file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
sequence | commit
stringclasses 4
values | url
stringclasses 4
values | start
sequence |
|---|---|---|---|---|---|---|
Mathlib/Computability/PartrecCode.lean | Nat.Partrec.Code.fixed_point₂ | [
{
"state_after": "no goals",
"state_before": "f : Code → ℕ →. ℕ\nhf : Partrec₂ f\ncf : Code\nef : eval cf = fun n => Part.bind ↑(decode n) fun a => Part.map encode ((fun p => f p.fst p.snd) a)\nc : Code\ne : eval (curry cf (encode c)) = eval c\nn : ℕ\n⊢ eval c n = f c n",
"tactic": "simp [e.symm, ef, Part.map_id']"
}
] | [
1192,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1189,
1
] |
Mathlib/Order/Height.lean | Set.chainHeight_insert_of_forall_gt | [
{
"state_after": "α : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ chainHeight (insert a s) + 0 = chainHeight s + 1",
"state_before": "α : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ chainHeight (insert a s) = chainHeight s + 1",
"tactic": "rw [← add_zero (insert a s).chainHeight]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ chainHeight (insert a s) + ↑0 = chainHeight s + ↑1",
"state_before": "α : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ chainHeight (insert a s) + 0 = chainHeight s + 1",
"tactic": "change (insert a s).chainHeight + (0 : ℕ) = s.chainHeight + (1 : ℕ)"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ ∀ (l : List α), l ∈ subchain (insert a s) → ∃ l', l' ∈ subchain s ∧ length l + 0 ≤ length l' + 1\n\ncase a\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ ∀ (l : List α), l ∈ subchain s → ∃ l', l' ∈ subchain (insert a s) ∧ length l + 1 ≤ length l' + 0",
"state_before": "α : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ chainHeight (insert a s) + ↑0 = chainHeight s + ↑1",
"tactic": "apply le_antisymm <;> rw [chainHeight_add_le_chainHeight_add]"
},
{
"state_after": "case a.nil\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nh : [] ∈ subchain (insert a s)\n⊢ ∃ l', l' ∈ subchain s ∧ length [] + 0 ≤ length l' + 1\n\ncase a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\n⊢ ∃ l', l' ∈ subchain s ∧ length (y :: ys) + 0 ≤ length l' + 1",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ ∀ (l : List α), l ∈ subchain (insert a s) → ∃ l', l' ∈ subchain s ∧ length l + 0 ≤ length l' + 1",
"tactic": "rintro (_ | ⟨y, ys⟩) h"
},
{
"state_after": "no goals",
"state_before": "case a.nil\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nh : [] ∈ subchain (insert a s)\n⊢ ∃ l', l' ∈ subchain s ∧ length [] + 0 ≤ length l' + 1",
"tactic": "exact ⟨[], nil_mem_subchain _, zero_le _⟩"
},
{
"state_after": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\nh' : y ∈ insert a s ∧ ys ∈ subchain (insert a s) ∧ ∀ (b : α), b ∈ head? ys → y < b\n⊢ ∃ l', l' ∈ subchain s ∧ length (y :: ys) + 0 ≤ length l' + 1",
"state_before": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\n⊢ ∃ l', l' ∈ subchain s ∧ length (y :: ys) + 0 ≤ length l' + 1",
"tactic": "have h' := cons_mem_subchain_iff.mp h"
},
{
"state_after": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\nh' : y ∈ insert a s ∧ ys ∈ subchain (insert a s) ∧ ∀ (b : α), b ∈ head? ys → y < b\ni : α\nhi : i ∈ ys\n⊢ i ∈ s",
"state_before": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\nh' : y ∈ insert a s ∧ ys ∈ subchain (insert a s) ∧ ∀ (b : α), b ∈ head? ys → y < b\n⊢ ∃ l', l' ∈ subchain s ∧ length (y :: ys) + 0 ≤ length l' + 1",
"tactic": "refine' ⟨ys, ⟨h'.2.1.1, fun i hi ↦ _⟩, by simp⟩"
},
{
"state_after": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\nh' : y ∈ insert a s ∧ ys ∈ subchain (insert a s) ∧ ∀ (b : α), b ∈ head? ys → y < b\ni : α\nhi : i ∈ ys\n⊢ ¬i = a",
"state_before": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\nh' : y ∈ insert a s ∧ ys ∈ subchain (insert a s) ∧ ∀ (b : α), b ∈ head? ys → y < b\ni : α\nhi : i ∈ ys\n⊢ i ∈ s",
"tactic": "apply (h'.2.1.2 i hi).resolve_left"
},
{
"state_after": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh' : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\n⊢ False",
"state_before": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\nh' : y ∈ insert a s ∧ ys ∈ subchain (insert a s) ∧ ∀ (b : α), b ∈ head? ys → y < b\ni : α\nhi : i ∈ ys\n⊢ ¬i = a",
"tactic": "rintro rfl"
},
{
"state_after": "case a.cons.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh' : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\na✝ : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : ∀ (a' : α), a' ∈ ys → y < a'\n⊢ False",
"state_before": "case a.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh' : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\n⊢ False",
"tactic": "cases' chain'_iff_pairwise.mp h.1 with _ _ hy"
},
{
"state_after": "case a.cons.cons.inl\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh'✝ : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\na✝ : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : ∀ (a' : α), a' ∈ ys → y < a'\nh' : y = i\n⊢ False\n\ncase a.cons.cons.inr\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh'✝ : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\na✝ : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : ∀ (a' : α), a' ∈ ys → y < a'\nh' : y ∈ s\n⊢ False",
"state_before": "case a.cons.cons\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh' : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\na✝ : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : ∀ (a' : α), a' ∈ ys → y < a'\n⊢ False",
"tactic": "cases' h'.1 with h' h'"
},
{
"state_after": "no goals",
"state_before": "case a.cons.cons.inl\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh'✝ : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\na✝ : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : ∀ (a' : α), a' ∈ ys → y < a'\nh' : y = i\n⊢ False\n\ncase a.cons.cons.inr\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\ny : α\nys : List α\ni : α\nhi : i ∈ ys\nhx : ∀ (b : α), b ∈ s → i < b\nh : y :: ys ∈ subchain (insert i s)\nh'✝ : y ∈ insert i s ∧ ys ∈ subchain (insert i s) ∧ ∀ (b : α), b ∈ head? ys → y < b\na✝ : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : ∀ (a' : α), a' ∈ ys → y < a'\nh' : y ∈ s\n⊢ False",
"tactic": "exacts [(hy _ hi).ne h', not_le_of_gt (hy _ hi) (hx _ h').le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\ny : α\nys : List α\nh : y :: ys ∈ subchain (insert a s)\nh' : y ∈ insert a s ∧ ys ∈ subchain (insert a s) ∧ ∀ (b : α), b ∈ head? ys → y < b\n⊢ length (y :: ys) + 0 ≤ length ys + 1",
"tactic": "simp"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ ∃ l', l' ∈ subchain (insert a s) ∧ length l + 1 ≤ length l' + 0",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\n⊢ ∀ (l : List α), l ∈ subchain s → ∃ l', l' ∈ subchain (insert a s) ∧ length l + 1 ≤ length l' + 0",
"tactic": "intro l hl"
},
{
"state_after": "case a.refine'_1\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ Chain' (fun x x_1 => x < x_1) (a :: l)\n\ncase a.refine'_2\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ ∀ (i : α), i ∈ a :: l → i ∈ insert a s",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ ∃ l', l' ∈ subchain (insert a s) ∧ length l + 1 ≤ length l' + 0",
"tactic": "refine' ⟨a::l, ⟨_, _⟩, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ length l + 1 ≤ length (a :: l) + 0",
"tactic": "simp"
},
{
"state_after": "case a.refine'_1\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ (∀ (y : α), y ∈ head? l → a < y) ∧ Chain' (fun x x_1 => x < x_1) l",
"state_before": "case a.refine'_1\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ Chain' (fun x x_1 => x < x_1) (a :: l)",
"tactic": "rw [chain'_cons']"
},
{
"state_after": "no goals",
"state_before": "case a.refine'_1\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ (∀ (y : α), y ∈ head? l → a < y) ∧ Chain' (fun x x_1 => x < x_1) l",
"tactic": "exact ⟨fun y hy ↦ hx _ (hl.2 _ (mem_of_mem_head? hy)), hl.1⟩"
},
{
"state_after": "case a.refine'_2.head\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ a ∈ insert a s\n\ncase a.refine'_2.tail\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\nx : α\na✝ : Mem x l\n⊢ x ∈ insert a s",
"state_before": "case a.refine'_2\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ ∀ (i : α), i ∈ a :: l → i ∈ insert a s",
"tactic": "rintro x (_ | _)"
},
{
"state_after": "no goals",
"state_before": "case a.refine'_2.head\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\n⊢ a ∈ insert a s\n\ncase a.refine'_2.tail\nα : Type u_1\nβ : Type ?u.34460\ns t : Set α\ninst✝ : Preorder α\na : α\nhx : ∀ (b : α), b ∈ s → a < b\nl : List α\nhl : l ∈ subchain s\nx : α\na✝ : Mem x l\n⊢ x ∈ insert a s",
"tactic": "exacts [Or.inl (Set.mem_singleton a), Or.inr (hl.2 x ‹_›)]"
}
] | [
309,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
284,
1
] |
Mathlib/Data/List/Nodup.lean | List.Nodup.pairwise_coe | [
{
"state_after": "case nil\nα : Type u\nβ : Type v\nl l₁ l₂ : List α\nr : α → α → Prop\na b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l\nhl : Nodup []\n⊢ Set.Pairwise {a | a ∈ []} r ↔ Pairwise r []\n\ncase cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l✝\na : α\nl : List α\nih : Nodup l → (Set.Pairwise {a | a ∈ l} r ↔ Pairwise r l)\nhl : Nodup (a :: l)\n⊢ Set.Pairwise {a_1 | a_1 ∈ a :: l} r ↔ Pairwise r (a :: l)",
"state_before": "α : Type u\nβ : Type v\nl l₁ l₂ : List α\nr : α → α → Prop\na b : α\ninst✝ : IsSymm α r\nhl : Nodup l\n⊢ Set.Pairwise {a | a ∈ l} r ↔ Pairwise r l",
"tactic": "induction' l with a l ih"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l✝\na : α\nl : List α\nih : Nodup l → (Set.Pairwise {a | a ∈ l} r ↔ Pairwise r l)\nhl : ¬a ∈ l ∧ Nodup l\n⊢ Set.Pairwise {a_1 | a_1 ∈ a :: l} r ↔ Pairwise r (a :: l)",
"state_before": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l✝\na : α\nl : List α\nih : Nodup l → (Set.Pairwise {a | a ∈ l} r ↔ Pairwise r l)\nhl : Nodup (a :: l)\n⊢ Set.Pairwise {a_1 | a_1 ∈ a :: l} r ↔ Pairwise r (a :: l)",
"tactic": "rw [List.nodup_cons] at hl"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l✝\na : α\nl : List α\nih : Nodup l → (Set.Pairwise {a | a ∈ l} r ↔ Pairwise r l)\nhl : ¬a ∈ l ∧ Nodup l\nthis : ∀ (b : α), b ∈ l → (¬a = b → r a b ↔ r a b)\n⊢ Set.Pairwise {a_1 | a_1 ∈ a :: l} r ↔ Pairwise r (a :: l)",
"state_before": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l✝\na : α\nl : List α\nih : Nodup l → (Set.Pairwise {a | a ∈ l} r ↔ Pairwise r l)\nhl : ¬a ∈ l ∧ Nodup l\n⊢ Set.Pairwise {a_1 | a_1 ∈ a :: l} r ↔ Pairwise r (a :: l)",
"tactic": "have : ∀ b ∈ l, ¬a = b → r a b ↔ r a b := fun b hb =>\n imp_iff_right (ne_of_mem_of_not_mem hb hl.1).symm"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l✝\na : α\nl : List α\nih : Nodup l → (Set.Pairwise {a | a ∈ l} r ↔ Pairwise r l)\nhl : ¬a ∈ l ∧ Nodup l\nthis : ∀ (b : α), b ∈ l → (¬a = b → r a b ↔ r a b)\n⊢ Set.Pairwise {a_1 | a_1 ∈ a :: l} r ↔ Pairwise r (a :: l)",
"tactic": "simp [Set.setOf_or, Set.pairwise_insert_of_symmetric (@symm_of _ r _), ih hl.2, and_comm,\n forall₂_congr this]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nl l₁ l₂ : List α\nr : α → α → Prop\na b : α\ninst✝ : IsSymm α r\nhl✝ : Nodup l\nhl : Nodup []\n⊢ Set.Pairwise {a | a ∈ []} r ↔ Pairwise r []",
"tactic": "simp"
}
] | [
449,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
441,
1
] |
Mathlib/Logic/Function/Conjugate.lean | Function.Semiconj.inverses_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1160\nf fab : α → β\nfbc : β → γ\nga ga' : α → α\ngb gb' : β → β\ngc gc' : γ → γ\nh : Semiconj f ga gb\nha : RightInverse ga' ga\nhb : LeftInverse gb' gb\nx : α\n⊢ f (ga' x) = gb' (f x)",
"tactic": "rw [← hb (f (ga' x)), ← h.eq, ha x]"
}
] | [
71,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
] |
Mathlib/Data/Set/Lattice.lean | iSup_iUnion | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.304567\nι : Sort u_3\nι' : Sort ?u.304573\nι₂ : Sort ?u.304576\nκ : ι → Sort ?u.304581\nκ₁ : ι → Sort ?u.304586\nκ₂ : ι → Sort ?u.304591\nκ' : ι' → Sort ?u.304596\ninst✝ : CompleteLattice β\ns : ι → Set α\nf : α → β\n⊢ (⨆ (a : α) (_ : a ∈ ⋃ (i : ι), s i), f a) = ⨆ (j : α) (i : ι) (_ : j ∈ s i), f j",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.304567\nι : Sort u_3\nι' : Sort ?u.304573\nι₂ : Sort ?u.304576\nκ : ι → Sort ?u.304581\nκ₁ : ι → Sort ?u.304586\nκ₂ : ι → Sort ?u.304591\nκ' : ι' → Sort ?u.304596\ninst✝ : CompleteLattice β\ns : ι → Set α\nf : α → β\n⊢ (⨆ (a : α) (_ : a ∈ ⋃ (i : ι), s i), f a) = ⨆ (i : ι) (a : α) (_ : a ∈ s i), f a",
"tactic": "rw [iSup_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.304567\nι : Sort u_3\nι' : Sort ?u.304573\nι₂ : Sort ?u.304576\nκ : ι → Sort ?u.304581\nκ₁ : ι → Sort ?u.304586\nκ₂ : ι → Sort ?u.304591\nκ' : ι' → Sort ?u.304596\ninst✝ : CompleteLattice β\ns : ι → Set α\nf : α → β\n⊢ (⨆ (a : α) (_ : a ∈ ⋃ (i : ι), s i), f a) = ⨆ (j : α) (i : ι) (_ : j ∈ s i), f j",
"tactic": "simp_rw [mem_iUnion, iSup_exists]"
}
] | [
2239,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2237,
