file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.subset_closure
|
[] |
[
903,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
903,
1
] |
Mathlib/LinearAlgebra/FreeModule/Rank.lean
|
rank_finsupp'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Free R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Free R N\nι : Type v\n⊢ Module.rank R (ι →₀ M) = (#ι) * Module.rank R M",
"tactic": "simp [rank_finsupp]"
}
] |
[
49,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.ker_mk'
|
[] |
[
130,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.map_aux
|
[
{
"state_after": "case nil\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ nil a₂\n⊢ Valid' (Option.map f a₁) nil (Option.map f a₂)",
"state_before": "case nil\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ nil a₂\n⊢ Valid' (Option.map f a₁) (map f nil) (Option.map f a₂) ∧ size (map f nil) = size nil",
"tactic": "simp [map]"
},
{
"state_after": "case nil.h\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ nil a₂\n⊢ Bounded nil (Option.map f a₁) (Option.map f a₂)",
"state_before": "case nil\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ nil a₂\n⊢ Valid' (Option.map f a₁) nil (Option.map f a₂)",
"tactic": "apply valid'_nil"
},
{
"state_after": "case nil.h.none\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₂ : WithTop α\nh : Valid' none nil a₂\n⊢ Bounded nil (Option.map f none) (Option.map f a₂)\n\ncase nil.h.some\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₂ : WithTop α\nval✝ : α\nh : Valid' (some val✝) nil a₂\n⊢ Bounded nil (Option.map f (some val✝)) (Option.map f a₂)",
"state_before": "case nil.h\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ nil a₂\n⊢ Bounded nil (Option.map f a₁) (Option.map f a₂)",
"tactic": "cases a₁"
},
{
"state_after": "case nil.h.some.none\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nval✝ : α\nh : Valid' (some val✝) nil none\n⊢ Bounded nil (Option.map f (some val✝)) (Option.map f none)\n\ncase nil.h.some.some\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nval✝¹ val✝ : α\nh : Valid' (some val✝¹) nil (some val✝)\n⊢ Bounded nil (Option.map f (some val✝¹)) (Option.map f (some val✝))",
"state_before": "case nil.h.some\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₂ : WithTop α\nval✝ : α\nh : Valid' (some val✝) nil a₂\n⊢ Bounded nil (Option.map f (some val✝)) (Option.map f a₂)",
"tactic": "cases a₂"
},
{
"state_after": "case nil.h.some.some\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nval✝¹ val✝ : α\nh : Valid' (some val✝¹) nil (some val✝)\n⊢ f val✝¹ < f val✝",
"state_before": "case nil.h.some.some\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nval✝¹ val✝ : α\nh : Valid' (some val✝¹) nil (some val✝)\n⊢ Bounded nil (Option.map f (some val✝¹)) (Option.map f (some val✝))",
"tactic": "simp [Bounded]"
},
{
"state_after": "no goals",
"state_before": "case nil.h.some.some\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nval✝¹ val✝ : α\nh : Valid' (some val✝¹) nil (some val✝)\n⊢ f val✝¹ < f val✝",
"tactic": "exact f_strict_mono h.ord"
},
{
"state_after": "no goals",
"state_before": "case nil.h.none\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\na₂ : WithTop α\nh : Valid' none nil a₂\n⊢ Bounded nil (Option.map f none) (Option.map f a₂)",
"tactic": "trivial"
},
{
"state_after": "no goals",
"state_before": "case nil.h.some.none\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nval✝ : α\nh : Valid' (some val✝) nil none\n⊢ Bounded nil (Option.map f (some val✝)) (Option.map f none)",
"tactic": "trivial"
},
{
"state_after": "case node\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\nt_ih_l :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ l✝ a₂ → Valid' (Option.map f a₁) (map f l✝) (Option.map f a₂) ∧ size (map f l✝) = size l✝\nt_ih_r :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ r✝ a₂ → Valid' (Option.map f a₁) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_l' : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝) ∧ size (map f l✝) = size l✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"state_before": "case node\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\nt_ih_l :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ l✝ a₂ → Valid' (Option.map f a₁) (map f l✝) (Option.map f a₂) ∧ size (map f l✝) = size l✝\nt_ih_r :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ r✝ a₂ → Valid' (Option.map f a₁) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"tactic": "have t_ih_l' := t_ih_l h.left"
},
{
"state_after": "case node\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\nt_ih_l :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ l✝ a₂ → Valid' (Option.map f a₁) (map f l✝) (Option.map f a₂) ∧ size (map f l✝) = size l✝\nt_ih_r :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ r✝ a₂ → Valid' (Option.map f a₁) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_l' : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝) ∧ size (map f l✝) = size l✝\nt_ih_r' : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"state_before": "case node\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\nt_ih_l :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ l✝ a₂ → Valid' (Option.map f a₁) (map f l✝) (Option.map f a₂) ∧ size (map f l✝) = size l✝\nt_ih_r :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ r✝ a₂ → Valid' (Option.map f a₁) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_l' : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝) ∧ size (map f l✝) = size l✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"tactic": "have t_ih_r' := t_ih_r h.right"
},
{
"state_after": "case node\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_l' : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝) ∧ size (map f l✝) = size l✝\nt_ih_r' : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"state_before": "case node\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\nt_ih_l :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ l✝ a₂ → Valid' (Option.map f a₁) (map f l✝) (Option.map f a₂) ∧ size (map f l✝) = size l✝\nt_ih_r :\n ∀ {a₁ : WithBot α} {a₂ : WithTop α},\n Valid' a₁ r✝ a₂ → Valid' (Option.map f a₁) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_l' : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝) ∧ size (map f l✝) = size l✝\nt_ih_r' : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"tactic": "clear t_ih_l t_ih_r"
},
{
"state_after": "case node.intro\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_r' : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"state_before": "case node\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_l' : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝) ∧ size (map f l✝) = size l✝\nt_ih_r' : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"tactic": "cases' t_ih_l' with t_l_valid t_l_size"
},
{
"state_after": "case node.intro.intro\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"state_before": "case node.intro\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_ih_r' : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂) ∧ size (map f r✝) = size r✝\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"tactic": "cases' t_ih_r' with t_r_valid t_r_size"
},
{
"state_after": "case node.intro.intro\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝)) (Option.map f a₂)",
"state_before": "case node.intro.intro\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (map f (Ordnode.node size✝ l✝ x✝ r✝)) (Option.map f a₂) ∧\n size (map f (Ordnode.node size✝ l✝ x✝ r✝)) = size (Ordnode.node size✝ l✝ x✝ r✝)",
"tactic": "simp [map]"
},
{
"state_after": "case node.intro.intro.ord\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Bounded (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝)) (Option.map f a₁) (Option.map f a₂)\n\ncase node.intro.intro.sz\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝))\n\ncase node.intro.intro.bal\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝))",
"state_before": "case node.intro.intro\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Valid' (Option.map f a₁) (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝)) (Option.map f a₂)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case node.intro.intro.ord\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Bounded (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝)) (Option.map f a₁) (Option.map f a₂)",
"tactic": "exact And.intro t_l_valid.ord t_r_valid.ord"
},
{
"state_after": "case node.intro.intro.sz.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ size✝ = size (map f l✝) + size (map f r✝) + 1\n\ncase node.intro.intro.sz.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (map f l✝) ∧ Sized (map f r✝)",
"state_before": "case node.intro.intro.sz\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝))",
"tactic": "constructor"
},
{
"state_after": "case node.intro.intro.sz.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ size✝ = size l✝ + size r✝ + 1",
"state_before": "case node.intro.intro.sz.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ size✝ = size (map f l✝) + size (map f r✝) + 1",
"tactic": "rw [t_l_size, t_r_size]"
},
{
"state_after": "no goals",
"state_before": "case node.intro.intro.sz.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ size✝ = size l✝ + size r✝ + 1",
"tactic": "exact h.sz.1"
},
{
"state_after": "case node.intro.intro.sz.right.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (map f l✝)\n\ncase node.intro.intro.sz.right.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (map f r✝)",
"state_before": "case node.intro.intro.sz.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (map f l✝) ∧ Sized (map f r✝)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case node.intro.intro.sz.right.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (map f l✝)",
"tactic": "exact t_l_valid.sz"
},
{
"state_after": "no goals",
"state_before": "case node.intro.intro.sz.right.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Sized (map f r✝)",
"tactic": "exact t_r_valid.sz"
},
{
"state_after": "case node.intro.intro.bal.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ BalancedSz (size (map f l✝)) (size (map f r✝))\n\ncase node.intro.intro.bal.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (map f l✝) ∧ Balanced (map f r✝)",
"state_before": "case node.intro.intro.bal\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (Ordnode.node size✝ (map f l✝) (f x✝) (map f r✝))",
"tactic": "constructor"
},
{
"state_after": "case node.intro.intro.bal.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ BalancedSz (size l✝) (size r✝)",
"state_before": "case node.intro.intro.bal.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ BalancedSz (size (map f l✝)) (size (map f r✝))",
"tactic": "rw [t_l_size, t_r_size]"
},
{
"state_after": "no goals",
"state_before": "case node.intro.intro.bal.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ BalancedSz (size l✝) (size r✝)",
"tactic": "exact h.bal.1"
},
{
"state_after": "case node.intro.intro.bal.right.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (map f l✝)\n\ncase node.intro.intro.bal.right.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (map f r✝)",
