file_path
stringlengths 11
79
| full_name
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100
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list | end
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stringclasses 4
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stringclasses 4
values | start
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|---|---|---|---|---|---|---|
Mathlib/Algebra/GCDMonoid/Basic.lean
|
Associated.gcd_eq_left
|
[] |
[
502,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_pair
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.12245\nα : Type u_1\nβ : Type ?u.12251\nγ : Type ?u.12254\ninst✝ : CommMonoid α\ns t : Multiset α\na✝ : α\nm : Multiset ι\nf g : ι → α\na b : α\n⊢ prod {a, b} = a * b",
"tactic": "rw [insert_eq_cons, prod_cons, prod_singleton]"
}
] |
[
113,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Order/Monotone/Union.lean
|
MonotoneOn.union_right
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\n⊢ f x ≤ f y",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\n⊢ MonotoneOn f (s ∪ t)",
"tactic": "intro x hx y hy hxy"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\n⊢ f x ≤ f y\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhcx : c ≤ x\n⊢ f x ≤ f y",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\n⊢ f x ≤ f y",
"tactic": "rcases lt_or_le x c with (hxc | hcx)"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhx : x ∈ s ∪ t\nhxc : x ≤ c\n⊢ x ∈ s",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\n⊢ ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s",
"tactic": "intro x hx hxc"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ s\n⊢ x ∈ s\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ t\n⊢ x ∈ s",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhx : x ∈ s ∪ t\nhxc : x ≤ c\n⊢ x ∈ s",
"tactic": "cases hx"
},
{
"state_after": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nx : α\nh✝ : x ∈ t\nhs : IsGreatest s x\nht : IsLeast t x\nhxc : x ≤ x\n⊢ x ∈ s\n\ncase inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ t\nh'x : x < c\n⊢ x ∈ s",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ t\n⊢ x ∈ s",
"tactic": "rcases eq_or_lt_of_le hxc with (rfl | h'x)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ t\nh'x : x < c\n⊢ x ∈ s",
"tactic": "exact (lt_irrefl _ (h'x.trans_le (ht.2 (by assumption)))).elim"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ s\n⊢ x ∈ s",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nx : α\nh✝ : x ∈ t\nhs : IsGreatest s x\nht : IsLeast t x\nhxc : x ≤ x\n⊢ x ∈ s",
"tactic": "exact hs.1"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ t\nh'x : x < c\n⊢ x ∈ t",
"tactic": "assumption"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nx : α\nhx : x ∈ s ∪ t\nhxc : c ≤ x\n⊢ x ∈ t",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\n⊢ ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t",
"tactic": "intro x hx hxc"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nx : α\nhx✝ : x ∈ s ∪ t\nhxc : c ≤ x\nhx : x ∈ t\n⊢ x ∈ t",
"tactic": "exact hx"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nhx✝ : c ∈ s ∪ t\nhxc : c ≤ c\nhx : c ∈ s\n⊢ c ∈ t\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nx : α\nhx✝ : x ∈ s ∪ t\nhxc : c ≤ x\nhx : x ∈ s\nh'x : c < x\n⊢ x ∈ t",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nx : α\nhx✝ : x ∈ s ∪ t\nhxc : c ≤ x\nhx : x ∈ s\n⊢ x ∈ t",
"tactic": "rcases eq_or_lt_of_le hxc with (rfl | h'x)"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nx : α\nhx✝ : x ∈ s ∪ t\nhxc : c ≤ x\nhx : x ∈ s\nh'x : c < x\n⊢ x ∈ t",
"tactic": "exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nhx✝ : c ∈ s ∪ t\nhxc : c ≤ c\nhx : c ∈ s\n⊢ c ∈ t",
"tactic": "exact ht.1"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\nxs : x ∈ s\n⊢ f x ≤ f y",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\n⊢ f x ≤ f y",
"tactic": "have xs : x ∈ s := A _ hx hxc.le"
},
{
"state_after": "case inl.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\nxs : x ∈ s\nhyc : y < c\n⊢ f x ≤ f y\n\ncase inl.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\nxs : x ∈ s\nhcy : c ≤ y\n⊢ f x ≤ f y",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\nxs : x ∈ s\n⊢ f x ≤ f y",
"tactic": "rcases lt_or_le y c with (hyc | hcy)"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\nxs : x ∈ s\nhyc : y < c\n⊢ f x ≤ f y",
"tactic": "exact h₁ xs (A _ hy hyc.le) hxy"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhxc : x < c\nxs : x ∈ s\nhcy : c ≤ y\n⊢ f x ≤ f y",
"tactic": "exact (h₁ xs hs.1 hxc.le).trans (h₂ ht.1 (B _ hy hcy) hcy)"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhcx : c ≤ x\nxt : x ∈ t\n⊢ f x ≤ f y",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhcx : c ≤ x\n⊢ f x ≤ f y",
"tactic": "have xt : x ∈ t := B _ hx hcx"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhcx : c ≤ x\nxt : x ∈ t\nyt : y ∈ t\n⊢ f x ≤ f y",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhcx : c ≤ x\nxt : x ∈ t\n⊢ f x ≤ f y",
"tactic": "have yt : y ∈ t := B _ hy (hcx.trans hxy)"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na : α\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : ∀ (x : α), x ∈ s ∪ t → x ≤ c → x ∈ s\nB : ∀ (x : α), x ∈ s ∪ t → c ≤ x → x ∈ t\nx : α\nhx : x ∈ s ∪ t\ny : α\nhy : y ∈ s ∪ t\nhxy : x ≤ y\nhcx : c ≤ x\nxt : x ∈ t\nyt : y ∈ t\n⊢ f x ≤ f y",
"tactic": "exact h₂ xt yt hxy"
}
] |
[
106,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
11
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Stream.foldl_eq_foldl_toList
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα✝ : Type u_2\nf : α✝ → α → α✝\ninit : α✝\nt : RBNode.Stream α\n⊢ foldl f init t = List.foldl f init (toList t)",
"tactic": "induction t generalizing init <;> simp [-List.foldl] <;> simp [*, RBNode.foldl_eq_foldl_toList]"
}
] |
[
383,
98
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
382,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.prime_dvd_card_quotient_normalizer
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\n⊢ Fintype.card (G ⧸ H) * Fintype.card { x // x ∈ H } = s * p * Fintype.card { x // x ∈ H }",
"tactic": "rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ Subgroup.comap (Subgroup.subtype (normalizer H)) H) % p\n⊢ Fintype.card ({ x // x ∈ normalizer H } ⧸ Subgroup.comap (Subgroup.subtype (normalizer H)) H) % p = 0",
"tactic": "rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm"
}
] |
[
580,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
div_sub'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.57003\nβ : Type ?u.57006\nK : Type u_1\ninst✝ : Field K\na b c : K\nhc : c ≠ 0\n⊢ a / c - b = (a - c * b) / c",
"tactic": "simpa using div_sub_div a b hc one_ne_zero"
}
] |
[
246,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
Filter.Tendsto.liminf_eq
|
[] |
[
207,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSubVSubWeights_self
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type ?u.418752\nP : Type ?u.418755\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.419411\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni : ι\n⊢ weightedVSubVSubWeights k i i = 0",
"tactic": "simp [weightedVSubVSubWeights]"
}
] |
[
681,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
680,
1
] |
Mathlib/CategoryTheory/NatIso.lean
|
CategoryTheory.NatIso.inv_inv_app
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF✝ G✝ : C ⥤ D\nα : F✝ ≅ G✝\nF G : C ⥤ D\ne : F ≅ G\nX : C\n⊢ inv (e.inv.app X) = e.hom.app X",
"tactic": "aesop_cat"
}
] |
[
166,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Equiv.map_rel
|
[] |
[
803,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq1.join_nil
|
[] |
[
941,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
940,
1
] |
Mathlib/Topology/Connected.lean
|
Continuous.image_connectedComponent_subset
|
[] |
[
712,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Cardinal.derivFamily_lt_ord_lift
|
[
{
"state_after": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ a < ord c → derivFamily f a < ord c",
"state_before": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\n⊢ a < ord c → derivFamily f a < ord c",
"tactic": "have hω : ℵ₀ < c.ord.cof := by\n rw [hc.cof_eq]\n exact lt_of_le_of_ne hc.1 hc'.symm"
},
{
"state_after": "case H₁\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ 0 < ord c → derivFamily f 0 < ord c\n\ncase H₂\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ ∀ (o : Ordinal), (o < ord c → derivFamily f o < ord c) → succ o < ord c → derivFamily f (succ o) < ord c\n\ncase H₃\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ ∀ (o : Ordinal),\n Ordinal.IsLimit o →\n (∀ (o' : Ordinal), o' < o → o' < ord c → derivFamily f o' < ord c) → o < ord c → derivFamily f o < ord c",
"state_before": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ a < ord c → derivFamily f a < ord c",
"tactic": "apply a.limitRecOn"
},
{
"state_after": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\n⊢ ℵ₀ < c",
"state_before": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\n⊢ ℵ₀ < Ordinal.cof (ord c)",
"tactic": "rw [hc.cof_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\n⊢ ℵ₀ < c",
"tactic": "exact lt_of_le_of_ne hc.1 hc'.symm"
