module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 15
} | [
{
"pp": "a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nt : p ∈ a.factorization.support ∩ b.factorization.support\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 737,
"column": 2
} | {
"line": 737,
"column": 86
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf : MvPolynomial σ R\n⊢ m.degree (m.leadingTerm f) = m.degree f",
"usedConstants": [
"Eq.mpr",
"MonomialOrder.degree_monomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Semiring.toMod... | simp only [leadingTerm, degree_monomial, leadingCoeff_eq_zero_iff, ite_eq_right_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 780,
"column": 2
} | {
"line": 780,
"column": 34
} | [
{
"pp": "σ✝ : Type u_1\nm✝ : MonomialOrder σ✝\nR✝ : Type u_2\ninst✝² : CommSemiring R✝\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np p' q : MvPolynomial σ R\nhp : p ≠ 0\nhq : q ≠ 0\nh : m.toSyn (m.degree (p * q)) < m.toSyn (m.degree p' + m.degree q)\n⊢ m.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 787,
"column": 2
} | {
"line": 787,
"column": 92
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np p' q : MvPolynomial σ R\nhp : p ≠ 0\nhq : q ≠ 0\nh : m.toSyn (m.degree p) < m.toSyn (m.degree p')\n⊢ m.toSyn (m.degree (p * q)) < m.toSyn (m.degree (p' * q))",
"usedConstants": [
"IsRightCanc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 13
} | [
{
"pp": "p a m n : ℕ\nhp : Prime p\nh : p ^ m = a ^ n\nthis : Finsupp.single p m = n • a.factorization\n⊢ m = n * a.factorization p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 197,
"column": 13
} | {
"line": 197,
"column": 27
} | [
{
"pp": "case h\nk x✝ : ℕ\n⊢ (ζ * pow k) x✝ = { toFun := fun n ↦ ∑ d ∈ n.divisors, d ^ k, map_zero' := ⋯ } x✝",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.instMul",
"congrArg",
"Arithmet... | zeta_mul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 32
} | [
{
"pp": "k n : ℕ\n⊢ ∑ i ∈ n.divisors, n ^ k ≤ n * n ^ k",
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Preorder.toLE",
"id",
"instMulNat",
"LE.le",
"instNatPowNat",
"Nat.divisors",
"HPow.hPow",
"Nat.instPreorder"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 64
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\nf g : ArithmeticFunction R\nN n : ℕ\nhn : 0 < n ∧ n ≤ N\n⊢ ∑ x ∈ n.divisorsAntidiagonal, f x.1 * g x.2 = ∑ x ∈ Ioc 0 N ×ˢ Ioc 0 N with x.1 * x.2 = n, f x.1 * g x.2",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"Nat.divisorsAnt... | rw [divisorsAntidiagonal_eq_prod_filter_of_le hn.1.ne' hn.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 396,
"column": 2
} | {
"line": 403,
"column": 47
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\nf g : ArithmeticFunction R\nN : ℕ\n⊢ ∑ n ∈ Ioc 0 N, (f * g) n = ∑ x ∈ Ioc 0 N ×ˢ Ioc 0 N with x.1 * x.2 ≤ N, f x.1 * g x.2",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Preorder.toLT",
"HMul.hMul",
"Nat.divisorsAntidiagonal",... | simp only [mul_apply]
trans ∑ n ∈ Ioc 0 N, ∑ x ∈ Ioc 0 N ×ˢ Ioc 0 N with x.1 * x.2 = n, f x.1 * g x.2
· refine sum_congr rfl fun n hn ↦ ?_
simp only [mem_Ioc] at hn
rw [divisorsAntidiagonal_eq_prod_filter_of_le hn.1.ne' hn.2]
· simp_rw [sum_filter]
rw [sum_comm]
exact sum_congr rfl fun _ _ ↦ (by s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 396,
"column": 2
} | {
"line": 403,
"column": 47
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\nf g : ArithmeticFunction R\nN : ℕ\n⊢ ∑ n ∈ Ioc 0 N, (f * g) n = ∑ x ∈ Ioc 0 N ×ˢ Ioc 0 N with x.1 * x.2 ≤ N, f x.1 * g x.2",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Preorder.toLT",
"HMul.hMul",
"Nat.divisorsAntidiagonal",... | simp only [mul_apply]
trans ∑ n ∈ Ioc 0 N, ∑ x ∈ Ioc 0 N ×ˢ Ioc 0 N with x.1 * x.2 = n, f x.1 * g x.2
· refine sum_congr rfl fun n hn ↦ ?_
simp only [mem_Ioc] at hn
rw [divisorsAntidiagonal_eq_prod_filter_of_le hn.1.ne' hn.2]
· simp_rw [sum_filter]
rw [sum_comm]
exact sum_congr rfl fun _ _ ↦ (by s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 272,
"column": 41
} | {
"line": 272,
"column": 64