1
] |
Mathlib/Data/Multiset/Nodup.lean | Multiset.nodup_singleton | [] | [
54,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Star.lean | HasFDerivAtFilter.star | [] | [
46,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
44,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean | AddMonoidAlgebra.mul_single_apply_of_not_exists_add | [] | [
1637,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1635,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineEquiv.coeFn_inj | [] | [
149,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean | MulChar.map_nonunit | [] | [
165,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
] |
Mathlib/Data/Fintype/Card.lean | Finite.of_injective | [
{
"state_after": "case intro\nα✝ : Type ?u.13191\nβ✝ : Type ?u.13194\nγ : Type ?u.13197\nα : Sort u_1\nβ : Sort u_2\ninst✝ : Finite β\nf : α → β\nH : Injective f\nval✝ : Fintype (PLift β)\n⊢ Finite α",
"state_before": "α✝ : Type ?u.13191\nβ✝ : Type ?u.13194\nγ : Type ?u.13197\nα : Sort u_1\nβ : Sort u_2\ninst✝ : Finite β\nf : α → β\nH : Injective f\n⊢ Finite α",
"tactic": "cases nonempty_fintype (PLift β)"
},
{
"state_after": "case intro\nα✝ : Type ?u.13191\nβ✝ : Type ?u.13194\nγ : Type ?u.13197\nα : Sort u_1\nβ : Sort u_2\ninst✝ : Finite β\nf : α → β\nH : Injective (↑Equiv.plift.symm ∘ f ∘ ↑Equiv.plift)\nval✝ : Fintype (PLift β)\n⊢ Finite α",
"state_before": "case intro\nα✝ : Type ?u.13191\nβ✝ : Type ?u.13194\nγ : Type ?u.13197\nα : Sort u_1\nβ : Sort u_2\ninst✝ : Finite β\nf : α → β\nH : Injective f\nval✝ : Fintype (PLift β)\n⊢ Finite α",
"tactic": "rw [← Equiv.injective_comp Equiv.plift f, ← Equiv.comp_injective _ Equiv.plift.symm] at H"
},
{
"state_after": "case intro\nα✝ : Type ?u.13191\nβ✝ : Type ?u.13194\nγ : Type ?u.13197\nα : Sort u_1\nβ : Sort u_2\ninst✝ : Finite β\nf : α → β\nH : Injective (↑Equiv.plift.symm ∘ f ∘ ↑Equiv.plift)\nval✝ : Fintype (PLift β)\nthis : Fintype (PLift α)\n⊢ Finite α",
"state_before": "case intro\nα✝ : Type ?u.13191\nβ✝ : Type ?u.13194\nγ : Type ?u.13197\nα : Sort u_1\nβ : Sort u_2\ninst✝ : Finite β\nf : α → β\nH : Injective (↑Equiv.plift.symm ∘ f ∘ ↑Equiv.plift)\nval✝ : Fintype (PLift β)\n⊢ Finite α",
"tactic": "haveI := Fintype.ofInjective _ H"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type ?u.13191\nβ✝ : Type ?u.13194\nγ : Type ?u.13197\nα : Sort u_1\nβ : Sort u_2\ninst✝ : Finite β\nf : α → β\nH : Injective (↑Equiv.plift.symm ∘ f ∘ ↑Equiv.plift)\nval✝ : Fintype (PLift β)\nthis : Fintype (PLift α)\n⊢ Finite α",
"tactic": "exact Finite.of_equiv _ Equiv.plift"
}
] | [
440,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
436,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean | Fin.prod_Ioi_zero | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.63689\nβ : Type ?u.63692\nM : Type u_1\ninst✝ : CommMonoid M\nn : ℕ\nv : Fin (Nat.succ n) → M\n⊢ ∏ i in Ioi 0, v i = ∏ j : Fin n, v (succ j)",
"tactic": "rw [Ioi_zero_eq_map, Finset.prod_map, RelEmbedding.coe_toEmbedding, val_succEmbedding]"
}
] | [
177,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.cycleFactorsFinset_noncommProd | [] | [
1404,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1400,
1
] |
Mathlib/Order/SupIndep.lean | Finset.SupIndep.subset | [] | [
74,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasSum.tendsto_sum_nat | [] | [
289,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.sUnion_diff_singleton_empty | [] | [
1150,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1149,
1
] |
Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean | CategoryTheory.AdditiveFunctor.ofExact_map | [] | [
325,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Order/CompleteLattice.lean | Monotone.iSup_nat_add | [] | [
1638,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1637,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.ofLists_moveLeft' | [] | [
195,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/Computability/TuringMachine.lean | Turing.TM2.stmts_supportsStmt | [
{
"state_after": "K : Type u_1\ninst✝¹ : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\nσ : Type u_4\ninst✝ : Inhabited Λ\nM : Λ → Stmt₂\nS : Finset Λ\nq : Stmt₂\nss : Supports M S\n⊢ ∀ (x : Λ), x ∈ S → q ∈ stmts₁ (M x) → SupportsStmt S q",
"state_before": "K : Type u_1\ninst✝¹ : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\nσ : Type u_4\ninst✝ : Inhabited Λ\nM : Λ → Stmt₂\nS : Finset Λ\nq : Stmt₂\nss : Supports M S\n⊢ some q ∈ stmts M S → SupportsStmt S q",
"tactic": "simp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq,\n forall_eq', exists_imp, and_imp]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\nσ : Type u_4\ninst✝ : Inhabited Λ\nM : Λ → Stmt₂\nS : Finset Λ\nq : Stmt₂\nss : Supports M S\n⊢ ∀ (x : Λ), x ∈ S → q ∈ stmts₁ (M x) → SupportsStmt S q",
"tactic": "exact fun l ls h ↦ stmts₁_supportsStmt_mono h (ss.2 _ ls)"
}
] | [
2249,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2245,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean | Polynomial.degree_pos_of_ne_zero_of_nonunit | [
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : DivisionRing R\np q : R[X]\nhp0 : ↑C (coeff p 0) ≠ 0\nhp : ¬IsUnit (↑C (coeff p 0))\nh : 0 ≥ degree p\n⊢ False",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : DivisionRing R\np q : R[X]\nhp0 : p ≠ 0\nhp : ¬IsUnit p\nh : 0 ≥ degree p\n⊢ False",
"tactic": "rw [eq_C_of_degree_le_zero h] at hp0 hp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : DivisionRing R\np q : R[X]\nhp0 : ↑C (coeff p 0) ≠ 0\nhp : ¬IsUnit (↑C (coeff p 0))\nh : 0 ≥ degree p\n⊢ False",
"tactic": "exact hp (IsUnit.map C (IsUnit.mk0 (coeff p 0) (mt C_inj.2 (by simpa using hp0))))"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : DivisionRing R\np q : R[X]\nhp0 : ↑C (coeff p 0) ≠ 0\nhp : ¬IsUnit (↑C (coeff p 0))\nh : 0 ≥ degree p\n⊢ ¬↑C (coeff p 0) = ↑C 0",
"tactic": "simpa using hp0"
}
] | [
124,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/Topology/Category/CompHaus/Basic.lean | CompHaus.isClosedMap | [] | [
116,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/SetTheory/Lists.lean | Lists'.subset_nil | [
{
"state_after": "α : Type u_1\nl : Lists' α true\n⊢ ofList (toList l) ⊆ nil → ofList (toList l) = nil",
"state_before": "α : Type u_1\nl : Lists' α true\n⊢ l ⊆ nil → l = nil",
"tactic": "rw [← of_toList l]"
},
{
"state_after": "case nil\nα : Type u_1\nl : Lists' α true\nh : ofList [] ⊆ nil\n⊢ ofList [] = nil\n\ncase cons\nα : Type u_1\nl : Lists' α true\nhead✝ : Lists α\ntail✝ : List (Lists α)\ntail_ih✝ : ofList tail✝ ⊆ nil → ofList tail✝ = nil\nh : ofList (head✝ :: tail✝) ⊆ nil\n⊢ ofList (head✝ :: tail✝) = nil",
"state_before": "α : Type u_1\nl : Lists' α true\n⊢ ofList (toList l) ⊆ nil → ofList (toList l) = nil",
"tactic": "induction toList l <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl : Lists' α true\nh : ofList [] ⊆ nil\n⊢ ofList [] = nil",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nl : Lists' α true\nhead✝ : Lists α\ntail✝ : List (Lists α)\ntail_ih✝ : ofList tail✝ ⊆ nil → ofList tail✝ = nil\nh : ofList (head✝ :: tail✝) ⊆ nil\n⊢ ofList (head✝ :: tail✝) = nil",
"tactic": "rcases cons_subset.1 h with ⟨⟨_, ⟨⟩, _⟩, _⟩"
}
] | [
193,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
189,
1
] |
Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | SimplicialObject.Splitting.IndexSet.eqId_iff_eq | [
{
"state_after": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ EqId A → A.fst = Δ\n\ncase mpr\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ A.fst = Δ → EqId A",
"state_before": "C : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ EqId A ↔ A.fst = Δ",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : EqId A\n⊢ A.fst = Δ",
"state_before": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ EqId A → A.fst = Δ",
"tactic": "intro h"
},
{
"state_after": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A = id Δ\n⊢ A.fst = Δ",
"state_before": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : EqId A\n⊢ A.fst = Δ",
"tactic": "dsimp at h"
},
{
"state_after": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A = id Δ\n⊢ (id Δ).fst = Δ",
"state_before": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A = id Δ\n⊢ A.fst = Δ",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case mp\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A = id Δ\n⊢ (id Δ).fst = Δ",
"tactic": "rfl"
},
{
"state_after": "case mpr\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A.fst = Δ\n⊢ EqId A",
"state_before": "case mpr\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ A.fst = Δ → EqId A",
"tactic": "intro h"
},
{
"state_after": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nΔ fst✝ : SimplexCategoryᵒᵖ\nf : Δ.unop ⟶ fst✝.unop\nhf : Epi f\nh : { fst := fst✝, snd := { val := f, property := hf } }.fst = Δ\n⊢ EqId { fst := fst✝, snd := { val := f, property := hf } }",
"state_before": "case mpr\nC : Type ?u.4324\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A.fst = Δ\n⊢ EqId A",
"tactic": "rcases A with ⟨_, ⟨f, hf⟩⟩"
},
{
"state_after": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nΔ fst✝ : SimplexCategoryᵒᵖ\nf : Δ.unop ⟶ fst✝.unop\nhf : Epi f\nh : fst✝ = Δ\n⊢ EqId { fst := fst✝, snd := { val := f, property := hf } }",
"state_before": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nΔ fst✝ : SimplexCategoryᵒᵖ\nf : Δ.unop ⟶ fst✝.unop\nhf : Epi f\nh : { fst := fst✝, snd := { val := f, property := hf } }.fst = Δ\n⊢ EqId { fst := fst✝, snd := { val := f, property := hf } }",
"tactic": "simp only at h"
},
{
"state_after": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf : Epi f\n⊢ EqId { fst := fst✝, snd := { val := f, property := hf } }",
"state_before": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nΔ fst✝ : SimplexCategoryᵒᵖ\nf : Δ.unop ⟶ fst✝.unop\nhf : Epi f\nh : fst✝ = Δ\n⊢ EqId { fst := fst✝, snd := { val := f, property := hf } }",
"tactic": "subst h"
},
{
"state_after": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf : Epi f\n⊢ e { fst := fst✝, snd := { val := f, property := hf } } ≫\n eqToHom (_ : { fst := fst✝, snd := { val := f, property := hf } }.fst.unop = (id fst✝).fst.unop) =\n e (id fst✝)",
"state_before": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf : Epi f\n⊢ EqId { fst := fst✝, snd := { val := f, property := hf } }",
"tactic": "refine' ext _ _ rfl _"
},
{
"state_after": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf this : Epi f\n⊢ e { fst := fst✝, snd := { val := f, property := hf } } ≫\n eqToHom (_ : { fst := fst✝, snd := { val := f, property := hf } }.fst.unop = (id fst✝).fst.unop) =\n e (id fst✝)",
"state_before": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf : Epi f\n⊢ e { fst := fst✝, snd := { val := f, property := hf } } ≫\n eqToHom (_ : { fst := fst✝, snd := { val := f, property := hf } }.fst.unop = (id fst✝).fst.unop) =\n e (id fst✝)",
"tactic": "haveI := hf"
},
{
"state_after": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf this : Epi f\n⊢ e { fst := fst✝, snd := { val := f, property := hf } } = e (id fst✝)",
"state_before": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf this : Epi f\n⊢ e { fst := fst✝, snd := { val := f, property := hf } } ≫\n eqToHom (_ : { fst := fst✝, snd := { val := f, property := hf } }.fst.unop = (id fst✝).fst.unop) =\n e (id fst✝)",
"tactic": "simp only [eqToHom_refl, comp_id]"
},
{
"state_after": "no goals",
"state_before": "case mpr.mk.mk\nC : Type ?u.4324\ninst✝ : Category C\nfst✝ : SimplexCategoryᵒᵖ\nf : fst✝.unop ⟶ fst✝.unop\nhf this : Epi f\n⊢ e { fst := fst✝, snd := { val := f, property := hf } } = e (id fst✝)",
"tactic": "exact eq_id_of_epi f"
}
] | [
144,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | CategoryTheory.reflects_epi_of_reflectsColimit | [
{
"state_after": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : ReflectsColimit (span f f) F\ninst✝ : Epi (F.map f)\nthis : IsColimit (PushoutCocone.mk (𝟙 (F.obj Y)) (𝟙 (F.obj Y)) (_ : F.map f ≫ 𝟙 (F.obj Y) = F.map f ≫ 𝟙 (F.obj Y)))\n⊢ Epi f",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : ReflectsColimit (span f f) F\ninst✝ : Epi (F.map f)\n⊢ Epi f",
"tactic": "have := PushoutCocone.isColimitMkIdId (F.map f)"
},
{
"state_after": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : ReflectsColimit (span f f) F\ninst✝ : Epi (F.map f)\nthis : IsColimit (PushoutCocone.mk (F.map (𝟙 Y)) (F.map (𝟙 Y)) (_ : F.map f ≫ F.map (𝟙 Y) = F.map f ≫ F.map (𝟙 Y)))\n⊢ Epi f",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : ReflectsColimit (span f f) F\ninst✝ : Epi (F.map f)\nthis : IsColimit (PushoutCocone.mk (𝟙 (F.obj Y)) (𝟙 (F.obj Y)) (_ : F.map f ≫ 𝟙 (F.obj Y) = F.map f ≫ 𝟙 (F.obj Y)))\n⊢ Epi f",
"tactic": "simp_rw [← F.map_id] at this"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝³ : Category C\ninst✝² : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝¹ : ReflectsColimit (span f f) F\ninst✝ : Epi (F.map f)\nthis : IsColimit (PushoutCocone.mk (F.map (𝟙 Y)) (F.map (𝟙 Y)) (_ : F.map f ≫ F.map (𝟙 Y) = F.map f ≫ F.map (𝟙 Y)))\n⊢ Epi f",
