module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Set.Prod | {
"line": 688,
"column": 63
} | {
"line": 688,
"column": 85
} | {
"line": 690,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ns : Set ι\nt : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), Nonempty (α i)\n⊢ s.pi t = ∅ ↔ ∃ i, i ∈ s ∧ t i = ∅",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Membership.mem",
"Exists",
"not_isEmpty_of... | [] | simp [pi_eq_empty_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Function | {
"line": 373,
"column": 25
} | {
"line": 373,
"column": 93
} | {
"line": 375,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nh : InjOn f s\ns₁ : Set α\nhs₁ : s₁ ∈ 𝒫 s\ns₂ : Set α\nhs₂ : s₂ ∈ 𝒫 s\nh' : f '' s₁ = f '' s₂\n⊢ s₁ = s₂",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Set.instInter",
"... | [] | by rw [← h.preimage_image_inter hs₁, h', h.preimage_image_inter hs₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Function | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 31
} | {
"line": 579,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Set α\nt : Set β\nf : α → β\ng₁ g₂ : β → γ\nhf : EqOn (g₁ ∘ f) (g₂ ∘ f) s\nhf' : SurjOn f s t\nb : β\nhb : b ∈ t\n⊢ g₁ b = g₂ b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Membership.mem",
"And.casesOn",
"And",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Set α\nt : Set β\nf : α → β\ng₁ g₂ : β → γ\nhf : EqOn (g₁ ∘ f) (g₂ ∘ f) s\nhf' : SurjOn f s t\na : α\nha : a ∈ s\nhb : f a ∈ t\n⊢ g₁ (f a) = g₂ (f a)"
] | obtain ⟨a, ha, rfl⟩ := hf' hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Set.Function | {
"line": 1202,
"column": 2
} | {
"line": 1205,
"column": 32
} | {
"line": 1207,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nfa : α → α\nfb : β → β\nf : α → β\nh : Semiconj f fa fb\ns : Set β\nhb : InjOn fb s\nhf : InjOn f (f ⁻¹' s)\n⊢ InjOn fa (f ⁻¹' s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"congrArg",
"Function.Semiconj.eq",
"Membership.mem",
... | [] | intro x hx y hy H
have := congr_arg f H
rw [h.eq, h.eq] at this
exact hf hx hy (hb hx hy this) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Function | {
"line": 1202,
"column": 2
} | {
"line": 1205,
"column": 32
} | {
"line": 1207,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nfa : α → α\nfb : β → β\nf : α → β\nh : Semiconj f fa fb\ns : Set β\nhb : InjOn fb s\nhf : InjOn f (f ⁻¹' s)\n⊢ InjOn fa (f ⁻¹' s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"congrArg",
"Function.Semiconj.eq",
"Membership.mem",
... | [] | intro x hx y hy H
have := congr_arg f H
rw [h.eq, h.eq] at this
exact hf hx hy (hb hx hy this) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Basic | {
"line": 754,
"column": 43
} | {
"line": 754,
"column": 90
} | {
"line": 756,
"column": 0
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na b : G\n⊢ a / b = a ↔ b = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"InvOneClass.toOne",
"inv_eq_one",
"HMul.hMul",
"DivInvOneMonoid.t... | [] | by rw [div_eq_mul_inv, mul_eq_left, inv_eq_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Basic | {
"line": 847,
"column": 18
} | {
"line": 847,
"column": 96
} | {
"line": 849,
"column": 0
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na b : G\nn : ℕ\n⊢ (a * b) ^ Int.negSucc n * a = a * (b * a) ^ Int.negSucc n",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"DivInvMonoid.toInv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [] | by simp [inv_mul_eq_iff_eq_mul, eq_mul_inv_iff_mul_eq, mul_assoc, mul_pow_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Prod | {
"line": 106,
"column": 21
} | {
"line": 106,
"column": 60
} | {
"line": 107,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹ : DivInvMonoid G\ninst✝ : DivInvMonoid H\nx✝ : G × H\n⊢ x✝ ^ 0 = 1",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"ZPow.zpow",
"DivInvMonoid.toZPow",
"DivInvMonoid.zpow_zero'",
"D... | [] | ext <;> exact DivInvMonoid.zpow_zero' _ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Group.Prod | {
"line": 106,
"column": 21
} | {
"line": 106,
"column": 60
} | {
"line": 107,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹ : DivInvMonoid G\ninst✝ : DivInvMonoid H\nx✝ : G × H\n⊢ x✝ ^ 0 = 1",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"ZPow.zpow",
"DivInvMonoid.toZPow",
"DivInvMonoid.zpow_zero'",
"D... | [] | ext <;> exact DivInvMonoid.zpow_zero' _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Prod | {
"line": 106,
"column": 21
} | {
"line": 106,
"column": 60
} | {
"line": 107,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹ : DivInvMonoid G\ninst✝ : DivInvMonoid H\nx✝ : G × H\n⊢ x✝ ^ 0 = 1",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"ZPow.zpow",
"DivInvMonoid.toZPow",
"DivInvMonoid.zpow_zero'",
"D... | [] | ext <;> exact DivInvMonoid.zpow_zero' _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Prod | {
"line": 108,
"column": 19
} | {
"line": 108,
"column": 61
} | {
"line": 110,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹ : DivInvMonoid G\ninst✝ : DivInvMonoid H\nx✝¹ : ℕ\nx✝ : G × H\n⊢ x✝ ^ Int.negSucc x✝¹ = (x✝ ^ ↑x✝¹.succ)⁻¹",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Prod.instInv",
"DivInvMonoid.toInv",... | [] | by ext <;> exact DivInvMonoid.zpow_neg' .. | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Cast.Defs | {
"line": 100,
"column": 2
} | {
"line": 102,
"column": 67
} | {
"line": 104,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : AddMonoidWithOne R\nm n : ℕ\n⊢ ↑(m + n) = ↑m + ↑n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Nat.recAux",
"AddMonoid.toAddSemigroup",
"congrArg",
"add_assoc",
"AddMonoid.toAddZeroClass... | [] | induction n with
| zero => simp
| succ n ih => rw [add_succ, cast_succ, ih, cast_succ, add_assoc] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.Nat.Cast.Defs | {
"line": 100,
"column": 2
} | {
"line": 102,
"column": 67
} | {
"line": 104,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : AddMonoidWithOne R\nm n : ℕ\n⊢ ↑(m + n) = ↑m + ↑n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Nat.recAux",
"AddMonoid.toAddSemigroup",
"congrArg",
"add_assoc",
"AddMonoid.toAddZeroClass... | [] | induction n with