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : ArithmeticFunction M\nx : ℕ\nx0 : ¬x = 0\nh : {(1, x)} ⊆ x.divisorsAntidiagonal\ny : ℕ × ℕ\nymem : y ∈ x.divisorsAntidiagonal\nynotMem : y ∉ {(1, x)}\ncon : y.1 = 1\n⊢ False",
"usedConstants": [
... | simp_all [Prod.ext_iff] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 272,
"column": 41
} | {
"line": 272,
"column": 64
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : ArithmeticFunction M\nx : ℕ\nx0 : ¬x = 0\nh : {(1, x)} ⊆ x.divisorsAntidiagonal\ny : ℕ × ℕ\nymem : y ∈ x.divisorsAntidiagonal\nynotMem : y ∉ {(1, x)}\ncon : y.1 = 1\n⊢ False",
"usedConstants": [
... | simp_all [Prod.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 272,
"column": 41
} | {
"line": 272,
"column": 64
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : ArithmeticFunction M\nx : ℕ\nx0 : ¬x = 0\nh : {(1, x)} ⊆ x.divisorsAntidiagonal\ny : ℕ × ℕ\nymem : y ∈ x.divisorsAntidiagonal\nynotMem : y ∉ {(1, x)}\ncon : y.1 = 1\n⊢ False",
"usedConstants": [
... | simp_all [Prod.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 13
} | [
{
"pp": "case h.e'_3\nN : ℕ\n⊢ ∑ n ∈ Ioc 0 N, N / n = ∑ n ∈ Ioc 0 N, zeta n * ↑(N / n)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"instHDiv",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 1002,
"column": 2
} | {
"line": 1002,
"column": 17
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\nh : m.degree f = m.degree g\nhs : m.sPolynomial f g ≠ 0\n⊢ m.toSyn (m.degree (m.sPolynomial f g)) < m.toSyn (m.degree f)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Preorder.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 428,
"column": 48
} | {
"line": 428,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : ArithmeticFunction R\nthis : Invertible (f 1) → Invertible f\n⊢ IsUnit (f 1) → IsUnit f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 428,
"column": 4
} | {
"line": 429,
"column": 86
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\nf : ArithmeticFunction R\n⊢ IsUnit (f 1) → IsUnit f",
"usedConstants": [
"MulOne.toOne",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.instMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring",
"AddGroupWit... | suffices Invertible (f 1) → Invertible f by simpa using Nonempty.map this
exact fun hf ↦ ⟨_, dirichletInverse_mul_self f hf, self_mul_dirichletInverse f hf⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 428,
"column": 4
} | {
"line": 429,
"column": 86
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\nf : ArithmeticFunction R\n⊢ IsUnit (f 1) → IsUnit f",
"usedConstants": [
"MulOne.toOne",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.instMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring",
"AddGroupWit... | suffices Invertible (f 1) → Invertible f by simpa using Nonempty.map this
exact fun hf ↦ ⟨_, dirichletInverse_mul_self f hf, self_mul_dirichletInverse f hf⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 523,
"column": 26
} | {
"line": 523,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\ncop : (((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2).Coprime (((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2... | cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 22
} | [
{
"pp": "n : ℕ\na : Fin 3 → ℕ\nha : a ∈ finMulAntidiag 3 n\nb : Fin 3 → ℕ\nhb : b ∈ finMulAntidiag 3 n\nhfab : f a ha = f b hb\nhfab1 : a 0 * a 1 = b 0 * b 1\nhfab2 : a 0 * a 2 = b 0 * b 2\nhprods : a 0 * a 1 * a 2 = a 0 * a 1 * b 2\nhab2 : a 2 = b 2\n⊢ a 0 = b 0",
"usedConstants": [
"HMul.hMul",
... | rw [hab2] at hfab2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 524,
"column": 23
} | {
"line": 524,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\nhb : ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 ≠ 0\nc1 c2 d1 d2 : ℕ\ncop : (((c1, c2), d1, d2).1.1 * ((c1,... | ← hcd.2.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 525,
"column": 42
} | {
"line": 525,
"column": 98
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\nhb : ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 ≠ 0\nc1 c2 d1 d2 : ℕ\ncop : (((c1, c2), d1, d2).1.1 * ((c1,... | cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 52