"tactic": "apply\n PushoutCocone.epi_of_isColimitMkIdId _\n (isColimitOfIsColimitPushoutCoconeMap F _ this)"
}
] | [
81,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.DominatedFinMeasAdditive.smul | [
{
"state_after": "α : Type u_3\nE : Type ?u.107083\nF : Type ?u.107086\nF' : Type ?u.107089\nG : Type ?u.107092\n𝕜 : Type u_1\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝² : SeminormedAddCommGroup β\nT T' : Set α → β\nC C' : ℝ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 β\nhT : DominatedFinMeasAdditive μ T C\nc : 𝕜\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ ‖(fun s => c • T s) s‖ ≤ ‖c‖ * C * ENNReal.toReal (↑↑μ s)",
"state_before": "α : Type u_3\nE : Type ?u.107083\nF : Type ?u.107086\nF' : Type ?u.107089\nG : Type ?u.107092\n𝕜 : Type u_1\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝² : SeminormedAddCommGroup β\nT T' : Set α → β\nC C' : ℝ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 β\nhT : DominatedFinMeasAdditive μ T C\nc : 𝕜\n⊢ DominatedFinMeasAdditive μ (fun s => c • T s) (‖c‖ * C)",
"tactic": "refine' ⟨hT.1.smul c, fun s hs hμs => _⟩"
},
{
"state_after": "α : Type u_3\nE : Type ?u.107083\nF : Type ?u.107086\nF' : Type ?u.107089\nG : Type ?u.107092\n𝕜 : Type u_1\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝² : SeminormedAddCommGroup β\nT T' : Set α → β\nC C' : ℝ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 β\nhT : DominatedFinMeasAdditive μ T C\nc : 𝕜\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ ‖c • T s‖ ≤ ‖c‖ * C * ENNReal.toReal (↑↑μ s)",
"state_before": "α : Type u_3\nE : Type ?u.107083\nF : Type ?u.107086\nF' : Type ?u.107089\nG : Type ?u.107092\n𝕜 : Type u_1\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝² : SeminormedAddCommGroup β\nT T' : Set α → β\nC C' : ℝ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 β\nhT : DominatedFinMeasAdditive μ T C\nc : 𝕜\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ ‖(fun s => c • T s) s‖ ≤ ‖c‖ * C * ENNReal.toReal (↑↑μ s)",
"tactic": "dsimp only"
},
{
"state_after": "α : Type u_3\nE : Type ?u.107083\nF : Type ?u.107086\nF' : Type ?u.107089\nG : Type ?u.107092\n𝕜 : Type u_1\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝² : SeminormedAddCommGroup β\nT T' : Set α → β\nC C' : ℝ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 β\nhT : DominatedFinMeasAdditive μ T C\nc : 𝕜\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ ‖c‖ * ‖T s‖ ≤ ‖c‖ * (C * ENNReal.toReal (↑↑μ s))",
"state_before": "α : Type u_3\nE : Type ?u.107083\nF : Type ?u.107086\nF' : Type ?u.107089\nG : Type ?u.107092\n𝕜 : Type u_1\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝² : SeminormedAddCommGroup β\nT T' : Set α → β\nC C' : ℝ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 β\nhT : DominatedFinMeasAdditive μ T C\nc : 𝕜\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ ‖c • T s‖ ≤ ‖c‖ * C * ENNReal.toReal (↑↑μ s)",
"tactic": "rw [norm_smul, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type ?u.107083\nF : Type ?u.107086\nF' : Type ?u.107089\nG : Type ?u.107092\n𝕜 : Type u_1\np : ℝ≥0∞\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝² : SeminormedAddCommGroup β\nT T' : Set α → β\nC C' : ℝ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 β\nhT : DominatedFinMeasAdditive μ T C\nc : 𝕜\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ ‖c‖ * ‖T s‖ ≤ ‖c‖ * (C * ENNReal.toReal (↑↑μ s))",
"tactic": "exact mul_le_mul le_rfl (hT.2 s hs hμs) (norm_nonneg _) (norm_nonneg _)"
}
] | [
232,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
] |
Mathlib/Computability/Encoding.lean | Computability.Encoding.encode_injective | [
{
"state_after": "α : Type u\ne : Encoding α\nx✝¹ x✝ : α\nh : encode e x✝¹ = encode e x✝\n⊢ some x✝¹ = some x✝",
"state_before": "α : Type u\ne : Encoding α\n⊢ Function.Injective e.encode",
"tactic": "refine' fun _ _ h => Option.some_injective _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ne : Encoding α\nx✝¹ x✝ : α\nh : encode e x✝¹ = encode e x✝\n⊢ some x✝¹ = some x✝",
"tactic": "rw [← e.decode_encode, ← e.decode_encode, h]"
}
] | [
47,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
] |
Mathlib/MeasureTheory/Measure/AEDisjoint.lean | MeasureTheory.AEDisjoint.eq | [] | [
54,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
11
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.infDist_empty | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.57575\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ infDist x ∅ = 0",
"tactic": "simp [infDist]"
}
] | [
485,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
485,
1
] |
Mathlib/Topology/Order/Basic.lean | Ico_mem_nhdsWithin_Iic | [] | [
574,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
573,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | Associates.dvd_of_mem_factors' | [
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : { val := p, property := hp } ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∣ Associates.mk a",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : { val := p, property := hp } ∈ factors' a\n⊢ p ∣ Associates.mk a",
"tactic": "haveI := Classical.decEq (Associates α)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : { val := p, property := hp } ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ factors (Associates.mk a)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : { val := p, property := hp } ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∣ Associates.mk a",
"tactic": "apply dvd_of_mem_factors (hp := hp)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : { val := p, property := hp } ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ ↑(factors' a)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : { val := p, property := hp } ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ factors (Associates.mk a)",
"tactic": "rw [factors_mk _ hz]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : { val := p, property := hp } ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ ↑(factors' a)",
"tactic": "apply mem_factorSet_some.2 h_mem"
}
] | [
1601,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1596,
1
] |
Mathlib/Algebra/Regular/Basic.lean | MulLECancellable.isLeftRegular | [] | [
81,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
11
] |
Mathlib/Probability/Kernel/Basic.lean | ProbabilityTheory.kernel.restrict_apply' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.1443714\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : { x // x ∈ kernel α β }\ns t : Set β\nκ : { x // x ∈ kernel α β }\nhs : MeasurableSet s\na : α\nht : MeasurableSet t\n⊢ ↑↑(↑(kernel.restrict κ hs) a) t = ↑↑(↑κ a) (t ∩ s)",
"tactic": "rw [restrict_apply κ hs a, Measure.restrict_apply ht]"
}
] | [
500,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
498,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | InnerProductSpace.Core.inner_mul_symm_re_eq_norm | [
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.606441\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑re (inner y x * ↑(starRingEnd 𝕜) (inner y x)) = ‖inner y x * ↑(starRingEnd 𝕜) (inner y x)‖",
"state_before": "𝕜 : Type u_1\nE : Type ?u.606441\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑re (inner x y * inner y x) = ‖inner x y * inner y x‖",
"tactic": "rw [← inner_conj_symm, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.606441\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑re (inner y x * ↑(starRingEnd 𝕜) (inner y x)) = ‖inner y x * ↑(starRingEnd 𝕜) (inner y x)‖",
"tactic": "exact re_eq_norm_of_mul_conj (inner y x)"
}
] | [
302,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean | Trivialization.map_proj_nhds | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.29960\nB : Type u_2\nF : Type u_3\nE : B → Type ?u.29971\nZ : Type u_1\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\nproj : Z → B\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace (TotalSpace E)\ne : Trivialization F proj\nx : Z\nex : x ∈ e.source\n⊢ map proj (𝓝 x) = 𝓝 (proj x)",
"tactic": "rw [← e.coe_fst ex, ← map_congr (e.coe_fst_eventuallyEq_proj ex), ← map_map, ← e.coe_coe,\n e.map_nhds_eq ex, map_fst_nhds]"
}
] | [
441,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
439,
1
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearEquiv.coe_toContinuousLinearEquiv_symm' | [] | [
387,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
385,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometryEquiv.antilipschitz | [] | [
664,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
663,
11
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean | Matrix.conjTranspose_zpow | [
{
"state_after": "no goals",
"state_before": "n' : Type u_2\ninst✝³ : DecidableEq n'\ninst✝² : Fintype n'\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : StarRing R\nA : M\nn : ℕ\n⊢ (A ^ ↑n)ᴴ = Aᴴ ^ ↑n",
"tactic": "rw [zpow_ofNat, zpow_ofNat, conjTranspose_pow]"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_2\ninst✝³ : DecidableEq n'\ninst✝² : Fintype n'\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : StarRing R\nA : M\nn : ℕ\n⊢ (A ^ -[n+1])ᴴ = Aᴴ ^ -[n+1]",
"tactic": "rw [zpow_negSucc, zpow_negSucc, conjTranspose_nonsing_inv, conjTranspose_pow]"
}
] | [
346,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
344,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean | ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm | [] | [
309,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
] |
Mathlib/CategoryTheory/Monoidal/End.lean | CategoryTheory.μ_naturalityₗ | [
{
"state_after": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm n m' : M\nf : m ⟶ m'\nX : C\n⊢ (F.obj n).map ((F.map f).app X) ≫ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m' n).app X =\n (F.map (𝟙 n)).app ((F.obj m).obj X) ≫\n (F.obj n).map ((F.map f).app X) ≫ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m' n).app X",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm n m' : M\nf : m ⟶ m'\nX : C\n⊢ (F.obj n).map ((F.map f).app X) ≫ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m' n).app X =\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m n).app X ≫ (F.map (f ⊗ 𝟙 n)).app X",
"tactic": "rw [← μ_naturality₂ F f (𝟙 n) X]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nm n m' : M\nf : m ⟶ m'\nX : C\n⊢ (F.obj n).map ((F.map f).app X) ≫ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m' n).app X =\n (F.map (𝟙 n)).app ((F.obj m).obj X) ≫\n (F.obj n).map ((F.map f).app X) ≫ (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor m' n).app X",
"tactic": "simp"
}
] | [
158,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean | Nat.dvd_left_injective | [] | [
253,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
252,
1
] |
Mathlib/Data/Set/Sups.lean | Set.infs_right_comm | [] | [
362,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/Data/Sum/Interval.lean | Sum.Ico_inr_inl | [] | [
184,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
183,
1
] |
Mathlib/Analysis/Convex/Topology.lean | Convex.combo_self_interior_mem_interior | [] | [
179,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean | IsCoprime.mul_add_right_left | [
{
"state_after": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime (x + z * y) y",
"state_before": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime (z * y + x) y",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime (x + z * y) y",
"tactic": "exact h.add_mul_right_left z"
}
] | [
310,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
308,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | volume_image_subtype_coe | [] | [
4266,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
4264,
1
] |
Mathlib/Analysis/Normed/Field/InfiniteSum.lean | summable_mul_of_summable_norm | [] | [
50,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean | VectorBundleCore.inCoordinates_eq | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.536498\nB : Type u_4\nF : Type u_5\nE : B → Type ?u.536509\ninst✝²² : NontriviallyNormedField R\ninst✝²¹ : (x : B) → AddCommMonoid (E x)\ninst✝²⁰ : (x : B) → Module R (E x)\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace R F\ninst✝¹⁷ : TopologicalSpace B\ninst✝¹⁶ : (x : B) → TopologicalSpace (E x)\n𝕜₁ : Type u_3\n𝕜₂ : Type u_6\ninst✝¹⁵ : NontriviallyNormedField 𝕜₁\ninst✝¹⁴ : NontriviallyNormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nB' : Type u_7\ninst✝¹³ : TopologicalSpace B'\ninst✝¹² : NormedSpace 𝕜₁ F\ninst✝¹¹ : (x : B) → Module 𝕜₁ (E x)\ninst✝¹⁰ : TopologicalSpace (TotalSpace E)\nF' : Type u_8\ninst✝⁹ : NormedAddCommGroup F'\ninst✝⁸ : NormedSpace 𝕜₂ F'\nE' : B' → Type ?u.537479\ninst✝⁷ : (x : B') → AddCommMonoid (E' x)\ninst✝⁶ : (x : B') → Module 𝕜₂ (E' x)\ninst✝⁵ : TopologicalSpace (TotalSpace E')\ninst✝⁴ : FiberBundle F E\ninst✝³ : VectorBundle 𝕜₁ F E\ninst✝² : (x : B') → TopologicalSpace (E' x)\ninst✝¹ : FiberBundle F' E'\ninst✝ : VectorBundle 𝕜₂ F' E'\nι : Type u_1\nι' : Type u_2\nZ : VectorBundleCore 𝕜₁ B F ι\nZ' : VectorBundleCore 𝕜₂ B' F' ι'\nx₀ x : B\ny₀ y : B'\nϕ : F →SL[σ] F'\nhx : x ∈ VectorBundleCore.baseSet Z (VectorBundleCore.indexAt Z x₀)\nhy : y ∈ VectorBundleCore.baseSet Z' (VectorBundleCore.indexAt Z' y₀)\n⊢ inCoordinates F (VectorBundleCore.Fiber Z) F' (VectorBundleCore.Fiber Z') x₀ x y₀ y ϕ =\n comp (VectorBundleCore.coordChange Z' (VectorBundleCore.indexAt Z' y) (VectorBundleCore.indexAt Z' y₀) y)\n (comp ϕ (VectorBundleCore.coordChange Z (VectorBundleCore.indexAt Z x₀) (VectorBundleCore.indexAt Z x) x))",