| zero => simp
| succ n ih => rw [add_succ, cast_succ, ih, cast_succ, add_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Cast.Defs | {
"line": 100,
"column": 2
} | {
"line": 102,
"column": 67
} | {
"line": 104,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : AddMonoidWithOne R\nm n : ℕ\n⊢ ↑(m + n) = ↑m + ↑n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Nat.recAux",
"AddMonoid.toAddSemigroup",
"congrArg",
"add_assoc",
"AddMonoid.toAddZeroClass... | [] | induction n with
| zero => simp
| succ n ih => rw [add_succ, cast_succ, ih, cast_succ, add_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Int.Basic | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 81
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case refine_3\nm n✝ : ℤ\nP : ℤ → Sort u_1\nlt : (n : ℤ) → n < m → P n\nge : (n : ℤ) → n ≥ m → ((k : ℤ) → k < n → P k) → P n\nn : ℤ\nx✝¹ : n ≥ m\nhn : m ≤ n\nk : ℤ\nhkm : k ≤ m\nih' :\n (∀ (k_1 : ℤ), k_1 < k → ∀ (hn : m ≤ k_1), Int.strongRec lt ge k_1 = ge k_1 hn fun k x ↦ Int.strongRec lt ge k) →\n ... | [] | exact fun l hlk hml ↦ (Int.not_lt.mpr hkm <| Int.lt_of_le_of_lt hml hlk).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Ring.Hom.Defs | {
"line": 619,
"column": 26
} | {
"line": 619,
"column": 40
} | {
"line": 619,
"column": 41
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), f (x * x) = f x * f x\nh_two : 2 ≠ 0\nh_one : f 1 = 1\nx y : β\nhxy : f x * f x + f (y * x) + (f (x * y) + f y * f y) = f x * f x + f x * f y + (f y * f x + f... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), f (x * x) = f x * f x\nh_two : 2 ≠ 0\nh_one : f 1 = 1\nx y : β\nhxy : f x * f x + f (y * x) + (f (x * y) + f y * f y) - (f x * f x + f x * f y + (f y * f x + f y * f y)) ... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Hom.Defs | {
"line": 622,
"column": 69
} | {
"line": 622,
"column": 91
} | {
"line": 623,
"column": 8
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), f (x * x) = f x * f x\nh_two : 2 ≠ 0\nh_one : f 1 = 1\nx y : β\nhxy : 2 * (f (x * y) - f x * f y) = 0\n⊢ (↑f).toFun (x * y) = (↑f).toFun x * (↑f).toFun y",
... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), f (x * x) = f x * f x\nh_two : 2 ≠ 0\nh_one : f 1 = 1\nx y : β\nhxy : 2 = 0 ∨ f (x * y) - f x * f y = 0\n⊢ (↑f).toFun (x * y) = (↑f).toFun x * (↑f).toFun y"
] | mul_eq_zero (M₀ := α), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Commute.Units | {
"line": 156,
"column": 41
} | {
"line": 157,
"column": 77
} | {
"line": 159,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : DivisionMonoid M\na b c d : M\nhbd : Commute b d\nhb : IsUnit b\nhd : IsUnit d\n⊢ a * b⁻¹ = c * d⁻¹ ↔ a * d = c * b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"DivInvO... | [] | by
rw [← div_eq_mul_inv, ← div_eq_mul_inv, hbd.div_eq_div_iff_of_isUnit hb hd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 298,
"column": 11
} | {
"line": 298,
"column": 22
} | {
"line": 298,
"column": 22
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\n⊢ a / a = 1 → a ≠ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.toOne",
"GroupWithZero.toDivInvMonoid",
... | [
"G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\n⊢ a = 0 → a / a ≠ 1"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Ring.Commute | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 20
} | {
"line": 57,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : NoZeroDivisors R\na b : R\nh : Commute a b\n⊢ a * a = b * b ↔ a = b ∨ a = -b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
... | [
"R : Type u\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : NoZeroDivisors R\na b : R\nh : Commute a b\n⊢ a * a - b * b = 0 ↔ a = b ∨ a = -b"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Commute | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 20
} | {
"line": 129,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\na b : R\ninst✝ : NoZeroDivisors R\nh : Commute a b\n⊢ a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
... | [
"R : Type u\ninst✝¹ : Ring R\na b : R\ninst✝ : NoZeroDivisors R\nh : Commute a b\n⊢ a ^ 2 - b ^ 2 = 0 ↔ a = b ∨ a = -b"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 67,
"column": 47
} | {
"line": 67,
"column": 61
} | {
"line": 67,
"column": 62
} | [
{
"pp": "α : Type u_3\ninst✝¹ : AddGroupWithOne α\ninst✝ : CharZero α\nm n : ℤ\n⊢ ↑m = ↑n ↔ m = n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"sub_eq_zero"... | [
"α : Type u_3\ninst✝¹ : AddGroupWithOne α\ninst✝ : CharZero α\nm n : ℤ\n⊢ ↑m - ↑n = 0 ↔ m = n"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 68
} | {
"line": 106,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝ : NonAssocRing α\na : α\nn : ℕ\n⊢ ↑n • a = ↑↑n * a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"instHSMul",
"HMul.hMul",
"AddGroup... | [] | rw [natCast_zsmul, nsmul_eq_mul, Int.cast_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 68
} | {
"line": 106,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝ : NonAssocRing α\na : α\nn : ℕ\n⊢ ↑n • a = ↑↑n * a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"instHSMul",
"HMul.hMul",
"AddGroup... | [] | rw [natCast_zsmul, nsmul_eq_mul, Int.cast_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 68
} | {
"line": 106,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝ : NonAssocRing α\na : α\nn : ℕ\n⊢ ↑n • a = ↑↑n * a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"instHSMul",
"HMul.hMul",
"AddGroup... | [] | rw [natCast_zsmul, nsmul_eq_mul, Int.cast_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Parity | {
"line": 411,
"column": 66
} | {
"line": 411,
"column": 77
} | {
"line": 411,
"column": 78
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DivisionMonoid α\ninst✝ : HasDistribNeg α\nn : ℤ\nh : Even n\n⊢ (-1) ^ n = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
... | [
"α : Type u_2\ninst✝¹ : DivisionMonoid α\ninst✝ : HasDistribNeg α\nn : ℤ\nh : Even n\n⊢ 1 ^ n = 1"
] | h.neg_zpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 13
} | {
"line": 704,
"column": 13
} | [
{
"pp": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : MulLeftMono α\na b c d : α\n⊢ a / d < b / c → a < b ∨ c < d",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instHDiv",