} | [
{
"pp": "case h\nn : ℕ\nhn : n ≠ 0\nb : ℕ × ℕ\nhb : b ∈ {x ∈ n.divisors ×ˢ n.divisors | x.1.lcm x.2 = n}\ng : ℕ := ⋯\na : Fin (succ 0).succ.succ → ℕ := ⋯\nha : a ∈ finMulAntidiag 3 n\n⊢ f a ha = b",
"usedConstants": [
"id",
"Prod.fst",
"Prod.ext",
"Nat",
"_private.Mathlib.Algeb... | apply Prod.ext <;> dsimp only [a, Matrix.cons_val] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 530,
"column": 23
} | {
"line": 530,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\nhb : ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 ≠ 0\nc1 c2 d1 d2 : ℕ\ncop : (((c1, c2), d1, d2).1.1 * ((c1,... | ← hcd.2.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 536,
"column": 23
} | {
"line": 536,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\nhb : ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 ≠ 0\nc1 c2 d1 d2 : ℕ\ncop : (((c1, c2), d1, d2).1.1 * ((c1,... | ← hcd.2.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 541,
"column": 23
} | {
"line": 541,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\na1 a2 b1 b2 : ℕ\nha : ((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 ≠ 0\nhb : ((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 ≠ 0\nc1 c2 d1 d2 : ℕ\ncop : (((c1, c2), d1, d2).1.1 * ((c1,... | ← hcd.2.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Disjointed | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 36
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁶ : GeneralizedBooleanAlgebra α\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : LocallyFiniteOrderBot ι\ninst✝³ : Add ι\ninst✝² : One ι\ninst✝¹ : SuccAddOrder ι\ninst✝ : NoMaxOrder ι\nf : ι → α\ni : ι\n⊢ disjointed f (i + 1) = f (i + 1) \\ (partialSups f) i",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Disjointed | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 38
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁵ : GeneralizedBooleanAlgebra α\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : Add ι\ninst✝¹ : One ι\ninst✝ : SuccAddOrder ι\nf : ι → α\nhf : Monotone f\ni : ι\n⊢ disjointed f (i + 1) ⊔ f i = f (i + 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Disjointed | {
"line": 76,
"column": 19
} | {
"line": 76,
"column": 77
} | [
{
"pp": "case succ\nα : Type u_1\nι : Type u_2\ninst✝ : GeneralizedBooleanAlgebra α\nf : ℕ → α\np : α → Sort u_3\nhdiff : ⦃t : α⦄ → ⦃i : ℕ⦄ → p t → p (t \\ f i)\nn : ℕ\nh : p (f (n + 1))\nk : ℕ\nih : p (f (n + 1) \\ (partialSups f) k)\n⊢ p (f (n + 1) \\ (partialSups f) (k + 1))",
"usedConstants": [
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Rearrangement | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 15
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁸ : Semiring α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : ExistsAddOfLE α\ninst✝⁴ : AddCommMonoid β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedCancelAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\ns : Finset ι\nσ : P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 17
} | [
{
"pp": "M✝ : Type u_1\ninst✝⁴ : Group M✝\ninst✝³ : Lattice M✝\nM : Type u_2\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b✝ : M\na b : MulArchimedeanOrder M\nthis : (∀ (n : ℕ), |val b|ₘ ^ n < |val a|ₘ) → ∃ n, |val b|ₘ ≤ |val a|ₘ ^ n\n⊢ (∀ (n : ℕ), |val b|ₘ ^ n < |val a|ₘ) ↔ (∃ n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 144,
"column": 17
} | {
"line": 144,
"column": 28
} | [
{
"pp": "M✝ : Type u_1\ninst✝⁴ : Group M✝\ninst✝³ : Lattice M✝\nM : Type u_2\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b✝ : M\na b : MulArchimedeanOrder M\nh✝ : ∀ (n : ℕ), |val b|ₘ ^ n < |val a|ₘ\nh : |val b|ₘ ^ 1 ≤ |val a|ₘ\n⊢ |val b|ₘ ≤ |val a|ₘ ^ 1",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 150,
"column": 24
} | {
"line": 150,
"column": 35
} | [
{
"pp": "M✝ : Type u_1\ninst✝⁴ : Group M✝\ninst✝³ : Lattice M✝\nM : Type u_2\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b✝ : M\na b : MulArchimedeanOrder M\nhab : |val a|ₘ ≤ |val b|ₘ\n⊢ |val a|ₘ ≤ |val b|ₘ ^ 1",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 151,
