nA : Type ?u.324794\nR : Type ?u.324797\ninst✝² : AddGroup A\ninst✝¹ : Ring R\nα : Type u_1\ninst✝ : Group α\na : α\nh : IsOfFinOrder a\n⊢ Finite { x // x ∈ zpowers a }",
"tactic": "rw [← orderOf_pos_iff, order_eq_card_zpowers'] at h"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nA : Type ?u.324794\nR : Type ?u.324797\ninst✝² : AddGroup A\ninst✝¹ : Ring R\nα : Type u_1\ninst✝ : Group α\na : α\nh : 0 < Nat.card { x // x ∈ zpowers a }\n⊢ Finite { x // x ∈ zpowers a }",
"tactic": "exact Nat.finite_of_card_ne_zero h.ne.symm"
}
] |
[
206,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_mulIndicator
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\n⊢ (if x ∈ s then if x ∈ t then f x else 1 else 1) = if x ∈ s ∩ t then f x else 1",
"state_before": "α : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\n⊢ mulIndicator s (mulIndicator t f) x = mulIndicator (s ∩ t) f x",
"tactic": "simp only [mulIndicator]"
},
{
"state_after": "case inl.inl.inl\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : x ∈ t\nh✝ : x ∈ s ∩ t\n⊢ f x = f x\n\ncase inl.inl.inr\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : x ∈ t\nh✝ : ¬x ∈ s ∩ t\n⊢ f x = 1\n\ncase inl.inr.inl\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : ¬x ∈ t\nh✝ : x ∈ s ∩ t\n⊢ 1 = f x\n\ncase inl.inr.inr\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : ¬x ∈ t\nh✝ : ¬x ∈ s ∩ t\n⊢ 1 = 1\n\ncase inr.inl\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝¹ : ¬x ∈ s\nh✝ : x ∈ s ∩ t\n⊢ 1 = f x\n\ncase inr.inr\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝¹ : ¬x ∈ s\nh✝ : ¬x ∈ s ∩ t\n⊢ 1 = 1",
"state_before": "α : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\n⊢ (if x ∈ s then if x ∈ t then f x else 1 else 1) = if x ∈ s ∩ t then f x else 1",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case inl.inl.inl\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : x ∈ t\nh✝ : x ∈ s ∩ t\n⊢ f x = f x\n\ncase inl.inl.inr\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : x ∈ t\nh✝ : ¬x ∈ s ∩ t\n⊢ f x = 1\n\ncase inl.inr.inl\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : ¬x ∈ t\nh✝ : x ∈ s ∩ t\n⊢ 1 = f x\n\ncase inl.inr.inr\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝² : x ∈ s\nh✝¹ : ¬x ∈ t\nh✝ : ¬x ∈ s ∩ t\n⊢ 1 = 1\n\ncase inr.inl\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝¹ : ¬x ∈ s\nh✝ : x ∈ s ∩ t\n⊢ 1 = f x\n\ncase inr.inr\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝¹ : ¬x ∈ s\nh✝ : ¬x ∈ s ∩ t\n⊢ 1 = 1",
"tactic": "repeat' simp_all (config := { contextual := true })"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nβ : Type ?u.12298\nι : Type ?u.12301\nM : Type u_2\nN : Type ?u.12307\ninst✝¹ : One M\ninst✝ : One N\ns✝ t✝ : Set α\nf✝ g : α → M\na : α\ns t : Set α\nf : α → M\nx : α\nh✝¹ : ¬x ∈ s\nh✝ : ¬x ∈ s ∩ t\n⊢ 1 = 1",
"tactic": "simp_all (config := { contextual := true })"
}
] |
[
240,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
Left.self_le_inv
|
[] |
[
453,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.succ_ne_self
|
[] |
[
16,
50
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
14,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.flip_iff
|
[] |
[
671,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
670,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.differentiableWithinAt
|
[] |
[
79,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
11
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.succ_descFactorial
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (n + 1 - 0) * descFactorial (n + 1) 0 = (n + 1) * descFactorial n 0",
"tactic": "rw [tsub_zero, descFactorial_zero, descFactorial_zero]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ (n + 1 - (k + 1)) * descFactorial (n + 1) (k + 1) = (n + 1) * descFactorial n (k + 1)",
"tactic": "rw [descFactorial, succ_descFactorial _ k, descFactorial_succ, succ_sub_succ, mul_left_comm]"
}
] |
[
388,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Computability/Primrec.lean
|
PrimrecRel.comp₂
|
[] |
[
505,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.ae_const_le_iff_forall_lt_measure_zero
|
[
{