},
{
"state_after": "case H₁\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ 0 < ord c → nfpFamily f 0 < ord c",
"state_before": "case H₁\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ 0 < ord c → derivFamily f 0 < ord c",
"tactic": "rw [derivFamily_zero]"
},
{
"state_after": "no goals",
"state_before": "case H₁\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ 0 < ord c → nfpFamily f 0 < ord c",
"tactic": "exact nfpFamily_lt_ord_lift hω (by rwa [hc.cof_eq]) hf"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ lift (#ι) < Ordinal.cof (ord c)",
"tactic": "rwa [hc.cof_eq]"
},
{
"state_after": "case H₂\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : b < ord c → derivFamily f b < ord c\nhb' : succ b < ord c\n⊢ derivFamily f (succ b) < ord c",
"state_before": "case H₂\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ ∀ (o : Ordinal), (o < ord c → derivFamily f o < ord c) → succ o < ord c → derivFamily f (succ o) < ord c",
"tactic": "intro b hb hb'"
},
{
"state_after": "case H₂\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : b < ord c → derivFamily f b < ord c\nhb' : succ b < ord c\n⊢ nfpFamily f (succ (derivFamily f b)) < ord c",
"state_before": "case H₂\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : b < ord c → derivFamily f b < ord c\nhb' : succ b < ord c\n⊢ derivFamily f (succ b) < ord c",
"tactic": "rw [derivFamily_succ]"
},
{
"state_after": "no goals",
"state_before": "case H₂\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : b < ord c → derivFamily f b < ord c\nhb' : succ b < ord c\n⊢ nfpFamily f (succ (derivFamily f b)) < ord c",
"tactic": "exact\n nfpFamily_lt_ord_lift hω (by rwa [hc.cof_eq]) hf\n ((ord_isLimit hc.1).2 _ (hb ((lt_succ b).trans hb')))"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : b < ord c → derivFamily f b < ord c\nhb' : succ b < ord c\n⊢ lift (#ι) < Ordinal.cof (ord c)",
"tactic": "rwa [hc.cof_eq]"
},
{
"state_after": "case H₃\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : Ordinal.IsLimit b\nH : ∀ (o' : Ordinal), o' < b → o' < ord c → derivFamily f o' < ord c\nhb' : b < ord c\n⊢ derivFamily f b < ord c",
"state_before": "case H₃\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\n⊢ ∀ (o : Ordinal),\n Ordinal.IsLimit o →\n (∀ (o' : Ordinal), o' < o → o' < ord c → derivFamily f o' < ord c) → o < ord c → derivFamily f o < ord c",
"tactic": "intro b hb H hb'"
},
{
"state_after": "case H₃\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : Ordinal.IsLimit b\nH : ∀ (o' : Ordinal), o' < b → o' < ord c → derivFamily f o' < ord c\nhb' : b < ord c\n⊢ (bsup b fun a x => derivFamily f a) < ord c",
"state_before": "case H₃\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : Ordinal.IsLimit b\nH : ∀ (o' : Ordinal), o' < b → o' < ord c → derivFamily f o' < ord c\nhb' : b < ord c\n⊢ derivFamily f b < ord c",
"tactic": "rw [derivFamily_limit f hb]"
},
{
"state_after": "no goals",
"state_before": "case H₃\nα : Type ?u.164801\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι) (b : Ordinal), b < ord c → f i b < ord c\na : Ordinal\nhω : ℵ₀ < Ordinal.cof (ord c)\nb : Ordinal\nhb : Ordinal.IsLimit b\nH : ∀ (o' : Ordinal), o' < b → o' < ord c → derivFamily f o' < ord c\nhb' : b < ord c\n⊢ (bsup b fun a x => derivFamily f a) < ord c",
"tactic": "exact\n bsup_lt_ord_of_isRegular.{u, v} hc (ord_lt_ord.1 ((ord_card_le b).trans_lt hb')) fun o' ho' =>\n H o' ho' (ho'.trans hb')"
}
] |
[
1179,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1160,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_iUnion
|
[] |
[
137,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/GroupTheory/Solvable.lean
|
derivedSeries_normal
|
[
{
"state_after": "case zero\nG : Type u_1\nG' : Type ?u.4100\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\n⊢ Normal (derivedSeries G Nat.zero)\n\ncase succ\nG : Type u_1\nG' : Type ?u.4100\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nn : ℕ\nih : Normal (derivedSeries G n)\n⊢ Normal (derivedSeries G (Nat.succ n))",
"state_before": "G : Type u_1\nG' : Type ?u.4100\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nn : ℕ\n⊢ Normal (derivedSeries G n)",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nG : Type u_1\nG' : Type ?u.4100\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\n⊢ Normal (derivedSeries G Nat.zero)",
"tactic": "exact (⊤ : Subgroup G).normal_of_characteristic"
},
{
"state_after": "no goals",
"state_before": "case succ\nG : Type u_1\nG' : Type ?u.4100\ninst✝¹ : Group G\ninst✝ : Group G'\nf : G →* G'\nn : ℕ\nih : Normal (derivedSeries G n)\n⊢ Normal (derivedSeries G (Nat.succ n))",
"tactic": "exact @Subgroup.commutator_normal G _ (derivedSeries G n) (derivedSeries G n) ih ih"
}
] |
[
62,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Algebra/Star/Prod.lean
|
Prod.snd_star
|
[] |
[
37,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.succ_one
|
[
{
"state_after": "α : Type ?u.178408\nβ : Type ?u.178411\nγ : Type ?u.178414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ succ One.toOfNat1.1 = { ofNat := ↑2 }.1",
"state_before": "α : Type ?u.178408\nβ : Type ?u.178411\nγ : Type ?u.178414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ succ 1 = 2",
"tactic": "unfold instOfNat OfNat.ofNat"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.178408\nβ : Type ?u.178411\nγ : Type ?u.178414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ succ One.toOfNat1.1 = { ofNat := ↑2 }.1",
"tactic": "simpa using by rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.178408\nβ : Type ?u.178411\nγ : Type ?u.178414\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\n⊢ One.toOfNat1.1 = 1",
"tactic": "rfl"
}
] |
[
1070,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1068,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.permCongr_def
|
[] |
[
432,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean
|
CategoryTheory.Monad.algebra_iso_of_iso
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nT : Monad C\nA B : Algebra T\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ T.map (inv f.f) ≫ T.map f.f ≫ B.a = B.a",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nT : Monad C\nA B : Algebra T\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ T.map (inv f.f) ≫ A.a = B.a ≫ inv f.f",
"tactic": "rw [IsIso.eq_comp_inv f.f, Category.assoc, ← f.h]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nT : Monad C\nA B : Algebra T\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ T.map (inv f.f) ≫ T.map f.f ≫ B.a = B.a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nT : Monad C\nA B : Algebra T\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ f ≫ Algebra.Hom.mk (inv f.f) = 𝟙 A ∧ Algebra.Hom.mk (inv f.f) ≫ f = 𝟙 B",
"tactic": "aesop_cat"
}
] |
[
215,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.val_eq_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na✝ b c u a : αˣ\n⊢ ↑a = 1 ↔ a = 1",
"tactic": "rw [← Units.val_one, eq_iff]"
}
] |
[
238,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.mem_of_mem_nsmul
|
[
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn : ℕ\nh✝ : a ∈ n • s\nh : a ∈ zero • s\n⊢ a ∈ s\n\ncase succ\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn✝ : ℕ\nh✝ : a ∈ n✝ • s\nn : ℕ\nih : a ∈ n • s → a ∈ s\nh : a ∈ succ n • s\n⊢ a ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn : ℕ\nh : a ∈ n • s\n⊢ a ∈ s",
"tactic": "induction' n with n ih"
},
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn : ℕ\nh✝ : a ∈ n • s\nh : a ∈ 0\n⊢ a ∈ s",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn : ℕ\nh✝ : a ∈ n • s\nh : a ∈ zero • s\n⊢ a ∈ s",
"tactic": "rw [zero_nsmul] at h"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn : ℕ\nh✝ : a ∈ n • s\nh : a ∈ 0\n⊢ a ∈ s",
"tactic": "exact absurd h (not_mem_zero _)"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn✝ : ℕ\nh✝ : a ∈ n✝ • s\nn : ℕ\nih : a ∈ n • s → a ∈ s\nh : a ∈ s ∨ a ∈ n • s\n⊢ a ∈ s",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn✝ : ℕ\nh✝ : a ∈ n✝ • s\nn : ℕ\nih : a ∈ n • s → a ∈ s\nh : a ∈ succ n • s\n⊢ a ∈ s",
"tactic": "rw [succ_nsmul, mem_add] at h"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.46161\nγ : Type ?u.46164\na : α\ns : Multiset α\nn✝ : ℕ\nh✝ : a ∈ n✝ • s\nn : ℕ\nih : a ∈ n • s → a ∈ s\nh : a ∈ s ∨ a ∈ n • s\n⊢ a ∈ s",
"tactic": "exact h.elim id ih"
}
] |
[
692,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
687,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_congr
|
[] |
[
1671,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1670,
1
] |
Mathlib/Order/Cover.lean
|
Wcovby.le_and_le_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.16828\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\n⊢ c = a ∨ c = b → a ≤ c ∧ c ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.16828\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\n⊢ a ≤ c ∧ c ≤ b ↔ c = a ∨ c = b",
"tactic": "refine' ⟨fun h2 => h.eq_or_eq h2.1 h2.2, _⟩"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.16828\ninst✝ : PartialOrder α\nb c : α\nh : c ⩿ b\n⊢ c ≤ c ∧ c ≤ b\n\ncase inr\nα : Type u_1\nβ : Type ?u.16828\ninst✝ : PartialOrder α\na c : α\nh : a ⩿ c\n⊢ a ≤ c ∧ c ≤ c",