"congrArg",
"PartialOrder.toPreorder",... | [
"α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : MulLeftMono α\na b c d : α\n⊢ b ≤ a ∧ d ≤ c → b / c ≤ a / d"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 736,
"column": 2
} | {
"line": 736,
"column": 45
} | {
"line": 737,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝³ : Group α\ninst✝² : LinearOrder α\ninst✝¹ : MulLeftMono α\na b : α\ninst✝ : MulRightMono α\n⊢ a / b ≤ a⁻¹ * b ↔ a ≤ b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"DivInvOn... | [
"α : Type u\ninst✝³ : Group α\ninst✝² : LinearOrder α\ninst✝¹ : MulLeftMono α\na b : α\ninst✝ : MulRightMono α\n⊢ a * a ≤ b * b ↔ a ≤ b"
] | rw [div_eq_mul_inv, mul_inv_le_inv_mul_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Monoid.Defs | {
"line": 124,
"column": 54
} | {
"line": 124,
"column": 65
} | {
"line": 124,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ a * a ≤ 1 ↔ a ≤ 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Preorder.toLT",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
... | [
"α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ 1 < a * a ↔ 1 < a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Monoid.Defs | {
"line": 127,
"column": 54
} | {
"line": 127,
"column": 65
} | {
"line": 127,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ a * a < 1 ↔ a < 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Preorder.toLT",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
... | [
"α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ 1 ≤ a * a ↔ 1 ≤ a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 13
} | {
"line": 85,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"congrArg"... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₂ ≤ a₁ ∧ b₂ ≤ b₁ → a₂ * b₂ ≤ a₁ * b₁"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | {
"line": 91,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightStrictMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"con... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightStrictMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₂ < a₁ ∧ b₂ ≤ b₁ → a₂ * b₂ < a₁ * b₁"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 13
} | {
"line": 97,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"con... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₂ ≤ a₁ ∧ b₂ < b₁ → a₂ * b₂ < a₁ * b₁"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 13
} | {
"line": 103,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₂ < a₁ ∧ b₂ < b₁ → a₂ * b₂ < a₁ * b₁"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Group.Defs | {
"line": 120,
"column": 48
} | {
"line": 120,
"column": 59
} | {
"line": 120,
"column": 59
} | [
{
"pp": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ a ≤ a⁻¹ ↔ a ≤ 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
... | [
"α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ a⁻¹ < a ↔ 1 < a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Group.Defs | {
"line": 123,
"column": 48
} | {
"line": 123,
"column": 59
} | {
"line": 123,
"column": 59
} | [
{
"pp": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ a < a⁻¹ ↔ a < 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
... | [
"α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na : α\n⊢ a⁻¹ ≤ a ↔ 1 ≤ a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 461,
"column": 96
} | {
"line": 462,
"column": 77
} | {
"line": 464,
"column": 0
} | [
{
"pp": "M₀ : Type u_2\ninst✝³ : MonoidWithZero M₀\ninst✝² : Preorder M₀\na : M₀\nn : ℕ\ninst✝¹ : ZeroLEOneClass M₀\ninst✝ : PosMulMono M₀\nha : 1 ≤ a\nhn : n ≠ 0\n⊢ a ≤ a ^ n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Nat.pos_iff_ne_zero",
... | [] | by
simpa only [pow_one] using pow_le_pow_right₀ ha <| Nat.pos_iff_ne_zero.2 hn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 297,
"column": 2
} | {
"line": 300,
"column": 12
} | {
"line": 302,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b c d : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddLeftMono R\ninst✝ : AddLeftReflectLE R\nhab : a ≤ b\nhcd : c ≤ d\n⊢ a * d + b * c ≤ a * c + b * d",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨d, hd, rfl⟩ := exists_nonneg_add_of_le hcd
rw [mul_add, add_right_comm, mul_add, ← add_assoc]
gcongr
assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 297,
"column": 2
} | {
"line": 300,
"column": 12
} | {
"line": 302,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b c d : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddLeftMono R\ninst✝ : AddLeftReflectLE R\nhab : a ≤ b\nhcd : c ≤ d\n⊢ a * d + b * c ≤ a * c + b * d",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨d, hd, rfl⟩ := exists_nonneg_add_of_le hcd
rw [mul_add, add_right_comm, mul_add, ← add_assoc]
gcongr
assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 46
} | {
"line": 336,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\na b : R\ninst✝¹ : MulPosStrictMono R\ninst✝ : PosMulStrictMono R\nhab : 0 ≤ a * b\n⊢ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Decidable.or_iff_not_not_and_not",
... | [
"R : Type u\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\na b : R\ninst✝¹ : MulPosStrictMono R\ninst✝ : PosMulStrictMono R\nhab : 0 ≤ a * b\n⊢ ¬(¬(0 ≤ a ∧ 0 ≤ b) ∧ ¬(a ≤ 0 ∧ b ≤ 0))"
] | refine Decidable.or_iff_not_not_and_not.2 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.Group.Unbundled.Abs | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 55
} | {
"line": 128,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : MulLeftMono α\na b : α\n⊢ a * b ≤ |a|ₘ * |b|ₘ",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"CommMonoid.toCommSemigroup",
"Monoid.toMulOneClass",
"PartialOrder.toPreorder",
"Preorder.to... | [] | exact mul_le_mul' (le_mabs_self a) (le_mabs_self b) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Group.Unbundled.Abs | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 55
} | {
"line": 128,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : MulLeftMono α\na b : α\n⊢ a * b ≤ |a|ₘ * |b|ₘ",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"CommMonoid.toCommSemigroup",
"Monoid.toMulOneClass",
"PartialOrder.toPreorder",