"state_after": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ ↑↑μ {a | ¬c ≤ f a} = 0 ↔ ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0",
"state_before": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ (∀ᵐ (x : α) ∂μ, c ≤ f x) ↔ ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0",
"tactic": "rw [ae_iff]"
},
{
"state_after": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ ↑↑μ {a | f a < c} = 0 ↔ ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0",
"state_before": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ ↑↑μ {a | ¬c ≤ f a} = 0 ↔ ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0",
"tactic": "push_neg"
},
{
"state_after": "case mp\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ ↑↑μ {a | f a < c} = 0 → ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\n\ncase mpr\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ (∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0) → ↑↑μ {a | f a < c} = 0",
"state_before": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ ↑↑μ {a | f a < c} = 0 ↔ ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0",
"tactic": "constructor"
},
{
"state_after": "case mpr\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\n⊢ ↑↑μ {a | f a < c} = 0",
"state_before": "case mpr\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\n⊢ (∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0) → ↑↑μ {a | f a < c} = 0",
"tactic": "intro hc"
},
{
"state_after": "case pos\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ∀ (b : β), c ≤ b\n⊢ ↑↑μ {a | f a < c} = 0\n\ncase neg\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\n⊢ ↑↑μ {a | f a < c} = 0",
"state_before": "case mpr\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\n⊢ ↑↑μ {a | f a < c} = 0",
"tactic": "by_cases h : ∀ b, c ≤ b"
},
{
"state_after": "case pos\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬IsLUB (Set.Iio c) c\n⊢ ↑↑μ {a | f a < c} = 0\n\ncase neg\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬¬IsLUB (Set.Iio c) c\n⊢ ↑↑μ {a | f a < c} = 0",
"state_before": "case neg\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\n⊢ ↑↑μ {a | f a < c} = 0",
"tactic": "by_cases H : ¬IsLUB (Set.Iio c) c"
},
{
"state_after": "case neg\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\n⊢ ↑↑μ {a | f a < c} = 0",
"state_before": "case neg\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬¬IsLUB (Set.Iio c) c\n⊢ ↑↑μ {a | f a < c} = 0",
"tactic": "push_neg at H h"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\n⊢ ↑↑μ {a | f a < c} = 0",
"state_before": "case neg\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\n⊢ ↑↑μ {a | f a < c} = 0",
"tactic": "obtain ⟨u, _, u_lt, u_lim, -⟩ :\n ∃ u : ℕ → β,\n StrictMono u ∧ (∀ n : ℕ, u n < c) ∧ Tendsto u atTop (nhds c) ∧ ∀ n : ℕ, u n ∈ Set.Iio c :=\n H.exists_seq_strictMono_tendsto_of_not_mem (lt_irrefl c) h"
},
{
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"tactic": "rw [h_Union, measure_iUnion_null_iff]"
},
{
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"tactic": "intro n"
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"tactic": "exact hc _ (u_lt n)"
},
{
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"tactic": "intro h b hb"
},
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"state_after": "no goals",
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"tactic": "exact measure_mono_null (fun y hy => (lt_of_le_of_lt hy hb : _)) h"
},
{
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"state_before": "case pos\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ∀ (b : β), c ≤ b\n⊢ ↑↑μ {a | f a < c} = 0",
"tactic": "have : {a : α | f a < c} = ∅ := by\n apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_\n exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim"
},
{
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"tactic": "simp [this]"
},
{
"state_after": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ∀ (b : β), c ≤ b\nx : α\nhx : x ∈ {a | f a < c}\n⊢ False",
"state_before": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ∀ (b : β), c ≤ b\n⊢ {a | f a < c} = ∅",
"tactic": "apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ∀ (b : β), c ≤ b\nx : α\nhx : x ∈ {a | f a < c}\n⊢ False",