"state_before": "case node.intro.intro.bal.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (map f l✝) ∧ Balanced (map f r✝)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case node.intro.intro.bal.right.left\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (map f l✝)",
"tactic": "exact t_l_valid.bal"
},
{
"state_after": "no goals",
"state_before": "case node.intro.intro.bal.right.right\nα : Type u_2\ninst✝¹ : Preorder α\nβ : Type u_1\ninst✝ : Preorder β\nf : α → β\nf_strict_mono : StrictMono f\nsize✝ : ℕ\nl✝ : Ordnode α\nx✝ : α\nr✝ : Ordnode α\na₁ : WithBot α\na₂ : WithTop α\nh : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂\nt_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ↑x✝)\nt_l_size : size (map f l✝) = size l✝\nt_r_valid : Valid' (Option.map f ↑x✝) (map f r✝) (Option.map f a₂)\nt_r_size : size (map f r✝) = size r✝\n⊢ Balanced (map f r✝)",
"tactic": "exact t_r_valid.bal"
}
] |
[
1592,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1564,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.eventuallyEq_one_of_mem_ball
|
[] |
[
427,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.strictMonoOn_sin
|
[] |
[
602,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.biInter_range
|
[] |
[
1688,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1687,
1
] |
Std/Data/List/Lemmas.lean
|
List.Sublist.append
|
[] |
[
399,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
398,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.count_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.369064\nγ : Type ?u.369067\nδ : Type ?u.369070\nι : Type ?u.369073\nR : Type ?u.369076\nR' : Type ?u.369079\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : MeasurableSpace α\n⊢ ↑↑count ∅ = 0",
"tactic": "rw [count_apply MeasurableSet.empty, tsum_empty]"
}
] |
[
2200,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2200,
1
] |
Mathlib/Control/Traversable/Equiv.lean
|
Equiv.traverse_eq_map_id
|
[
{
"state_after": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : α → β\nx : t' α\n⊢ ↑(eqv β) (id.mk (f <$> ↑(eqv α).symm x)) = Equiv.map eqv f x",
"state_before": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : α → β\nx : t' α\n⊢ Equiv.traverse eqv (pure ∘ f) x = pure (Equiv.map eqv f x)",
"tactic": "simp [Equiv.traverse, traverse_eq_map_id, functor_norm]"
},
{
"state_after": "no goals",
"state_before": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : α → β\nx : t' α\n⊢ ↑(eqv β) (id.mk (f <$> ↑(eqv α).symm x)) = Equiv.map eqv f x",
"tactic": "rfl"
}
] |
[
144,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
11
] |
Mathlib/Computability/EpsilonNFA.lean
|
NFA.toεNFA_evalFrom_match
|
[
{
"state_after": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\nstart : Set σ\n⊢ List.foldl (εNFA.stepSet (toεNFA M)) start = List.foldl (stepSet M) start",
"state_before": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\nstart : Set σ\n⊢ εNFA.evalFrom (toεNFA M) start = evalFrom M start",
"tactic": "rw [evalFrom, εNFA.evalFrom, toεNFA_εClosure]"
},
{
"state_after": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\nstart : Set σ\n⊢ εNFA.stepSet (toεNFA M) = stepSet M",
"state_before": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\nstart : Set σ\n⊢ List.foldl (εNFA.stepSet (toεNFA M)) start = List.foldl (stepSet M) start",
"tactic": "suffices εNFA.stepSet (toεNFA M) = stepSet M by rw [this]"
},
{
"state_after": "case h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ x✝ ∈ εNFA.stepSet (toεNFA M) S s ↔ x✝ ∈ stepSet M S s",
"state_before": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\nstart : Set σ\n⊢ εNFA.stepSet (toεNFA M) = stepSet M",
"tactic": "ext S s"
},
{
"state_after": "case h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ (∃ i, i ∈ S ∧ x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) i (some s))) ↔ ∃ i, i ∈ S ∧ x✝ ∈ step M i s",
"state_before": "case h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ x✝ ∈ εNFA.stepSet (toεNFA M) S s ↔ x✝ ∈ stepSet M S s",
"tactic": "simp only [stepSet, εNFA.stepSet, exists_prop, Set.mem_iUnion]"
},
{
"state_after": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ ∀ (a : σ), a ∈ S ∧ x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) a (some s)) ↔ a ∈ S ∧ x✝ ∈ step M a s",
"state_before": "case h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ (∃ i, i ∈ S ∧ x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) i (some s))) ↔ ∃ i, i ∈ S ∧ x✝ ∈ step M i s",
"tactic": "apply exists_congr"
},
{
"state_after": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ ∀ (a : σ), a ∈ S → (x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) a (some s)) ↔ x✝ ∈ step M a s)",
"state_before": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ ∀ (a : σ), a ∈ S ∧ x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) a (some s)) ↔ a ∈ S ∧ x✝ ∈ step M a s",
"tactic": "simp only [and_congr_right_iff]"
},
{
"state_after": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ a✝¹ : σ\na✝ : a✝¹ ∈ S\n⊢ x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) a✝¹ (some s)) ↔ x✝ ∈ step M a✝¹ s",
"state_before": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ : σ\n⊢ ∀ (a : σ), a ∈ S → (x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) a (some s)) ↔ x✝ ∈ step M a s)",
"tactic": "intro _ _"
},
{
"state_after": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ a✝¹ : σ\na✝ : a✝¹ ∈ S\n⊢ x✝ ∈ εNFA.step (toεNFA M) a✝¹ (some s) ↔ x✝ ∈ step M a✝¹ s",
"state_before": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ a✝¹ : σ\na✝ : a✝¹ ∈ S\n⊢ x✝ ∈ εNFA.εClosure (toεNFA M) (εNFA.step (toεNFA M) a✝¹ (some s)) ↔ x✝ ∈ step M a✝¹ s",
"tactic": "rw [M.toεNFA_εClosure]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h\nα : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS✝ : Set σ\nx : List α\ns✝ : σ\na : α\nM : NFA α σ\nstart S : Set σ\ns : α\nx✝ a✝¹ : σ\na✝ : a✝¹ ∈ S\n⊢ x✝ ∈ εNFA.step (toεNFA M) a✝¹ (some s) ↔ x✝ ∈ step M a✝¹ s",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nσ σ' : Type v\nM✝ : εNFA α σ\nS : Set σ\nx : List α\ns : σ\na : α\nM : NFA α σ\nstart : Set σ\nthis : εNFA.stepSet (toεNFA M) = stepSet M\n⊢ List.foldl (εNFA.stepSet (toεNFA M)) start = List.foldl (stepSet M) start",
"tactic": "rw [this]"
}
] |
[
206,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean
|
isGLB_of_tendsto_atTop
|
[] |
[
301,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.Bounded.subset_ball
|
[] |
[
2345,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2344,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
SubgroupClass.subtype_comp_inclusion
|
[
{
"state_after": "case h\nG : Type u_2\nG' : Type ?u.68703\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.68712\ninst✝⁴ : AddGroup A\nM : Type ?u.68718\nS : Type u_1\ninst✝³ : DivInvMonoid M\ninst✝² : SetLike S M\nhSM : SubgroupClass S M\nH✝ K✝ : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nH K : S\nhH : H ≤ K\nx✝ : { x // x ∈ H }\n⊢ ↑(MonoidHom.comp (↑K) (inclusion hH)) x✝ = ↑↑H x✝",
"state_before": "G : Type u_2\nG' : Type ?u.68703\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.68712\ninst✝⁴ : AddGroup A\nM : Type ?u.68718\nS : Type u_1\ninst✝³ : DivInvMonoid M\ninst✝² : SetLike S M\nhSM : SubgroupClass S M\nH✝ K✝ : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nH K : S\nhH : H ≤ K\n⊢ MonoidHom.comp (↑K) (inclusion hH) = ↑H",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nG : Type u_2\nG' : Type ?u.68703\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\nA : Type ?u.68712\ninst✝⁴ : AddGroup A\nM : Type ?u.68718\nS : Type u_1\ninst✝³ : DivInvMonoid M\ninst✝² : SetLike S M\nhSM : SubgroupClass S M\nH✝ K✝ : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nH K : S\nhH : H ≤ K\nx✝ : { x // x ∈ H }\n⊢ ↑(MonoidHom.comp (↑K) (inclusion hH)) x✝ = ↑↑H x✝",
"tactic": "simp only [MonoidHom.comp_apply, coeSubtype, coe_inclusion]"
}
] |
[
355,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Topology/Connected.lean
|
IsPreconnected.union
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.3583\nπ : ι → Type ?u.3588\ninst✝ : TopologicalSpace α\ns✝ t✝ u v : Set α\nx : α\ns t : Set α\nH1 : x ∈ s\nH2 : x ∈ t\nH3 : IsPreconnected s\nH4 : IsPreconnected t\n⊢ ∀ (s_1 : Set α), s_1 ∈ {s, t} → x ∈ s_1",
"tactic": "rintro r (rfl | rfl | h) <;> assumption"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.3583\nπ : ι → Type ?u.3588\ninst✝ : TopologicalSpace α\ns✝ t✝ u v : Set α\nx : α\ns t : Set α\nH1 : x ∈ s\nH2 : x ∈ t\nH3 : IsPreconnected s\nH4 : IsPreconnected t\n⊢ ∀ (s_1 : Set α), s_1 ∈ {s, t} → IsPreconnected s_1",
"tactic": "rintro r (rfl | rfl | h) <;> assumption"
}
] |
[
145,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Control/Functor.lean
|
Functor.map_id
|
[
{
"state_after": "no goals",
"state_before": "F : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F\ninst✝ : LawfulFunctor F\n⊢ (fun x x_1 => x <$> x_1) id = id",
"tactic": "apply funext <;> apply id_map"
}
] |
[
46,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.withDensitySMulLI_apply
|
[] |
[
980,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
976,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
lowerClosure_empty
|
[] |
[
1397,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1396,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.leftLim_le_rightLim
|
[] |
[
148,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum_mul_add
|
[
{
"state_after": "α : Type u\ninst✝ : Semiring α\nx : α\nn : ℕ\nthis : (∑ i in range n, (x + 1) ^ i * 1 ^ (n - 1 - i)) * x + 1 ^ n = (x + 1) ^ n\n⊢ (∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n",
"state_before": "α : Type u\ninst✝ : Semiring α\nx : α\nn : ℕ\n⊢ (∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n",
"tactic": "have := (Commute.one_right x).geom_sum₂_mul_add n"
},
{
"state_after": "α : Type u\ninst✝ : Semiring α\nx : α\nn : ℕ\nthis : (∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n\n⊢ (∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n",
"state_before": "α : Type u\ninst✝ : Semiring α\nx : α\nn : ℕ\nthis : (∑ i in range n, (x + 1) ^ i * 1 ^ (n - 1 - i)) * x + 1 ^ n = (x + 1) ^ n\n⊢ (∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n",
"tactic": "rw [one_pow, geom_sum₂_with_one] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Semiring α\nx : α\nn : ℕ\nthis : (∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n\n⊢ (∑ i in range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n",
"tactic": "exact this"
}
] |
[
169,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
TopologicalGroup.continuous_conj_prod
|
[] |
[
461,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.eq_false_of_not_eq_true'
|
[
{
"state_after": "no goals",
"state_before": "a : Bool\n⊢ (!decide (a = true)) = true → a = false",
"tactic": "cases a <;> decide"
}
] |
[
244,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.sieveOfSubfunctor_functorInclusion
|
[
{
"state_after": "case h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (sieveOfSubfunctor (functorInclusion S)).arrows f✝ ↔ S.arrows f✝",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\n⊢ sieveOfSubfunctor (functorInclusion S) = S",
"tactic": "ext"
},
{