"Preorder.to... | [] | exact mul_le_mul' (le_mabs_self a) (le_mabs_self b) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Unbundled.Abs | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 55
} | {
"line": 128,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : MulLeftMono α\na b : α\n⊢ a * b ≤ |a|ₘ * |b|ₘ",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"CommMonoid.toCommSemigroup",
"Monoid.toMulOneClass",
"PartialOrder.toPreorder",
"Preorder.to... | [] | exact mul_le_mul' (le_mabs_self a) (le_mabs_self b) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Unbundled.Abs | {
"line": 173,
"column": 39
} | {
"line": 174,
"column": 69
} | {
"line": 175,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : MulLeftMono α\na b c : α\nthis : DistribLattice α := CommGroup.toDistribLattice α\n⊢ (b ⊔ a ⊔ c) * ((b ⊔ a) ⊓ c) / ((b ⊓ a ⊔ c) * (b ⊓ a ⊓ c)) = (b ⊔ a) * c / ((b ⊓ a) * c)",
"ppTerm": "?m.228",
"assigned": true,
"usedConstants... | [] | by
rw [mul_comm, inf_mul_sup, mul_comm (b ⊓ a ⊔ c), inf_mul_sup] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Int.Parity | {
"line": 156,
"column": 67
} | {
"line": 156,
"column": 78
} | {
"line": 156,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DivisionMonoid α\ninst✝ : HasDistribNeg α\nn : ℤ\nh : Odd n\n⊢ (-1) ^ n = -1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
... | [
"α : Type u_1\ninst✝¹ : DivisionMonoid α\ninst✝ : HasDistribNeg α\nn : ℤ\nh : Odd n\n⊢ -1 ^ n = -1"
] | h.neg_zpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Unbundled.Abs | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 21
} | {
"line": 276,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝³ : Group α\ninst✝² : LinearOrder α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\na b : α\nba : b ≤ a\n⊢ 1 ≤ a / b",
"ppTerm": "?inr✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.to... | [] | rwa [one_le_div'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Order.Ring.Int | {
"line": 72,
"column": 17
} | {
"line": 72,
"column": 54
} | {
"line": 72,
"column": 54
} | [
{
"pp": "q n : ℕ\ndvd : gcd 0 q ∣ n\nle : pred 0 * q.pred ≤ n\nb : ℕ\neq : n = q * b\n⊢ 0 * 0 + b * q = n",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"... | [] | by simpa [mul_comm, eq_comm] using eq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Int | {
"line": 75,
"column": 17
} | {
"line": 75,
"column": 54
} | {
"line": 75,
"column": 54
} | [
{
"pp": "n p : ℕ\ndvd : (p + 1).gcd 0 ∣ n\nle : (p + 1).pred * pred 0 ≤ n\na : ℕ\neq : n = (p.gcd 0 + 1).gcd 0 * a\n⊢ a * (p + 1) + 0 * 0 = n",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Nat.gcd_ze... | [] | by simpa [mul_comm, eq_comm] using eq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Defs | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 62
} | {
"line": 149,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝³ : MulOneClass α\ninst✝² : PartialOrder α\ninst✝¹ : CanonicallyOrderedMul α\na b : α\ninst✝ : MulLeftStrictMono α\n⊢ a < b ↔ ∃ c, c > 1 ∧ b = a * c",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"lt_iff_le_and_ne",
"Eq.mpr",
"MulOne.toOne",
... | [
"α : Type u\ninst✝³ : MulOneClass α\ninst✝² : PartialOrder α\ninst✝¹ : CanonicallyOrderedMul α\na b : α\ninst✝ : MulLeftStrictMono α\n⊢ (∃ x, b = a * x ∧ a ≠ b) ↔ ∃ c, c > 1 ∧ b = a * c"
] | rw [lt_iff_le_and_ne, le_iff_exists_mul, ← exists_and_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Int.GCD | {
"line": 304,
"column": 2
} | {
"line": 309,
"column": 64
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case neg.refine_1\nα : Type u_1\ninst✝ : GroupWithZero α\na b : α\nm n : ℕ\nhab : Commute a b\nhmn : m.Coprime n\nh : a ^ m = b ^ n\nh✝¹ : ¬m = 0\nh✝ : ¬n = 0\nhb : ¬b = 0\nha : ¬a = 0\n⊢ a = (a ^ m.gcdB n * b ^ m.gcdA n) ^ n",
"ppTerm": "?neg.refine_1✝",
"assigned": true,
"usedConstants": ... | [] | · refine (pow_one _).symm.trans ?_
conv_lhs => rw [← zpow_natCast, ← hmn, Nat.gcd_eq_gcd_ab]
simp only [zpow_add₀ ha, zpow_add₀ hb, ← zpow_natCast, (hab.zpow_zpow₀ _ _).mul_zpow,
← zpow_mul, mul_comm (Nat.gcdB m n), mul_comm (Nat.gcdA m n)]
simp only [zpow_mul, zpow_natCast, h]
exact ((Commute.pow... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Int.GCD | {
"line": 304,
"column": 2
} | {
"line": 309,
"column": 64
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\ninst✝ : GroupWithZero α\na b : α\nm n : ℕ\nhab : Commute a b\nhmn : m.Coprime n\nh : a ^ m = b ^ n\nh✝¹ : ¬m = 0\nh✝ : ¬n = 0\nhb : ¬b = 0\nha : ¬a = 0\n⊢ b = (a ^ m.gcdB n * b ^ m.gcdA n) ^ m",
"ppTerm": "?neg.refine_2✝",
"assigned": true,
"usedConstants": ... | [] | · refine (pow_one _).symm.trans ?_
conv_lhs => rw [← zpow_natCast, ← hmn, Nat.gcd_eq_gcd_ab]
simp only [zpow_add₀ ha, zpow_add₀ hb, ← zpow_natCast, (hab.zpow_zpow₀ _ _).mul_zpow,
← zpow_mul, mul_comm (Nat.gcdB m n), mul_comm (Nat.gcdA m n)]
simp only [zpow_mul, zpow_natCast, h]
exact ((Commute.pow... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.GroupWithZero.WithZero | {
"line": 256,
"column": 17
} | {
"line": 256,
"column": 71
} | {
"line": 257,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DivInvMonoid α\nn : ℕ\n⊢ none ^ ↑n.succ = none ^ ↑n * none",
"ppTerm": "?m.133",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"With... | [] | by change 0 ^ _ = 0 ^ _ * 0; simp only [mul_zero]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.WithZero | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 13
} | {
"line": 475,
"column": 2
} | [
{
"pp": "G₀ : Type u_1\nM₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : MulZeroOneClass M₀\ninst✝ : Nontrivial M₀\nf : G₀ →*₀ M₀\nx : G₀\n⊢ f x = 0 → x = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"MonoidWithZeroHom.funLike",
"i... | [
"G₀ : Type u_1\nM₀ : Type u_2\ninst✝² : GroupWithZero G₀\ninst✝¹ : MulZeroOneClass M₀\ninst✝ : Nontrivial M₀\nf : G₀ →*₀ M₀\nx : G₀\n⊢ x ≠ 0 → f x ≠ 0"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 24
} | {
"line": 128,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : Add α\na : α\nha : IsAddLeftRegular a\n⊢ IsAddLeftRegular ↑a",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Option.casesOn",
"Option.some",
"WithTop.some",
"Option.none",