"tactic": "simp_rw [inCoordinates, Z'.trivializationAt_continuousLinearMapAt hy,\n Z.trivializationAt_symmL hx]"
}
] | [
1059,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1052,
11
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.eapprox_comp | [] | [
929,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
927,
1
] |
Mathlib/Data/Sign.lean | SignType.not_lt_neg_one | [] | [
216,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.union_le_union_right | [] | [
1711,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1710,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.IsTrail.of_cons | [
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\nh : Adj G u v\np : Walk G v w\n⊢ IsTrail (cons h p) → IsTrail p",
"tactic": "simp [isTrail_def]"
}
] | [
923,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
922,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | CategoryTheory.Limits.IsZero.of_mono | [
{
"state_after": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\ni : IsZero Y\nhf : f = 0\n⊢ IsZero X",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\ni : IsZero Y\n⊢ IsZero X",
"tactic": "have hf := i.eq_zero_of_tgt f"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ni : IsZero Y\ninst✝ : Mono 0\n⊢ IsZero X",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : Mono f\ni : IsZero Y\nhf : f = 0\n⊢ IsZero X",
"tactic": "subst hf"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ni : IsZero Y\ninst✝ : Mono 0\n⊢ IsZero X",
"tactic": "exact IsZero.of_mono_zero X Y"
}
] | [
238,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/RingTheory/Polynomial/Chebyshev.lean | Polynomial.Chebyshev.add_one_mul_T_eq_poly_in_U | [
{
"state_after": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\nh :\n ↑derivative (T R (n + 2)) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)\n⊢ (↑n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ (↑n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"tactic": "have h : derivative (T R (n + 2)) = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) +\n 2 * X * U R n - (1 - X ^ 2) * derivative (U R n) := by\n conv_lhs => rw [T_eq_X_mul_T_sub_pol_U]\n simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,\n one_mul, T_derivative_eq_U]\n rw [T_eq_U_sub_X_mul_U, C_eq_nat_cast]\n ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\nh :\n ↑derivative (T R (n + 2)) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)\n⊢ (↑n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"tactic": "calc\n ((n : R[X]) + 1) * T R (n + 1) =\n ((n : R[X]) + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((n + 1 : R[X]) * U R n) -\n (X * U R n + T R (n + 1)) :=\n by ring\n _ = derivative (T R (n + 2)) - X * derivative (T R (n + 1)) - U R (n + 1) := by\n rw [← U_eq_X_mul_U_add_T, ← T_derivative_eq_U, ← Nat.cast_one, ← Nat.cast_add, Nat.cast_one, ←\n T_derivative_eq_U (n + 1)]\n _ = U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n -\n (1 - X ^ 2) * derivative (U R n) -\n X * derivative (T R (n + 1)) -\n U R (n + 1) :=\n by rw [h]\n _ = X * U R n - (1 - X ^ 2) * derivative (U R n) := by ring"
},
{
"state_after": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ ↑derivative (X * T R (n + 1) - (1 - X ^ 2) * U R n) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ ↑derivative (T R (n + 2)) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"tactic": "conv_lhs => rw [T_eq_X_mul_T_sub_pol_U]"
},
{
"state_after": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ T R (n + 1) + X * ((↑n + 1) * U R n) - ((0 - ↑C ↑2 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * ↑derivative (U R n)) =\n U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ ↑derivative (X * T R (n + 1) - (1 - X ^ 2) * U R n) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"tactic": "simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,\n one_mul, T_derivative_eq_U]"
},
{
"state_after": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) -\n ((0 - ↑2 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * ↑derivative (U R n)) =\n U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ T R (n + 1) + X * ((↑n + 1) * U R n) - ((0 - ↑C ↑2 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * ↑derivative (U R n)) =\n U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"tactic": "rw [T_eq_U_sub_X_mul_U, C_eq_nat_cast]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) -\n ((0 - ↑2 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * ↑derivative (U R n)) =\n U R (n + 1) - X * U R n + X * ((↑n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\nh :\n ↑derivative (T R (n + 2)) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)\n⊢ (↑n + 1) * T R (n + 1) = (↑n + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((↑n + 1) * U R n) - (X * U R n + T R (n + 1))",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\nh :\n ↑derivative (T R (n + 2)) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)\n⊢ (↑n + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((↑n + 1) * U R n) - (X * U R n + T R (n + 1)) =\n ↑derivative (T R (n + 2)) - X * ↑derivative (T R (n + 1)) - U R (n + 1)",
"tactic": "rw [← U_eq_X_mul_U_add_T, ← T_derivative_eq_U, ← Nat.cast_one, ← Nat.cast_add, Nat.cast_one, ←\n T_derivative_eq_U (n + 1)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\nh :\n ↑derivative (T R (n + 2)) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)\n⊢ ↑derivative (T R (n + 2)) - X * ↑derivative (T R (n + 1)) - U R (n + 1) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n) -\n X * ↑derivative (T R (n + 1)) -\n U R (n + 1)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.79032\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\nh :\n ↑derivative (T R (n + 2)) =\n U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n)\n⊢ U R (n + 1) - X * U R n + X * ↑derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * ↑derivative (U R n) -\n X * ↑derivative (T R (n + 1)) -\n U R (n + 1) =\n X * U R n - (1 - X ^ 2) * ↑derivative (U R n)",
"tactic": "ring"
}
] | [
236,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
] |
Mathlib/Order/RelIso/Basic.lean | RelIso.cast_symm | [] | [
758,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
756,
11
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | csInf_univ | [] | [
1029,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1028,
1
] |
Mathlib/RingTheory/Polynomial/Pochhammer.lean | pochhammer_map | [
{
"state_after": "case zero\nS : Type u\ninst✝¹ : Semiring S\nT : Type v\ninst✝ : Semiring T\nf : S →+* T\n⊢ map f (pochhammer S Nat.zero) = pochhammer T Nat.zero\n\ncase succ\nS : Type u\ninst✝¹ : Semiring S\nT : Type v\ninst✝ : Semiring T\nf : S →+* T\nn : ℕ\nih : map f (pochhammer S n) = pochhammer T n\n⊢ map f (pochhammer S (Nat.succ n)) = pochhammer T (Nat.succ n)",
"state_before": "S : Type u\ninst✝¹ : Semiring S\nT : Type v\ninst✝ : Semiring T\nf : S →+* T\nn : ℕ\n⊢ map f (pochhammer S n) = pochhammer T n",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nS : Type u\ninst✝¹ : Semiring S\nT : Type v\ninst✝ : Semiring T\nf : S →+* T\n⊢ map f (pochhammer S Nat.zero) = pochhammer T Nat.zero",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nS : Type u\ninst✝¹ : Semiring S\nT : Type v\ninst✝ : Semiring T\nf : S →+* T\nn : ℕ\nih : map f (pochhammer S n) = pochhammer T n\n⊢ map f (pochhammer S (Nat.succ n)) = pochhammer T (Nat.succ n)",
"tactic": "simp [ih, pochhammer_succ_left, map_comp]"
}
] | [
73,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/NumberTheory/Multiplicity.lean | Int.sq_mod_four_eq_one_of_odd | [
{
"state_after": "R : Type ?u.849581\nn : ℕ\nx : ℤ\nhx : Odd x\n⊢ x ^ 2 % 4 = 1",
"state_before": "R : Type ?u.849581\nn : ℕ\nx : ℤ\n⊢ Odd x → x ^ 2 % 4 = 1",
"tactic": "intro hx"
},
{
"state_after": "R : Type ?u.849581\nn : ℕ\nx : ℤ\nhx : ∃ k, x = 2 * k + 1\n⊢ x ^ 2 % 4 = 1",
"state_before": "R : Type ?u.849581\nn : ℕ\nx : ℤ\nhx : Odd x\n⊢ x ^ 2 % 4 = 1",
"tactic": "unfold Odd at hx"
},
{
"state_after": "case intro\nR : Type ?u.849581\nn : ℕ\nw✝ : ℤ\n⊢ (2 * w✝ + 1) ^ 2 % 4 = 1",
"state_before": "R : Type ?u.849581\nn : ℕ\nx : ℤ\nhx : ∃ k, x = 2 * k + 1\n⊢ x ^ 2 % 4 = 1",
"tactic": "rcases hx with ⟨_, rfl⟩"
},
{
"state_after": "case intro\nR : Type ?u.849581\nn : ℕ\nw✝ : ℤ\n⊢ (1 + w✝ * 4 + w✝ ^ 2 * 4) % 4 = 1",
"state_before": "case intro\nR : Type ?u.849581\nn : ℕ\nw✝ : ℤ\n⊢ (2 * w✝ + 1) ^ 2 % 4 = 1",
"tactic": "ring_nf"
},
{
"state_after": "case intro\nR : Type ?u.849581\nn : ℕ\nw✝ : ℤ\n⊢ 1 % 4 = 1",
"state_before": "case intro\nR : Type ?u.849581\nn : ℕ\nw✝ : ℤ\n⊢ (1 + w✝ * 4 + w✝ ^ 2 * 4) % 4 = 1",
"tactic": "rw [add_assoc, ← add_mul, Int.add_mul_emod_self]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type ?u.849581\nn : ℕ\nw✝ : ℤ\n⊢ 1 % 4 = 1",
"tactic": "norm_num"
}
] | [
275,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
269,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.le_toReal_sub | [
{
"state_after": "case intro\nα : Type ?u.791480\nβ : Type ?u.791483\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nb : ℝ≥0\n⊢ ENNReal.toReal a - ENNReal.toReal ↑b ≤ ENNReal.toReal (a - ↑b)",
"state_before": "α : Type ?u.791480\nβ : Type ?u.791483\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nhb : b ≠ ⊤\n⊢ ENNReal.toReal a - ENNReal.toReal b ≤ ENNReal.toReal (a - b)",
"tactic": "lift b to ℝ≥0 using hb"
},
{
"state_after": "case intro.top\nα : Type ?u.791480\nβ : Type ?u.791483\na b✝ c d : ℝ≥0∞\nr p q b : ℝ≥0\n⊢ ENNReal.toReal ⊤ - ENNReal.toReal ↑b ≤ ENNReal.toReal (⊤ - ↑b)\n\ncase intro.coe\nα : Type ?u.791480\nβ : Type ?u.791483\na b✝ c d : ℝ≥0∞\nr p q b x✝ : ℝ≥0\n⊢ ENNReal.toReal ↑x✝ - ENNReal.toReal ↑b ≤ ENNReal.toReal (↑x✝ - ↑b)",
"state_before": "case intro\nα : Type ?u.791480\nβ : Type ?u.791483\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nb : ℝ≥0\n⊢ ENNReal.toReal a - ENNReal.toReal ↑b ≤ ENNReal.toReal (a - ↑b)",
"tactic": "induction a using recTopCoe"
},
{
"state_after": "no goals",
"state_before": "case intro.top\nα : Type ?u.791480\nβ : Type ?u.791483\na b✝ c d : ℝ≥0∞\nr p q b : ℝ≥0\n⊢ ENNReal.toReal ⊤ - ENNReal.toReal ↑b ≤ ENNReal.toReal (⊤ - ↑b)",
"tactic": "simp"
},
{
"state_after": "case intro.coe\nα : Type ?u.791480\nβ : Type ?u.791483\na b✝ c d : ℝ≥0∞\nr p q b x✝ : ℝ≥0\n⊢ ↑x✝ - ↑b ≤ max (↑x✝ - ↑b) 0",
"state_before": "case intro.coe\nα : Type ?u.791480\nβ : Type ?u.791483\na b✝ c d : ℝ≥0∞\nr p q b x✝ : ℝ≥0\n⊢ ENNReal.toReal ↑x✝ - ENNReal.toReal ↑b ≤ ENNReal.toReal (↑x✝ - ↑b)",
"tactic": "simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal]"
},
{
"state_after": "no goals",
"state_before": "case intro.coe\nα : Type ?u.791480\nβ : Type ?u.791483\na b✝ c d : ℝ≥0∞\nr p q b x✝ : ℝ≥0\n⊢ ↑x✝ - ↑b ≤ max (↑x✝ - ↑b) 0",
"tactic": "exact le_max_left _ _"
}
] | [
1963,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1958,
1
] |
Mathlib/Data/Set/Prod.lean | Set.prod_mk_mem_set_prod_eq | [] | [
65,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
64,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean | LinearPMap.mem_domain_iff | [
{
"state_after": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nx : E\nh : x ∈ f.domain\n⊢ ∃ y, (x, y) ∈ graph f\n\ncase mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nx : E\nh : ∃ y, (x, y) ∈ graph f\n⊢ x ∈ f.domain",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nx : E\n⊢ x ∈ f.domain ↔ ∃ y, (x, y) ∈ graph f",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nh : (x, y) ∈ graph f\n⊢ x ∈ f.domain",
"state_before": "case mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nx : E\nh : ∃ y, (x, y) ∈ graph f\n⊢ x ∈ f.domain",
"tactic": "cases' h with y h"
},
{
"state_after": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nh : ∃ y_1, ↑y_1 = (x, y).fst ∧ ↑f y_1 = (x, y).snd\n⊢ x ∈ f.domain",
"state_before": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nh : (x, y) ∈ graph f\n⊢ x ∈ f.domain",
"tactic": "rw [mem_graph_iff] at h"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nx' : { x // x ∈ f.domain }\nh : ↑x' = (x, y).fst ∧ ↑f x' = (x, y).snd\n⊢ x ∈ f.domain",
"state_before": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nh : ∃ y_1, ↑y_1 = (x, y).fst ∧ ↑f y_1 = (x, y).snd\n⊢ x ∈ f.domain",