"state_before": "α : Type u_1\nβ : Type ?u.16828\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\n⊢ c = a ∨ c = b → a ≤ c ∧ c ≤ b",
"tactic": "rintro (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.16828\ninst✝ : PartialOrder α\nb c : α\nh : c ⩿ b\n⊢ c ≤ c ∧ c ≤ b\n\ncase inr\nα : Type u_1\nβ : Type ?u.16828\ninst✝ : PartialOrder α\na c : α\nh : a ⩿ c\n⊢ a ≤ c ∧ c ≤ c",
"tactic": "exacts [⟨le_rfl, h.le⟩, ⟨h.le, le_rfl⟩]"
}
] |
[
172,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_lt_inv_iff
|
[] |
[
326,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/RingTheory/Coprime/Lemmas.lean
|
IsCoprime.prod_left
|
[
{
"state_after": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt✝ : Finset I\nb : I\nt : Finset I\nhbt : ¬b ∈ t\nih : (∀ (i : I), i ∈ t → IsCoprime (s i) x) → IsCoprime (∏ i in t, s i) x\nH : ∀ (i : I), i ∈ insert b t → IsCoprime (s i) x\n⊢ IsCoprime (s b * ∏ x in t, s x) x",
"state_before": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt✝ : Finset I\nb : I\nt : Finset I\nhbt : ¬b ∈ t\nih : (∀ (i : I), i ∈ t → IsCoprime (s i) x) → IsCoprime (∏ i in t, s i) x\nH : ∀ (i : I), i ∈ insert b t → IsCoprime (s i) x\n⊢ IsCoprime (∏ i in insert b t, s i) x",
"tactic": "rw [Finset.prod_insert hbt]"
},
{
"state_after": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt✝ : Finset I\nb : I\nt : Finset I\nhbt : ¬b ∈ t\nih : (∀ (i : I), i ∈ t → IsCoprime (s i) x) → IsCoprime (∏ i in t, s i) x\nH : IsCoprime (s b) x ∧ ∀ (x_1 : I), x_1 ∈ t → IsCoprime (s x_1) x\n⊢ IsCoprime (s b * ∏ x in t, s x) x",
"state_before": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt✝ : Finset I\nb : I\nt : Finset I\nhbt : ¬b ∈ t\nih : (∀ (i : I), i ∈ t → IsCoprime (s i) x) → IsCoprime (∏ i in t, s i) x\nH : ∀ (i : I), i ∈ insert b t → IsCoprime (s i) x\n⊢ IsCoprime (s b * ∏ x in t, s x) x",
"tactic": "rw [Finset.forall_mem_insert] at H"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt✝ : Finset I\nb : I\nt : Finset I\nhbt : ¬b ∈ t\nih : (∀ (i : I), i ∈ t → IsCoprime (s i) x) → IsCoprime (∏ i in t, s i) x\nH : IsCoprime (s b) x ∧ ∀ (x_1 : I), x_1 ∈ t → IsCoprime (s x_1) x\n⊢ IsCoprime (s b * ∏ x in t, s x) x",
"tactic": "exact H.1.mul_left (ih H.2)"
}
] |
[
57,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.symm_conj_apply
|
[] |
[
2380,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2378,
1
] |
Mathlib/Data/Fintype/BigOperators.lean
|
Function.Bijective.prod_comp
|
[] |
[
210,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Topology/Algebra/ContinuousAffineMap.lean
|
ContinuousAffineMap.zero_apply
|
[] |
[
190,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.pmap_eq_map_attach
|
[] |
[
1530,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1528,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.Tendsto.mul_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.98424\nγ : Type ?u.98427\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf : Filter α\nm : α → ℝ≥0∞\na b : ℝ≥0∞\nhm : Tendsto m f (𝓝 a)\nha : a ≠ 0 ∨ b ≠ ⊤\n⊢ Tendsto (fun x => m x * b) f (𝓝 (a * b))",
"tactic": "simpa only [mul_comm] using ENNReal.Tendsto.const_mul hm ha"
}
] |
[
384,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
11
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.coeff_sub
|
[
{
"state_after": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nq : R[X]\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff ({ toFinsupp := toFinsupp✝ } - q) n = coeff { toFinsupp := toFinsupp✝ } n - coeff q n",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\np q : R[X]\nn : ℕ\n⊢ coeff (p - q) n = coeff p n - coeff q n",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nn : ℕ\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff ({ toFinsupp := toFinsupp✝¹ } - { toFinsupp := toFinsupp✝ }) n =\n coeff { toFinsupp := toFinsupp✝¹ } n - coeff { toFinsupp := toFinsupp✝ } n",
"state_before": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nq : R[X]\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff ({ toFinsupp := toFinsupp✝ } - q) n = coeff { toFinsupp := toFinsupp✝ } n - coeff q n",
"tactic": "rcases q with ⟨⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nn : ℕ\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ (match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝¹ - toFinsupp✝ } with\n | { toFinsupp := p } => ↑p)\n n =\n (match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝¹ } with\n | { toFinsupp := p } => ↑p)\n n -\n (match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝ } with\n | { toFinsupp := p } => ↑p)\n n",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nn : ℕ\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff ({ toFinsupp := toFinsupp✝¹ } - { toFinsupp := toFinsupp✝ }) n =\n coeff { toFinsupp := toFinsupp✝¹ } n - coeff { toFinsupp := toFinsupp✝ } n",
"tactic": "rw [← ofFinsupp_sub, coeff, coeff, coeff]"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Ring R\nn : ℕ\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ (match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝¹ - toFinsupp✝ } with\n | { toFinsupp := p } => ↑p)\n n =\n (match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝¹ } with\n | { toFinsupp := p } => ↑p)\n n -\n (match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝ } with\n | { toFinsupp := p } => ↑p)\n n",
"tactic": "apply Finsupp.sub_apply"
}
] |
[
1168,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1164,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
disjoint_compl_right_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.60077\nw x y z : α\ninst✝ : BooleanAlgebra α\n⊢ Disjoint x (yᶜ) ↔ x ≤ y",
"tactic": "rw [← le_compl_iff_disjoint_right, compl_compl]"
}
] |
[
739,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
738,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Iic_union_Ioo_eq_Iio
|
[] |
[
1431,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1429,
1
] |
Mathlib/Order/Disjoint.lean
|
Disjoint.sup_right
|
[] |
[
201,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_comp
|
[] |
[
150,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.card_toMultiset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Cycle α\n⊢ ∀ (a : List α), ↑Multiset.card (toMultiset (Quotient.mk'' a)) = length (Quotient.mk'' a)",
"tactic": "simp"
}
] |
[
704,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometry.isometry
|
[] |
[
160,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
11
] |
Mathlib/Analysis/Convex/Star.lean
|
StarConvex.mem_smul
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.156386\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht : 1 ≤ t\n⊢ t⁻¹ • x ∈ s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.156386\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht : 1 ≤ t\n⊢ x ∈ t • s",
"tactic": "rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne']"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.156386\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht : 1 ≤ t\n⊢ t⁻¹ • x ∈ s",
"tactic": "exact hs.smul_mem hx (by positivity) (inv_le_one ht)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.156386\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht : 1 ≤ t\n⊢ 0 ≤ t⁻¹",
"tactic": "positivity"
}
] |
[
411,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/Algebra/Free.lean
|
FreeMagma.toFreeSemigroup_map
|
[] |
[
738,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
736,
1
] |
Mathlib/Order/Hom/Set.lean
|
OrderIso.image_preimage
|
[] |
[
60,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.map_ne_zero
|
[] |
[
146,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
|
Polynomial.IsSplittingField.splits
|
[] |
[
62,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.map_zero
|
[] |
[
98,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.sum_monomial_eq
|
[] |
[
361,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
LocalRing.of_isUnit_or_isUnit_one_sub_self
|
[] |
[
167,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/CategoryTheory/Shift/Basic.lean
|
CategoryTheory.shift_shiftFunctorCompIsoId_neg_add_self_inv_app
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\nA : Type u_1\ninst✝² : Category C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nn : A\nX : C\n⊢ (shiftFunctor C (-n)).map ((shiftFunctorCompIsoId C (-n) n (_ : -n + n = 0)).inv.app X) =\n (shiftFunctorCompIsoId C n (-n) (_ : n + -n = 0)).inv.app ((shiftFunctor C (-n)).obj X)",
"tactic": "apply shift_shiftFunctorCompIsoId_inv_app"
}
] |
[
544,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
541,
1
] |
Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean
|
HasFDerivAt.restrictScalars
|
[] |
[
67,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
hasFPowerSeriesAt_exp_zero_of_radius_pos
|
[] |
[
248,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
Rel.card_interedges_finpartition_right
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.24112\nι : Type ?u.24115\nκ : Type ?u.24118\nα : Type u_2\nβ : Type u_1\ninst✝² : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝¹ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ninst✝ : DecidableEq β\ns : Finset α\nP : Finpartition t\n⊢ card (interedges r s t) = ∑ b in P.parts, card (interedges r s b)",