"instHAdd",
"HAdd.hAdd",
"Eq",
"WithTop.a... | [
"case none.none\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddLeftRegular a\n⊢ (fun x ↦ ↑a + x) none = (fun x ↦ ↑a + x) none → none = none",
"case none.some\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddLeftRegular a\nc : α\n⊢ (fun x ↦ ↑a + x) none = (fun x ↦ ↑a + x) (Option.some c) → none = Option.some c",
"case... | rintro (_ | b) (_ | c) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 24
} | {
"line": 132,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : Add α\na : α\nha : IsAddRightRegular a\n⊢ IsAddRightRegular ↑a",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Option.casesOn",
"Option.some",
"WithTop.some",
"Option.none",
"instHAdd",
"HAdd.hAdd",
"Eq",
"WithTop... | [
"case none.none\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddRightRegular a\n⊢ (fun x ↦ x + ↑a) none = (fun x ↦ x + ↑a) none → none = none",
"case none.some\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddRightRegular a\nc : α\n⊢ (fun x ↦ x + ↑a) none = (fun x ↦ x + ↑a) (Option.some c) → none = Option.some c",
"ca... | rintro (_ | b) (_ | c) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 24
} | {
"line": 138,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\n⊢ AddLECancellable ↑a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Option.casesOn",
"Option.some",
"LE.le",
"WithTop.some",
"Option.none",
"instHAdd",
"HAdd.hAdd"... | [
"case none.none\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\n⊢ ↑a + none ≤ ↑a + none → none ≤ none",
"case none.some\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\nc : α\n⊢ ↑a + none ≤ ↑a + Option.some c → none ≤ Option.some c",
"case some.none\nα : Type... | rintro (_ | b) (_ | c) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 22
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case none.none\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\n⊢ ↑a + none ≤ ↑a + none → none ≤ none",
"ppTerm": "?none.none",
"assigned": true,
"usedConstants": [
"congrArg",
"le_top._simp_2",
"LE.le",
"WithTop.add_top",
"imp_self._s... | [] | simp [none_eq_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 22
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case none.none\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\n⊢ ↑a + none ≤ ↑a + none → none ≤ none",
"ppTerm": "?none.none",
"assigned": true,
"usedConstants": [
"congrArg",
"le_top._simp_2",
"LE.le",
"WithTop.add_top",
"imp_self._s... | [] | simp [none_eq_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 22
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case none.none\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\n⊢ ↑a + none ≤ ↑a + none → none ≤ none",
"ppTerm": "?none.none",
"assigned": true,
"usedConstants": [
"congrArg",
"le_top._simp_2",
"LE.le",
"WithTop.add_top",
"imp_self._s... | [] | simp [none_eq_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 22
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case none.some\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\nc : α\n⊢ ↑a + none ≤ ↑a + Option.some c → none ≤ Option.some c",
"ppTerm": "?none.some",
"assigned": true,
"usedConstants": [
"WithTop.add_eq_top._simp_1",
"False",
"Option.ctorIdx",
... | [] | simp [none_eq_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 22
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case none.some\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\nc : α\n⊢ ↑a + none ≤ ↑a + Option.some c → none ≤ Option.some c",
"ppTerm": "?none.some",
"assigned": true,
"usedConstants": [
"WithTop.add_eq_top._simp_1",
"False",
"Option.ctorIdx",
... | [] | simp [none_eq_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 22
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case none.some\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\nc : α\n⊢ ↑a + none ≤ ↑a + Option.some c → none ≤ Option.some c",
"ppTerm": "?none.some",
"assigned": true,
"usedConstants": [
"WithTop.add_eq_top._simp_1",
"False",
"Option.ctorIdx",
... | [] | simp [none_eq_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Sub.Unbundled.Basic | {
"line": 92,
"column": 35
} | {
"line": 92,
"column": 65
} | {
"line": 92,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : AddLeftMono α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhd : AddLECancellable d\nhba : b ≤ a\nhdc : d ≤ c\n⊢ a + (c - d) - b = a + c - (b + d)",
"ppTerm": "?m.54... | [
"α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : AddLeftMono α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhd : AddLECancellable d\nhba : b ≤ a\nhdc : d ≤ c\n⊢ a + c - d - b = a + c - (b + d)"
] | ← hd.add_tsub_assoc_of_le hdc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 24
} | {
"line": 489,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : Add α\na : α\nha : IsAddLeftRegular a\n⊢ IsAddLeftRegular ↑a",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"WithBot.some",
"WithBot",
"Option.casesOn",
"Option.some",
"Option.none",
"instHAdd",
"HAdd.hAdd",
"With... | [
"case none.none\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddLeftRegular a\n⊢ (fun x ↦ ↑a + x) none = (fun x ↦ ↑a + x) none → none = none",
"case none.some\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddLeftRegular a\nc : α\n⊢ (fun x ↦ ↑a + x) none = (fun x ↦ ↑a + x) (Option.some c) → none = Option.some c",
"case... | rintro (_ | b) (_ | c) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 24
} | {
"line": 493,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : Add α\na : α\nha : IsAddRightRegular a\n⊢ IsAddRightRegular ↑a",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"WithBot.some",
"WithBot",
"Option.casesOn",
"Option.some",
"Option.none",
"instHAdd",
"HAdd.hAdd",
"Wi... | [
"case none.none\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddRightRegular a\n⊢ (fun x ↦ x + ↑a) none = (fun x ↦ x + ↑a) none → none = none",
"case none.some\nα : Type u\ninst✝ : Add α\na : α\nha : IsAddRightRegular a\nc : α\n⊢ (fun x ↦ x + ↑a) none = (fun x ↦ x + ↑a) (Option.some c) → none = Option.some c",
"ca... | rintro (_ | b) (_ | c) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Monoid.Unbundled.WithTop | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 24