"state_after": "case h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (∃ t, ↑t = f✝) ↔ S.arrows f✝",
"state_before": "case h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (sieveOfSubfunctor (functorInclusion S)).arrows f✝ ↔ S.arrows f✝",
"tactic": "simp only [functorInclusion_app, sieveOfSubfunctor_apply]"
},
{
"state_after": "case h.mp\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (∃ t, ↑t = f✝) → S.arrows f✝\n\ncase h.mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ S.arrows f✝ → ∃ t, ↑t = f✝",
"state_before": "case h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (∃ t, ↑t = f✝) ↔ S.arrows f✝",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.mk\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf : Y✝.op.unop ⟶ X\nhf : S.arrows f\n⊢ S.arrows ↑{ val := f, property := hf }",
"state_before": "case h.mp\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (∃ t, ↑t = f✝) → S.arrows f✝",
"tactic": "rintro ⟨⟨f, hf⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.mk\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf : Y✝.op.unop ⟶ X\nhf : S.arrows f\n⊢ S.arrows ↑{ val := f, property := hf }",
"tactic": "exact hf"
},
{
"state_after": "case h.mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\nhf : S.arrows f✝\n⊢ ∃ t, ↑t = f✝",
"state_before": "case h.mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ S.arrows f✝ → ∃ t, ↑t = f✝",
"tactic": "intro hf"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\nhf : S.arrows f✝\n⊢ ∃ t, ↑t = f✝",
"tactic": "exact ⟨⟨_, hf⟩, rfl⟩"
}
] |
[
808,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.indicator_const_smul_apply
|
[] |
[
506,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.isLocallyFiniteMeasure_of_isFiniteMeasureOnCompacts
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.836440\nγ : Type ?u.836443\nδ : Type ?u.836446\nι : Type ?u.836449\nR : Type ?u.836452\nR' : Type ?u.836455\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : LocallyCompactSpace α\ninst✝ : IsFiniteMeasureOnCompacts μ\nx : α\n⊢ FiniteAtFilter μ (𝓝 x)",
"state_before": "α : Type u_1\nβ : Type ?u.836440\nγ : Type ?u.836443\nδ : Type ?u.836446\nι : Type ?u.836449\nR : Type ?u.836452\nR' : Type ?u.836455\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : LocallyCompactSpace α\ninst✝ : IsFiniteMeasureOnCompacts μ\n⊢ ∀ (x : α), FiniteAtFilter μ (𝓝 x)",
"tactic": "intro x"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.836440\nγ : Type ?u.836443\nδ : Type ?u.836446\nι : Type ?u.836449\nR : Type ?u.836452\nR' : Type ?u.836455\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : LocallyCompactSpace α\ninst✝ : IsFiniteMeasureOnCompacts μ\nx : α\nK : Set α\nK_compact : IsCompact K\nK_mem : K ∈ 𝓝 x\n⊢ FiniteAtFilter μ (𝓝 x)",
"state_before": "α : Type u_1\nβ : Type ?u.836440\nγ : Type ?u.836443\nδ : Type ?u.836446\nι : Type ?u.836449\nR : Type ?u.836452\nR' : Type ?u.836455\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : LocallyCompactSpace α\ninst✝ : IsFiniteMeasureOnCompacts μ\nx : α\n⊢ FiniteAtFilter μ (𝓝 x)",
"tactic": "rcases exists_compact_mem_nhds x with ⟨K, K_compact, K_mem⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.836440\nγ : Type ?u.836443\nδ : Type ?u.836446\nι : Type ?u.836449\nR : Type ?u.836452\nR' : Type ?u.836455\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : TopologicalSpace α\ninst✝¹ : LocallyCompactSpace α\ninst✝ : IsFiniteMeasureOnCompacts μ\nx : α\nK : Set α\nK_compact : IsCompact K\nK_mem : K ∈ 𝓝 x\n⊢ FiniteAtFilter μ (𝓝 x)",
"tactic": "exact ⟨K, K_mem, K_compact.measure_lt_top⟩"
}
] |
[
3991,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3986,
1
] |
src/lean/Init/SimpLemmas.lean
|
Bool.or_true
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ (b || true) = true",
"tactic": "cases b <;> rfl"
}
] |
[
103,
83
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
103,
9
] |
Mathlib/Order/Filter/Bases.lean
|
FilterBasis.mem_filter_iff
|
[] |
[
178,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BoundedOrderHom.id_comp
|
[] |
[
687,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
src/lean/Init/Data/Nat/SOM.lean
|
Nat.SOM.Mon.mul_denote
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\nm₁✝ m₂✝ : Mon\nfuel : Nat\nm₁ m₂ : Mon\n⊢ denote ctx (mul.go fuel m₁ m₂) = denote ctx m₁ * denote ctx m₂",
"tactic": "induction fuel generalizing m₁ m₂ with\n| zero => simp! [append_denote]\n| succ _ ih =>\n simp!\n split <;> simp!\n next v₁ m₁ v₂ m₂ =>\n by_cases hlt : Nat.blt v₁ v₂ <;> simp! [hlt, Nat.mul_assoc, ih]\n by_cases hgt : Nat.blt v₂ v₁ <;> simp! [hgt, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm, ih]"
},
{
"state_after": "no goals",
"state_before": "case zero\nctx : Context\nm₁✝ m₂✝ m₁ m₂ : Mon\n⊢ denote ctx (mul.go zero m₁ m₂) = denote ctx m₁ * denote ctx m₂",
"tactic": "simp! [append_denote]"
},
{
"state_after": "case succ\nctx : Context\nm₁✝ m₂✝ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁ m₂ : Mon\n⊢ denote ctx\n (match m₁, m₂ with\n | m₁, [] => m₁\n | [], m₁ => m₁\n | v₁ :: m₁, v₂ :: m₂ =>\n bif blt v₁ v₂ then v₁ :: mul.go n✝ m₁ (v₂ :: m₂)\n else bif blt v₂ v₁ then v₂ :: mul.go n✝ (v₁ :: m₁) m₂ else v₁ :: v₂ :: mul.go n✝ m₁ m₂) =\n denote ctx m₁ * denote ctx m₂",
"state_before": "case succ\nctx : Context\nm₁✝ m₂✝ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁ m₂ : Mon\n⊢ denote ctx (mul.go (succ n✝) m₁ m₂) = denote ctx m₁ * denote ctx m₂",
"tactic": "simp!"
},
{
"state_after": "case succ.h_3\nctx : Context\nm₁ m₂ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁✝¹ m₂✝¹ : Mon\nv₁✝ : Var\nm₁✝ : List Var\nv₂✝ : Var\nm₂✝ : List Var\n⊢ denote ctx\n (bif blt v₁✝ v₂✝ then v₁✝ :: mul.go n✝ m₁✝ (v₂✝ :: m₂✝)\n else bif blt v₂✝ v₁✝ then v₂✝ :: mul.go n✝ (v₁✝ :: m₁✝) m₂✝ else v₁✝ :: v₂✝ :: mul.go n✝ m₁✝ m₂✝) =\n Var.denote ctx v₁✝ * denote ctx m₁✝ * (Var.denote ctx v₂✝ * denote ctx m₂✝)",
"state_before": "case succ\nctx : Context\nm₁✝ m₂✝ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁ m₂ : Mon\n⊢ denote ctx\n (match m₁, m₂ with\n | m₁, [] => m₁\n | [], m₁ => m₁\n | v₁ :: m₁, v₂ :: m₂ =>\n bif blt v₁ v₂ then v₁ :: mul.go n✝ m₁ (v₂ :: m₂)\n else bif blt v₂ v₁ then v₂ :: mul.go n✝ (v₁ :: m₁) m₂ else v₁ :: v₂ :: mul.go n✝ m₁ m₂) =\n denote ctx m₁ * denote ctx m₂",
"tactic": "split <;> simp!"
},
{
"state_after": "no goals",
"state_before": "case succ.h_3\nctx : Context\nm₁ m₂ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁✝¹ m₂✝¹ : Mon\nv₁✝ : Var\nm₁✝ : List Var\nv₂✝ : Var\nm₂✝ : List Var\n⊢ denote ctx\n (bif blt v₁✝ v₂✝ then v₁✝ :: mul.go n✝ m₁✝ (v₂✝ :: m₂✝)\n else bif blt v₂✝ v₁✝ then v₂✝ :: mul.go n✝ (v₁✝ :: m₁✝) m₂✝ else v₁✝ :: v₂✝ :: mul.go n✝ m₁✝ m₂✝) =\n Var.denote ctx v₁✝ * denote ctx m₁✝ * (Var.denote ctx v₂✝ * denote ctx m₂✝)",
"tactic": "next v₁ m₁ v₂ m₂ =>\n by_cases hlt : Nat.blt v₁ v₂ <;> simp! [hlt, Nat.mul_assoc, ih]\n by_cases hgt : Nat.blt v₂ v₁ <;> simp! [hgt, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm, ih]"
},
{
"state_after": "case inr\nctx : Context\nm₁✝¹ m₂✝¹ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁✝ m₂✝ : Mon\nv₁ : Var\nm₁ : List Var\nv₂ : Var\nm₂ : List Var\nhlt : ¬blt v₁ v₂ = true\n⊢ denote ctx (bif blt v₂ v₁ then v₂ :: mul.go n✝ (v₁ :: m₁) m₂ else v₁ :: v₂ :: mul.go n✝ m₁ m₂) =\n Var.denote ctx v₁ * (denote ctx m₁ * (Var.denote ctx v₂ * denote ctx m₂))",
"state_before": "ctx : Context\nm₁✝¹ m₂✝¹ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁✝ m₂✝ : Mon\nv₁ : Var\nm₁ : List Var\nv₂ : Var\nm₂ : List Var\n⊢ denote ctx\n (bif blt v₁ v₂ then v₁ :: mul.go n✝ m₁ (v₂ :: m₂)\n else bif blt v₂ v₁ then v₂ :: mul.go n✝ (v₁ :: m₁) m₂ else v₁ :: v₂ :: mul.go n✝ m₁ m₂) =\n Var.denote ctx v₁ * denote ctx m₁ * (Var.denote ctx v₂ * denote ctx m₂)",
"tactic": "by_cases hlt : Nat.blt v₁ v₂ <;> simp! [hlt, Nat.mul_assoc, ih]"
},
{
"state_after": "no goals",
"state_before": "case inr\nctx : Context\nm₁✝¹ m₂✝¹ : Mon\nn✝ : Nat\nih : ∀ (m₁ m₂ : Mon), denote ctx (mul.go n✝ m₁ m₂) = denote ctx m₁ * denote ctx m₂\nm₁✝ m₂✝ : Mon\nv₁ : Var\nm₁ : List Var\nv₂ : Var\nm₂ : List Var\nhlt : ¬blt v₁ v₂ = true\n⊢ denote ctx (bif blt v₂ v₁ then v₂ :: mul.go n✝ (v₁ :: m₁) m₂ else v₁ :: v₂ :: mul.go n✝ m₁ m₂) =\n Var.denote ctx v₁ * (denote ctx m₁ * (Var.denote ctx v₂ * denote ctx m₂))",
"tactic": "by_cases hgt : Nat.blt v₂ v₁ <;> simp! [hgt, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm, ih]"
}
] |
[
120,
105
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
109,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.fG_of_fG_toSubalgebra
|
[
{
"state_after": "case intro\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : IntermediateField F E\nt : Finset E\nht : Algebra.adjoin F ↑t = S.toSubalgebra\n⊢ FG S",
"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\nh : Subalgebra.FG S.toSubalgebra\n⊢ FG S",
"tactic": "cases' h with t ht"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : IntermediateField F E\nt : Finset E\nht : Algebra.adjoin F ↑t = S.toSubalgebra\n⊢ FG S",
"tactic": "exact ⟨t, (eq_adjoin_of_eq_algebra_adjoin _ _ _ ht.symm).symm⟩"
}
] |
[
938,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
936,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.260169\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns✝ : Set α\nt : Set β\nf : Filter α\nm : α → β\nh : Injective m\ns : Set α\nhs : s ∈ f\n⊢ m ⁻¹' (m '' s) ⊆ s",
"tactic": "simp only [preimage_image_eq s h, Subset.rfl]"
}
] |
[
2302,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2297,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.InjOn.image_biInter_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.188767\nι : Sort u_2\nι' : Sort ?u.188773\nι₂ : Sort ?u.188776\nκ : ι → Sort ?u.188781\nκ₁ : ι → Sort ?u.188786\nκ₂ : ι → Sort ?u.188791\nκ' : ι' → Sort ?u.188796\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ (i : ι) (hi : p i), s i hi)\n⊢ (f '' ⨅ (x : { i // p i }), s ↑x (_ : p ↑x)) = ⨅ (x : { i // p i }), f '' s ↑x (_ : p ↑x)",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.188767\nι : Sort u_2\nι' : Sort ?u.188773\nι₂ : Sort ?u.188776\nκ : ι → Sort ?u.188781\nκ₁ : ι → Sort ?u.188786\nκ₂ : ι → Sort ?u.188791\nκ' : ι' → Sort ?u.188796\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ (i : ι) (hi : p i), s i hi)\n⊢ (f '' ⋂ (i : ι) (hi : p i), s i hi) = ⋂ (i : ι) (hi : p i), f '' s i hi",
"tactic": "simp only [iInter, iInf_subtype']"
},
{
"state_after": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.188767\nι : Sort u_2\nι' : Sort ?u.188773\nι₂ : Sort ?u.188776\nκ : ι → Sort ?u.188781\nκ₁ : ι → Sort ?u.188786\nκ₂ : ι → Sort ?u.188791\nκ' : ι' → Sort ?u.188796\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ (i : ι) (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n⊢ (f '' ⨅ (x : { i // p i }), s ↑x (_ : p ↑x)) = ⨅ (x : { i // p i }), f '' s ↑x (_ : p ↑x)",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.188767\nι : Sort u_2\nι' : Sort ?u.188773\nι₂ : Sort ?u.188776\nκ : ι → Sort ?u.188781\nκ₁ : ι → Sort ?u.188786\nκ₂ : ι → Sort ?u.188791\nκ' : ι' → Sort ?u.188796\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ (i : ι) (hi : p i), s i hi)\n⊢ (f '' ⨅ (x : { i // p i }), s ↑x (_ : p ↑x)) = ⨅ (x : { i // p i }), f '' s ↑x (_ : p ↑x)",