"tactic": "classical\nsimp_rw [← P.biUnion_parts, interedges_biUnion_right, id]\nrw [card_biUnion]\nexact fun x hx y hy h ↦ interedges_disjoint_right r _ (P.disjoint hx hy h)"
},
{
"state_after": "𝕜 : Type ?u.24112\nι : Type ?u.24115\nκ : Type ?u.24118\nα : Type u_2\nβ : Type u_1\ninst✝² : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝¹ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ninst✝ : DecidableEq β\ns : Finset α\nP : Finpartition t\n⊢ card (Finset.biUnion P.parts fun b => interedges r s b) = ∑ b in P.parts, card (interedges r s b)",
"state_before": "𝕜 : Type ?u.24112\nι : Type ?u.24115\nκ : Type ?u.24118\nα : Type u_2\nβ : Type u_1\ninst✝² : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝¹ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ninst✝ : DecidableEq β\ns : Finset α\nP : Finpartition t\n⊢ card (interedges r s t) = ∑ b in P.parts, card (interedges r s b)",
"tactic": "simp_rw [← P.biUnion_parts, interedges_biUnion_right, id]"
},
{
"state_after": "𝕜 : Type ?u.24112\nι : Type ?u.24115\nκ : Type ?u.24118\nα : Type u_2\nβ : Type u_1\ninst✝² : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝¹ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ninst✝ : DecidableEq β\ns : Finset α\nP : Finpartition t\n⊢ ∀ (x : Finset β), x ∈ P.parts → ∀ (y : Finset β), y ∈ P.parts → x ≠ y → Disjoint (interedges r s x) (interedges r s y)",
"state_before": "𝕜 : Type ?u.24112\nι : Type ?u.24115\nκ : Type ?u.24118\nα : Type u_2\nβ : Type u_1\ninst✝² : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝¹ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ninst✝ : DecidableEq β\ns : Finset α\nP : Finpartition t\n⊢ card (Finset.biUnion P.parts fun b => interedges r s b) = ∑ b in P.parts, card (interedges r s b)",
"tactic": "rw [card_biUnion]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.24112\nι : Type ?u.24115\nκ : Type ?u.24118\nα : Type u_2\nβ : Type u_1\ninst✝² : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝¹ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ninst✝ : DecidableEq β\ns : Finset α\nP : Finpartition t\n⊢ ∀ (x : Finset β), x ∈ P.parts → ∀ (y : Finset β), y ∈ P.parts → x ≠ y → Disjoint (interedges r s x) (interedges r s y)",
"tactic": "exact fun x hx y hy h ↦ interedges_disjoint_right r _ (P.disjoint hx hy h)"
}
] |
[
175,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.pow_apply
|
[
{
"state_after": "case zero\nR : Type u_1\nR₁ : Type ?u.141510\nR₂ : Type ?u.141513\nR₃ : Type ?u.141516\nR₄ : Type ?u.141519\nS : Type ?u.141522\nK : Type ?u.141525\nK₂ : Type ?u.141528\nM : Type u_2\nM' : Type ?u.141534\nM₁ : Type ?u.141537\nM₂ : Type ?u.141540\nM₃ : Type ?u.141543\nM₄ : Type ?u.141546\nN : Type ?u.141549\nN₂ : Type ?u.141552\nι : Type ?u.141555\nV : Type ?u.141558\nV₂ : Type ?u.141561\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf : M →ₗ[R] M\nm : M\n⊢ ↑(f ^ Nat.zero) m = (↑f^[Nat.zero]) m\n\ncase succ\nR : Type u_1\nR₁ : Type ?u.141510\nR₂ : Type ?u.141513\nR₃ : Type ?u.141516\nR₄ : Type ?u.141519\nS : Type ?u.141522\nK : Type ?u.141525\nK₂ : Type ?u.141528\nM : Type u_2\nM' : Type ?u.141534\nM₁ : Type ?u.141537\nM₂ : Type ?u.141540\nM₃ : Type ?u.141543\nM₄ : Type ?u.141546\nN : Type ?u.141549\nN₂ : Type ?u.141552\nι : Type ?u.141555\nV : Type ?u.141558\nV₂ : Type ?u.141561\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf : M →ₗ[R] M\nm : M\nn : ℕ\nih : ↑(f ^ n) m = (↑f^[n]) m\n⊢ ↑(f ^ Nat.succ n) m = (↑f^[Nat.succ n]) m",
"state_before": "R : Type u_1\nR₁ : Type ?u.141510\nR₂ : Type ?u.141513\nR₃ : Type ?u.141516\nR₄ : Type ?u.141519\nS : Type ?u.141522\nK : Type ?u.141525\nK₂ : Type ?u.141528\nM : Type u_2\nM' : Type ?u.141534\nM₁ : Type ?u.141537\nM₂ : Type ?u.141540\nM₃ : Type ?u.141543\nM₄ : Type ?u.141546\nN : Type ?u.141549\nN₂ : Type ?u.141552\nι : Type ?u.141555\nV : Type ?u.141558\nV₂ : Type ?u.141561\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf : M →ₗ[R] M\nn : ℕ\nm : M\n⊢ ↑(f ^ n) m = (↑f^[n]) m",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\nR₁ : Type ?u.141510\nR₂ : Type ?u.141513\nR₃ : Type ?u.141516\nR₄ : Type ?u.141519\nS : Type ?u.141522\nK : Type ?u.141525\nK₂ : Type ?u.141528\nM : Type u_2\nM' : Type ?u.141534\nM₁ : Type ?u.141537\nM₂ : Type ?u.141540\nM₃ : Type ?u.141543\nM₄ : Type ?u.141546\nN : Type ?u.141549\nN₂ : Type ?u.141552\nι : Type ?u.141555\nV : Type ?u.141558\nV₂ : Type ?u.141561\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf : M →ₗ[R] M\nm : M\n⊢ ↑(f ^ Nat.zero) m = (↑f^[Nat.zero]) m",
"tactic": "rfl"
},
{
"state_after": "case succ\nR : Type u_1\nR₁ : Type ?u.141510\nR₂ : Type ?u.141513\nR₃ : Type ?u.141516\nR₄ : Type ?u.141519\nS : Type ?u.141522\nK : Type ?u.141525\nK₂ : Type ?u.141528\nM : Type u_2\nM' : Type ?u.141534\nM₁ : Type ?u.141537\nM₂ : Type ?u.141540\nM₃ : Type ?u.141543\nM₄ : Type ?u.141546\nN : Type ?u.141549\nN₂ : Type ?u.141552\nι : Type ?u.141555\nV : Type ?u.141558\nV₂ : Type ?u.141561\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf : M →ₗ[R] M\nm : M\nn : ℕ\nih : ↑(f ^ n) m = (↑f^[n]) m\n⊢ ↑f ((↑f^[n]) m) = (↑f^[n]) (↑f m)",
"state_before": "case succ\nR : Type u_1\nR₁ : Type ?u.141510\nR₂ : Type ?u.141513\nR₃ : Type ?u.141516\nR₄ : Type ?u.141519\nS : Type ?u.141522\nK : Type ?u.141525\nK₂ : Type ?u.141528\nM : Type u_2\nM' : Type ?u.141534\nM₁ : Type ?u.141537\nM₂ : Type ?u.141540\nM₃ : Type ?u.141543\nM₄ : Type ?u.141546\nN : Type ?u.141549\nN₂ : Type ?u.141552\nι : Type ?u.141555\nV : Type ?u.141558\nV₂ : Type ?u.141561\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf : M →ₗ[R] M\nm : M\nn : ℕ\nih : ↑(f ^ n) m = (↑f^[n]) m\n⊢ ↑(f ^ Nat.succ n) m = (↑f^[Nat.succ n]) m",
"tactic": "simp only [Function.comp_apply, Function.iterate_succ, LinearMap.mul_apply, pow_succ, ih]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u_1\nR₁ : Type ?u.141510\nR₂ : Type ?u.141513\nR₃ : Type ?u.141516\nR₄ : Type ?u.141519\nS : Type ?u.141522\nK : Type ?u.141525\nK₂ : Type ?u.141528\nM : Type u_2\nM' : Type ?u.141534\nM₁ : Type ?u.141537\nM₂ : Type ?u.141540\nM₃ : Type ?u.141543\nM₄ : Type ?u.141546\nN : Type ?u.141549\nN₂ : Type ?u.141552\nι : Type ?u.141555\nV : Type ?u.141558\nV₂ : Type ?u.141561\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\nf : M →ₗ[R] M\nm : M\nn : ℕ\nih : ↑(f ^ n) m = (↑f^[n]) m\n⊢ ↑f ((↑f^[n]) m) = (↑f^[n]) (↑f m)",
"tactic": "exact (Function.Commute.iterate_self _ _ m).symm"
}
] |
[
331,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Data/Finite/Basic.lean
|
Finite.sum_right
|
[] |
[
94,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/MeasureTheory/Integral/ExpDecay.lean
|
exp_neg_integrableOn_Ioi
|
[
{
"state_after": "a b : ℝ\nh : 0 < b\nthis : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b))\n⊢ IntegrableOn (fun x => exp (-b * x)) (Ioi a)",
"state_before": "a b : ℝ\nh : 0 < b\n⊢ IntegrableOn (fun x => exp (-b * x)) (Ioi a)",
"tactic": "have : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b)) := by\n refine' Tendsto.div_const (Tendsto.neg _) _\n exact tendsto_exp_atBot.comp (tendsto_id.neg_const_mul_atTop (Right.neg_neg_iff.2 h))"
},
{
"state_after": "a b : ℝ\nh : 0 < b\nthis : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b))\nx : ℝ\nx✝ : x ∈ Ici a\n⊢ HasDerivAt (fun x => -exp (-b * x) / b) (exp (-b * x)) x",
"state_before": "a b : ℝ\nh : 0 < b\nthis : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b))\n⊢ IntegrableOn (fun x => exp (-b * x)) (Ioi a)",
"tactic": "refine' integrableOn_Ioi_deriv_of_nonneg' (fun x _ => _) (fun x _ => (exp_pos _).le) this"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nh : 0 < b\nthis : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b))\nx : ℝ\nx✝ : x ∈ Ici a\n⊢ HasDerivAt (fun x => -exp (-b * x) / b) (exp (-b * x)) x",
"tactic": "simpa [h.ne'] using ((hasDerivAt_id x).const_mul b).neg.exp.neg.div_const b"
},
{
"state_after": "a b : ℝ\nh : 0 < b\n⊢ Tendsto (fun x => exp (-b * x)) atTop (𝓝 0)",
"state_before": "a b : ℝ\nh : 0 < b\n⊢ Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b))",
"tactic": "refine' Tendsto.div_const (Tendsto.neg _) _"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nh : 0 < b\n⊢ Tendsto (fun x => exp (-b * x)) atTop (𝓝 0)",
"tactic": "exact tendsto_exp_atBot.comp (tendsto_id.neg_const_mul_atTop (Right.neg_neg_iff.2 h))"
}
] |
[
38,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
32,
1
] |
Mathlib/Topology/Algebra/Order/Compact.lean
|
ContinuousOn.exists_forall_le'
|
[] |
[
286,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/CategoryTheory/Limits/Cones.lean
|
CategoryTheory.Limits.CoconeMorphism.ext
|
[
{
"state_after": "case mk\nJ : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₃\ninst✝¹ : Category C\nD : Type u₄\ninst✝ : Category D\nF : J ⥤ C\nc c' : Cocone F\ng : c ⟶ c'\nHom✝ : c.pt ⟶ c'.pt\nw✝ : ∀ (j : J), c.ι.app j ≫ Hom✝ = c'.ι.app j\nw : (mk Hom✝).Hom = g.Hom\n⊢ mk Hom✝ = g",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₃\ninst✝¹ : Category C\nD : Type u₄\ninst✝ : Category D\nF : J ⥤ C\nc c' : Cocone F\nf g : c ⟶ c'\nw : f.Hom = g.Hom\n⊢ f = g",
"tactic": "cases f"
},
{