"column": 24
} | {
"line": 151,
"column": 35
} | [
{
"pp": "M✝ : Type u_1\ninst✝⁴ : Group M✝\ninst✝³ : Lattice M✝\nM : Type u_2\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b✝ : M\na b : MulArchimedeanOrder M\nhab : |val b|ₘ ≤ |val a|ₘ\n⊢ |val b|ₘ ≤ |val a|ₘ ^ 1",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 13
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\nha : a ∈ Set.Ici 1\nb : M\nhb : b ∈ Set.Ici 1\nhab : ∀ (n : ℕ), |b|ₘ ^ n < |a|ₘ\nh : b ^ 1 < a\n⊢ b < a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 13
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na : M\nha : a ∈ Set.Iic 1\nb : M\nhb : b ∈ Set.Iic 1\nhab : ∀ (n : ℕ), |a|ₘ ^ n < |b|ₘ\nh : a⁻¹ ^ 1 < b⁻¹\n⊢ b < a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Field.GeomSum | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 15
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nx : K\nm n : ℕ\nhx : 0 ≤ x\nh'x : x < 1\nhmn : m ≤ n\n⊢ x ^ m - x ^ n ≤ x ^ m",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddLeftCancelSemigroup.toIsLeftCanc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Field.GeomSum | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 17
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nx : K\nm n : ℕ\nhx : 0 ≤ x\nh'x : x < 1\nhmn : n < m\n⊢ 0 ≤ 1 - x",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Field.GeomSum | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 17
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nx : K\nm n : ℕ\nhx : 0 ≤ x\nh'x : x < 1\nhmn : n < m\n⊢ ¬m < n",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 30
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\n⊢ min (mk a) (mk b) ≤ mk (a / b)",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Lattice.toSemilatticeSup",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 19
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nhab : mk a ≤ mk b\n⊢ mk a ≤ mk (a * b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 19
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nhba : mk b ≤ mk a\n⊢ mk b ≤ mk (a * b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 35
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nhab : mk a ≤ mk b\n⊢ mk a ≤ mk (a / b)",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Partial... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 35
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nhba : mk b ≤ mk a\n⊢ mk b ≤ mk (a / b)",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Partial... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 384,
"column": 13
} | {
"line": 384,
"column": 24
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk a ≤ mk (a * b)\n⊢ mk a ≤ mk b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 393,
"column": 13
} | {
"line": 393,
"column": 24
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk a ≤ mk (a / b)\n⊢ mk a ≤ mk b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 432,
"column": 2
} | {
"line": 432,
"column": 33
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk a < mk b\n⊢ mk (a / b) = mk a",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 33
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh : mk b < mk a\n⊢ mk (a / b) = mk b",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 494,
"column": 4
} | {
"line": 494,
"column": 15
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nha : 1 < a\nhab : mk a < mk (b / a)\nthis : a⁻¹ < b / a\n⊢ 1 < b",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 496,
"column": 4
} | {
"line": 496,
"column": 15
} | [
{
"pp": "case h\nM : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nha : 1 < a\nhab : mk a < mk (b / a)\n⊢ mk a⁻¹ < mk (b / a)",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instHDiv",
"DivisionCommMonoid.toDivisionMonoid",
"DivInv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 15
} | [
{
"pp": "case hneg\nM : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nha : 1 < a\nhab : mk a < mk (b / a)\n⊢ a⁻¹ ≤ 1",
"usedConstants": [
"Eq.mpr",
"Left.inv_le_one_iff._simp_2",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 505,