"tactic": "haveI : Nonempty { i // p i } := nonempty_subtype.2 hp"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.188767\nι : Sort u_2\nι' : Sort ?u.188773\nι₂ : Sort ?u.188776\nκ : ι → Sort ?u.188781\nκ₁ : ι → Sort ?u.188786\nκ₂ : ι → Sort ?u.188791\nκ' : ι' → Sort ?u.188796\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ (i : ι) (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n⊢ InjOn f (⋃ (i : { i // p i }), s ↑i (_ : p ↑i))",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type ?u.188767\nι : Sort u_2\nι' : Sort ?u.188773\nι₂ : Sort ?u.188776\nκ : ι → Sort ?u.188781\nκ₁ : ι → Sort ?u.188786\nκ₂ : ι → Sort ?u.188791\nκ' : ι' → Sort ?u.188796\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ (i : ι) (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n⊢ (f '' ⨅ (x : { i // p i }), s ↑x (_ : p ↑x)) = ⨅ (x : { i // p i }), f '' s ↑x (_ : p ↑x)",
"tactic": "apply InjOn.image_iInter_eq"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_3\nγ : Type ?u.188767\nι : Sort u_2\nι' : Sort ?u.188773\nι₂ : Sort ?u.188776\nκ : ι → Sort ?u.188781\nκ₁ : ι → Sort ?u.188786\nκ₂ : ι → Sort ?u.188791\nκ' : ι' → Sort ?u.188796\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ (i : ι) (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n⊢ InjOn f (⋃ (i : { i // p i }), s ↑i (_ : p ↑i))",
"tactic": "simpa only [iUnion, iSup_subtype'] using h"
}
] |
[
1553,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1547,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
HasFDerivWithinAt.mul'
|
[] |
[
300,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
LatticeHom.copy_eq
|
[] |
[
1074,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1073,
1
] |
Mathlib/Order/Filter/Pi.lean
|
Filter.coprodᵢ_eq_bot_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), Nonempty (α i)\n⊢ Filter.coprodᵢ f = ⊥ ↔ f = ⊥",
"tactic": "simpa [funext_iff] using coprodᵢ_neBot_iff.not"
}
] |
[
230,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
CategoryTheory.Abelian.Pseudoelement.eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : f = 0\na : Pseudoelement P\n⊢ pseudoApply f a = 0",
"tactic": "simp [h]"
}
] |
[
300,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/RingTheory/Subsemiring/Pointwise.lean
|
Subsemiring.smul_mem_pointwise_smul_iff₀
|
[] |
[
148,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.map_zero
|
[] |
[
663,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/Data/PFunctor/Multivariate/M.lean
|
MvPFunctor.M.dest_corec'
|
[] |
[
249,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.radius_eq_top_of_summable_norm
|
[] |
[
313,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
MonotoneOn.quasiconvexOn
|
[] |
[
185,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Std/Data/List/Basic.lean
|
List.intercalate_eq_intercalateTR
|
[
{
"state_after": "case h.h.h\nα : Type u_1\nsep : List α\nl : List (List α)\n⊢ intercalate sep l = intercalateTR sep l",
"state_before": "⊢ @intercalate = @intercalateTR",
"tactic": "funext α sep l"
},
{
"state_after": "case h.h.h\nα : Type u_1\nsep : List α\nl : List (List α)\n⊢ join (intersperse sep l) =\n match l with\n | [] => []\n | [x] => x\n | x :: xs => intercalateTR.go (toArray sep) x xs #[]",
"state_before": "case h.h.h\nα : Type u_1\nsep : List α\nl : List (List α)\n⊢ intercalate sep l = intercalateTR sep l",
"tactic": "simp [intercalate, intercalateTR]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u_1\nsep : List α\nl : List (List α)\n⊢ join (intersperse sep l) =\n match l with\n | [] => []\n | [x] => x\n | x :: xs => intercalateTR.go (toArray sep) x xs #[]",
"tactic": "match l with\n| [] => rfl\n| [_] => simp\n| x::y::xs =>\n let rec go {acc x} : ∀ xs,\n intercalateTR.go sep.toArray x xs acc = acc.data ++ join (intersperse sep (x::xs))\n | [] => by simp [intercalateTR.go]\n | _::_ => by simp [intercalateTR.go, go]\n simp [intersperse, go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsep : List α\nl : List (List α)\n⊢ join (intersperse sep []) =\n match [] with\n | [] => []\n | [x] => x\n | x :: xs => intercalateTR.go (toArray sep) x xs #[]",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsep : List α\nl : List (List α)\nhead✝ : List α\n⊢ join (intersperse sep [head✝]) =\n match [head✝] with\n | [] => []\n | [x] => x\n | x :: xs => intercalateTR.go (toArray sep) x xs #[]",
"tactic": "simp"
},
{
"state_after": "α : Type u_1\nsep : List α\nl : List (List α)\nx y : List α\nxs : List (List α)\n⊢ join (intersperse sep (x :: y :: xs)) =\n match x :: y :: xs with\n | [] => []\n | [x] => x\n | x :: xs => intercalateTR.go (toArray sep) x xs #[]",
"state_before": "α : Type u_1\nsep : List α\nl : List (List α)\nx y : List α\nxs : List (List α)\n⊢ join (intersperse sep (x :: y :: xs)) =\n match x :: y :: xs with\n | [] => []\n | [x] => x\n | x :: xs => intercalateTR.go (toArray sep) x xs #[]",
"tactic": "let rec go {acc x} : ∀ xs,\n intercalateTR.go sep.toArray x xs acc = acc.data ++ join (intersperse sep (x::xs))\n| [] => by simp [intercalateTR.go]\n| _::_ => by simp [intercalateTR.go, go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsep : List α\nl : List (List α)\nx y : List α\nxs : List (List α)\n⊢ join (intersperse sep (x :: y :: xs)) =\n match x :: y :: xs with\n | [] => []\n | [x] => x\n | x :: xs => intercalateTR.go (toArray sep) x xs #[]",
"tactic": "simp [intersperse, go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsep : List α\nl : List (List α)\nx✝ y : List α\nxs : List (List α)\nacc : Array α\nx : List α\n⊢ intercalateTR.go (toArray sep) x [] acc = acc.data ++ join (intersperse sep [x])",
"tactic": "simp [intercalateTR.go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsep : List α\nl : List (List α)\nx✝ y : List α\nxs : List (List α)\nacc : Array α\nx head✝ : List α\ntail✝ : List (List α)\n⊢ intercalateTR.go (toArray sep) x (head✝ :: tail✝) acc = acc.data ++ join (intersperse sep (x :: head✝ :: tail✝))",
"tactic": "simp [intercalateTR.go, go]"
}
] |
[
253,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
243,
10
] |
Mathlib/Order/SymmDiff.lean
|
Codisjoint.bihimp_eq_inf
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.44592\nα : Type u_1\nβ : Type ?u.44598\nπ : ι → Type ?u.44603\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c d a b : α\nh : Codisjoint a b\n⊢ a ⇔ b = a ⊓ b",
"tactic": "rw [bihimp, h.himp_eq_left, h.himp_eq_right]"
}
] |
[
287,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.map_iSup_comap_of_surjective
|
[] |
[
489,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
MonoidHomClass.isometry_iff_norm
|
[
{
"state_after": "𝓕 : Type u_1\n𝕜 : Type ?u.280047\nα : Type ?u.280050\nι : Type ?u.280053\nκ : Type ?u.280056\nE : Type u_2\nF : Type u_3\nG : Type ?u.280065\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\n⊢ (∀ (x y : E), ‖↑f (x / y)‖ = ‖x / y‖) ↔ ∀ (x : E), ‖↑f x‖ = ‖x‖",
"state_before": "𝓕 : Type u_1\n𝕜 : Type ?u.280047\nα : Type ?u.280050\nι : Type ?u.280053\nκ : Type ?u.280056\nE : Type u_2\nF : Type u_3\nG : Type ?u.280065\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\n⊢ Isometry ↑f ↔ ∀ (x : E), ‖↑f x‖ = ‖x‖",
"tactic": "simp only [isometry_iff_dist_eq, dist_eq_norm_div, ← map_div]"
},
{
"state_after": "𝓕 : Type u_1\n𝕜 : Type ?u.280047\nα : Type ?u.280050\nι : Type ?u.280053\nκ : Type ?u.280056\nE : Type u_2\nF : Type u_3\nG : Type ?u.280065\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nh : ∀ (x y : E), ‖↑f (x / y)‖ = ‖x / y‖\nx : E\n⊢ ‖↑f x‖ = ‖x‖",
"state_before": "𝓕 : Type u_1\n𝕜 : Type ?u.280047\nα : Type ?u.280050\nι : Type ?u.280053\nκ : Type ?u.280056\nE : Type u_2\nF : Type u_3\nG : Type ?u.280065\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\n⊢ (∀ (x y : E), ‖↑f (x / y)‖ = ‖x / y‖) ↔ ∀ (x : E), ‖↑f x‖ = ‖x‖",
"tactic": "refine' ⟨fun h x => _, fun h x y => h _⟩"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type u_1\n𝕜 : Type ?u.280047\nα : Type ?u.280050\nι : Type ?u.280053\nκ : Type ?u.280056\nE : Type u_2\nF : Type u_3\nG : Type ?u.280065\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nh : ∀ (x y : E), ‖↑f (x / y)‖ = ‖x / y‖\nx : E\n⊢ ‖↑f x‖ = ‖x‖",
"tactic": "simpa using h x 1"
}
] |
[
868,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarAlgHom.ext_adjoin
|
[
{
"state_after": "case refine'_1\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na : { x // x ∈ adjoin R s }\nx : A\nhx : x ∈ s\n⊢ (fun y => ↑f y = ↑g y) { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\n\ncase refine'_2\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na : { x // x ∈ adjoin R s }\nr : R\n⊢ (fun y => ↑f y = ↑g y) (↑(algebraMap R { x // x ∈ adjoin R s }) r)\n\ncase refine'_3\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na x y : { x // x ∈ adjoin R s }\nhx : (fun y => ↑f y = ↑g y) x\nhy : (fun y => ↑f y = ↑g y) y\n⊢ (fun y => ↑f y = ↑g y) (x + y)\n\ncase refine'_4\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na x y : { x // x ∈ adjoin R s }\nhx : (fun y => ↑f y = ↑g y) x\nhy : (fun y => ↑f y = ↑g y) y\n⊢ (fun y => ↑f y = ↑g y) (x * y)\n\ncase refine'_5\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na x : { x // x ∈ adjoin R s }\nhx : (fun y => ↑f y = ↑g y) x\n⊢ (fun y => ↑f y = ↑g y) (star x)",
"state_before": "F : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\n⊢ f = g",
"tactic": "refine' FunLike.ext f g fun a =>\n adjoin_induction' (p := fun y => f y = g y) a (fun x hx => _) (fun r => _)\n (fun x y hx hy => _) (fun x y hx hy => _) fun x hx => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na : { x // x ∈ adjoin R s }\nx : A\nhx : x ∈ s\n⊢ (fun y => ↑f y = ↑g y) { val := x, property := (_ : x ∈ ↑(adjoin R s)) }",
"tactic": "exact h ⟨x, subset_adjoin R s hx⟩ hx"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na : { x // x ∈ adjoin R s }\nr : R\n⊢ (fun y => ↑f y = ↑g y) (↑(algebraMap R { x // x ∈ adjoin R s }) r)",
"tactic": "simp only [AlgHomClass.commutes]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na x y : { x // x ∈ adjoin R s }\nhx : (fun y => ↑f y = ↑g y) x\nhy : (fun y => ↑f y = ↑g y) y\n⊢ (fun y => ↑f y = ↑g y) (x + y)",
"tactic": "simp only [map_add, map_add, hx, hy]"
},
{
"state_after": "no goals",
"state_before": "case refine'_4\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na x y : { x // x ∈ adjoin R s }\nhx : (fun y => ↑f y = ↑g y) x\nhy : (fun y => ↑f y = ↑g y) y\n⊢ (fun y => ↑f y = ↑g y) (x * y)",
"tactic": "simp only [map_mul, map_mul, hx, hy]"
},
{
"state_after": "no goals",
"state_before": "case refine'_5\nF : Type u_2\nR : Type u_3\nA : Type u_1\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\nhF : StarAlgHomClass F R A B\nf✝ g✝ : F\ns : Set A\ninst✝ : StarAlgHomClass F R { x // x ∈ adjoin R s } B\nf g : F\nh : ∀ (x : { x // x ∈ adjoin R s }), ↑x ∈ s → ↑f x = ↑g x\na x : { x // x ∈ adjoin R s }\nhx : (fun y => ↑f y = ↑g y) x\n⊢ (fun y => ↑f y = ↑g y) (star x)",
"tactic": "simp only [map_star, hx]"
}
] |
[
767,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
758,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.nfpFamily_eq_self
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nf✝ : ι → Ordinal → Ordinal\nf : ι → Ordinal → Ordinal\na : Ordinal\nh : ∀ (i : ι), f i a = a\nl : List ι\n⊢ List.foldr f a l ≤ a",