"state_after": "case mk.mk\nJ : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₃\ninst✝¹ : Category C\nD : Type u₄\ninst✝ : Category D\nF : J ⥤ C\nc c' : Cocone F\nHom✝¹ : c.pt ⟶ c'.pt\nw✝¹ : ∀ (j : J), c.ι.app j ≫ Hom✝¹ = c'.ι.app j\nHom✝ : c.pt ⟶ c'.pt\nw✝ : ∀ (j : J), c.ι.app j ≫ Hom✝ = c'.ι.app j\nw : (mk Hom✝¹).Hom = (mk Hom✝).Hom\n⊢ mk Hom✝¹ = mk Hom✝",
"state_before": "case mk\nJ : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₃\ninst✝¹ : Category C\nD : Type u₄\ninst✝ : Category D\nF : J ⥤ C\nc c' : Cocone F\ng : c ⟶ c'\nHom✝ : c.pt ⟶ c'.pt\nw✝ : ∀ (j : J), c.ι.app j ≫ Hom✝ = c'.ι.app j\nw : (mk Hom✝).Hom = g.Hom\n⊢ mk Hom✝ = g",
"tactic": "cases g"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nJ : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₃\ninst✝¹ : Category C\nD : Type u₄\ninst✝ : Category D\nF : J ⥤ C\nc c' : Cocone F\nHom✝¹ : c.pt ⟶ c'.pt\nw✝¹ : ∀ (j : J), c.ι.app j ≫ Hom✝¹ = c'.ι.app j\nHom✝ : c.pt ⟶ c'.pt\nw✝ : ∀ (j : J), c.ι.app j ≫ Hom✝ = c'.ι.app j\nw : (mk Hom✝¹).Hom = (mk Hom✝).Hom\n⊢ mk Hom✝¹ = mk Hom✝",
"tactic": "congr"
}
] |
[
505,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Data/W/Basic.lean
|
WType.elim_injective
|
[
{
"state_after": "case intro\nα : Type u_2\nβ : α → Type u_3\nγ : Type u_1\nfγ : (a : α) × (β a → γ) → γ\nfγ_injective : Function.Injective fγ\na₁ : α\nf₁ f₂ : β a₁ → WType fun a => β a\nh✝ : elim γ fγ (mk a₁ f₁) = elim γ fγ (mk a₁ f₂)\nh : HEq (fun b => elim γ fγ (f₁ b)) fun b => elim γ fγ (f₂ b)\n⊢ mk a₁ f₁ = mk a₁ f₂",
"state_before": "α : Type u_2\nβ : α → Type u_3\nγ : Type u_1\nfγ : (a : α) × (β a → γ) → γ\nfγ_injective : Function.Injective fγ\na₁ : α\nf₁ : β a₁ → WType fun a => β a\na₂ : α\nf₂ : β a₂ → WType fun a => β a\nh : elim γ fγ (mk a₁ f₁) = elim γ fγ (mk a₂ f₂)\n⊢ mk a₁ f₁ = mk a₂ f₂",
"tactic": "obtain ⟨rfl, h⟩ := Sigma.mk.inj_iff.mp (fγ_injective h)"
},
{
"state_after": "case intro.e_f.h\nα : Type u_2\nβ : α → Type u_3\nγ : Type u_1\nfγ : (a : α) × (β a → γ) → γ\nfγ_injective : Function.Injective fγ\na₁ : α\nf₁ f₂ : β a₁ → WType fun a => β a\nh✝ : elim γ fγ (mk a₁ f₁) = elim γ fγ (mk a₁ f₂)\nh : HEq (fun b => elim γ fγ (f₁ b)) fun b => elim γ fγ (f₂ b)\nx : β a₁\n⊢ f₁ x = f₂ x",
"state_before": "case intro\nα : Type u_2\nβ : α → Type u_3\nγ : Type u_1\nfγ : (a : α) × (β a → γ) → γ\nfγ_injective : Function.Injective fγ\na₁ : α\nf₁ f₂ : β a₁ → WType fun a => β a\nh✝ : elim γ fγ (mk a₁ f₁) = elim γ fγ (mk a₁ f₂)\nh : HEq (fun b => elim γ fγ (f₁ b)) fun b => elim γ fγ (f₂ b)\n⊢ mk a₁ f₁ = mk a₁ f₂",
"tactic": "congr with x"
},
{
"state_after": "no goals",
"state_before": "case intro.e_f.h\nα : Type u_2\nβ : α → Type u_3\nγ : Type u_1\nfγ : (a : α) × (β a → γ) → γ\nfγ_injective : Function.Injective fγ\na₁ : α\nf₁ f₂ : β a₁ → WType fun a => β a\nh✝ : elim γ fγ (mk a₁ f₁) = elim γ fγ (mk a₁ f₂)\nh : HEq (fun b => elim γ fγ (f₁ b)) fun b => elim γ fγ (f₂ b)\nx : β a₁\n⊢ f₁ x = f₂ x",
"tactic": "exact elim_injective γ fγ fγ_injective (congr_fun (eq_of_heq h) x : _)"
}
] |
[
101,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Data/PFun.lean
|
PFun.dom_prodMap
|
[] |
[
670,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
ofAdd_multiset_prod
|
[
{
"state_after": "ι : Type ?u.999738\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : AddCommMonoid α\ns : Multiset α\n⊢ Multiset.sum s = Multiset.prod s",
"state_before": "ι : Type ?u.999738\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : AddCommMonoid α\ns : Multiset α\n⊢ ↑ofAdd (Multiset.sum s) = Multiset.prod (Multiset.map (↑ofAdd) s)",
"tactic": "simp [ofAdd]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.999738\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : AddCommMonoid α\ns : Multiset α\n⊢ Multiset.sum s = Multiset.prod s",
"tactic": "rfl"
}
] |
[
2350,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2349,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.partiallyWellOrderedOn_singleton
|
[] |
[
307,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Combinatorics/Colex.lean
|
Colex.colex_lt_of_ssubset
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrder α\nA B : Finset α\nh : A ⊂ B\n⊢ toColex ∅ < toColex (B \\ A)",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\nA B : Finset α\nh : A ⊂ B\n⊢ toColex A < toColex B",
"tactic": "rw [← sdiff_lt_sdiff_iff_lt, sdiff_eq_empty_iff_subset.2 h.1]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\nA B : Finset α\nh : A ⊂ B\n⊢ toColex ∅ < toColex (B \\ A)",
"tactic": "exact empty_toColex_lt (by simpa [Finset.Nonempty] using exists_of_ssubset h)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\nA B : Finset α\nh : A ⊂ B\n⊢ Finset.Nonempty (B \\ A)",
"tactic": "simpa [Finset.Nonempty] using exists_of_ssubset h"
}
] |
[
351,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioi_inj
|
[] |
[
1124,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1123,
1
] |
Mathlib/RingTheory/Coprime/Lemmas.lean
|
IsCoprime.of_prod_right
|
[] |
[
80,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
sInfHom.cancel_right
|
[] |
[
489,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.set_biUnion_singleton
|
[] |
[
2007,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2006,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_quadratic_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.936788\n⊢ degree (↑C a * X ^ 2 + ↑C b * X + ↑C c) ≤ 2",
"tactic": "simpa only [add_assoc] using\n degree_add_le_of_degree_le (degree_C_mul_X_pow_le 2 a)\n (le_trans degree_linear_le <| WithBot.coe_le_coe.mpr one_le_two)"
}
] |
[
1176,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1173,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.finrank_adjoin_eq_one_iff
|
[] |
[
740,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
739,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_const_add_Ioc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a + x) ⁻¹' Ioc b c = Ioc (b - a) (c - a)",
"tactic": "simp [← Ioi_inter_Iic]"
}
] |
[
74,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisInsertion.u_injective
|
[] |
[
530,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.eq_pow_of_mul_eq_pow
|
[
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : k = 0\n⊢ ∃ d, a = d ^ k\n\ncase neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\n⊢ ∃ d, a = d ^ k",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\n⊢ ∃ d, a = d ^ k",
"tactic": "by_cases hk0 : k = 0"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : k = 0\n⊢ a = 1 ^ k",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : k = 0\n⊢ ∃ d, a = d ^ k",
"tactic": "use 1"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = 1\nhk0 : k = 0\n⊢ a = 1",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : k = 0\n⊢ a = 1 ^ k",
"tactic": "rw [hk0, pow_zero] at h⊢"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = 1\nhk0 : k = 0\n⊢ a = 1",
"tactic": "apply (mul_eq_one_iff.1 h).1"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\n⊢ ∀ (p : Associates α), Irreducible p → k ∣ count p (factors a)",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\n⊢ ∃ d, a = d ^ k",
"tactic": "refine' is_pow_of_dvd_count ha _"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ k ∣ count p (factors a)",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\n⊢ ∀ (p : Associates α), Irreducible p → k ∣ count p (factors a)",
"tactic": "intro p hp"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ k ∣ count p (factors (a * b))",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ k ∣ count p (factors a)",
"tactic": "apply dvd_count_of_dvd_count_mul hb hp hab"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ k ∣ count p (factors (c ^ k))",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ k ∣ count p (factors (a * b))",
"tactic": "rw [h]"
},
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ c ≠ 0",
"state_before": "case neg\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ k ∣ count p (factors (c ^ k))",
"tactic": "apply dvd_count_pow _ hp"
},
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\nh : a * b = 0 ^ k\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b c : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nh : a * b = c ^ k\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\n⊢ c ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\nh : a * b = 0\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\nh : a * b = 0 ^ k\n⊢ False",
"tactic": "rw [zero_pow' _ hk0] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\na b : Associates α\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhk0 : ¬k = 0\np : Associates α\nhp : Irreducible p\nh : a * b = 0\n⊢ False",
"tactic": "cases mul_eq_zero.mp h <;> contradiction"
}
] |
[
1885,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1870,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.type_uLift
|
[
{
"state_after": "α : Type u\nβ : Type ?u.103331\nγ : Type ?u.103334\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ (type fun x y => r x.down y.down) = lift (type r)",
"state_before": "α : Type u\nβ : Type ?u.103331\nγ : Type ?u.103334\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ type (ULift.down ⁻¹'o r) = lift (type r)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.103331\nγ : Type ?u.103334\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\n⊢ (type fun x y => r x.down y.down) = lift (type r)",