"tactic": "exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim"
},
{
"state_after": "case pos\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬IsLUB (Set.Iio c) c\nthis : c ∈ upperBounds (Set.Iio c)\n⊢ ↑↑μ {a | f a < c} = 0",
"state_before": "case pos\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬IsLUB (Set.Iio c) c\n⊢ ↑↑μ {a | f a < c} = 0",
"tactic": "have : c ∈ upperBounds (Set.Iio c) := fun y hy => le_of_lt hy"
},
{
"state_after": "case pos.intro.intro\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬IsLUB (Set.Iio c) c\nthis : c ∈ upperBounds (Set.Iio c)\nb : β\nb_up : b ∈ upperBounds (Set.Iio c)\nbc : b < c\n⊢ ↑↑μ {a | f a < c} = 0",
"state_before": "case pos\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬IsLUB (Set.Iio c) c\nthis : c ∈ upperBounds (Set.Iio c)\n⊢ ↑↑μ {a | f a < c} = 0",
"tactic": "obtain ⟨b, b_up, bc⟩ : ∃ b : β, b ∈ upperBounds (Set.Iio c) ∧ b < c := by\n simpa [IsLUB, IsLeast, this, lowerBounds] using H"
},
{
"state_after": "no goals",
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"tactic": "exact measure_mono_null (fun x hx => b_up hx) (hc b bc)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nh : ¬∀ (b : β), c ≤ b\nH : ¬IsLUB (Set.Iio c) c\nthis : c ∈ upperBounds (Set.Iio c)\n⊢ ∃ b, b ∈ upperBounds (Set.Iio c) ∧ b < c",
"tactic": "simpa [IsLUB, IsLeast, this, lowerBounds] using H"
},
{
"state_after": "case h\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\n⊢ x ∈ {x | f x < c} ↔ x ∈ ⋃ (n : ℕ), {x | f x ≤ u n}",
"state_before": "α : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\n⊢ {x | f x < c} = ⋃ (n : ℕ), {x | f x ≤ u n}",
"tactic": "ext1 x"
},
{
"state_after": "case h\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\n⊢ f x < c ↔ ∃ i, f x ≤ u i",
"state_before": "case h\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\n⊢ x ∈ {x | f x < c} ↔ x ∈ ⋃ (n : ℕ), {x | f x ≤ u n}",
"tactic": "simp_rw [Set.mem_iUnion, Set.mem_setOf_eq]"
},
{
"state_after": "case h.mp\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh✝ : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nh : f x < c\n⊢ ∃ i, f x ≤ u i\n\ncase h.mpr\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh✝ : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nh : ∃ i, f x ≤ u i\n⊢ f x < c",
"state_before": "case h\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\n⊢ f x < c ↔ ∃ i, f x ≤ u i",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case h.mp.intro\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh✝ : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nh : f x < c\nn : ℕ\nhn : f x < u n\n⊢ ∃ i, f x ≤ u i",
"state_before": "case h.mp\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh✝ : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nh : f x < c\n⊢ ∃ i, f x ≤ u i",
"tactic": "obtain ⟨n, hn⟩ := ((tendsto_order.1 u_lim).1 _ h).exists"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh✝ : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nh : f x < c\nn : ℕ\nhn : f x < u n\n⊢ ∃ i, f x ≤ u i",
"tactic": "exact ⟨n, hn.le⟩"
},
{
"state_after": "case h.mpr.intro\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nn : ℕ\nhn : f x ≤ u n\n⊢ f x < c",
"state_before": "case h.mpr\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh✝ : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nh : ∃ i, f x ≤ u i\n⊢ f x < c",
"tactic": "obtain ⟨n, hn⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro\nα : Type u_2\nE : Type ?u.30634\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\np : ℝ≥0∞\nβ : Type u_1\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : FirstCountableTopology β\nf : α → β\nc : β\nhc : ∀ (b : β), b < c → ↑↑μ {x | f x ≤ b} = 0\nH : IsLUB (Set.Iio c) c\nh : ∃ b, b < c\nu : ℕ → β\nleft✝ : StrictMono u\nu_lt : ∀ (n : ℕ), u n < c\nu_lim : Tendsto u atTop (nhds c)\nx : α\nn : ℕ\nhn : f x ≤ u n\n⊢ f x < c",
"tactic": "exact hn.trans_lt (u_lt _)"
}
] |
[
161,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasurableEquiv.map_measurableEquiv_injective