} | {
"line": 499,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\n⊢ AddLECancellable ↑a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"WithBot.some",
"WithBot",
"Option.casesOn",
"Option.some",
"LE.le",
"Option.none",
"instHAdd",
... | [
"case none.none\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\n⊢ ↑a + none ≤ ↑a + none → none ≤ none",
"case none.some\nα : Type u\ninst✝¹ : Add α\na : α\ninst✝ : LE α\nha : AddLECancellable a\nc : α\n⊢ ↑a + none ≤ ↑a + Option.some c → none ≤ Option.some c",
"case some.none\nα : Type... | rintro (_ | b) (_ | c) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Floor.Defs | {
"line": 306,
"column": 46
} | {
"line": 306,
"column": 91
} | {
"line": 308,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : Ring α\ninst✝¹ : LinearOrder α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ z < ⌊a⌋ ↔ ↑z + 1 ≤ a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.floor",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",... | [] | by rw [← add_one_le_iff, le_floor]; norm_cast | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.GroupWithZero.Canonical | {
"line": 511,
"column": 41
} | {
"line": 511,
"column": 88
} | {
"line": 513,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedCancelMonoid α\na : α\nx✝ : 0 < ↑a\nb c : α\nhbc : ↑b < ↑c\n⊢ ↑a * ↑b < ↑a * ↑c",
"ppTerm": "?m.218",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
... | [] | by norm_cast at *; exact mul_lt_mul_right hbc _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Floor.Semiring | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 20
} | {
"line": 162,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorSemiring R\ninst✝ : IsStrictOrderedRing R\nn : ℕ\nhn : n ≠ 0\na : R\n⊢ ⌊↑n * a⌋₊ / n = ⌊a⌋₊",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Nat.cast_comm",
"Eq.mpr",
"NonAssocSemiring.toAdd... | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorSemiring R\ninst✝ : IsStrictOrderedRing R\nn : ℕ\nhn : n ≠ 0\na : R\n⊢ ⌊a * ↑n⌋₊ / n = ⌊a⌋₊"
] | Nat.cast_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Floor.Semiring | {
"line": 291,
"column": 95
} | {
"line": 293,
"column": 25
} | {
"line": 295,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\ninst✝ : FloorSemiring R\na : R\nha : 0 ≤ a\n⊢ Nat.cast ⁻¹' Iic a = Iic ⌊a⌋₊",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ext",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"Partial... | [] | by
ext
simp [le_floor_iff, ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Floor.Semiring | {
"line": 341,
"column": 4
} | {
"line": 341,
"column": 15
} | {
"line": 342,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorSemiring R\na : R\ninst✝ : IsStrictOrderedRing R\nha : 0 ≤ a\nn b : ℕ\n⊢ ⌈a + ↑n⌉₊ ≤ b ↔ ⌈a⌉₊ + n ≤ b",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWit... | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorSemiring R\na : R\ninst✝ : IsStrictOrderedRing R\nha : 0 ≤ a\nn b : ℕ\n⊢ b < ⌈a + ↑n⌉₊ ↔ b < ⌈a⌉₊ + n"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.Order.Floor.Semiring | {
"line": 446,
"column": 6
} | {
"line": 446,
"column": 44
} | {
"line": 447,
"column": 6
} | [
{
"pp": "case inr\nR : Type u_3\ninst✝¹ : Semiring R\ninst✝ : LinearOrder R\nH₁ H₂ : FloorSemiring R\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : R\nh : 0 ≤ a\n⊢ FloorSemiring.floor a = FloorSemiring.floor a",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"SemilatticeInf.toPar... | [
"case inr\nR : Type u_3\ninst✝¹ : Semiring R\ninst✝ : LinearOrder R\nH₁ H₂ : FloorSemiring R\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : R\nh : 0 ≤ a\nn : ℕ\n⊢ n ≤ FloorSemiring.floor a ↔ n ≤ FloorSemiring.floor a"
] | refine eq_of_forall_le_iff fun n => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Field.Basic | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 54
} | {
"line": 123,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionRing K\na b : K\nha : a ≠ 0\nhb : b ≠ 0\n⊢ 1 / a * (b - a) * (1 / b) = 1 / a - 1 / b",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul... | [] | simpa only [one_div] using (inv_sub_inv' ha hb).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Field.Basic | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 54
} | {
"line": 123,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionRing K\na b : K\nha : a ≠ 0\nhb : b ≠ 0\n⊢ 1 / a * (b - a) * (1 / b) = 1 / a - 1 / b",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul... | [] | simpa only [one_div] using (inv_sub_inv' ha hb).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Field.Basic | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 54
} | {
"line": 123,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionRing K\na b : K\nha : a ≠ 0\nhb : b ≠ 0\n⊢ 1 / a * (b - a) * (1 / b) = 1 / a - 1 / b",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul... | [] | simpa only [one_div] using (inv_sub_inv' ha hb).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.NAry | {
"line": 39,
"column": 2
} | {
"line": 40,
"column": 41
} | {
"line": 42,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nf : α → β → γ\ns s' : Set α\nt t' : Set β\nhs : s ⊆ s'\nht : t ⊆ t'\n⊢ image2 f s t ⊆ image2 f s' t'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exists",
"And.casesOn",
"And",
"Exists.ca... | [] | rintro _ ⟨a, ha, b, hb, rfl⟩
exact mem_image2_of_mem (hs ha) (ht hb) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.NAry | {
"line": 39,
"column": 2
} | {
"line": 40,
"column": 41
} | {
"line": 42,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nf : α → β → γ\ns s' : Set α\nt t' : Set β\nhs : s ⊆ s'\nht : t ⊆ t'\n⊢ image2 f s t ⊆ image2 f s' t'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exists",
"And.casesOn",
"And",
"Exists.ca... | [] | rintro _ ⟨a, ha, b, hb, rfl⟩
exact mem_image2_of_mem (hs ha) (ht hb) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Abs | {
"line": 32,
"column": 20
} | {
"line": 32,
"column": 31
} | {
"line": 32,
"column": 32
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\nn : ℤ\na : α\nn0 : n ≤ 0\nm : ℕ\nh : -n = ↑m\n⊢ |(a ^ n)⁻¹|ₘ = |a|ₘ ^ |n|",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