"tactic": "cases' h with x' h"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nx' : { x // x ∈ f.domain }\nh : ↑x' = x ∧ ↑f x' = y\n⊢ x ∈ f.domain",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nx' : { x // x ∈ f.domain }\nh : ↑x' = (x, y).fst ∧ ↑f x' = (x, y).snd\n⊢ x ∈ f.domain",
"tactic": "simp only at h"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nx' : { x // x ∈ f.domain }\nh : ↑x' = x ∧ ↑f x' = y\n⊢ ↑x' ∈ f.domain",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nx' : { x // x ∈ f.domain }\nh : ↑x' = x ∧ ↑f x' = y\n⊢ x ∈ f.domain",
"tactic": "rw [← h.1]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\nx : E\ny : F\nx' : { x // x ∈ f.domain }\nh : ↑x' = x ∧ ↑f x' = y\n⊢ ↑x' ∈ f.domain",
"tactic": "simp"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nx : E\nh : x ∈ f.domain\n⊢ (x, ↑f { val := x, property := h }) ∈ graph f",
"state_before": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nx : E\nh : x ∈ f.domain\n⊢ ∃ y, (x, y) ∈ graph f",
"tactic": "use f ⟨x, h⟩"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.569553\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.570069\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny : M\nf : E →ₗ.[R] F\nx : E\nh : x ∈ f.domain\n⊢ (x, ↑f { val := x, property := h }) ∈ graph f",
"tactic": "exact f.mem_graph ⟨x, h⟩"
}
] | [
840,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
831,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.mem_closure_iff | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\n⊢ 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\n⊢ 0 * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "rw [zero_mul q]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\n⊢ 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "apply AddSubgroup.zero_mem _"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\np₁ p₂ : R\nihp₁ : p₁ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nihp₂ : p₂ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ p₁ * q + p₂ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\np₁ p₂ : R\nihp₁ : p₁ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nihp₂ : p₂ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ (p₁ + p₂) * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "rw [add_mul p₁ p₂ q]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\np₁ p₂ : R\nihp₁ : p₁ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nihp₂ : p₂ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ p₁ * q + p₂ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "apply AddSubgroup.add_mem _ ihp₁ ihp₂"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx✝ y : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\nx : R\nhx : x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : -x * q = -(x * q)\n⊢ -x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx✝ y : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\nx : R\nhx : x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ -x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "have f : -x * q = -(x * q) := by simp"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx✝ y : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\nx : R\nhx : x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : -x * q = -(x * q)\n⊢ -(x * q) ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx✝ y : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\nx : R\nhx : x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : -x * q = -(x * q)\n⊢ -x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "rw [f]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx✝ y : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\nx : R\nhx : x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : -x * q = -(x * q)\n⊢ -(x * q) ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "apply AddSubgroup.neg_mem _ hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx✝ y : R\nhx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq : R\nhq : q ∈ ↑(Submonoid.closure s)\nx : R\nhx : x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ -x * q = -(x * q)",
"tactic": "simp"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x * 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "rw [mul_zero x]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "apply AddSubgroup.zero_mem _"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq₁ q₂ : R\nihq₁ : x * q₁ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nihq₂ : x * q₂ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x * q₁ + x * q₂ ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq₁ q₂ : R\nihq₁ : x * q₁ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nihq₂ : x * q₂ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x * (q₁ + q₂) ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "rw [mul_add x q₁ q₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nq₁ q₂ : R\nihq₁ : x * q₁ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nihq₂ : x * q₂ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x * q₁ + x * q₂ ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "apply AddSubgroup.add_mem _ ihq₁ ihq₂"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nz : R\nhz : x * z ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : x * -z = -(x * z)\n⊢ x * -z ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nz : R\nhz : x * z ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x * -z ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "have f : x * -z = -(x * z) := by simp"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nz : R\nhz : x * z ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : x * -z = -(x * z)\n⊢ -(x * z) ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nz : R\nhz : x * z ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : x * -z = -(x * z)\n⊢ x * -z ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "rw [f]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nz : R\nhz : x * z ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nf : x * -z = -(x * z)\n⊢ -(x * z) ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"tactic": "apply AddSubgroup.neg_mem _ hz"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh : x✝ ∈ closure s\nx y : R\nhx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nhy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)\nz : R\nhz : x * z ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ x * -z = -(x * z)",
"tactic": "simp"
}
] | [
982,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
955,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.measurable | [] | [
216,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
11
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | NNReal.rpow_lt_rpow_of_exponent_gt | [] | [
180,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.L1.setToL1_eq_setToL1SCLM | [] | [
1033,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1030,
1
] |
Mathlib/Data/Finmap.lean | Finmap.mem_erase | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na a' : α\ns✝ : Finmap β\ns : AList β\n⊢ a' ∈ erase a ⟦s⟧ ↔ a' ≠ a ∧ a' ∈ ⟦s⟧",
"tactic": "simp"
}
] | [
439,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
438,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean | MeasureTheory.ae_le_trim_of_stronglyMeasurable | [
{
"state_after": "α : Type ?u.1731454\nE : Type ?u.1731457\nF : Type ?u.1731460\n𝕜 : Type ?u.1731463\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nH : Type ?u.1734134\nβ : Type u_2\nγ : Type u_1\ninst✝⁴ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\ninst✝³ : LinearOrder γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderClosedTopology γ\ninst✝ : PseudoMetrizableSpace γ\nhm : m ≤ m0\nf g : β → γ\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nhfg : f ≤ᵐ[μ] g\n⊢ MeasurableSet {a | ¬f a ≤ g a}",
"state_before": "α : Type ?u.1731454\nE : Type ?u.1731457\nF : Type ?u.1731460\n𝕜 : Type ?u.1731463\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nH : Type ?u.1734134\nβ : Type u_2\nγ : Type u_1\ninst✝⁴ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\ninst✝³ : LinearOrder γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderClosedTopology γ\ninst✝ : PseudoMetrizableSpace γ\nhm : m ≤ m0\nf g : β → γ\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nhfg : f ≤ᵐ[μ] g\n⊢ f ≤ᵐ[Measure.trim μ hm] g",
"tactic": "rwa [EventuallyLE, @ae_iff _ m, trim_measurableSet_eq hm _]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1731454\nE : Type ?u.1731457\nF : Type ?u.1731460\n𝕜 : Type ?u.1731463\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : SMulCommClass ℝ 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : CompleteSpace F\nH : Type ?u.1734134\nβ : Type u_2\nγ : Type u_1\ninst✝⁴ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\ninst✝³ : LinearOrder γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderClosedTopology γ\ninst✝ : PseudoMetrizableSpace γ\nhm : m ≤ m0\nf g : β → γ\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nhfg : f ≤ᵐ[μ] g\n⊢ MeasurableSet {a | ¬f a ≤ g a}",
"tactic": "exact (hf.measurableSet_le hg).compl"
}
] | [
1815,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1810,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean | LocalizedModule.divBy_mul_by | [
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ ↑(divBy s) (↑(↑(algebraMap R (Module.End R (LocalizedModule S M))) ↑s) (Quotient.mk (r.setoid S M) (m, t))) =\n Quotient.mk (r.setoid S M) (m, t)",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\n⊢ ∀ (a : M × { x // x ∈ S }),\n ↑(divBy s) (↑(↑(algebraMap R (Module.End R (LocalizedModule S M))) ↑s) (Quotient.mk (r.setoid S M) a)) =\n Quotient.mk (r.setoid S M) a",
"tactic": "intro ⟨m, t⟩"
},
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ liftOn (↑s • Quotient.mk (r.setoid S M) (m, t)) (fun p => mk p.fst (s * p.snd))\n (_ :\n ∀ (x x_1 : M × { x // x ∈ S }),\n x ≈ x_1 → (fun p => mk p.fst (s * p.snd)) x = (fun p => mk p.fst (s * p.snd)) x_1) =\n Quotient.mk (r.setoid S M) (m, t)",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ ↑(divBy s) (↑(↑(algebraMap R (Module.End R (LocalizedModule S M))) ↑s) (Quotient.mk (r.setoid S M) (m, t))) =\n Quotient.mk (r.setoid S M) (m, t)",
"tactic": "simp only [Module.algebraMap_end_apply, smul'_mk, divBy_apply]"
},
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ mk ((↑(algebraMap R R) ↑s, 1).fst • (m, t).fst, (↑(algebraMap R R) ↑s, 1).snd * (m, t).snd).fst\n (s * ((↑(algebraMap R R) ↑s, 1).fst • (m, t).fst, (↑(algebraMap R R) ↑s, 1).snd * (m, t).snd).snd) =\n Quotient.mk (r.setoid S M) (m, t)",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ liftOn (↑s • Quotient.mk (r.setoid S M) (m, t)) (fun p => mk p.fst (s * p.snd))\n (_ :\n ∀ (x x_1 : M × { x // x ∈ S }),\n x ≈ x_1 → (fun p => mk p.fst (s * p.snd)) x = (fun p => mk p.fst (s * p.snd)) x_1) =\n Quotient.mk (r.setoid S M) (m, t)",
"tactic": "erw [LocalizedModule.liftOn_mk]"
},
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ mk (↑(algebraMap R R) ↑s • m) (s * t) = Quotient.mk (r.setoid S M) (m, t)",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ mk ((↑(algebraMap R R) ↑s, 1).fst • (m, t).fst, (↑(algebraMap R R) ↑s, 1).snd * (m, t).snd).fst\n (s * ((↑(algebraMap R R) ↑s, 1).fst • (m, t).fst, (↑(algebraMap R R) ↑s, 1).snd * (m, t).snd).snd) =\n Quotient.mk (r.setoid S M) (m, t)",
"tactic": "simp only [one_mul]"
},
{
"state_after": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ mk (s • m) (s * t) = mk m t",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ mk (↑(algebraMap R R) ↑s • m) (s * t) = Quotient.mk (r.setoid S M) (m, t)",
"tactic": "change mk (s • m) (s * t) = mk m t"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\np : LocalizedModule S M\nm : M\nt : { x // x ∈ S }\n⊢ mk (s • m) (s * t) = mk m t",
"tactic": "rw [mk_cancel_common_left s t]"
}
] | [
519,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
510,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.Lf.not_ge | [] | [
418,
5
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
417,
1
] |
Mathlib/Algebra/Lie/Submodule.lean | LieIdeal.comap_map_eq | [
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑(map f I) = ↑f '' ↑I\n⊢ comap f (map f I) = I ⊔ LieHom.ker f",
"tactic": "rw [← LieSubmodule.coe_toSubmodule_eq_iff, comap_coeSubmodule, I.map_coeSubmodule f h,\n LieSubmodule.sup_coe_toSubmodule, f.ker_coeSubmodule, Submodule.comap_map_eq]"
}
] | [
1098,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1096,
1
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean | Basis.coe_constrL | [] | [
450,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
449,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean | Real.logb_lt_logb_iff | [
{
"state_after": "b x y : ℝ\nhb : 1 < b\nhx : 0 < x\nhy : 0 < y\n⊢ log x < log y ↔ x < y",
"state_before": "b x y : ℝ\nhb : 1 < b\nhx : 0 < x\nhy : 0 < y\n⊢ logb b x < logb b y ↔ x < y",
"tactic": "rw [logb, logb, div_lt_div_right (log_pos hb)]"
},
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nhb : 1 < b\nhx : 0 < x\nhy : 0 < y\n⊢ log x < log y ↔ x < y",