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2439120\nδ : Type ?u.2439123\nι : Type ?u.2439126\nR : Type ?u.2439129\nR' : Type ?u.2439132\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : MeasureTheory.Measure α\nν : MeasureTheory.Measure β\ne : α ≃ᵐ β\nμ₁ μ₂ : MeasureTheory.Measure α\nhμ : Measure.map (↑e) μ₁ = Measure.map (↑e) μ₂\n⊢ μ₁ = μ₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2439120\nδ : Type ?u.2439123\nι : Type ?u.2439126\nR : Type ?u.2439129\nR' : Type ?u.2439132\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : MeasureTheory.Measure α\nν : MeasureTheory.Measure β\ne : α ≃ᵐ β\n⊢ Injective (Measure.map ↑e)",
"tactic": "intro μ₁ μ₂ hμ"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2439120\nδ : Type ?u.2439123\nι : Type ?u.2439126\nR : Type ?u.2439129\nR' : Type ?u.2439132\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : MeasureTheory.Measure α\nν : MeasureTheory.Measure β\ne : α ≃ᵐ β\nμ₁ μ₂ : MeasureTheory.Measure α\nhμ : Measure.map (↑(symm e)) (Measure.map (↑e) μ₁) = Measure.map (↑(symm e)) (Measure.map (↑e) μ₂)\n⊢ μ₁ = μ₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2439120\nδ : Type ?u.2439123\nι : Type ?u.2439126\nR : Type ?u.2439129\nR' : Type ?u.2439132\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : MeasureTheory.Measure α\nν : MeasureTheory.Measure β\ne : α ≃ᵐ β\nμ₁ μ₂ : MeasureTheory.Measure α\nhμ : Measure.map (↑e) μ₁ = Measure.map (↑e) μ₂\n⊢ μ₁ = μ₂",
"tactic": "apply_fun Measure.map e.symm at hμ"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2439120\nδ : Type ?u.2439123\nι : Type ?u.2439126\nR : Type ?u.2439129\nR' : Type ?u.2439132\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : MeasureTheory.Measure α\nν : MeasureTheory.Measure β\ne : α ≃ᵐ β\nμ₁ μ₂ : MeasureTheory.Measure α\nhμ : Measure.map (↑(symm e)) (Measure.map (↑e) μ₁) = Measure.map (↑(symm e)) (Measure.map (↑e) μ₂)\n⊢ μ₁ = μ₂",
"tactic": "simpa [map_symm_map e] using hμ"
}
] |
[
4309,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4306,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mul_lt_aleph0_iff_of_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\na b : Cardinal\nha : a ≠ 0\nhb : b ≠ 0\n⊢ a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀",
"tactic": "simp [mul_lt_aleph0_iff, ha, hb]"
}
] |
[
1590,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1589,
1
] |
Mathlib/Algebra/Invertible.lean
|
map_invOf
|
[
{
"state_after": "α : Type u\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝³ : MulOneClass R\ninst✝² : Monoid S\ninst✝¹ : MonoidHomClass F R S\nf : F\nr : R\ninst✝ : Invertible r\n⊢ ↑f ⅟r = ⅟(↑f r)",
"state_before": "α : Type u\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝³ : MulOneClass R\ninst✝² : Monoid S\ninst✝¹ : MonoidHomClass F R S\nf : F\nr : R\ninst✝ : Invertible r\nifr : Invertible (↑f r)\nh : ifr = Invertible.map f r\n⊢ ↑f ⅟r = ⅟(↑f r)",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝³ : MulOneClass R\ninst✝² : Monoid S\ninst✝¹ : MonoidHomClass F R S\nf : F\nr : R\ninst✝ : Invertible r\n⊢ ↑f ⅟r = ⅟(↑f r)",
"tactic": "rfl"
}
] |
[
445,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
441,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.coinduction2
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\n⊢ IsBisimulation fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\n⊢ f s = g s",
"tactic": "refine' eq_of_bisim (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) _ ⟨s, rfl, rfl⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq β\nh : ∃ s, s1 = f s ∧ s2 = g s\n⊢ BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct s1) (destruct s2)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\n⊢ IsBisimulation fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s",
"tactic": "intro s1 s2 h"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq β\ns : Seq α\nh1 : s1 = f s\nh2 : s2 = g s\n⊢ BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct s1) (destruct s2)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq β\nh : ∃ s, s1 = f s ∧ s2 = g s\n⊢ BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct s1) (destruct s2)",