"tactic": "rfl"
}
] |
[
659,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.Infinite.not_infinitesimal
|
[] |
[
486,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.monotone_upper
|
[] |
[
236,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.and_xor_distrib_left
|
[
{
"state_after": "no goals",
"state_before": "a b c : Bool\n⊢ (a && xor b c) = xor (a && b) (a && c)",
"tactic": "cases a <;> simp"
}
] |
[
294,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.replicate_left_injective
|
[] |
[
949,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
947,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.codeSupp_fix
|
[
{
"state_after": "no goals",
"state_before": "f : Code\nk : Cont'\n⊢ codeSupp (Code.fix f) k = trStmts₁ (trNormal (Code.fix f) k) ∪ codeSupp f (Cont'.fix f k)",
"tactic": "simp [codeSupp, codeSupp', contSupp, Finset.union_assoc, Finset.union_left_comm,\n Finset.union_left_idem]"
}
] |
[
1859,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1856,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
gramSchmidt_orthogonal
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\n⊢ ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a ≠ b\n⊢ ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"tactic": "clear h₀ a b"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\n⊢ ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"tactic": "intro a b h₀"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb : ι\n⊢ ∀ (a : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"tactic": "revert a"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb : ι\n⊢ ∀ (x : ι),\n (∀ (y : ι), y < x → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0) →\n ∀ (a : ι), a < x → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f x) = 0",
"tactic": "intro b ih a h₀"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n ∑ x in Iio b,\n inner (gramSchmidt 𝕜 f x) (f b) / ↑(‖gramSchmidt 𝕜 f x‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f x) =\n 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"tactic": "simp only [gramSchmidt_def 𝕜 f b, inner_sub_right, inner_sum, orthogonalProjection_singleton,\n inner_smul_right]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n inner (gramSchmidt 𝕜 f a) (f b) / ↑(‖gramSchmidt 𝕜 f a‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) =\n 0\n\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ ∀ (b_1 : ι),\n b_1 ∈ Iio b →\n b_1 ≠ a →\n inner (gramSchmidt 𝕜 f b_1) (f b) / ↑(‖gramSchmidt 𝕜 f b_1‖ ^ 2) *\n inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b_1) =\n 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n ∑ x in Iio b,\n inner (gramSchmidt 𝕜 f x) (f b) / ↑(‖gramSchmidt 𝕜 f x‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f x) =\n 0",
"tactic": "rw [Finset.sum_eq_single_of_mem a (Finset.mem_Iio.mpr h₀)]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\n⊢ inner (gramSchmidt 𝕜 f i) (f b) / ↑(‖gramSchmidt 𝕜 f i‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ ∀ (b_1 : ι),\n b_1 ∈ Iio b →\n b_1 ≠ a →\n inner (gramSchmidt 𝕜 f b_1) (f b) / ↑(‖gramSchmidt 𝕜 f b_1‖ ^ 2) *\n inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b_1) =\n 0",
"tactic": "intro i hi hia"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\n⊢ (inner (gramSchmidt 𝕜 f i) (f b) = 0 ∨ ↑(‖gramSchmidt 𝕜 f i‖ ^ 2) = 0) ∨\n inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\n⊢ inner (gramSchmidt 𝕜 f i) (f b) / ↑(‖gramSchmidt 𝕜 f i‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"tactic": "simp only [mul_eq_zero, div_eq_zero_iff, inner_self_eq_zero]"
},
{
"state_after": "case h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\n⊢ (inner (gramSchmidt 𝕜 f i) (f b) = 0 ∨ ↑(‖gramSchmidt 𝕜 f i‖ ^ 2) = 0) ∨\n inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"tactic": "right"
},
{
"state_after": "case h.inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\nhia₁ : i < a\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0\n\ncase h.inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\nhia₂ : a < i\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"state_before": "case h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"tactic": "cases' hia.lt_or_lt with hia₁ hia₂"
},
{
"state_after": "case inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0\nha : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0\n\ncase inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0\nhb : b < a\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
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"tactic": "cases' h₀.lt_or_lt with ha hb"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0\nha : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"tactic": "exact this _ _ ha"
},
{
"state_after": "case inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0\nhb : b < a\n⊢ inner (gramSchmidt 𝕜 f b) (gramSchmidt 𝕜 f a) = 0",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0\nhb : b < a\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0",
"tactic": "rw [inner_eq_zero_symm]"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f b) = 0\nhb : b < a\n⊢ inner (gramSchmidt 𝕜 f b) (gramSchmidt 𝕜 f a) = 0",
"tactic": "exact this _ _ hb"
},
{
"state_after": "case pos\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\nh : gramSchmidt 𝕜 f a = 0\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n inner (gramSchmidt 𝕜 f a) (f b) / ↑(‖gramSchmidt 𝕜 f a‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) =\n 0\n\ncase neg\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\nh : ¬gramSchmidt 𝕜 f a = 0\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n inner (gramSchmidt 𝕜 f a) (f b) / ↑(‖gramSchmidt 𝕜 f a‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) =\n 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n inner (gramSchmidt 𝕜 f a) (f b) / ↑(‖gramSchmidt 𝕜 f a‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) =\n 0",
"tactic": "by_cases h : gramSchmidt 𝕜 f a = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\nh : gramSchmidt 𝕜 f a = 0\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n inner (gramSchmidt 𝕜 f a) (f b) / ↑(‖gramSchmidt 𝕜 f a‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) =\n 0",
"tactic": "simp only [h, inner_zero_left, zero_div, MulZeroClass.zero_mul, sub_zero]"
},
{
"state_after": "case neg.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\nh : ¬gramSchmidt 𝕜 f a = 0\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) ≠ 0",
"state_before": "case neg\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\nh : ¬gramSchmidt 𝕜 f a = 0\n⊢ inner (gramSchmidt 𝕜 f a) (f b) -\n inner (gramSchmidt 𝕜 f a) (f b) / ↑(‖gramSchmidt 𝕜 f a‖ ^ 2) * inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) =\n 0",
"tactic": "rw [IsROrC.ofReal_pow, ← inner_self_eq_norm_sq_to_K, div_mul_cancel, sub_self]"
},
{
"state_after": "no goals",
"state_before": "case neg.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\nh : ¬gramSchmidt 𝕜 f a = 0\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f a) ≠ 0",
"tactic": "rwa [inner_self_ne_zero]"
},
{
"state_after": "case h.inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\nhia₁ : i < a\n⊢ inner (gramSchmidt 𝕜 f i) (gramSchmidt 𝕜 f a) = 0",
"state_before": "case h.inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\nhia₁ : i < a\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"tactic": "rw [inner_eq_zero_symm]"
},
{
"state_after": "no goals",
"state_before": "case h.inl\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\nhia₁ : i < a\n⊢ inner (gramSchmidt 𝕜 f i) (gramSchmidt 𝕜 f a) = 0",
"tactic": "exact ih a h₀ i hia₁"
},
{
"state_after": "no goals",
"state_before": "case h.inr\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\nb✝ b : ι\nih : ∀ (y : ι), y < b → ∀ (a : ι), a < y → inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0\na : ι\nh₀ : a < b\ni : ι\nhi : i ∈ Iio b\nhia : i ≠ a\nhia₂ : a < i\n⊢ inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f i) = 0",
"tactic": "exact ih i (mem_Iio.1 hi) a hia₂"
}
] |
[
114,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
|
Matrix.Represents.eq
|
[] |
[
154,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/RingTheory/AlgebraicIndependent.lean
|
AlgHom.algebraicIndependent_iff
|
[] |
[
176,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean
|
Real.hasStrictDerivAt_arcsin
|
[] |
[
57,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.Eventually.diag_of_prod
|
[
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.9533\nγ : Type ?u.9536\nδ : Type ?u.9539\nι : Sort ?u.9542\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\np : α × α → Prop\nh : ∀ᶠ (i : α × α) in f ×ˢ f, p i\nt : α → Prop\nht : ∀ᶠ (x : α) in f, t x\ns : α → Prop\nhs : ∀ᶠ (y : α) in f, s y\nhst : ∀ {x : α}, t x → ∀ {y : α}, s y → p (x, y)\n⊢ ∀ᶠ (i : α) in f, p (i, i)",
"state_before": "α : Type u_1\nβ : Type ?u.9533\nγ : Type ?u.9536\nδ : Type ?u.9539\nι : Sort ?u.9542\ns : Set α\nt : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\np : α × α → Prop\nh : ∀ᶠ (i : α × α) in f ×ˢ f, p i\n⊢ ∀ᶠ (i : α) in f, p (i, i)",