"tactic": "exact log_lt_log_iff hx hy"
}
] | [
160,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | MeasureTheory.Measure.add_haar_ball_center | [
{
"state_after": "no goals",
"state_before": "E✝ : Type ?u.1965249\ninst✝¹² : NormedAddCommGroup E✝\ninst✝¹¹ : NormedSpace ℝ E✝\ninst✝¹⁰ : MeasurableSpace E✝\ninst✝⁹ : BorelSpace E✝\ninst✝⁸ : FiniteDimensional ℝ E✝\nμ✝ : Measure E✝\ninst✝⁷ : IsAddHaarMeasure μ✝\nF : Type ?u.1965918\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\ns : Set E✝\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nx : E\nr : ℝ\nthis : ball 0 r = (fun x x_1 => x + x_1) x ⁻¹' ball x r\n⊢ ↑↑μ (ball x r) = ↑↑μ (ball 0 r)",
"tactic": "rw [this, measure_preimage_add]"
},
{
"state_after": "no goals",
"state_before": "E✝ : Type ?u.1965249\ninst✝¹² : NormedAddCommGroup E✝\ninst✝¹¹ : NormedSpace ℝ E✝\ninst✝¹⁰ : MeasurableSpace E✝\ninst✝⁹ : BorelSpace E✝\ninst✝⁸ : FiniteDimensional ℝ E✝\nμ✝ : Measure E✝\ninst✝⁷ : IsAddHaarMeasure μ✝\nF : Type ?u.1965918\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : CompleteSpace F\ns : Set E✝\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nx : E\nr : ℝ\n⊢ ball 0 r = (fun x x_1 => x + x_1) x ⁻¹' ball x r",
"tactic": "simp [preimage_add_ball]"
}
] | [
406,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
403,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Metric.ball_subset_interior_closedBall | [] | [
1894,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1893,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | min_div_div_right | [] | [
605,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
604,
1
] |
Mathlib/GroupTheory/Nilpotent.lean | comap_upperCentralSeries_quotient_center | [
{
"state_after": "case zero\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) Nat.zero) = upperCentralSeries G (Nat.succ Nat.zero)\n\ncase succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nih : comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = upperCentralSeries G (Nat.succ n)\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) (Nat.succ n)) = upperCentralSeries G (Nat.succ (Nat.succ n))",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = upperCentralSeries G (Nat.succ n)",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) Nat.zero) = upperCentralSeries G (Nat.succ Nat.zero)",
"tactic": "simp only [Nat.zero_eq, upperCentralSeries_zero, MonoidHom.comap_bot, ker_mk',\n (upperCentralSeries_one G).symm]"
},
{
"state_after": "case succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nih : comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = upperCentralSeries G (Nat.succ n)\nHn : Subgroup (G ⧸ center G) := upperCentralSeries (G ⧸ center G) n\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) (Nat.succ n)) = upperCentralSeries G (Nat.succ (Nat.succ n))",
"state_before": "case succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nih : comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = upperCentralSeries G (Nat.succ n)\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) (Nat.succ n)) = upperCentralSeries G (Nat.succ (Nat.succ n))",
"tactic": "let Hn := upperCentralSeries (G ⧸ center G) n"
},
{
"state_after": "no goals",
"state_before": "case succ\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nih : comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = upperCentralSeries G (Nat.succ n)\nHn : Subgroup (G ⧸ center G) := upperCentralSeries (G ⧸ center G) n\n⊢ comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) (Nat.succ n)) = upperCentralSeries G (Nat.succ (Nat.succ n))",
"tactic": "calc\n comap (mk' (center G)) (upperCentralSeriesStep Hn) =\n comap (mk' (center G)) (comap (mk' Hn) (center ((G ⧸ center G) ⧸ Hn))) :=\n by rw [upperCentralSeriesStep_eq_comap_center]\n _ = comap (mk' (comap (mk' (center G)) Hn)) (center (G ⧸ comap (mk' (center G)) Hn)) :=\n QuotientGroup.comap_comap_center\n _ = comap (mk' (upperCentralSeries G n.succ)) (center (G ⧸ upperCentralSeries G n.succ)) :=\n (comap_center_subst ih)\n _ = upperCentralSeriesStep (upperCentralSeries G n.succ) :=\n symm (upperCentralSeriesStep_eq_comap_center _)"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nih : comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = upperCentralSeries G (Nat.succ n)\nHn : Subgroup (G ⧸ center G) := upperCentralSeries (G ⧸ center G) n\n⊢ comap (mk' (center G)) (upperCentralSeriesStep Hn) =\n comap (mk' (center G)) (comap (mk' Hn) (center ((G ⧸ center G) ⧸ Hn)))",
"tactic": "rw [upperCentralSeriesStep_eq_comap_center]"
}
] | [
610,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
595,
1
] |
Mathlib/Topology/Support.lean | hasCompactMulSupport_comp_left | [
{
"state_after": "no goals",
"state_before": "X : Type ?u.14903\nα : Type u_3\nα' : Type ?u.14909\nβ : Type u_2\nγ : Type u_1\nδ : Type ?u.14918\nM : Type ?u.14921\nE : Type ?u.14924\nR : Type ?u.14927\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : One β\ninst✝¹ : One γ\ninst✝ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\nhg : ∀ {x : β}, g x = 1 ↔ x = 1\n⊢ HasCompactMulSupport (g ∘ f) ↔ HasCompactMulSupport f",
"tactic": "simp_rw [hasCompactMulSupport_def, mulSupport_comp_eq g (@hg) f]"
}
] | [
212,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/Data/Set/BoolIndicator.lean | Set.mem_iff_boolIndicator | [
{
"state_after": "α : Type u_1\ns : Set α\nx : α\n⊢ x ∈ s ↔ (if x ∈ s then true else false) = true",
"state_before": "α : Type u_1\ns : Set α\nx : α\n⊢ x ∈ s ↔ boolIndicator s x = true",
"tactic": "unfold boolIndicator"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Set α\nx : α\n⊢ x ∈ s ↔ (if x ∈ s then true else false) = true",
"tactic": "split_ifs with h <;> simp [h]"
}
] | [
32,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
30,
1
] |
Mathlib/Data/Int/GCD.lean | Nat.xgcdAux_zero | [] | [
52,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean | CategoryTheory.StructuredArrow.mk_right | [] | [
75,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
] |
Mathlib/Analysis/InnerProductSpace/Orientation.lean | OrthonormalBasis.orientation_adjustToOrientation | [
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.orientation (Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) = x",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.orientation (OrthonormalBasis.toBasis (adjustToOrientation e x)) = x",
"tactic": "rw [e.toBasis_adjustToOrientation]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.orientation (Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) = x",
"tactic": "exact e.toBasis.orientation_adjustToOrientation x"
}
] | [
127,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/NumberTheory/BernoulliPolynomials.lean | Polynomial.bernoulli_generating_function | [
{
"state_after": "case h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ↑(PowerSeries.coeff A n) ((PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1)) =\n ↑(PowerSeries.coeff A n) (PowerSeries.X * ↑(rescale t) (exp A))",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\n⊢ (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1) = PowerSeries.X * ↑(rescale t) (exp A)",
"tactic": "ext n"
},
{
"state_after": "case h.zero\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\n⊢ ↑(PowerSeries.coeff A Nat.zero) ((PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1)) =\n ↑(PowerSeries.coeff A Nat.zero) (PowerSeries.X * ↑(rescale t) (exp A))\n\ncase h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ↑(PowerSeries.coeff A (succ n)) ((PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1)) =\n ↑(PowerSeries.coeff A (succ n)) (PowerSeries.X * ↑(rescale t) (exp A))",
"state_before": "case h\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ↑(PowerSeries.coeff A n) ((PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1)) =\n ↑(PowerSeries.coeff A n) (PowerSeries.X * ↑(rescale t) (exp A))",
"tactic": "cases' n with n"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A (x, succ n - x).fst) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (x, succ n - x).snd) (exp A - 1) +\n ↑(PowerSeries.coeff A (n + 1, succ n - (n + 1)).fst)\n (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (n + 1, succ n - (n + 1)).snd) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !)",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ↑(PowerSeries.coeff A (succ n)) ((PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1)) =\n ↑(PowerSeries.coeff A (succ n)) (PowerSeries.X * ↑(rescale t) (exp A))",
"tactic": "rw [coeff_succ_X_mul, coeff_rescale, coeff_exp, PowerSeries.coeff_mul,\n Nat.sum_antidiagonal_eq_sum_range_succ_mk, sum_range_succ]"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !)",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A (x, succ n - x).fst) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (x, succ n - x).snd) (exp A - 1) +\n ↑(PowerSeries.coeff A (n + 1, succ n - (n + 1)).fst)\n (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (n + 1, succ n - (n + 1)).snd) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !)",
"tactic": "simp only [RingHom.map_sub, tsub_self, constantCoeff_one, constantCoeff_exp,\n coeff_zero_eq_constantCoeff, MulZeroClass.mul_zero, sub_self, add_zero]"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !)",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !)",
"tactic": "have hnp1 : IsUnit ((n + 1)! : ℚ) := IsUnit.mk0 _ (by exact_mod_cast factorial_ne_zero (n + 1))"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n ↑(algebraMap ℚ A) ↑(n + 1)! * (t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !))",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !)",
"tactic": "rw [← (hnp1.map (algebraMap ℚ A)).mul_right_inj]"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (↑(n + 1)! * (1 / ↑n !))",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n ↑(algebraMap ℚ A) ↑(n + 1)! * (t ^ n * ↑(algebraMap ℚ A) (1 / ↑n !))",
"tactic": "rw [mul_left_comm, ← RingHom.map_mul]"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (↑((n + 1) * n !) * (1 / ↑n !))",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (↑(n + 1)! * (1 / ↑n !))",
"tactic": "change _ = t ^ n * algebraMap ℚ A (((n + 1) * n ! : ℕ) * (1 / n !))"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n ↑(aeval t) (↑(monomial n) (↑n + 1))",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n t ^ n * ↑(algebraMap ℚ A) (↑((n + 1) * n !) * (1 / ↑n !))",
"tactic": "rw [cast_mul, mul_assoc,\n mul_one_div_cancel (show (n ! : ℚ) ≠ 0 from cast_ne_zero.2 (factorial_ne_zero n)), mul_one,\n mul_comm (t ^ n), ← aeval_monomial, cast_add, cast_one]"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ∑ x in range (n + 1),\n ↑(algebraMap ℚ A) ↑(n + 1)! *\n (↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1)) =\n ∑ x in range (n + 1), ↑(aeval t) (↑(Nat.choose (n + 1) x) • bernoulli x)",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n ∑ x in range (n + 1),\n ↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1) =\n ↑(aeval t) (↑(monomial n) (↑n + 1))",
"tactic": "rw [← sum_bernoulli, Finset.mul_sum, AlgHom.map_sum]"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ∀ (x : ℕ),\n x ∈ range (n + 1) →\n ↑(algebraMap ℚ A) ↑(n + 1)! *\n (↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1)) =\n ↑(aeval t) (↑(Nat.choose (n + 1) x) • bernoulli x)",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ∑ x in range (n + 1),\n ↑(algebraMap ℚ A) ↑(n + 1)! *\n (↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1)) =\n ∑ x in range (n + 1), ↑(aeval t) (↑(Nat.choose (n + 1) x) • bernoulli x)",
"tactic": "apply Finset.sum_congr rfl"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\ni : ℕ\nhi : i ∈ range (n + 1)\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n (↑(PowerSeries.coeff A i) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - i)) (exp A - 1)) =\n ↑(aeval t) (↑(Nat.choose (n + 1) i) • bernoulli i)",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\n⊢ ∀ (x : ℕ),\n x ∈ range (n + 1) →\n ↑(algebraMap ℚ A) ↑(n + 1)! *\n (↑(PowerSeries.coeff A x) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - x)) (exp A - 1)) =\n ↑(aeval t) (↑(Nat.choose (n + 1) x) • bernoulli x)",
"tactic": "intro i hi"
},
{
"state_after": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\ni : ℕ\nhi : i ∈ range (n + 1)\n⊢ ↑(algebraMap ℚ A) (↑(n + 1)! * (↑i !)⁻¹ * (↑(n + 1 - i)!)⁻¹) * ↑(aeval t) (bernoulli i) =\n ↑(algebraMap ℚ A) (↑(n + 1)! / (↑i ! * ↑(n + 1 - i)!)) * ↑(aeval t) (bernoulli i)",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\ni : ℕ\nhi : i ∈ range (n + 1)\n⊢ ↑(algebraMap ℚ A) ↑(n + 1)! *\n (↑(PowerSeries.coeff A i) (PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) *\n ↑(PowerSeries.coeff A (succ n - i)) (exp A - 1)) =\n ↑(aeval t) (↑(Nat.choose (n + 1) i) • bernoulli i)",