"tactic": "rcases h with ⟨s, h1, h2⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq β\ns : Seq α\nh1 : s1 = f s\nh2 : s2 = g s\n⊢ BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq β\ns : Seq α\nh1 : s1 = f s\nh2 : s2 = g s\n⊢ BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct s1) (destruct s2)",
"tactic": "rw [h1, h2]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : Seq α\nf g : Seq α → Seq β\nH : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq β\ns : Seq α\nh1 : s1 = f s\nh2 : s2 = g s\n⊢ BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (destruct (f s)) (destruct (g s))",
"tactic": "apply H"
}
] |
[
433,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_mul_re
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.1980449\ninst✝ : IsROrC K\nr : ℝ\nz : K\n⊢ ↑re (↑r * z) = r * ↑re z",
"tactic": "simp only [mul_re, ofReal_im, zero_mul, ofReal_re, sub_zero]"
}
] |
[
275,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Data/Nat/Sqrt.lean
|
Nat.le_sqrt
|
[] |
[
89,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
nhds_eq_uniformity'
|
[] |
[
793,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
792,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.limsSup_le_limsSup
|
[] |
[
501,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.restrict_biInf
|
[
{
"state_after": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\ns : Set α\nm : ι → OuterMeasure α\nthis : Nonempty ↑I\n⊢ ↑(restrict s) (⨅ (i : ι) (_ : i ∈ I), m i) = ⨅ (i : ι) (_ : i ∈ I), ↑(restrict s) (m i)",
"state_before": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\ns : Set α\nm : ι → OuterMeasure α\n⊢ ↑(restrict s) (⨅ (i : ι) (_ : i ∈ I), m i) = ⨅ (i : ι) (_ : i ∈ I), ↑(restrict s) (m i)",
"tactic": "haveI := hI.to_subtype"
},
{
"state_after": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\ns : Set α\nm : ι → OuterMeasure α\nthis : Nonempty ↑I\n⊢ ↑(restrict s) (⨅ (i : ↑I), m ↑i) = ⨅ (i : ↑I), ↑(restrict s) (m ↑i)",
"state_before": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\ns : Set α\nm : ι → OuterMeasure α\nthis : Nonempty ↑I\n⊢ ↑(restrict s) (⨅ (i : ι) (_ : i ∈ I), m i) = ⨅ (i : ι) (_ : i ∈ I), ↑(restrict s) (m i)",
"tactic": "rw [← iInf_subtype'', ← iInf_subtype'']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\ns : Set α\nm : ι → OuterMeasure α\nthis : Nonempty ↑I\n⊢ ↑(restrict s) (⨅ (i : ↑I), m ↑i) = ⨅ (i : ↑I), ↑(restrict s) (m ↑i)",
"tactic": "exact restrict_iInf _ _"
}
] |
[
1282,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1278,
1
] |
src/lean/Init/Classical.lean
|
Classical.byContradiction
|
[] |
[
124,
55
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
123,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_le_snorm_mul_snorm'_of_norm
|
[] |
[
1450,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1446,
1
] |
Mathlib/Order/Antisymmetrization.lean
|
OrderHom.antisymmetrization_apply_mk
|
[] |
[
238,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.lift_spec_mul
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_3\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_1\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nz : N\nw v : (fun x => P) z\n⊢ ↑(lift f hg) z * w = v ↔ ↑g (sec f z).fst * w = ↑g ↑(sec f z).snd * v",
"tactic": "erw [mul_comm, ← mul_assoc, mul_inv_left hg, mul_comm]"
}
] |
[
975,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
974,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.splits_zero
|
[] |
[
63,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoDiv_apply_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a - 0 • p ∈ Set.Ico a (a + p)",
"tactic": "simp [hp]"
}
] |
[
196,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_right_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.34821\nG : Type ?u.34824\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a / b / c = a / c / b",
"tactic": "simp"
}
] |
[
535,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.of_mul
|
[] |
[
422,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.toLinearEquiv_injective
|
[] |
[
564,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Order/Minimal.lean
|
IsAntichain.minimals_eq
|
[] |
[
184,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Std/Data/List/Lemmas.lean
|
List.mem_append_right
|
[] |
[