"tactic": "rw [List.foldr_fixed' h l]"
}
] |
[
142,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Algebra/Ring/Units.lean
|
Units.add_divp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : Ring α\na✝ b✝ a b : α\nu : αˣ\n⊢ a + b /ₚ u = (a * ↑u + b) /ₚ u",
"tactic": "simp only [divp, add_mul, Units.mul_inv_cancel_right]"
}
] |
[
75,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.setCurr
|
[] |
[
585,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
584,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.mem_span_singleton_sup
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b : α\nS : Type u_1\ninst✝ : CommSemiring S\nx y : S\nI : Ideal S\n⊢ (∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = x) ↔ ∃ a b, b ∈ I ∧ a * y + b = x",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b : α\nS : Type u_1\ninst✝ : CommSemiring S\nx y : S\nI : Ideal S\n⊢ x ∈ span {y} ⊔ I ↔ ∃ a b, b ∈ I ∧ a * y + b = x",
"tactic": "rw [Submodule.mem_sup]"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b : α\nS : Type u_1\ninst✝ : CommSemiring S\nx y : S\nI : Ideal S\n⊢ (∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = x) → ∃ a b, b ∈ I ∧ a * y + b = x\n\ncase mpr\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b : α\nS : Type u_1\ninst✝ : CommSemiring S\nx y : S\nI : Ideal S\n⊢ (∃ a b, b ∈ I ∧ a * y + b = x) → ∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = x",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b : α\nS : Type u_1\ninst✝ : CommSemiring S\nx y : S\nI : Ideal S\n⊢ (∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = x) ↔ ∃ a b, b ∈ I ∧ a * y + b = x",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b✝ : α\nS : Type u_1\ninst✝ : CommSemiring S\ny : S\nI : Ideal S\nya : S\nhya : ya ∈ span {y}\nb : S\nhb : b ∈ I\n⊢ ∃ a b_1, b_1 ∈ I ∧ a * y + b_1 = ya + b",
"state_before": "case mp\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b : α\nS : Type u_1\ninst✝ : CommSemiring S\nx y : S\nI : Ideal S\n⊢ (∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = x) → ∃ a b, b ∈ I ∧ a * y + b = x",
"tactic": "rintro ⟨ya, hya, b, hb, rfl⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na✝ b✝ : α\nS : Type u_1\ninst✝ : CommSemiring S\ny : S\nI : Ideal S\nb : S\nhb : b ∈ I\na : S\nhya : a * y ∈ span {y}\n⊢ ∃ a_1 b_1, b_1 ∈ I ∧ a_1 * y + b_1 = a * y + b",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b✝ : α\nS : Type u_1\ninst✝ : CommSemiring S\ny : S\nI : Ideal S\nya : S\nhya : ya ∈ span {y}\nb : S\nhb : b ∈ I\n⊢ ∃ a b_1, b_1 ∈ I ∧ a * y + b_1 = ya + b",
"tactic": "obtain ⟨a, rfl⟩ := mem_span_singleton'.mp hya"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na✝ b✝ : α\nS : Type u_1\ninst✝ : CommSemiring S\ny : S\nI : Ideal S\nb : S\nhb : b ∈ I\na : S\nhya : a * y ∈ span {y}\n⊢ ∃ a_1 b_1, b_1 ∈ I ∧ a_1 * y + b_1 = a * y + b",
"tactic": "exact ⟨a, b, hb, rfl⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na✝ b✝ : α\nS : Type u_1\ninst✝ : CommSemiring S\ny : S\nI : Ideal S\na b : S\nhb : b ∈ I\n⊢ ∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = a * y + b",
"state_before": "case mpr\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na b : α\nS : Type u_1\ninst✝ : CommSemiring S\nx y : S\nI : Ideal S\n⊢ (∃ a b, b ∈ I ∧ a * y + b = x) → ∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = x",
"tactic": "rintro ⟨a, b, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI✝ : Ideal α\na✝ b✝ : α\nS : Type u_1\ninst✝ : CommSemiring S\ny : S\nI : Ideal S\na b : S\nhb : b ∈ I\n⊢ ∃ y_1, y_1 ∈ span {y} ∧ ∃ z, z ∈ I ∧ y_1 + z = a * y + b",
"tactic": "exact ⟨a * y, Ideal.mem_span_singleton'.mpr ⟨a, rfl⟩, b, hb, rfl⟩"
}
] |
[
227,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Ico.coe_zero
|
[] |
[
193,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.Integrable.dist_integralSum_sum_integral_le_of_memBaseSet
|
[] |
[
628,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.appendFun_aux
|
[] |
[
298,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Infinite.of_surjective_from_set
|
[] |
[
997,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
995,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.superpolynomialDecay_iff_norm_tendsto_zero
|
[] |
[
323,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/LinearAlgebra/Matrix/Basis.lean
|
basis_toMatrix_basisFun_mul
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.555917\nκ : Type ?u.555920\nκ' : Type ?u.555923\nR : Type u_2\nM : Type ?u.555929\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₂ : Type ?u.556116\nM₂ : Type ?u.556119\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type ?u.556864\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² : Fintype κ\ninst✝¹ : Fintype κ'\ninst✝ : Fintype ι\nb : Basis ι R (ι → R)\nA : Matrix ι ι R\n⊢ Basis.toMatrix b ↑(Pi.basisFun R ι) ⬝ A = ↑of fun i j => ↑(↑b.repr (Aᵀ j)) i",
"tactic": "classical\n simp only [basis_toMatrix_mul _ _ (Pi.basisFun R ι), Matrix.toLin_eq_toLin']\n ext (i j)\n rw [LinearMap.toMatrix_apply, Matrix.toLin'_apply, Pi.basisFun_apply,\n Matrix.mulVec_stdBasis_apply, Matrix.of_apply]"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.555917\nκ : Type ?u.555920\nκ' : Type ?u.555923\nR : Type u_2\nM : Type ?u.555929\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₂ : Type ?u.556116\nM₂ : Type ?u.556119\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type ?u.556864\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² : Fintype κ\ninst✝¹ : Fintype κ'\ninst✝ : Fintype ι\nb : Basis ι R (ι → R)\nA : Matrix ι ι R\n⊢ ↑(toMatrix (Pi.basisFun R ι) b) (↑toLin' A) = ↑of fun i j => ↑(↑b.repr (Aᵀ j)) i",
"state_before": "ι : Type u_1\nι' : Type ?u.555917\nκ : Type ?u.555920\nκ' : Type ?u.555923\nR : Type u_2\nM : Type ?u.555929\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₂ : Type ?u.556116\nM₂ : Type ?u.556119\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type ?u.556864\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² : Fintype κ\ninst✝¹ : Fintype κ'\ninst✝ : Fintype ι\nb : Basis ι R (ι → R)\nA : Matrix ι ι R\n⊢ Basis.toMatrix b ↑(Pi.basisFun R ι) ⬝ A = ↑of fun i j => ↑(↑b.repr (Aᵀ j)) i",
"tactic": "simp only [basis_toMatrix_mul _ _ (Pi.basisFun R ι), Matrix.toLin_eq_toLin']"
},
{
"state_after": "case a.h\nι : Type u_1\nι' : Type ?u.555917\nκ : Type ?u.555920\nκ' : Type ?u.555923\nR : Type u_2\nM : Type ?u.555929\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₂ : Type ?u.556116\nM₂ : Type ?u.556119\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni✝ : ι\nj✝ : ι'\nN : Type ?u.556864\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² : Fintype κ\ninst✝¹ : Fintype κ'\ninst✝ : Fintype ι\nb : Basis ι R (ι → R)\nA : Matrix ι ι R\ni j : ι\n⊢ ↑(toMatrix (Pi.basisFun R ι) b) (↑toLin' A) i j = ↑of (fun i j => ↑(↑b.repr (Aᵀ j)) i) i j",
"state_before": "ι : Type u_1\nι' : Type ?u.555917\nκ : Type ?u.555920\nκ' : Type ?u.555923\nR : Type u_2\nM : Type ?u.555929\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₂ : Type ?u.556116\nM₂ : Type ?u.556119\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type ?u.556864\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² : Fintype κ\ninst✝¹ : Fintype κ'\ninst✝ : Fintype ι\nb : Basis ι R (ι → R)\nA : Matrix ι ι R\n⊢ ↑(toMatrix (Pi.basisFun R ι) b) (↑toLin' A) = ↑of fun i j => ↑(↑b.repr (Aᵀ j)) i",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case a.h\nι : Type u_1\nι' : Type ?u.555917\nκ : Type ?u.555920\nκ' : Type ?u.555923\nR : Type u_2\nM : Type ?u.555929\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\nR₂ : Type ?u.556116\nM₂ : Type ?u.556119\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni✝ : ι\nj✝ : ι'\nN : Type ?u.556864\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝³ : Fintype ι'\ninst✝² : Fintype κ\ninst✝¹ : Fintype κ'\ninst✝ : Fintype ι\nb : Basis ι R (ι → R)\nA : Matrix ι ι R\ni j : ι\n⊢ ↑(toMatrix (Pi.basisFun R ι) b) (↑toLin' A) i j = ↑of (fun i j => ↑(↑b.repr (Aᵀ j)) i) i j",
"tactic": "rw [LinearMap.toMatrix_apply, Matrix.toLin'_apply, Pi.basisFun_apply,\n Matrix.mulVec_stdBasis_apply, Matrix.of_apply]"
}
] |
[
222,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.get_think
|
[] |
[
479,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.isBoundedUnder_const
|
[] |
[
118,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
BiheytingHom.coe_copy
|
[] |
[
537,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
EuclideanSpace.nnnorm_single
|
[] |
[
292,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Analysis/LocallyConvex/AbsConvex.lean
|
AbsConvexOpenSets.coe_isOpen
|
[] |
[
111,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Order/Disjoint.lean
|
disjoint_bot_left
|
[] |
[
61,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean
|
Measurable.sqrt
|
[] |
[
173,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.mul_comm
|
[
{
"state_after": "case h_real.h_real\nx y : ℝ\n⊢ ↑x * ↑y = ↑y * ↑x",
"state_before": "x y : EReal\n⊢ x * y = y * x",
"tactic": "induction' x using EReal.rec with x <;> induction' y using EReal.rec with y <;>\n try { rfl }"
},
{
"state_after": "no goals",
"state_before": "case h_real.h_real\nx y : ℝ\n⊢ ↑x * ↑y = ↑y * ↑x",
"tactic": "rw [← coe_mul, ← coe_mul, mul_comm]"
}
] |
[
224,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
11
] |
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
SimpleGraph.mem_cliqueSet_iff
|
[] |
[
249,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Algebra/DirectLimit.lean
|
Module.DirectLimit.toModule_totalize_of_le
|
[
{
"state_after": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\n⊢ ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j) (Dfinsupp.sum x Dfinsupp.single) =\n ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i) (Dfinsupp.sum x Dfinsupp.single))",
"state_before": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\n⊢ ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j) x =\n ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i) x)",
"tactic": "rw [← @Dfinsupp.sum_single ι G _ _ _ x]"
},
{
"state_after": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\n⊢ ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j)\n (Finset.sum (Dfinsupp.support x) fun i => Dfinsupp.single i (↑x i)) =\n ↑(f i j hij)\n (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i)\n (Finset.sum (Dfinsupp.support x) fun i => Dfinsupp.single i (↑x i)))",
"state_before": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\n⊢ ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j) (Dfinsupp.sum x Dfinsupp.single) =\n ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i) (Dfinsupp.sum x Dfinsupp.single))",
"tactic": "unfold Dfinsupp.sum"
},
{
"state_after": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\n⊢ (Finset.sum (Dfinsupp.support x) fun x_1 =>\n ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j) (Dfinsupp.single x_1 (↑x x_1))) =\n Finset.sum (Dfinsupp.support x) fun x_1 =>\n ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i) (Dfinsupp.single x_1 (↑x x_1)))",
"state_before": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\n⊢ ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j)\n (Finset.sum (Dfinsupp.support x) fun i => Dfinsupp.single i (↑x i)) =\n ↑(f i j hij)\n (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i)\n (Finset.sum (Dfinsupp.support x) fun i => Dfinsupp.single i (↑x i)))",