"tactic": "simp only [Nat.cast_choose ℚ (mem_range_le hi), coeff_mk, if_neg (mem_range_sub_ne_zero hi),\n one_div, AlgHom.map_smul, PowerSeries.coeff_one, coeff_exp, sub_zero, LinearMap.map_sub,\n Algebra.smul_mul_assoc, Algebra.smul_def, mul_right_comm _ ((aeval t) _), ← mul_assoc, ←\n RingHom.map_mul, succ_eq_add_one, ← Polynomial.C_eq_algebraMap, Polynomial.aeval_mul,\n Polynomial.aeval_C]"
},
{
"state_after": "no goals",
"state_before": "case h.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\nhnp1 : IsUnit ↑(n + 1)!\ni : ℕ\nhi : i ∈ range (n + 1)\n⊢ ↑(algebraMap ℚ A) (↑(n + 1)! * (↑i !)⁻¹ * (↑(n + 1 - i)!)⁻¹) * ↑(aeval t) (bernoulli i) =\n ↑(algebraMap ℚ A) (↑(n + 1)! / (↑i ! * ↑(n + 1 - i)!)) * ↑(aeval t) (bernoulli i)",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case h.zero\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\n⊢ ↑(PowerSeries.coeff A Nat.zero) ((PowerSeries.mk fun n => ↑(aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1)) =\n ↑(PowerSeries.coeff A Nat.zero) (PowerSeries.X * ↑(rescale t) (exp A))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nt : A\nn : ℕ\n⊢ ↑(n + 1)! ≠ 0",
"tactic": "exact_mod_cast factorial_ne_zero (n + 1)"
}
] | [
260,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.coe_castPred | [
{
"state_after": "case mk\nn✝ m n a : ℕ\nha : a < n + 2\nhx : { val := a, isLt := ha } < last (n + 1)\n⊢ ↑(castPred { val := a, isLt := ha }) = ↑{ val := a, isLt := ha }",
"state_before": "n✝ m n : ℕ\na : Fin (n + 2)\nhx : a < last (n + 1)\n⊢ ↑(castPred a) = ↑a",
"tactic": "rcases a with ⟨a, ha⟩"
},
{
"state_after": "case mk.h\nn✝ m n a : ℕ\nha : a < n + 2\nhx : { val := a, isLt := ha } < last (n + 1)\n⊢ a < n + 1",
"state_before": "case mk\nn✝ m n a : ℕ\nha : a < n + 2\nhx : { val := a, isLt := ha } < last (n + 1)\n⊢ ↑(castPred { val := a, isLt := ha }) = ↑{ val := a, isLt := ha }",
"tactic": "rw [castPred_mk]"
},
{
"state_after": "no goals",
"state_before": "case mk.h\nn✝ m n a : ℕ\nha : a < n + 2\nhx : { val := a, isLt := ha } < last (n + 1)\n⊢ a < n + 1",
"tactic": "exact hx"
}
] | [
2338,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2334,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean | PFunctor.M.iselect_eq_default | [
{
"state_after": "case nil\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nx : M F\nh : ¬IsPath [] x\n⊢ iselect [] x = head default\n\ncase cons\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nps_hd : IdxCat F\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → iselect ps_tail x = head default\nx : M F\nh : ¬IsPath (ps_hd :: ps_tail) x\n⊢ iselect (ps_hd :: ps_tail) x = head default",
"state_before": "F : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx : M F\nh : ¬IsPath ps x\n⊢ iselect ps x = head default",
"tactic": "induction' ps with ps_hd ps_tail ps_ih generalizing x"
},
{
"state_after": "case nil.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nx : M F\nh : ¬IsPath [] x\n⊢ False",
"state_before": "case nil\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nx : M F\nh : ¬IsPath [] x\n⊢ iselect [] x = head default",
"tactic": "exfalso"
},
{
"state_after": "case nil.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nx : M F\nh : ¬IsPath [] x\n⊢ IsPath [] x",
"state_before": "case nil.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nx : M F\nh : ¬IsPath [] x\n⊢ False",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case nil.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nx : M F\nh : ¬IsPath [] x\n⊢ IsPath [] x",
"tactic": "constructor"
},
{
"state_after": "case cons.mk\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → iselect ps_tail x = head default\nx : M F\na : F.A\ni : B F a\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\n⊢ iselect ({ fst := a, snd := i } :: ps_tail) x = head default",
"state_before": "case cons\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nps_hd : IdxCat F\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → iselect ps_tail x = head default\nx : M F\nh : ¬IsPath (ps_hd :: ps_tail) x\n⊢ iselect (ps_hd :: ps_tail) x = head default",
"tactic": "cases' ps_hd with a i"
},
{
"state_after": "case cons.mk.f\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → iselect ps_tail x = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\n⊢ iselect ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f }) = head default",
"state_before": "case cons.mk\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → iselect ps_tail x = head default\nx : M F\na : F.A\ni : B F a\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\n⊢ iselect ({ fst := a, snd := i } :: ps_tail) x = head default",
"tactic": "induction' x using PFunctor.M.casesOn' with x_a x_f"
},
{
"state_after": "case cons.mk.f\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default",
"state_before": "case cons.mk.f\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → iselect ps_tail x = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\n⊢ iselect ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f }) = head default",
"tactic": "simp only [iselect, isubtree] at ps_ih⊢"
},
{
"state_after": "case pos\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\nh'' : a = x_a\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default\n\ncase neg\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\nh'' : ¬a = x_a\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default",
"state_before": "case cons.mk.f\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default",
"tactic": "by_cases h'' : a = x_a"
},
{
"state_after": "case pos\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\n⊢ head\n (M.casesOn' (M.mk { fst := a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default\n\ncase neg\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\nh'' : ¬a = x_a\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default",
"state_before": "case pos\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\nh'' : a = x_a\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default\n\ncase neg\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\nh'' : ¬a = x_a\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default",
"tactic": "subst x_a"
},
{
"state_after": "case pos\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\n⊢ head (isubtree ps_tail (x_f (cast (_ : B F a = B F a) i))) = head default",
"state_before": "case pos\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\n⊢ head\n (M.casesOn' (M.mk { fst := a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default",
"tactic": "simp only [dif_pos, eq_self_iff_true, casesOn_mk']"
},
{
"state_after": "case pos.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\n⊢ ¬IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))",
"state_before": "case pos\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\n⊢ head (isubtree ps_tail (x_f (cast (_ : B F a = B F a) i))) = head default",
"tactic": "rw [ps_ih]"
},
{
"state_after": "case pos.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ False",
"state_before": "case pos.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\n⊢ ¬IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))",
"tactic": "intro h'"
},
{
"state_after": "case pos.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })",
"state_before": "case pos.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ False",
"tactic": "apply h"
},
{
"state_after": "case pos.h.a\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ IsPath ps_tail (x_f i)",
"state_before": "case pos.h\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })",
"tactic": "constructor <;> try rfl"
},
{
"state_after": "no goals",
"state_before": "case pos.h.a\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ IsPath ps_tail (x_f i)",
"tactic": "apply h'"
},
{
"state_after": "case pos.h.a\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ IsPath ps_tail (x_f i)",
"state_before": "case pos.h.a\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_f : B F a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := a, snd := x_f })\nh' : IsPath ps_tail (x_f (cast (_ : B F a = B F a) i))\n⊢ IsPath ps_tail (x_f i)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case neg\nF : PFunctor\nX : Type ?u.22988\nf : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\nx✝ : M F\nh✝¹ : ¬IsPath ps x✝\nps_tail : List (IdxCat F)\nps_ih : ∀ (x : M F), ¬IsPath ps_tail x → head (isubtree ps_tail x) = head default\nx : M F\na : F.A\ni : B F a\nh✝ : ¬IsPath ({ fst := a, snd := i } :: ps_tail) x\nx_a : F.A\nx_f : B F x_a → M F\nh : ¬IsPath ({ fst := a, snd := i } :: ps_tail) (M.mk { fst := x_a, snd := x_f })\nh'' : ¬a = x_a\n⊢ head\n (M.casesOn' (M.mk { fst := x_a, snd := x_f }) fun a' f =>\n if h : a = a' then isubtree ps_tail (f (cast (_ : B F a = B F a') i)) else default) =\n head default",
"tactic": "simp [*]"
}
] | [
532,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
515,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | SemiconjBy.cast_int_mul_cast_int_mul | [] | [
1076,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1074,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | HasFTaylorSeriesUpToOn.restrictScalars | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁵ : NormedAddCommGroup D\ninst✝¹⁴ : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedSpace 𝕜 G\nX : Type ?u.3124593\ninst✝⁷ : NormedAddCommGroup X\ninst✝⁶ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\n𝕜' : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜'\ninst✝⁴ : NormedAlgebra 𝕜 𝕜'\ninst✝³ : NormedSpace 𝕜' E\ninst✝² : IsScalarTower 𝕜 𝕜' E\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\np' : E → FormalMultilinearSeries 𝕜' E F\nh : HasFTaylorSeriesUpToOn n f p' s\nm : ℕ\nhm : ↑m < n\nx : E\nhx : x ∈ s\n⊢ HasFDerivWithinAt (fun x => FormalMultilinearSeries.restrictScalars 𝕜 (p' x) m)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.restrictScalars 𝕜 (p' x) (Nat.succ m))) s x",
"tactic": "simpa only using (ContinuousMultilinearMap.restrictScalarsLinear 𝕜).hasFDerivAt.comp_hasFDerivWithinAt x <|\n (h.fderivWithin m hm x hx).restrictScalars 𝕜"
}
] | [
2214,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2207,
1
] |
Mathlib/Topology/Sheaves/Limits.lean | TopCat.limit_isSheaf | [] | [
58,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | self_sub_toIocDiv_zsmul | [] | [
121,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.interpolate_empty | [
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\n⊢ ↑(interpolate ∅ v) r = 0",
"tactic": "rw [interpolate_apply, sum_empty]"
}
] | [
309,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/Order/Lattice.lean | SemilatticeInf.dual_dual | [] | [
582,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
580,
1
] |
Mathlib/Algebra/Group/Pi.lean | Function.update_div | [] | [
638,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
636,
1
] |
Std/Data/List/Lemmas.lean | List.get?_set | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : α\nm n : Nat\nl : List α\n⊢ get? (set l m a) n = if m = n then (fun x => a) <$> get? l n else get? l n",
"tactic": "by_cases m = n <;> simp [*, get?_set_eq, get?_set_ne]"
}
] | [
807,
56
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
805,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineEquiv.injective_pointReflection_left_of_injective_bit0 | [] | [
596,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
594,
1
] |
Mathlib/Topology/LocalExtr.lean | IsLocalMinOn.inf | [] | [
487,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
485,
8
] |
Mathlib/Data/Stream/Init.lean | Stream'.eq_or_mem_of_mem_cons | [
{
"state_after": "case zero\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nh : a = nth (b :: s) zero\n⊢ a = b ∨ a ∈ s\n\ncase succ\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = nth (b :: s) (succ n')\n⊢ a = b ∨ a ∈ s",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn : ℕ\nh : (fun b => a = b) (nth (b :: s) n)\n⊢ a = b ∨ a ∈ s",
"tactic": "cases' n with n'"
},
{
"state_after": "case zero.h\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nh : a = nth (b :: s) zero\n⊢ a = b",
"state_before": "case zero\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nh : a = nth (b :: s) zero\n⊢ a = b ∨ a ∈ s",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case zero.h\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nh : a = nth (b :: s) zero\n⊢ a = b",
"tactic": "exact h"
},
{
"state_after": "case succ.h\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = nth (b :: s) (succ n')\n⊢ a ∈ s",
"state_before": "case succ\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = nth (b :: s) (succ n')\n⊢ a = b ∨ a ∈ s",
"tactic": "right"
},
{
"state_after": "case succ.h\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = nth s n'\n⊢ a ∈ s",
"state_before": "case succ.h\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = nth (b :: s) (succ n')\n⊢ a ∈ s",
"tactic": "rw [nth_succ, tail_cons] at h"
},
{
"state_after": "no goals",