150,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
149,
1
] |
Mathlib/Data/Int/Cast/Prod.lean
|
Prod.fst_intCast
|
[] |
[
31,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/FieldTheory/Fixed.lean
|
FixedPoints.minpoly.irreducible_aux
|
[
{
"state_after": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\n⊢ f = 1 ∨ g = 1",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\n⊢ f = 1 ∨ g = 1",
"tactic": "have hf2 : f ∣ minpoly G F x := by rw [← hfg]; exact dvd_mul_right _ _"
},
{
"state_after": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\n⊢ f = 1 ∨ g = 1",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\n⊢ f = 1 ∨ g = 1",
"tactic": "have hg2 : g ∣ minpoly G F x := by rw [← hfg]; exact dvd_mul_left _ _"
},
{
"state_after": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x (minpoly G F x) = 0\n⊢ f = 1 ∨ g = 1",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\n⊢ f = 1 ∨ g = 1",
"tactic": "have := eval₂ G F x"
},
{
"state_after": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis :\n Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0 ∨\n Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\n⊢ f = 1 ∨ g = 1",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x (minpoly G F x) = 0\n⊢ f = 1 ∨ g = 1",
"tactic": "rw [← hfg, Polynomial.eval₂_mul, mul_eq_zero] at this"
},
{
"state_after": "case inl\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0\n⊢ f = 1 ∨ g = 1\n\ncase inr\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\n⊢ f = 1 ∨ g = 1",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis :\n Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0 ∨\n Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\n⊢ f = 1 ∨ g = 1",
"tactic": "cases' this with this this"
},
{
"state_after": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\n⊢ f ∣ f * g",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\n⊢ f ∣ minpoly G F x",
"tactic": "rw [← hfg]"
},
{
"state_after": "no goals",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\n⊢ f ∣ f * g",
"tactic": "exact dvd_mul_right _ _"
},
{
"state_after": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\n⊢ g ∣ f * g",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\n⊢ g ∣ minpoly G F x",
"tactic": "rw [← hfg]"
},
{
"state_after": "no goals",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\n⊢ g ∣ f * g",
"tactic": "exact dvd_mul_left _ _"
},
{
"state_after": "case inl.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0\n⊢ g = 1",
"state_before": "case inl\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0\n⊢ f = 1 ∨ g = 1",
"tactic": "right"
},
{
"state_after": "case inl.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0\nhf3 : f = minpoly G F x\n⊢ g = 1",
"state_before": "case inl.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0\n⊢ g = 1",
"tactic": "have hf3 : f = minpoly G F x :=\n Polynomial.eq_of_monic_of_associated hf (monic G F x)\n (associated_of_dvd_dvd hf2 <| @of_eval₂ G _ F _ _ _ x f this)"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x f = 0\nhf3 : f = minpoly G F x\n⊢ g = 1",
"tactic": "rwa [← mul_one (minpoly G F x), hf3, mul_right_inj' (monic G F x).ne_zero] at hfg"
},
{
"state_after": "case inr.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\n⊢ f = 1",
"state_before": "case inr\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\n⊢ f = 1 ∨ g = 1",
"tactic": "left"
},
{
"state_after": "case inr.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\nhg3 : g = minpoly G F x\n⊢ f = 1",
"state_before": "case inr.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\n⊢ f = 1",
"tactic": "have hg3 : g = minpoly G F x :=\n Polynomial.eq_of_monic_of_associated hg (monic G F x)\n (associated_of_dvd_dvd hg2 <| @of_eval₂ G _ F _ _ _ x g this)"
},
{
"state_after": "no goals",
"state_before": "case inr.h\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\nf g : Polynomial { x // x ∈ subfield G F }\nhf : Polynomial.Monic f\nhg : Polynomial.Monic g\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (Subring.subtype (subfield G F).toSubring) x g = 0\nhg3 : g = minpoly G F x\n⊢ f = 1",
"tactic": "rwa [← one_mul (minpoly G F x), hg3, mul_left_inj' (monic G F x).ne_zero] at hfg"