"tactic": "obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.9533\nγ : Type ?u.9536\nδ : Type ?u.9539\nι : Sort ?u.9542\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\np : α × α → Prop\nh : ∀ᶠ (i : α × α) in f ×ˢ f, p i\nt : α → Prop\nht : ∀ᶠ (x : α) in f, t x\ns : α → Prop\nhs : ∀ᶠ (y : α) in f, s y\nhst : ∀ {x : α}, t x → ∀ {y : α}, s y → p (x, y)\n⊢ ∀ᶠ (i : α) in f, p (i, i)",
"tactic": "apply (ht.and hs).mono fun x hx => hst hx.1 hx.2"
}
] |
[
195,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
src/lean/Init/SimpLemmas.lean
|
eq_true
|
[] |
[
15,
41
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
14,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
RingHom.mem_range
|
[] |
[
659,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/LinearAlgebra/Matrix/Polynomial.lean
|
Polynomial.coeff_det_X_add_C_zero
|
[
{
"state_after": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∑ b : Equiv.Perm n, coeff (↑sign b • ∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑b i) i) 0 =\n ∑ σ : Equiv.Perm n, ↑sign σ • ∏ i : n, B (↑σ i) i",
"state_before": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ coeff (det (X • Matrix.map A ↑C + Matrix.map B ↑C)) 0 = det B",
"tactic": "rw [det_apply, finset_sum_coeff, det_apply]"
},
{
"state_after": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∀ (x : Equiv.Perm n),\n x ∈ Finset.univ →\n coeff (↑sign x • ∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑x i) i) 0 = ↑sign x • ∏ i : n, B (↑x i) i",
"state_before": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∑ b : Equiv.Perm n, coeff (↑sign b • ∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑b i) i) 0 =\n ∑ σ : Equiv.Perm n, ↑sign σ • ∏ i : n, B (↑σ i) i",
"tactic": "refine' Finset.sum_congr rfl _"
},
{
"state_after": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ coeff (↑sign g • ∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g i) i) 0 = ↑sign g • ∏ i : n, B (↑g i) i",
"state_before": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ ∀ (x : Equiv.Perm n),\n x ∈ Finset.univ →\n coeff (↑sign x • ∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑x i) i) 0 = ↑sign x • ∏ i : n, B (↑x i) i",
"tactic": "rintro g -"
},
{
"state_after": "case h.e'_3.h.e'_6\nn : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ ∏ i : n, B (↑g i) i = coeff (∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g i) i) 0",
"state_before": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ coeff (↑sign g • ∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g i) i) 0 = ↑sign g • ∏ i : n, B (↑g i) i",
"tactic": "convert coeff_smul (R := α) (sign g) _ 0"
},
{
"state_after": "case h.e'_3.h.e'_6\nn : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ ∏ i : n, B (↑g i) i = ∏ i : n, coeff ((X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g i) i) 0",
"state_before": "case h.e'_3.h.e'_6\nn : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ ∏ i : n, B (↑g i) i = coeff (∏ i : n, (X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g i) i) 0",
"tactic": "rw [coeff_zero_prod]"
},
{
"state_after": "case h.e'_3.h.e'_6\nn : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ ∀ (x : n), x ∈ Finset.univ → B (↑g x) x = coeff ((X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g x) x) 0",
"state_before": "case h.e'_3.h.e'_6\nn : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ ∏ i : n, B (↑g i) i = ∏ i : n, coeff ((X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g i) i) 0",
"tactic": "refine' Finset.prod_congr rfl _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6\nn : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ ∀ (x : n), x ∈ Finset.univ → B (↑g x) x = coeff ((X • Matrix.map A ↑C + Matrix.map B ↑C) (↑g x) x) 0",
"tactic": "simp"
}
] |
[
80,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_compl_eq_image_compl
|
[] |
[
365,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_eq_ofBijective
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ Finset.card (Finite.toFinset hs) = n",
"state_before": "α : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ncard s = n",
"tactic": "rw [ncard_eq_toFinset_card _ hs]"
},
{
"state_after": "case hf\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ∀ (a : α), a ∈ Finite.toFinset hs → ∃ i h, ?f i h = a\n\ncase hf'\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ∀ (i : ℕ) (h : i < n), ?f i h ∈ Finite.toFinset hs\n\ncase f_inj\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ∀ (i j : ℕ) (hi : i < n) (hj : j < n), ?f i hi = ?f j hj → i = j\n\ncase f\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ (i : ℕ) → i < n → α",
"state_before": "α : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ Finset.card (Finite.toFinset hs) = n",
"tactic": "apply Finset.card_eq_of_bijective"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ∀ (a : α), a ∈ Finite.toFinset hs → ∃ i h, ?f i h = a\n\ncase hf'\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ∀ (i : ℕ) (h : i < n), ?f i h ∈ Finite.toFinset hs\n\ncase f_inj\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ∀ (i j : ℕ) (hi : i < n) (hj : j < n), ?f i hi = ?f j hj → i = j\n\ncase f\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ (i : ℕ) → i < n → α",
"tactic": "all_goals simpa"
},
{
"state_after": "no goals",
"state_before": "case f_inj\nα : Type u_1\nβ : Type ?u.71917\ns t : Set α\na b x y : α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j",
"tactic": "simpa"
}
] |
[
360,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Topology/Algebra/FilterBasis.lean
|
GroupFilterBasis.conj
|
[] |
[
118,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral
|
[
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ (∫⁻ (a : α), f a ∂μ) + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\n⊢ (∫⁻ (a : α), f a ∂μ) + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ",
"tactic": "rcases exists_measurable_le_lintegral_eq μ f with ⟨φ, hφm, hφ_le, hφ_eq⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\n⊢ φ x + indicator {x | φ x + ε ≤ g x} (fun x => ε) x ≤ g x",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ (∫⁻ (a : α), f a ∂μ) + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ",
"tactic": "calc\n (∫⁻ x, f x ∂μ) + ε * μ { x | f x + ε ≤ g x } = (∫⁻ x, φ x ∂μ) + ε * μ { x | f x + ε ≤ g x } :=\n by rw [hφ_eq]\n _ ≤ (∫⁻ x, φ x ∂μ) + ε * μ { x | φ x + ε ≤ g x } := by\n gcongr\n exact measure_mono fun x => (add_le_add_right (hφ_le _) _).trans\n _ = ∫⁻ x, φ x + indicator { x | φ x + ε ≤ g x } (fun _ => ε) x ∂μ := by\n rw [lintegral_add_left hφm, lintegral_indicator₀, set_lintegral_const]\n exact measurableSet_le (hφm.nullMeasurable.measurable'.add_const _) hg.nullMeasurable\n _ ≤ ∫⁻ x, g x ∂μ := lintegral_mono_ae (hle.mono fun x hx₁ => ?_)"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\n⊢ (φ x + if x ∈ {x | φ x + ε ≤ g x} then ε else 0) ≤ g x",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\n⊢ φ x + indicator {x | φ x + ε ≤ g x} (fun x => ε) x ≤ g x",
"tactic": "simp only [indicator_apply]"
},
{
"state_after": "case intro.intro.intro.inl\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\nhx₂ : x ∈ {x | φ x + ε ≤ g x}\n⊢ φ x + ε ≤ g x\n\ncase intro.intro.intro.inr\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\nhx₂ : ¬x ∈ {x | φ x + ε ≤ g x}\n⊢ φ x + 0 ≤ g x",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\n⊢ (φ x + if x ∈ {x | φ x + ε ≤ g x} then ε else 0) ≤ g x",
"tactic": "split_ifs with hx₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.inl\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\nhx₂ : x ∈ {x | φ x + ε ≤ g x}\n⊢ φ x + ε ≤ g x\n\ncase intro.intro.intro.inr\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\nx : α\nhx₁ : f x ≤ g x\nhx₂ : ¬x ∈ {x | φ x + ε ≤ g x}\n⊢ φ x + 0 ≤ g x",
"tactic": "exacts [hx₂, (add_zero _).trans_le <| (hφ_le x).trans hx₁]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ (∫⁻ (x : α), f x ∂μ) + ε * ↑↑μ {x | f x + ε ≤ g x} = (∫⁻ (x : α), φ x ∂μ) + ε * ↑↑μ {x | f x + ε ≤ g x}",
"tactic": "rw [hφ_eq]"
},
{
"state_after": "case bc.bc\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ ↑↑μ {x | f x + ε ≤ g x} ≤ ↑↑μ {x | φ x + ε ≤ g x}",
"state_before": "α : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ (∫⁻ (x : α), φ x ∂μ) + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ (∫⁻ (x : α), φ x ∂μ) + ε * ↑↑μ {x | φ x + ε ≤ g x}",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case bc.bc\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ ↑↑μ {x | f x + ε ≤ g x} ≤ ↑↑μ {x | φ x + ε ≤ g x}",
"tactic": "exact measure_mono fun x => (add_le_add_right (hφ_le _) _).trans"
},
{
"state_after": "case hs\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ NullMeasurableSet {x | φ x + ε ≤ g x}",
"state_before": "α : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ (∫⁻ (x : α), φ x ∂μ) + ε * ↑↑μ {x | φ x + ε ≤ g x} = ∫⁻ (x : α), φ x + indicator {x | φ x + ε ≤ g x} (fun x => ε) x ∂μ",
"tactic": "rw [lintegral_add_left hφm, lintegral_indicator₀, set_lintegral_const]"
},
{
"state_after": "no goals",