"tactic": "simp only [LinearMap.map_sum]"
},
{
"state_after": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\nk : ι\nhk : k ∈ Dfinsupp.support x\n⊢ ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j) (Dfinsupp.single k (↑x k)) =\n ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i) (Dfinsupp.single k (↑x k)))",
"state_before": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\n⊢ (Finset.sum (Dfinsupp.support x) fun x_1 =>\n ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j) (Dfinsupp.single x_1 (↑x x_1))) =\n Finset.sum (Dfinsupp.support x) fun x_1 =>\n ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i) (Dfinsupp.single x_1 (↑x x_1)))",
"tactic": "refine' Finset.sum_congr rfl fun k hk => _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : Ring R\nι : Type v\ndec_ι : DecidableEq ι\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → AddCommGroup (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\nP : Type u₁\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑(g j) (↑(f i j hij) x) = ↑(g i) x\ninst✝ : DirectedSystem G fun i j h => ↑(f i j h)\nx : DirectSum ι G\ni j : ι\nhij : i ≤ j\nhx : ∀ (k : ι), k ∈ Dfinsupp.support x → k ≤ i\nk : ι\nhk : k ∈ Dfinsupp.support x\n⊢ ↑(DirectSum.toModule R ι (G j) fun k => totalize G f k j) (Dfinsupp.single k (↑x k)) =\n ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => totalize G f k i) (Dfinsupp.single k (↑x k)))",
"tactic": "rw [DirectSum.single_eq_lof R k (x k), DirectSum.toModule_lof, DirectSum.toModule_lof,\n totalize_of_le (hx k hk), totalize_of_le (le_trans (hx k hk) hij), DirectedSystem.map_map]"
}
] |
[
212,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_col
|
[
{
"state_after": "n : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) ⬝ M)\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) (inr ()) (inl i) = 0",
"state_before": "n : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) (inr ()) (inl i) = 0",
"tactic": "have : listTransvecRow M = listTransvecRow ((listTransvecCol M).prod ⬝ M) := by\n simp [listTransvecRow, listTransvecCol_mul_last_row]"
},
{
"state_after": "n : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) ⬝ M)\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow (List.prod (listTransvecCol M) ⬝ M))) (inr ())\n (inl i) =\n 0",
"state_before": "n : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) ⬝ M)\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) (inr ()) (inl i) = 0",
"tactic": "rw [this]"
},
{
"state_after": "case hM\nn : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) ⬝ M)\n⊢ (List.prod (listTransvecCol M) ⬝ M) (inr ()) (inr ()) ≠ 0",
"state_before": "n : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) ⬝ M)\n⊢ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow (List.prod (listTransvecCol M) ⬝ M))) (inr ())\n (inl i) =\n 0",
"tactic": "apply mul_listTransvecRow_last_row"
},
{
"state_after": "no goals",
"state_before": "case hM\nn : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) ⬝ M)\n⊢ (List.prod (listTransvecCol M) ⬝ M) (inr ()) (inr ()) ≠ 0",
"tactic": "simpa [listTransvecCol_mul_last_row] using hM"
},
{
"state_after": "no goals",
"state_before": "n : Type ?u.204767\np : Type ?u.204770\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\n⊢ listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) ⬝ M)",
"tactic": "simp [listTransvecRow, listTransvecCol_mul_last_row]"
}
] |
[
507,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.ssubset_univ_iff
|
[] |
[
723,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
722,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
Submodule.IsPrincipal.mem_iff_eq_smul_generator
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Ring R\ninst✝¹ : Module R M\nS : Submodule R M\ninst✝ : IsPrincipal S\nx : M\n⊢ x ∈ S ↔ ∃ s, x = s • generator S",
"tactic": "simp_rw [@eq_comm _ x, ← mem_span_singleton, span_singleton_generator]"
}
] |
[
122,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/ModelTheory/ElementaryMaps.lean
|
FirstOrder.Language.ElementarySubstructure.isElementary
|
[] |
[
415,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.continuous_re
|
[] |
[
900,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
src/lean/Init/Core.lean
|
Subtype.eta
|
[
{
"state_after": "case mk\na b c d : Prop\nα : Type u\np : α → Prop\nval✝ : α\nproperty✝ : p val✝\nh : p { val := val✝, property := property✝ }.val\n⊢ { val := { val := val✝, property := property✝ }.val, property := h } = { val := val✝, property := property✝ }",
"state_before": "a✝ b c d : Prop\nα : Type u\np : α → Prop\na : Subtype fun x => p x\nh : p a.val\n⊢ { val := a.val, property := h } = a",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case mk\na b c d : Prop\nα : Type u\np : α → Prop\nval✝ : α\nproperty✝ : p val✝\nh : p { val := val✝, property := property✝ }.val\n⊢ { val := { val := val✝, property := property✝ }.val, property := h } = { val := val✝, property := property✝ }",
"tactic": "exact rfl"
}
] |
[
965,
12
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
963,
1
] |
Mathlib/Data/Part.lean
|
Part.le_total_of_le_of_le
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nh : x = none\n⊢ x ≤ y ∨ y ≤ x\n\ncase inr.intro\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : x = some b\n⊢ x ≤ y ∨ y ≤ x",
"state_before": "α : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\n⊢ x ≤ y ∨ y ≤ x",
"tactic": "rcases Part.eq_none_or_eq_some x with (h | ⟨b, h₀⟩)"
},
{
"state_after": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : x = some b\n⊢ y ≤ x",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : x = some b\n⊢ x ≤ y ∨ y ≤ x",
"tactic": "right"
},
{
"state_after": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : x = some b\nb' : α\nh₁ : b' ∈ y\n⊢ b' ∈ x",
"state_before": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : x = some b\n⊢ y ≤ x",
"tactic": "intro b' h₁"
},
{
"state_after": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\n⊢ b' ∈ x",
"state_before": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : x = some b\nb' : α\nh₁ : b' ∈ y\n⊢ b' ∈ x",
"tactic": "rw [Part.eq_some_iff] at h₀"
},
{
"state_after": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝ : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\nhx : b ∈ z\n⊢ b' ∈ x",
"state_before": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\n⊢ b' ∈ x",
"tactic": "have hx := hx _ h₀"
},
{
"state_after": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝ : x ≤ z\nhy✝ : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\nhx : b ∈ z\nhy : b' ∈ z\n⊢ b' ∈ x",
"state_before": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝ : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\nhx : b ∈ z\n⊢ b' ∈ x",
"tactic": "have hy := hy _ h₁"
},
{
"state_after": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝¹ : x ≤ z\nhy✝ : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\nhx✝ : b ∈ z\nhy : b' ∈ z\nhx : b = b'\n⊢ b' ∈ x",
"state_before": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝ : x ≤ z\nhy✝ : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\nhx : b ∈ z\nhy : b' ∈ z\n⊢ b' ∈ x",
"tactic": "have hx := Part.mem_unique hx hy"
},
{
"state_after": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝ : x ≤ z\nhy✝ : y ≤ z\nb : α\nh₀ : b ∈ x\nhx : b ∈ z\nh₁ : b ∈ y\nhy : b ∈ z\n⊢ b ∈ x",
"state_before": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝¹ : x ≤ z\nhy✝ : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\nhx✝ : b ∈ z\nhy : b' ∈ z\nhx : b = b'\n⊢ b' ∈ x",
"tactic": "subst hx"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx✝ : x ≤ z\nhy✝ : y ≤ z\nb : α\nh₀ : b ∈ x\nhx : b ∈ z\nh₁ : b ∈ y\nhy : b ∈ z\n⊢ b ∈ x",
"tactic": "exact h₀"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nh : x = none\n⊢ none ≤ y ∨ y ≤ none",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nh : x = none\n⊢ x ≤ y ∨ y ≤ x",
"tactic": "rw [h]"
},
{
"state_after": "case inl.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nh : x = none\n⊢ none ≤ y",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nh : x = none\n⊢ none ≤ y ∨ y ≤ none",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nα : Type u_1\nβ : Type ?u.12580\nγ : Type ?u.12583\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nh : x = none\n⊢ none ≤ y",
"tactic": "apply OrderBot.bot_le _"
}
] |
[
419,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty
|
[
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\n⊢ vectorSpan k s = ⊤ → affineSpan k s = ⊤",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\n⊢ affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤",
"tactic": "refine' ⟨vectorSpan_eq_top_of_affineSpan_eq_top k V P, _⟩"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = ⊤\n⊢ affineSpan k s = ⊤",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\n⊢ vectorSpan k s = ⊤ → affineSpan k s = ⊤",
"tactic": "intro h"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = ⊤\n⊢ Nonempty { x // x ∈ affineSpan k s }",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = ⊤\n⊢ affineSpan k s = ⊤",
"tactic": "suffices Nonempty (affineSpan k s) by\n obtain ⟨p, hp : p ∈ affineSpan k s⟩ := this\n rw [eq_iff_direction_eq_of_mem hp (mem_top k V p), direction_affineSpan, h, direction_top]"
},
{
"state_after": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : vectorSpan k s = ⊤\nx : P\nhx : x ∈ s\n⊢ Nonempty { x // x ∈ affineSpan k s }",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = ⊤\n⊢ Nonempty { x // x ∈ affineSpan k s }",
"tactic": "obtain ⟨x, hx⟩ := hs"
},
{
"state_after": "no goals",
"state_before": "case intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nh : vectorSpan k s = ⊤\nx : P\nhx : x ∈ s\n⊢ Nonempty { x // x ∈ affineSpan k s }",
"tactic": "exact ⟨⟨x, mem_affineSpan k hx⟩⟩"
},
{
"state_after": "case intro.mk\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = ⊤\np : P\nhp : p ∈ affineSpan k s\n⊢ affineSpan k s = ⊤",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = ⊤\nthis : Nonempty { x // x ∈ affineSpan k s }\n⊢ affineSpan k s = ⊤",
"tactic": "obtain ⟨p, hp : p ∈ affineSpan k s⟩ := this"
},
{
"state_after": "no goals",
"state_before": "case intro.mk\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = ⊤\np : P\nhp : p ∈ affineSpan k s\n⊢ affineSpan k s = ⊤",
"tactic": "rw [eq_iff_direction_eq_of_mem hp (mem_top k V p), direction_affineSpan, h, direction_top]"
}
] |
[
801,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
inv_eq_of_mul
|
[] |
[
1091,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1090,
9
] |
Mathlib/Algebra/Star/Pointwise.lean
|
Set.nonempty_star
|
[] |
[
54,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/Lie/DirectSum.lean
|
DirectSum.lie_module_bracket_apply
|
[] |
[
65,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.toDual_symm_apply
|
[] |
[
663,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/Data/Set/Enumerate.lean
|
Set.enumerate_eq_none_of_sel
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsel : Set α → Option α\ns : Set α\nh : sel s = none\n⊢ enumerate sel s 0 = none",