"state_before": "case succ.h\nα : Type u\nβ : Type v\nδ : Type w\na b : α\ns : Stream' α\nx✝ : a ∈ b :: s\nn' : ℕ\nh : a = nth s n'\n⊢ a ∈ s",
"tactic": "exact ⟨n', h⟩"
}
] | [
136,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Topology/GDelta.lean | Set.Finite.isGδ_compl | [] | [
150,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
] |
Mathlib/Algebra/GCDMonoid/Finset.lean | Finset.normalize_lcm | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5029\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf : β → α\n⊢ ↑normalize (lcm s f) = lcm s f",
"tactic": "simp [lcm_def]"
}
] | [
95,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean | Set.projIcc_of_le_left | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.573\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nx : α\nhx : x ≤ a\n⊢ projIcc a b h x = { val := a, property := (_ : a ∈ Icc a b) }",
"tactic": "simp [projIcc, hx, hx.trans h]"
}
] | [
42,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/Algebra/Regular/SMul.lean | isRightRegular_iff | [] | [
59,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
57,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean | intervalIntegral.intervalIntegrable_cpow | [
{
"state_after": "case pos\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re ∨ ¬0 ∈ [[a, b]]\nh2 : ¬0 ∈ [[a, b]]\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b\n\ncase neg\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re ∨ ¬0 ∈ [[a, b]]\nh2 : ¬¬0 ∈ [[a, b]]\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"state_before": "a b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re ∨ ¬0 ∈ [[a, b]]\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"tactic": "by_cases h2 : (0 : ℝ) ∉ [[a, b]]"
},
{
"state_after": "case neg\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"state_before": "case neg\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re ∨ ¬0 ∈ [[a, b]]\nh2 : ¬¬0 ∈ [[a, b]]\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"tactic": "rw [eq_false h2, or_false_iff] at h"
},
{
"state_after": "case neg.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 < r.re\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b\n\ncase neg.inr\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"state_before": "case neg\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"tactic": "rcases lt_or_eq_of_le h with (h' | h')"
},
{
"state_after": "case neg.inr.refine'_1\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ AEStronglyMeasurable (fun x => ↑x ^ r) (Measure.restrict μ (Ι a b))\n\ncase neg.inr.refine'_2\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ IntervalIntegrable (fun t => ‖↑t ^ r‖) μ a b",
"state_before": "case neg.inr\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"tactic": "refine' (IntervalIntegrable.intervalIntegrable_norm_iff _).mp _"
},
{
"state_after": "case neg.inr.refine'_2\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nthis : ∀ (c : ℝ), IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c\n⊢ IntervalIntegrable (fun t => ‖↑t ^ r‖) μ a b\n\ncase this\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ ∀ (c : ℝ), IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"state_before": "case neg.inr.refine'_2\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ IntervalIntegrable (fun t => ‖↑t ^ r‖) μ a b",
"tactic": "suffices : ∀ c : ℝ, IntervalIntegrable (fun x : ℝ => ‖(x:ℂ) ^ r‖) μ 0 c"
},
{
"state_after": "case this\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ ∀ (c : ℝ), IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"state_before": "case neg.inr.refine'_2\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nthis : ∀ (c : ℝ), IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c\n⊢ IntervalIntegrable (fun t => ‖↑t ^ r‖) μ a b\n\ncase this\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ ∀ (c : ℝ), IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"tactic": "exact (this a).symm.trans (this b)"
},
{
"state_after": "case this\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"state_before": "case this\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ ∀ (c : ℝ), IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"tactic": "intro c"
},
{
"state_after": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c\n\ncase this.inr\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"state_before": "case this\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"tactic": "rcases le_or_lt 0 c with (hc | hc)"
},
{
"state_after": "case pos\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re ∨ ¬0 ∈ [[a, b]]\nh2 : ¬0 ∈ [[a, b]]\nx : ℝ\nhx : x ∈ [[a, b]]\n⊢ ContinuousAt (fun x => ↑x ^ r) x",
"state_before": "case pos\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re ∨ ¬0 ∈ [[a, b]]\nh2 : ¬0 ∈ [[a, b]]\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"tactic": "refine' (ContinuousAt.continuousOn fun x hx => _).intervalIntegrable"
},
{
"state_after": "no goals",
"state_before": "case pos\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re ∨ ¬0 ∈ [[a, b]]\nh2 : ¬0 ∈ [[a, b]]\nx : ℝ\nhx : x ∈ [[a, b]]\n⊢ ContinuousAt (fun x => ↑x ^ r) x",
"tactic": "exact Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_mem_of_not_mem hx h2)"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 < r.re\n⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b",
"tactic": "exact (Complex.continuous_ofReal_cpow_const h').intervalIntegrable _ _"
},
{
"state_after": "case neg.inr.refine'_1\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ ContinuousOn (fun x => ↑x ^ r) ({0}ᶜ)",
"state_before": "case neg.inr.refine'_1\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ AEStronglyMeasurable (fun x => ↑x ^ r) (Measure.restrict μ (Ι a b))",
"tactic": "refine' (measurable_of_continuousOn_compl_singleton (0 : ℝ) _).aestronglyMeasurable"
},
{
"state_after": "no goals",
"state_before": "case neg.inr.refine'_1\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\n⊢ ContinuousOn (fun x => ↑x ^ r) ({0}ᶜ)",
"tactic": "exact ContinuousAt.continuousOn fun x hx =>\n Complex.continuousAt_ofReal_cpow_const x r (Or.inr hx)"
},
{
"state_after": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntervalIntegrable (fun x => 1) μ 0 c\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"state_before": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"tactic": "have : IntervalIntegrable (fun _ => 1 : ℝ → ℝ) μ 0 c := intervalIntegrable_const"
},
{
"state_after": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntegrableOn (fun x => 1) (Set.Ioc 0 c)\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioc 0 c)",
"state_before": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntervalIntegrable (fun x => 1) μ 0 c\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"tactic": "rw [intervalIntegrable_iff_integrable_Ioc_of_le hc] at this ⊢"
},
{
"state_after": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntegrableOn (fun x => 1) (Set.Ioc 0 c)\nx : ℝ\nhx : x ∈ Set.Ioc 0 c\n⊢ 1 = ‖↑x ^ r‖",
"state_before": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntegrableOn (fun x => 1) (Set.Ioc 0 c)\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioc 0 c)",
"tactic": "refine' IntegrableOn.congr_fun this (fun x hx => _) measurableSet_Ioc"
},
{
"state_after": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntegrableOn (fun x => 1) (Set.Ioc 0 c)\nx : ℝ\nhx : x ∈ Set.Ioc 0 c\n⊢ 1 = ‖↑x ^ r‖",
"state_before": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntegrableOn (fun x => 1) (Set.Ioc 0 c)\nx : ℝ\nhx : x ∈ Set.Ioc 0 c\n⊢ 1 = ‖↑x ^ r‖",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case this.inl\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : 0 ≤ c\nthis : IntegrableOn (fun x => 1) (Set.Ioc 0 c)\nx : ℝ\nhx : x ∈ Set.Ioc 0 c\n⊢ 1 = ‖↑x ^ r‖",
"tactic": "rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos hx.1, ← h', rpow_zero]"
},
{
"state_after": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ c 0",
"state_before": "case this.inr\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ 0 c",
"tactic": "apply IntervalIntegrable.symm"
},
{
"state_after": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioc c 0)",
"state_before": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ IntervalIntegrable (fun x => ‖↑x ^ r‖) μ c 0",
"tactic": "rw [intervalIntegrable_iff_integrable_Ioc_of_le hc.le]"
},
{
"state_after": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioc c 0)",
"state_before": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioc c 0)",
"tactic": "have : Ioc c 0 = Ioo c 0 ∪ {(0 : ℝ)} := by\n rw [← Ioo_union_Icc_eq_Ioc hc (le_refl 0), ← Icc_def]\n simp_rw [← le_antisymm_iff, setOf_eq_eq_singleton']"
},
{
"state_after": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) {0} ∧ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioo c 0)",
"state_before": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioc c 0)",
"tactic": "rw [this, integrableOn_union, and_comm]"
},
{
"state_after": "case this.inr.h.left\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) {0}\n\ncase this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioo c 0)",
"state_before": "case this.inr.h\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) {0} ∧ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioo c 0)",
"tactic": "constructor"
},
{
"state_after": "a b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ Set.Ioo c 0 ∪ {x | 0 ≤ x ∧ x ≤ 0} = Set.Ioo c 0 ∪ {0}",
"state_before": "a b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}",
"tactic": "rw [← Ioo_union_Icc_eq_Ioc hc (le_refl 0), ← Icc_def]"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\n⊢ Set.Ioo c 0 ∪ {x | 0 ≤ x ∧ x ≤ 0} = Set.Ioo c 0 ∪ {0}",
"tactic": "simp_rw [← le_antisymm_iff, setOf_eq_eq_singleton']"
},
{
"state_after": "case this.inr.h.left\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ ↑↑μ {0} < ⊤",
"state_before": "case this.inr.h.left\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) {0}",
"tactic": "refine' integrableOn_singleton_iff.mpr (Or.inr _)"
},
{
"state_after": "no goals",
"state_before": "case this.inr.h.left\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ ↑↑μ {0} < ⊤",
"tactic": "exact isFiniteMeasureOnCompacts_of_isLocallyFiniteMeasure.lt_top_of_isCompact\n isCompact_singleton"
},
{
"state_after": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioo c 0)",
"state_before": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioo c 0)",
"tactic": "have : ∀ x : ℝ, x ∈ Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖(x : ℂ) ^ r‖ := by\n intro x hx\n rw [Complex.ofReal_cpow_of_nonpos hx.2.le, norm_mul, ← Complex.ofReal_neg,\n Complex.norm_eq_abs (_ ^ _), Complex.abs_cpow_eq_rpow_re_of_pos (neg_pos.mpr hx.2), ← h',\n rpow_zero, one_mul]"
},
{
"state_after": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ IntegrableOn (fun x => ‖Complex.exp (↑π * Complex.I * r)‖) (Set.Ioo c 0)",
"state_before": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ IntegrableOn (fun x => ‖↑x ^ r‖) (Set.Ioo c 0)",
"tactic": "refine' IntegrableOn.congr_fun _ this measurableSet_Ioo"
},
{
"state_after": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ ‖Complex.exp (↑π * Complex.I * r)‖ = 0 ∨ ↑↑μ (Set.Ioo c 0) < ⊤",
"state_before": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ IntegrableOn (fun x => ‖Complex.exp (↑π * Complex.I * r)‖) (Set.Ioo c 0)",
"tactic": "rw [integrableOn_const]"
},
{
"state_after": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ ↑↑μ (Set.Icc c 0) < ⊤",
"state_before": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ ‖Complex.exp (↑π * Complex.I * r)‖ = 0 ∨ ↑↑μ (Set.Ioo c 0) < ⊤",
"tactic": "refine' Or.inr ((measure_mono Set.Ioo_subset_Icc_self).trans_lt _)"
},
{
"state_after": "no goals",
"state_before": "case this.inr.h.right\na b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis✝ : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nthis : ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖\n⊢ ↑↑μ (Set.Icc c 0) < ⊤",
"tactic": "exact isFiniteMeasureOnCompacts_of_isLocallyFiniteMeasure.lt_top_of_isCompact isCompact_Icc"
},
{
"state_after": "a b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nx : ℝ\nhx : x ∈ Set.Ioo c 0\n⊢ ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖",
"state_before": "a b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\n⊢ ∀ (x : ℝ), x ∈ Set.Ioo c 0 → ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\nf : ℝ → ℝ\nμ ν : MeasureTheory.Measure ℝ\ninst✝ : IsLocallyFiniteMeasure μ\nc✝ d : ℝ\nr : ℂ\nh : 0 ≤ r.re\nh2 : ¬¬0 ∈ [[a, b]]\nh' : 0 = r.re\nc : ℝ\nhc : c < 0\nthis : Set.Ioc c 0 = Set.Ioo c 0 ∪ {0}\nx : ℝ\nhx : x ∈ Set.Ioo c 0\n⊢ ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖",
"tactic": "rw [Complex.ofReal_cpow_of_nonpos hx.2.le, norm_mul, ← Complex.ofReal_neg,\n Complex.norm_eq_abs (_ ^ _), Complex.abs_cpow_eq_rpow_re_of_pos (neg_pos.mpr hx.2), ← h',\n rpow_zero, one_mul]"
}
] | [
147,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.