"state_before": "case hs\nα : Type u_1\nβ : Type ?u.953853\nγ : Type ?u.953856\nδ : Type ?u.953859\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhle : f ≤ᵐ[μ] g\nhg : AEMeasurable g\nε : ℝ≥0∞\nφ : α → ℝ≥0∞\nhφm : Measurable φ\nhφ_le : φ ≤ f\nhφ_eq : (∫⁻ (a : α), f a ∂μ) = ∫⁻ (a : α), φ a ∂μ\n⊢ NullMeasurableSet {x | φ x + ε ≤ g x}",
"tactic": "exact measurableSet_le (hφm.nullMeasurable.measurable'.add_const _) hg.nullMeasurable"
}
] |
[
822,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/Algebra/Order/Module.lean
|
smul_le_smul_iff_of_neg
|
[
{
"state_after": "k : Type u_1\nM : Type u_2\nN : Type ?u.89573\ninst✝³ : LinearOrderedField k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ -c • b ≤ -c • a ↔ b ≤ a",
"state_before": "k : Type u_1\nM : Type u_2\nN : Type ?u.89573\ninst✝³ : LinearOrderedField k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ c • a ≤ c • b ↔ b ≤ a",
"tactic": "rw [← neg_neg c, neg_smul, neg_smul (-c), neg_le_neg_iff]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nM : Type u_2\nN : Type ?u.89573\ninst✝³ : LinearOrderedField k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ -c • b ≤ -c • a ↔ b ≤ a",
"tactic": "exact smul_le_smul_iff_of_pos (neg_pos_of_neg hc)"
}
] |
[
163,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.cast_ofZNum
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\n⊢ ↑(ofZNum n) = ↑(toNat ↑n)",
"tactic": "rw [← cast_to_nat, ofZNum_toNat]"
}
] |
[
1307,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1306,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.coeff_eq_zero
|
[] |
[
101,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.zero_lt_one
|
[] |
[
1882,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1881,
11
] |
Std/Data/Int/Lemmas.lean
|
Int.lt_of_add_lt_add_left
|
[
{
"state_after": "a b c : Int\nh : a + b < a + c\nthis : -a + (a + b) < -a + (a + c)\n⊢ b < c",
"state_before": "a b c : Int\nh : a + b < a + c\n⊢ b < c",
"tactic": "have : -a + (a + b) < -a + (a + c) := Int.add_lt_add_left h _"
},
{
"state_after": "a b c : Int\nh : a + b < a + c\nthis : b < c\n⊢ b < c",
"state_before": "a b c : Int\nh : a + b < a + c\nthis : -a + (a + b) < -a + (a + c)\n⊢ b < c",
"tactic": "simp [Int.neg_add_cancel_left] at this"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh : a + b < a + c\nthis : b < c\n⊢ b < c",
"tactic": "assumption"
}
] |
[
771,
13
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
768,
11
] |
Mathlib/Data/Int/Cast/Lemmas.lean
|
toLex_intCast
|
[] |
[
409,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/Data/Int/ConditionallyCompleteOrder.lean
|
Int.csInf_of_not_bdd_below
|
[
{
"state_after": "no goals",
"state_before": "s : Set ℤ\nh : ¬BddBelow s\n⊢ ¬(Set.Nonempty s ∧ BddBelow s)",
"tactic": "simp [h]"
}
] |
[
96,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/CategoryTheory/Yoneda.lean
|
CategoryTheory.yonedaSectionsSmall_hom
|
[] |
[
416,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_hom₂
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nα : Type u_4\nβ : Type u_1\nγ : Type u_2\ninst✝² : CommMonoid α\ns✝ t : Multiset α\na : α\nm : Multiset ι\nf✝ g : ι → α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\ns : Multiset ι\nf : α → β → γ\nhf : ∀ (a b : α) (c d : β), f (a * b) (c * d) = f a c * f b d\nhf' : f 1 1 = 1\nf₁ : ι → α\nf₂ : ι → β\nl : List ι\n⊢ prod (map (fun i => f (f₁ i) (f₂ i)) (Quotient.mk (List.isSetoid ι) l)) =\n f (prod (map f₁ (Quotient.mk (List.isSetoid ι) l))) (prod (map f₂ (Quotient.mk (List.isSetoid ι) l)))",
"tactic": "simp only [l.prod_hom₂ f hf hf', quot_mk_to_coe, coe_map, coe_prod]"
}
] |
[
178,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_insert
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1731961\nγ : Type ?u.1731964\nδ : Type ?u.1731967\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSingletonClass α\na : α\ns : Set α\nh : ¬a ∈ s\nf : α → ℝ≥0∞\n⊢ Disjoint s {a}",
"state_before": "α : Type u_1\nβ : Type ?u.1731961\nγ : Type ?u.1731964\nδ : Type ?u.1731967\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSingletonClass α\na : α\ns : Set α\nh : ¬a ∈ s\nf : α → ℝ≥0∞\n⊢ (∫⁻ (x : α) in insert a s, f x ∂μ) = f a * ↑↑μ {a} + ∫⁻ (x : α) in s, f x ∂μ",
"tactic": "rw [← union_singleton, lintegral_union (measurableSet_singleton a), lintegral_singleton,\n add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1731961\nγ : Type ?u.1731964\nδ : Type ?u.1731967\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSingletonClass α\na : α\ns : Set α\nh : ¬a ∈ s\nf : α → ℝ≥0∞\n⊢ Disjoint s {a}",
"tactic": "rwa [disjoint_singleton_right]"
}
] |
[
1474,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1470,
1
] |
Std/Data/PairingHeap.lean
|
Std.PairingHeapImp.Heap.size_tail?_lt
|
[
{
"state_after": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\n⊢ Option.map (fun x => x.snd) (deleteMin le s) = some s' → size s' < size s",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\n⊢ tail? le s = some s' → size s' < size s",
"tactic": "simp only [Heap.tail?]"
},
{
"state_after": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n⊢ size s' < size s",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\n⊢ Option.map (fun x => x.snd) (deleteMin le s) = some s' → size s' < size s",
"tactic": "intro eq"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\n⊢ size s' < size s",
"tactic": "match eq₂ : s.deleteMin le, eq with\n| some (a, tl), rfl => exact size_deleteMin_lt eq₂"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\ns' s : Heap α\neq : Option.map (fun x => x.snd) (deleteMin le s) = some s'\na : α\ntl : Heap α\neq₂ : deleteMin le s = some (a, tl)\n⊢ size ((fun x => x.snd) (a, tl)) < size s",
"tactic": "exact size_deleteMin_lt eq₂"
}
] |
[
163,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
159,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.mul_equiv_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\ng f : CauSeq β abv\nhf : f ≈ 0\nthis : LimZero (f - 0)\n⊢ LimZero f",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\ng f : CauSeq β abv\nhf : f ≈ 0\nthis✝ : LimZero (f - 0)\nthis : LimZero (g * f)\n⊢ LimZero (g * f - 0)",
"tactic": "simpa"
}
] |
[
531,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
528,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.exists_update_iff
|
[
{
"state_after": "α : Sort u\nβ : α → Sort v\nα' : Sort w\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq α'\nf✝ g : (a : α) → β a\na✝ : α\nb✝ : β a✝\nf : (a : α) → β a\na : α\nb : β a\np : (a : α) → β a → Prop\n⊢ ¬(¬p a b ∧ ∀ (x : α), x ≠ a → ¬p x (f x)) ↔ p a b ∨ ∃ x x_1, p x (f x)",
"state_before": "α : Sort u\nβ : α → Sort v\nα' : Sort w\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq α'\nf✝ g : (a : α) → β a\na✝ : α\nb✝ : β a✝\nf : (a : α) → β a\na : α\nb : β a\np : (a : α) → β a → Prop\n⊢ (∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ x x_1, p x (f x)",
"tactic": "rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u\nβ : α → Sort v\nα' : Sort w\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq α'\nf✝ g : (a : α) → β a\na✝ : α\nb✝ : β a✝\nf : (a : α) → β a\na : α\nb : β a\np : (a : α) → β a → Prop\n⊢ ¬(¬p a b ∧ ∀ (x : α), x ≠ a → ¬p x (f x)) ↔ p a b ∨ ∃ x x_1, p x (f x)",
"tactic": "simp [-not_and, not_and_or]"
}
] |
[
591,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
588,
1
] |
Mathlib/Data/Semiquot.lean
|
Semiquot.ext_s
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.885\nq₁ q₂ : Semiquot α\nh : q₁.s = q₂.s\n⊢ q₁ = q₂",
"state_before": "α : Type u_1\nβ : Type ?u.885\nq₁ q₂ : Semiquot α\n⊢ q₁ = q₂ ↔ q₁.s = q₂.s",
"tactic": "refine' ⟨congr_arg _, fun h => _⟩"
},
{
"state_after": "case mk'\nα : Type u_1\nβ : Type ?u.885\nq₂ : Semiquot α\ns✝ : Set α\nv₁ : Trunc ↑s✝\nh : { s := s✝, val := v₁ }.s = q₂.s\n⊢ { s := s✝, val := v₁ } = q₂",
"state_before": "α : Type u_1\nβ : Type ?u.885\nq₁ q₂ : Semiquot α\nh : q₁.s = q₂.s\n⊢ q₁ = q₂",
"tactic": "cases' q₁ with _ v₁"
},
{
"state_after": "case mk'.mk'\nα : Type u_1\nβ : Type ?u.885\ns✝¹ : Set α\nv₁ : Trunc ↑s✝¹\ns✝ : Set α\nv₂ : Trunc ↑s✝\nh : { s := s✝¹, val := v₁ }.s = { s := s✝, val := v₂ }.s\n⊢ { s := s✝¹, val := v₁ } = { s := s✝, val := v₂ }",
"state_before": "case mk'\nα : Type u_1\nβ : Type ?u.885\nq₂ : Semiquot α\ns✝ : Set α\nv₁ : Trunc ↑s✝\nh : { s := s✝, val := v₁ }.s = q₂.s\n⊢ { s := s✝, val := v₁ } = q₂",
"tactic": "cases' q₂ with _ v₂"
},
{
"state_after": "case mk'.mk'.h.e_3\nα : Type u_1\nβ : Type ?u.885\ns✝¹ : Set α\nv₁ : Trunc ↑s✝¹\ns✝ : Set α\nv₂ : Trunc ↑s✝\nh : { s := s✝¹, val := v₁ }.s = { s := s✝, val := v₂ }.s\n⊢ HEq v₁ v₂",
"state_before": "case mk'.mk'\nα : Type u_1\nβ : Type ?u.885\ns✝¹ : Set α\nv₁ : Trunc ↑s✝¹\ns✝ : Set α\nv₂ : Trunc ↑s✝\nh : { s := s✝¹, val := v₁ }.s = { s := s✝, val := v₂ }.s\n⊢ { s := s✝¹, val := v₁ } = { s := s✝, val := v₂ }",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case mk'.mk'.h.e_3\nα : Type u_1\nβ : Type ?u.885\ns✝¹ : Set α\nv₁ : Trunc ↑s✝¹\ns✝ : Set α\nv₂ : Trunc ↑s✝\nh : { s := s✝¹, val := v₁ }.s = { s := s✝, val := v₂ }.s\n⊢ HEq v₁ v₂",
"tactic": "exact Subsingleton.helim (congrArg Trunc (congrArg Set.Elem h)) v₁ v₂"
}
] |
[
53,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
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