"tactic": "simp [h, enumerate]"
},
{
"state_after": "α : Type u_1\nsel : Set α → Option α\ns : Set α\nh : sel s = none\nn : ℕ\n⊢ (do\n let a ← none\n enumerate sel (s \\ {a}) n) =\n none",
"state_before": "α : Type u_1\nsel : Set α → Option α\ns : Set α\nh : sel s = none\nn : ℕ\n⊢ enumerate sel s (n + 1) = none",
"tactic": "simp [h, enumerate]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsel : Set α → Option α\ns : Set α\nh : sel s = none\nn : ℕ\n⊢ (do\n let a ← none\n enumerate sel (s \\ {a}) n) =\n none",
"tactic": "rfl"
}
] |
[
45,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
AlgHom.mapMatrix_comp
|
[] |
[
1607,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1605,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.coeFn_inf
|
[] |
[
501,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
hnot_hnot_hnot
|
[] |
[
1092,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1091,
1
] |
Mathlib/Data/MvPolynomial/Rename.lean
|
MvPolynomial.rename_rename
|
[
{
"state_after": "σ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\n⊢ ↑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) (eval₂ C (X ∘ f) p) =\n ↑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g ∘ f)) p",
"state_before": "σ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\n⊢ ↑(rename g) (eval₂ C (X ∘ f) p) = ↑(rename (g ∘ f)) p",
"tactic": "simp only [rename, aeval_eq_eval₂Hom]"
},
{
"state_after": "σ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\n⊢ eval₂ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C)\n (↑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) ∘ X ∘ f) p =\n ↑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g ∘ f)) p",
"state_before": "σ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\n⊢ ↑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) (eval₂ C (X ∘ f) p) =\n ↑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g ∘ f)) p",
"tactic": "rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]"
},
{
"state_after": "σ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\n⊢ RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C = algebraMap R (MvPolynomial α R)",
"state_before": "σ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\n⊢ eval₂ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C) (fun x => X (g (f x))) p =\n eval₂ (algebraMap R (MvPolynomial α R)) (fun x => X (g (f x))) p",
"tactic": "refine' eval₂Hom_congr _ rfl rfl"
},
{
"state_after": "case a\nσ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\nx✝ : R\n⊢ ↑(RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C) x✝ =\n ↑(algebraMap R (MvPolynomial α R)) x✝",
"state_before": "σ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\n⊢ RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C = algebraMap R (MvPolynomial α R)",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case a\nσ : Type u_1\nτ : Type u_4\nα : Type u_3\nR : Type u_2\nS : Type ?u.149749\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : σ → τ\ng : τ → α\np : MvPolynomial σ R\nx✝ : R\n⊢ ↑(RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C) x✝ =\n ↑(algebraMap R (MvPolynomial α R)) x✝",
"tactic": "simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]"
}
] |
[
93,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.pos_iff_ne_zero
|
[] |
[
39,
94
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
39,
11
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.isSuccLimit_iff
|
[] |
[
226,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.mem_spanningSets_of_index_le
|
[] |
[
3522,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3520,
1
] |
Mathlib/Data/Int/Lemmas.lean
|
Int.strictMonoOn_natAbs
|
[] |
[
89,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
Asymptotics.IsBigO.hasFDerivWithinAt
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.366356\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.366451\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\ns : Set E\nx₀ : E\nn : ℕ\nh : f =O[𝓝[s] x₀] fun x => ‖x - x₀‖ ^ n\nhx₀ : x₀ ∈ s\nhn : 1 < n\n⊢ HasFDerivWithinAt f 0 s x₀",
"tactic": "simp_rw [HasFDerivWithinAt, HasFDerivAtFilter,\n h.eq_zero_of_norm_pow_within hx₀ <| zero_lt_one.trans hn, zero_apply, sub_zero,\n h.trans_isLittleO ((isLittleO_pow_sub_sub x₀ hn).mono nhdsWithin_le_nhds)]"
}
] |
[
681,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_lt_univ
|
[] |
[
156,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
src/lean/Init/Data/AC.lean
|
Lean.Data.AC.Context.evalList_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx : Nat\nxs : List Nat\n⊢ evalList α ctx (insert x xs) = evalList α ctx (x :: xs)",
"tactic": "induction xs using List.two_step_induction with\n| empty => rfl\n| single =>\n simp [insert]\n split\n . rfl\n . simp [evalList, h.1, EvalInformation.evalOp]\n| step y z zs ih =>\n simp [insert] at *; split\n case inl => rfl\n case inr =>\n split\n case inl => simp [evalList, EvalInformation.evalOp]; rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]\n case inr => simp_all [evalList, EvalInformation.evalOp]; rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx : Nat\n⊢ evalList α ctx (insert x []) = evalList α ctx [x]",
"tactic": "rfl"
},
{
"state_after": "case single\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\n⊢ evalList α ctx (if x < a✝ then [x, a✝] else [a✝, x]) = evalList α ctx [x, a✝]",
"state_before": "case single\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\n⊢ evalList α ctx (insert x [a✝]) = evalList α ctx [x, a✝]",
"tactic": "simp [insert]"
},
{
"state_after": "case single.inl\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : x < a✝\n⊢ evalList α ctx [x, a✝] = evalList α ctx [x, a✝]\n\ncase single.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : ¬x < a✝\n⊢ evalList α ctx [a✝, x] = evalList α ctx [x, a✝]",
"state_before": "case single\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\n⊢ evalList α ctx (if x < a✝ then [x, a✝] else [a✝, x]) = evalList α ctx [x, a✝]",
"tactic": "split"
},
{
"state_after": "case single.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : ¬x < a✝\n⊢ evalList α ctx [a✝, x] = evalList α ctx [x, a✝]",
"state_before": "case single.inl\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : x < a✝\n⊢ evalList α ctx [x, a✝] = evalList α ctx [x, a✝]\n\ncase single.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : ¬x < a✝\n⊢ evalList α ctx [a✝, x] = evalList α ctx [x, a✝]",
"tactic": ". rfl"
},
{
"state_after": "no goals",
"state_before": "case single.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : ¬x < a✝\n⊢ evalList α ctx [a✝, x] = evalList α ctx [x, a✝]",
"tactic": ". simp [evalList, h.1, EvalInformation.evalOp]"
},
{
"state_after": "no goals",
"state_before": "case single.inl\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : x < a✝\n⊢ evalList α ctx [x, a✝] = evalList α ctx [x, a✝]",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case single.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx a✝ : Nat\nh✝ : ¬x < a✝\n⊢ evalList α ctx [a✝, x] = evalList α ctx [x, a✝]",
"tactic": "simp [evalList, h.1, EvalInformation.evalOp]"
},
{
"state_after": "case step\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\n⊢ evalList α ctx (if x < y then x :: y :: z :: zs else y :: if x < z then x :: z :: zs else z :: insert x zs) =\n evalList α ctx (x :: y :: z :: zs)",
"state_before": "case step\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (insert x (z :: zs)) = evalList α ctx (x :: z :: zs)\n⊢ evalList α ctx (insert x (y :: z :: zs)) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "simp [insert] at *"
},
{
"state_after": "case step.inl\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : x < y\n⊢ evalList α ctx (x :: y :: z :: zs) = evalList α ctx (x :: y :: z :: zs)\n\ncase step.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : ¬x < y\n⊢ evalList α ctx (y :: if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"state_before": "case step\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\n⊢ evalList α ctx (if x < y then x :: y :: z :: zs else y :: if x < z then x :: z :: zs else z :: insert x zs) =\n evalList α ctx (x :: y :: z :: zs)",
"tactic": "split"
},
{
"state_after": "case step.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : ¬x < y\n⊢ evalList α ctx (y :: if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"state_before": "case step.inl\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : x < y\n⊢ evalList α ctx (x :: y :: z :: zs) = evalList α ctx (x :: y :: z :: zs)\n\ncase step.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : ¬x < y\n⊢ evalList α ctx (y :: if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "case inl => rfl"
},
{
"state_after": "no goals",
"state_before": "case step.inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : ¬x < y\n⊢ evalList α ctx (y :: if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "case inr =>\n split\n case inl => simp [evalList, EvalInformation.evalOp]; rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]\n case inr => simp_all [evalList, EvalInformation.evalOp]; rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : x < y\n⊢ evalList α ctx (x :: y :: z :: zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "rfl"
},
{
"state_after": "case inl\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : x < z\n⊢ evalList α ctx (y :: x :: z :: zs) = evalList α ctx (x :: y :: z :: zs)\n\ncase inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : ¬x < z\n⊢ evalList α ctx (y :: z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"state_before": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝ : ¬x < y\n⊢ evalList α ctx (y :: if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "split"
},
{
"state_after": "case inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : ¬x < z\n⊢ evalList α ctx (y :: z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"state_before": "case inl\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : x < z\n⊢ evalList α ctx (y :: x :: z :: zs) = evalList α ctx (x :: y :: z :: zs)\n\ncase inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : ¬x < z\n⊢ evalList α ctx (y :: z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "case inl => simp [evalList, EvalInformation.evalOp]; rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : ¬x < z\n⊢ evalList α ctx (y :: z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "case inr => simp_all [evalList, EvalInformation.evalOp]; rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]"
},
{
"state_after": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : x < z\n⊢ op ctx (EvalInformation.evalVar ctx y) (op ctx (EvalInformation.evalVar ctx x) (evalList α ctx (z :: zs))) =\n op ctx (EvalInformation.evalVar ctx x) (op ctx (EvalInformation.evalVar ctx y) (evalList α ctx (z :: zs)))",
"state_before": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : x < z\n⊢ evalList α ctx (y :: x :: z :: zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "simp [evalList, EvalInformation.evalOp]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : x < z\n⊢ op ctx (EvalInformation.evalVar ctx y) (op ctx (EvalInformation.evalVar ctx x) (evalList α ctx (z :: zs))) =\n op ctx (EvalInformation.evalVar ctx x) (op ctx (EvalInformation.evalVar ctx y) (evalList α ctx (z :: zs)))",