}
] |
[
249,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.Semiconj.mapsTo_periodicPts
|
[] |
[
264,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.ConnectedComponent.map_mk
|
[] |
[
2057,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2055,
1
] |
Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean
|
GeneralizedContinuedFraction.of_correctness_of_terminatedAt
|
[] |
[
253,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.eq_C_of_degree_le_zero
|
[
{
"state_after": "case a.zero\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.469754\nh : degree p ≤ 0\n⊢ coeff p Nat.zero = coeff (↑C (coeff p 0)) Nat.zero\n\ncase a.succ\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.469754\nh : degree p ≤ 0\nn : ℕ\n⊢ coeff p (Nat.succ n) = coeff (↑C (coeff p 0)) (Nat.succ n)",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.469754\nh : degree p ≤ 0\n⊢ p = ↑C (coeff p 0)",
"tactic": "ext (_ | n)"
},
{
"state_after": "case a.succ\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.469754\nh : degree p ≤ 0\nn : ℕ\n⊢ degree p < ↑(Nat.succ n)",
"state_before": "case a.succ\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.469754\nh : degree p ≤ 0\nn : ℕ\n⊢ coeff p (Nat.succ n) = coeff (↑C (coeff p 0)) (Nat.succ n)",
"tactic": "rw [coeff_C, if_neg (Nat.succ_ne_zero _), coeff_eq_zero_of_degree_lt]"
},
{
"state_after": "no goals",
"state_before": "case a.succ\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.469754\nh : degree p ≤ 0\nn : ℕ\n⊢ degree p < ↑(Nat.succ n)",
"tactic": "exact h.trans_lt (WithBot.some_lt_some.2 n.succ_pos)"
},
{
"state_after": "no goals",
"state_before": "case a.zero\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.469754\nh : degree p ≤ 0\n⊢ coeff p Nat.zero = coeff (↑C (coeff p 0)) Nat.zero",
"tactic": "simp"
}
] |
[
623,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
620,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.commute_add_nat
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ↑f fun x => x + ↑n",
"tactic": "simp only [add_comm _ (n : ℝ), f.commute_nat_add n]"
}
] |
[
339,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.subset_union_left
|
[] |
[
1374,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1374,
1
] |
Mathlib/Topology/Basic.lean
|
closure_empty
|
[] |
[
490,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Data/Finset/PImage.lean
|
Finset.mem_pimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝² : DecidableEq β\nf g : α →. β\ninst✝¹ : (x : α) → Decidable (f x).Dom\ninst✝ : (x : α) → Decidable (g x).Dom\ns t : Finset α\nb : β\n⊢ b ∈ pimage f s ↔ ∃ a, a ∈ s ∧ b ∈ f a",
"tactic": "simp [pimage]"
}
] |
[
70,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.coprime_div_gcd_div_gcd
|
[
{
"state_after": "no goals",
"state_before": "m n : Nat\nH : 0 < gcd m n\n⊢ coprime (m / gcd m n) (n / gcd m n)",
"tactic": "rw [coprime_iff_gcd_eq_one, gcd_div (gcd_dvd_left m n) (gcd_dvd_right m n), Nat.div_self H]"
}
] |
[
275,
94
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
273,
1
] |
src/lean/Init/Data/Nat/SOM.lean
|
Nat.SOM.Expr.eq_of_toPoly_eq
|
[
{
"state_after": "ctx : Context\na b : Expr\nh✝ : (toPoly a == toPoly b) = true\nh : Poly.denote ctx (toPoly a) = Poly.denote ctx (toPoly b)\n⊢ denote ctx a = denote ctx b",
"state_before": "ctx : Context\na b : Expr\nh : (toPoly a == toPoly b) = true\n⊢ denote ctx a = denote ctx b",
"tactic": "have h := congrArg (Poly.denote ctx) (eq_of_beq h)"
},
{
"state_after": "ctx : Context\na b : Expr\nh✝ : (toPoly a == toPoly b) = true\nh : denote ctx a = denote ctx b\n⊢ denote ctx a = denote ctx b",
"state_before": "ctx : Context\na b : Expr\nh✝ : (toPoly a == toPoly b) = true\nh : Poly.denote ctx (toPoly a) = Poly.denote ctx (toPoly b)\n⊢ denote ctx a = denote ctx b",
"tactic": "simp [toPoly_denote] at h"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\na b : Expr\nh✝ : (toPoly a == toPoly b) = true\nh : denote ctx a = denote ctx b\n⊢ denote ctx a = denote ctx b",
"tactic": "assumption"
}
] |
[
183,
13
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
180,
1
] |
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