... | [
"case inr\nα : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\nn : ℤ\na : α\nn0 : n ≤ 0\nm : ℕ\nh : -n = ↑m\n⊢ |a ^ (-n)|ₘ = |a|ₘ ^ |n|"
] | ← zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.NAry | {
"line": 323,
"column": 2
} | {
"line": 324,
"column": 51
} | {
"line": 326,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nf : α → β → γ\ns s' : Set α\nt t' : Set β\n⊢ image2 f (s ∪ s') (t ∩ t') ⊆ image2 f s t ∪ image2 f s' t'",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Set.image2_subset",
"Set.union_subset_uni... | [] | rw [image2_union_left]
nth_grw 1 [inter_subset_left, inter_subset_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.NAry | {
"line": 323,
"column": 2
} | {
"line": 324,
"column": 51
} | {
"line": 326,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nf : α → β → γ\ns s' : Set α\nt t' : Set β\n⊢ image2 f (s ∪ s') (t ∩ t') ⊆ image2 f s t ∪ image2 f s' t'",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Set.image2_subset",
"Set.union_subset_uni... | [] | rw [image2_union_left]
nth_grw 1 [inter_subset_left, inter_subset_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.WellFounded | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | {
"line": 45,
"column": 2
} | [
{
"pp": "α : Sort u_1\nr : α → α → Prop\nx : α\n⊢ Acc r x ↔ IsEmpty { f // f 0 = x ∧ ∀ (n : ℕ), r (f (n + 1)) (f n) }",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"congrArg",
"id",
"Subtype",
"ins... | [
"α : Sort u_1\nr : α → α → Prop\nx : α\n⊢ ¬Acc r x ↔ Nonempty { f // f 0 = x ∧ ∀ (n : ℕ), r (f (n + 1)) (f n) }"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.WellFounded | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 49
} | {
"line": 153,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\nh : ∀ (s : Set α), s.Nonempty → ∃ m, m ∈ s ∧ ∀ (x : α), x ∈ s → ¬r x m\nx : α\nhx : ¬Acc r x\n⊢ False",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"False",
"Set.ofPred",
"Membership.mem",
"Exists",
"And.casesOn",
... | [
"α : Type u_1\nr : α → α → Prop\nh : ∀ (s : Set α), s.Nonempty → ∃ m, m ∈ s ∧ ∀ (x : α), x ∈ s → ¬r x m\nx : α\nhx : ¬Acc r x\nm : α\nhm : m ∈ {x | ¬Acc r x}\nhm' : ∀ (x : α), x ∈ {x | ¬Acc r x} → ¬r x m\n⊢ False"
] | obtain ⟨m, hm, hm'⟩ := h {x | ¬Acc r x} ⟨x, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Logic.Equiv.Set | {
"line": 481,
"column": 33
} | {
"line": 481,
"column": 67
} | {
"line": 482,
"column": 2
} | [
{
"pp": "α✝¹ : Sort u\nβ✝¹ : Sort v\nγ : Sort w\nα✝ : Type u_1\nβ✝ : Type u_2\nα : Type u_3\nβ : Type u_4\nval✝ : β\nh : Sum.inr val✝ ∈ range Sum.inl\n⊢ False",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"False",
"Sum.ctorIdx",
"HEq.refl",
"False.elim",
"n... | [] | by rcases h with ⟨x, h'⟩; cases h' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Equiv.Set | {
"line": 493,
"column": 33
} | {
"line": 493,
"column": 67
} | {
"line": 494,
"column": 2
} | [
{
"pp": "α✝¹ : Sort u\nβ✝¹ : Sort v\nγ : Sort w\nα✝ : Type u_1\nβ✝ : Type u_2\nα : Type u_3\nβ : Type u_4\nval✝ : α\nh : Sum.inl val✝ ∈ range Sum.inr\n⊢ False",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"False",
"Sum.ctorIdx",
"HEq.refl",
"False.elim",
"n... | [] | by rcases h with ⟨x, h'⟩; cases h' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.Basic | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 13
} | {
"line": 377,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ Icc a b = ∅ ↔ ¬a ≤ b",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Preorder.toLE",
"id",
"LE.le",
"Iff",
"Set.Nonempty",
"Set.Icc",
"Set.instEmptyCollection",... | [
"α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ (Icc a b).Nonempty ↔ a ≤ b"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.Interval.Set.Basic | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 13
} | {
"line": 381,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ Ico a b = ∅ ↔ ¬a < b",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"id",
"Set.Ico",
"Iff",
"Set.Nonempty",
"LT.lt",
"Set.instEmptyCollection",... | [
"α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ (Ico a b).Nonempty ↔ a < b"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.Interval.Set.Basic | {
"line": 385,
"column": 2
} | {
"line": 385,
"column": 13
} | {
"line": 385,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\na b : α\ninst✝ : DenselyOrdered α\n⊢ Ioo a b = ∅ ↔ ¬a < b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"id",
"Iff",
"Set.Nonempty",
"LT.lt",
"Set.instEmptyCo... | [
"α : Type u_1\ninst✝¹ : Preorder α\na b : α\ninst✝ : DenselyOrdered α\n⊢ (Ioo a b).Nonempty ↔ a < b"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Order.Bounds.Basic | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 16
} | {
"line": 145,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : Preorder α\ns t : Set α\nhc : IsCofinalFor s t\na : α\nhas : a ∈ s\n⊢ ∃ b ∈ t, a ≤ b",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hc",
"a",
"has"
],
"usedGoals": []
},
{
"pp": "case inr\nα : ... | [
"case inr\nα : Type u_1\ninst✝ : Preorder α\ns t : Set α\nhc : IsCofinalFor s t\na : α\nhat : a ∈ t\n⊢ ∃ b ∈ t, a ≤ b"
] | · exact hc has | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Bounds.Basic | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 11
} | {
"line": 196,
"column": 4
} | [
{
"pp": "case inl.inl\nα : Type u_1\ninst✝ : Preorder α\nt : Set α\nhn : (∅ ∪ t).Nonempty\nh✝ : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) (∅ ∪ t)\nh : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) ∅\n⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) ∅ ∧ ∅.Nonempty ∨ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) t ∧ t.Nonempty",
"ppTerm": "?inl.inl",
"a... | [
"case inl.inr\nα : Type u_1\ninst✝ : Preorder α\ns t : Set α\nhn : (s ∪ t).Nonempty\nh✝ : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) (s ∪ t)\nh : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\nhs : s.Nonempty\n⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s ∧ s.Nonempty ∨ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) t ∧ t.Nonempty"
] | · aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Bounds.Basic | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 11
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝ : Preorder α\ns : Set α\nhn : (s ∪ ∅).Nonempty\nh✝ : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) (s ∪ ∅)\nh : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) ∅\n⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s ∧ s.Nonempty ∨ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) ∅ ∧ ∅.Nonempty",
"ppTerm": "?inr.inl",
"a... | [
"case inr.inr\nα : Type u_1\ninst✝ : Preorder α\ns t : Set α\nhn : (s ∪ t).Nonempty\nh✝ : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) (s ∪ t)\nh : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) t\nht : t.Nonempty\n⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s ∧ s.Nonempty ∨ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) t ∧ t.Nonempty"
] | · aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Tactic.Positivity.Core | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 35
} | {
"line": 190,
"column": 0
} | [
{
"pp": "A : Type u_1\ne : A\nn d : ℕ\ninst✝² : Semiring A\ninst✝¹ : LinearOrder A\ninst✝ : IsStrictOrderedRing A\ninv : Invertible ↑d\neq : e = ↑n * ⅟↑d\nh : decide (0 < n) = true\npos_invOf_d : 0 < ⅟↑d\npos_n : 0 < ↑n\n⊢ 0 < ↑n * ⅟↑d",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
... | [] | exact mul_pos pos_n pos_invOf_d | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Tactic.Positivity.Core | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 35
} | {
"line": 198,
"column": 0
} | [
{
"pp": "A : Type u_1\ne : A\nn : ℤ\nd : ℕ\ninst✝² : Ring A\ninst✝¹ : LinearOrder A\ninst✝ : IsStrictOrderedRing A\ninv : Invertible ↑d\neq : e = ↑n * ⅟↑d\nh : decide (0 < n) = true\npos_invOf_d : 0 < ⅟↑d\npos_n : 0 < ↑n\n⊢ 0 < ↑n * ⅟↑d",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
... | [] | exact mul_pos pos_n pos_invOf_d | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Field.Basic | {
"line": 326,
"column": 41
} | {
"line": 327,
"column": 39
} | {
"line": 329,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : Field α\ninst✝² : PartialOrder α\ninst✝¹ : PosMulReflectLT α\ninst✝ : IsStrictOrderedRing α\na : α\n⊢ a⁻¹ ≤ 0 ↔ a ≤ 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra.Order.Field.Basic.0.inv_nonpos'._proof_1_1"
],
"used... | [] | by
grind [inv_lt_zero', le_iff_eq_or_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.TakeDrop | {
"line": 190,
"column": 69
} | {
"line": 191,
"column": 67
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u\ni j : ℕ\nxs : List α\n⊢ (dropSlice i j xs).length = xs.length - min j (xs.length - i)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.List.TakeDrop.0.List.length_dropSlice._proof_1_1",
"_private.Mathlib.Data.List.TakeDrop.0.List.lengt... | [] | by
induction xs generalizing i j with cases i with grind [dropSlice] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.Forall2 | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 47
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR S : α → β → Prop\nH : ∀ (a : α) (b : β), R a b → S a b\nl₁ : List α\nl₂ : List β\nh : Forall₂ R l₁ l₂\n⊢ Forall₂ S l₁ l₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"List.Forall₂.cons",
"List.Forall₂.rec",
"List.Forall₂.nil",
... | [] | induction h <;> constructor <;> solve_by_elim | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.List.Forall2 | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 47
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR S : α → β → Prop\nH : ∀ (a : α) (b : β), R a b → S a b\nl₁ : List α\nl₂ : List β\nh : Forall₂ R l₁ l₂\n⊢ Forall₂ S l₁ l₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"List.Forall₂.cons",
"List.Forall₂.rec",
"List.Forall₂.nil",
... | [] | induction h <;> constructor <;> solve_by_elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Forall2 | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 47
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR S : α → β → Prop\nH : ∀ (a : α) (b : β), R a b → S a b\nl₁ : List α\nl₂ : List β\nh : Forall₂ R l₁ l₂\n⊢ Forall₂ S l₁ l₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"List.Forall₂.cons",
"List.Forall₂.rec",
"List.Forall₂.nil",
... | [] | induction h <;> constructor <;> solve_by_elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Perm.Basic | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 40
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type u_2\nf g : α → List β\na : α\nl : List α\nIH : flatMap f l ++ flatMap g l ~ flatMap (fun x ↦ f x ++ g x) l\n⊢ flatMap f (a :: l) ++ flatMap g (a :: l) ~ flatMap (fun x ↦ f x ++ g x) (a :: l)",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"E... | [
"case cons\nα : Type u_1\nβ : Type u_2\nf g : α → List β\na : α\nl : List α\nIH : flatMap f l ++ flatMap g l ~ flatMap (fun x ↦ f x ++ g x) l\n⊢ f a ++ (flatMap f l ++ (g a ++ flatMap g l)) ~ f a ++ (g a ++ flatMap (fun x ↦ f x ++ g x) l)"
] | simp only [flatMap_cons, append_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.FieldSimp.Lemmas | {
"line": 163,
"column": 29
} | {
"line": 166,
"column": 32
} | {
"line": 168,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : MonoidWithZero M\ninst✝² : PartialOrder M\ninst✝¹ : PosMulMono M\ninst✝ : PosMulReflectLE M\ne₁ e₂ f₁ f₂ L : M\nH₁ : e₁ = L * f₁\nH₂ : e₂ = L * f₂\nHL : 0 < L\n⊢ (e₁ ≤ e₂) = (f₁ ≤ f₂)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"... | [] | by
subst H₁ H₂
apply Iff.eq
exact mul_le_mul_iff_right₀ HL | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.FieldSimp.Lemmas | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 16
} | {
"line": 278,
"column": 17
} | [
{
"pp": "M : Type u_1\ninst✝ : CommGroupWithZero M\nn : ℤ\ne : M\nL l l' : NF M\nh : L.eval * l.eval = l'.eval\n⊢ L.eval * ((n, e) ::ᵣ l).eval = ((n, e) ::ᵣ l').eval",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"HMul.hMul",
... | [
"M : Type u_1\ninst✝ : CommGroupWithZero M\nn : ℤ\ne : M\nL l l' : NF M\nh : L.eval * l.eval = l'.eval\n⊢ L.eval * (l.eval * zpow' (n, e).2 (n, e).1) = ((n, e) ::ᵣ l').eval"
] | eval_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
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