"tactic": "rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]"
},
{
"state_after": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (z :: insert x zs) = op ctx (EvalInformation.evalVar ctx x) (evalList α ctx (z :: zs))\nh✝¹ : ¬x < y\nh✝ : ¬x < z\n⊢ op ctx (EvalInformation.evalVar ctx y) (op ctx (EvalInformation.evalVar ctx x) (evalList α ctx (z :: zs))) =\n op ctx (EvalInformation.evalVar ctx x) (op ctx (EvalInformation.evalVar ctx y) (evalList α ctx (z :: zs)))",
"state_before": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (if x < z then x :: z :: zs else z :: insert x zs) = evalList α ctx (x :: z :: zs)\nh✝¹ : ¬x < y\nh✝ : ¬x < z\n⊢ evalList α ctx (y :: z :: insert x zs) = evalList α ctx (x :: y :: z :: zs)",
"tactic": "simp_all [evalList, EvalInformation.evalOp]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nctx : Context α\nh : IsCommutative ctx.op\nx y z : Nat\nzs : List Nat\nih : evalList α ctx (z :: insert x zs) = op ctx (EvalInformation.evalVar ctx x) (evalList α ctx (z :: zs))\nh✝¹ : ¬x < y\nh✝ : ¬x < z\n⊢ op ctx (EvalInformation.evalVar ctx y) (op ctx (EvalInformation.evalVar ctx x) (evalList α ctx (z :: zs))) =\n op ctx (EvalInformation.evalVar ctx x) (op ctx (EvalInformation.evalVar ctx y) (evalList α ctx (z :: zs)))",
"tactic": "rw [h.1, ctx.assoc.1, h.1 (evalList _ _ _)]"
}
] |
[
189,
107
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
170,
1
] |
Mathlib/Algebra/Ring/Prod.lean
|
RingHom.prodMap_def
|
[] |
[
258,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.liftRel_dropn_destruct
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (do\n let x ← destruct (drop s n)\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n do\n let x ← destruct (drop t n)\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct (drop s (n + 1))) (destruct (drop t (n + 1)))",
"tactic": "simp only [LiftRelO, drop, Nat.add_eq, add_zero, destruct_tail, tail.aux]"
},
{
"state_after": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ Computation.LiftRel ?R (destruct (drop s n)) (destruct (drop t n))\n\ncase h2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ ∀ {a : Option (α × WSeq α)} {b : Option (β × WSeq β)},\n ?R a b →\n Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (match a with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n (match b with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n\ncase R\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ Option (α × WSeq α) → Option (β × WSeq β) → Prop",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (do\n let x ← destruct (drop s n)\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n do\n let x ← destruct (drop t n)\n match x with\n | none => Computation.pure none\n | some (fst, s) => destruct s",
"tactic": "apply liftRel_bind"
},
{
"state_after": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ ∀ {a : Option (α × WSeq α)} {b : Option (β × WSeq β)},\n LiftRelO R (LiftRel R) a b →\n Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (match a with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n (match b with\n | none => Computation.pure none\n | some (fst, s) => destruct s)",
"state_before": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ Computation.LiftRel ?R (destruct (drop s n)) (destruct (drop t n))\n\ncase h2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ ∀ {a : Option (α × WSeq α)} {b : Option (β × WSeq β)},\n ?R a b →\n Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (match a with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n (match b with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n\ncase R\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ Option (α × WSeq α) → Option (β × WSeq β) → Prop",
"tactic": "apply liftRel_dropn_destruct H n"
},
{
"state_after": "no goals",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\n⊢ ∀ {a : Option (α × WSeq α)} {b : Option (β × WSeq β)},\n LiftRelO R (LiftRel R) a b →\n Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (match a with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n (match b with\n | none => Computation.pure none\n | some (fst, s) => destruct s)",
"tactic": "exact fun {a b} o =>\n match a, b, o with\n | none, none, _ => by\n simp [-liftRel_pure_left, -liftRel_pure_right]\n | some (a, s), some (b, t), ⟨_, h2⟩ => by simp [tail.aux]; apply liftRel_destruct h2"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns : WSeq α\nt : WSeq β\nH : LiftRel R s t\nn : ℕ\na : Option (α × WSeq α)\nb : Option (β × WSeq β)\no : LiftRelO R (LiftRel R) a b\nx✝ : LiftRelO R (LiftRel R) none none\n⊢ Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (match none with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n (match none with\n | none => Computation.pure none\n | some (fst, s) => destruct s)",
"tactic": "simp [-liftRel_pure_left, -liftRel_pure_right]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nH : LiftRel R s✝ t✝\nn : ℕ\na✝ : Option (α × WSeq α)\nb✝ : Option (β × WSeq β)\no : LiftRelO R (LiftRel R) a✝ b✝\na : α\ns : WSeq α\nb : β\nt : WSeq β\nleft✝ : R a b\nh2 : LiftRel R s t\n⊢ Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (destruct s) (destruct t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nH : LiftRel R s✝ t✝\nn : ℕ\na✝ : Option (α × WSeq α)\nb✝ : Option (β × WSeq β)\no : LiftRelO R (LiftRel R) a✝ b✝\na : α\ns : WSeq α\nb : β\nt : WSeq β\nleft✝ : R a b\nh2 : LiftRel R s t\n⊢ Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (match some (a, s) with\n | none => Computation.pure none\n | some (fst, s) => destruct s)\n (match some (b, t) with\n | none => Computation.pure none\n | some (fst, s) => destruct s)",
"tactic": "simp [tail.aux]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nH : LiftRel R s✝ t✝\nn : ℕ\na✝ : Option (α × WSeq α)\nb✝ : Option (β × WSeq β)\no : LiftRelO R (LiftRel R) a✝ b✝\na : α\ns : WSeq α\nb : β\nt : WSeq β\nleft✝ : R a b\nh2 : LiftRel R s t\n⊢ Computation.LiftRel\n (fun x x_1 =>\n match x, x_1 with\n | none, none => True\n | some (a, s), some (b, t) => R a b ∧ LiftRel R s t\n | x, x_2 => False)\n (destruct s) (destruct t)",
"tactic": "apply liftRel_destruct h2"
}
] |
[
1051,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1039,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.moveRight_mk
|
[] |
[
157,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Substructure.map_injective_of_injective
|
[] |
[
576,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
575,
1
] |
Mathlib/Topology/List.lean
|
Vector.continuous_insertNth'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.66987\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\nx✝ : α × Vector α n\na : α\nl : Vector α n\n⊢ Tendsto (fun p => insertNth p.fst i p.snd) (𝓝 a ×ˢ 𝓝 l) (𝓝 (insertNth (a, l).fst i (a, l).snd))",
"state_before": "α : Type u_1\nβ : Type ?u.66987\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\nx✝ : α × Vector α n\na : α\nl : Vector α n\n⊢ ContinuousAt (fun p => insertNth p.fst i p.snd) (a, l)",
"tactic": "rw [ContinuousAt, nhds_prod_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.66987\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\nx✝ : α × Vector α n\na : α\nl : Vector α n\n⊢ Tendsto (fun p => insertNth p.fst i p.snd) (𝓝 a ×ˢ 𝓝 l) (𝓝 (insertNth (a, l).fst i (a, l).snd))",
"tactic": "exact tendsto_insertNth"
}
] |
[
210,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Data/Polynomial/IntegralNormalization.lean
|
Polynomial.integralNormalization_coeff_degree
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\ni : ℕ\nhi : degree f = ↑i\n⊢ coeff (integralNormalization f) i = 1",
"tactic": "rw [integralNormalization_coeff, if_pos hi]"
}
] |
[
66,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul
|
[
{
"state_after": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"tactic": "have hpd : p ^ (k + l) * p ∣ m * n := by\n let hpmn' : p ^ (succ (k + l)) ∣ m * n := hpmn\n rwa [pow_succ'] at hpmn'"
},
{
"state_after": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"tactic": "have hpd2 : p ∣ m * n / p ^ (k + l) := dvd_div_of_mul_dvd hpd"
},
{
"state_after": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"tactic": "have hpd3 : p ∣ m * n / (p ^ k * p ^ l) := by simpa [pow_add] using hpd2"
},
{
"state_after": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\nhpd4 : p ∣ m / p ^ k * (n / p ^ l)\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"tactic": "have hpd4 : p ∣ m / p ^ k * (n / p ^ l) := by simpa [Nat.div_mul_div_comm hpm hpn] using hpd3"
},
{
"state_after": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\nhpd4 : p ∣ m / p ^ k * (n / p ^ l)\nhpd5 : p ∣ m / p ^ k ∨ p ∣ n / p ^ l\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\nhpd4 : p ∣ m / p ^ k * (n / p ^ l)\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"tactic": "have hpd5 : p ∣ m / p ^ k ∨ p ∣ n / p ^ l :=\n (Prime.dvd_mul p_prime).1 hpd4"
},
{
"state_after": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\nhpd4 : p ∣ m / p ^ k * (n / p ^ l)\nhpd5 : p ∣ m / p ^ k ∨ p ∣ n / p ^ l\n⊢ p ^ k * p ∣ m ∨ p ^ l * p ∣ n",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\nhpd4 : p ∣ m / p ^ k * (n / p ^ l)\nhpd5 : p ∣ m / p ^ k ∨ p ∣ n / p ^ l\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"tactic": "suffices p ^ k * p ∣ m ∨ p ^ l * p ∣ n by rwa [_root_.pow_succ', _root_.pow_succ']"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\nhpd4 : p ∣ m / p ^ k * (n / p ^ l)\nhpd5 : p ∣ m / p ^ k ∨ p ∣ n / p ^ l\n⊢ p ^ k * p ∣ m ∨ p ^ l * p ∣ n",
"tactic": "exact hpd5.elim (fun h : p ∣ m / p ^ k => Or.inl <| mul_dvd_of_dvd_div hpm h)\n fun h : p ∣ n / p ^ l => Or.inr <| mul_dvd_of_dvd_div hpn h"
},
{
"state_after": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpmn' : p ^ succ (k + l) ∣ m * n := hpmn\n⊢ p ^ (k + l) * p ∣ m * n",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\n⊢ p ^ (k + l) * p ∣ m * n",
"tactic": "let hpmn' : p ^ (succ (k + l)) ∣ m * n := hpmn"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpmn' : p ^ succ (k + l) ∣ m * n := hpmn\n⊢ p ^ (k + l) * p ∣ m * n",
"tactic": "rwa [pow_succ'] at hpmn'"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\n⊢ p ∣ m * n / (p ^ k * p ^ l)",
"tactic": "simpa [pow_add] using hpd2"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\n⊢ p ∣ m / p ^ k * (n / p ^ l)",
"tactic": "simpa [Nat.div_mul_div_comm hpm hpn] using hpd3"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\nhpd4 : p ∣ m / p ^ k * (n / p ^ l)\nhpd5 : p ∣ m / p ^ k ∨ p ∣ n / p ^ l\nthis : p ^ k * p ∣ m ∨ p ^ l * p ∣ n\n⊢ p ^ (k + 1) ∣ m ∨ p ^ (l + 1) ∣ n",
"tactic": "rwa [_root_.pow_succ', _root_.pow_succ']"
}
] |
[
743,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsGreatest.bddAbove
|
[] |
[
340,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
smulAddHom_apply
|
[] |
[
201,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Topology/UniformSpace/CompactConvergence.lean
|
ContinuousMap.hasBasis_nhds_compactConvergence
|
[] |
[
190,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
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