"column": 6
} | {
"line": 505,
"column": 17
} | [
{
"pp": "case h\nM : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\nh : ∀ (a : M), a ≠ 1 → ∀ (b : M), b ≠ 1 → mk a = mk b\nx y : M\nhy : 1 < y\nhx : x ≤ 1\n⊢ x ≤ y ^ 0",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 514,
"column": 58
} | {
"line": 514,
"column": 69
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na b : M\ninst✝ : MulArchimedean M\nha : a ≠ 1\nhb : b ≠ 1\n⊢ 1 < |a|ₘ",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"Di... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 515,
"column": 58
} | {
"line": 515,
"column": 69
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na b : M\ninst✝ : MulArchimedean M\nha : a ≠ 1\nhb : b ≠ 1\nhm : ∃ n, |b|ₘ ≤ |a|ₘ ^ n\n⊢ 1 < |b|ₘ",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"DivisionCommMonoid.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 70
} | [
{
"pp": "case mk.mk\nM : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : Function.Injective ⇑f\na b : M\n⊢ ((∃ m, f |b|ₘ ≤ f (|a|ₘ ^ m)) ∧ ∃ n, f |a|ₘ ≤ f (|b|ₘ ^ n)) → (∃ ... | obtain hmono := (OrderHomClass.monotone f).strictMono_of_injective h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 570,
"column": 4
} | {
"line": 570,
"column": 15
} | [
{
"pp": "case mk.mk\nM : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na✝ b✝ : M\nα : Type u_2\ninst✝ : PartialOrder α\nf : M → α\nh : ∀ (a b : M), mk a ≤ mk b → f a ≤ f b\na b : M\nhle : mk a ≤ mk b\n⊢ lift f ⋯ (mk a) ≤ lift f ⋯ (mk b)",
"usedConstants": [
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 600,
"column": 2
} | {
"line": 600,
"column": 83
} | [
{
"pp": "case hab\nM : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\ns t : UpperSet (MulArchimedeanClass M)\nheq : ↑(subsemigroup t) = ↑(subsemigroup s)\n⊢ t = s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 616,
"column": 6
} | {
"line": 616,
"column": 17
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\ns : UpperSet (MulArchimedeanClass M)\nhs : ¬s = ⊤\nu : MulArchimedeanClass M\nhu : u ∈ ↑s\n⊢ mk 1 ∈ ↑s",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"MulOne.toOne",
"U... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 642,
"column": 43
} | {
"line": 642,
"column": 69
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\ns : UpperSet (MulArchimedeanClass M)\nhs : s ∈ Set.Iio ⊤\nt : UpperSet (MulArchimedeanClass M)\nht : t ∈ Set.Iio ⊤\nhst : s < t\n⊢ ↑(subsemigroup t) ⊆ ↑(subsemigroup s)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 645,
"column": 2
} | {
"line": 645,
"column": 43
} | [
{
"pp": "case hab\nM : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\ns : UpperSet (MulArchimedeanClass M)\nhs : s ∈ Set.Iio ⊤\nt : UpperSet (MulArchimedeanClass M)\nht : t ∈ Set.Iio ⊤\nheq : ↑(subsemigroup t) = ↑(subsemigroup s)\n⊢ t = s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 752,
"column": 31
} | {
"line": 752,
"column": 40
} | [
{
"pp": "case mk.mk\nM : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na✝ b✝ : M\ninst✝ : MulArchimedean M\na : M\nha : a ≠ 1\nb : M\nhb : b ≠ 1\n⊢ mk a ha = mk b hb",
"usedConstants": []
}
] | | mk b hb
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 753,
"column": 4
} | {
"line": 753,
"column": 20
} | [
{
"pp": "case mk.mk\nM : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na✝ b✝ : M\ninst✝ : MulArchimedean M\na : M\nha : a ≠ 1\nb : M\nhb : b ≠ 1\n⊢ mk a ha = mk b hb",
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Semi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 758,
"column": 21
} | {
"line": 758,
"column": 32
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na b : M\ninst✝ : Nontrivial M\nx : M\nhx : x ≠ 1\n⊢ ↑(mk x hx) ≠ ⊤",
"usedConstants": [
"Eq.mpr",
"MulArchimedeanClass.mk_eq_top_iff._simp_2",
"InvOneClass.toOne",
"DivisionCommMonoid.toD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 772,
"column": 32
} | {
"line": 772,
"column": 48
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b✝ : M\nα : Type u_2\nf : { a // a ≠ 1 } → α\nh : ∀ (a b : { a // a ≠ 1 }), mk ↑a ⋯ = mk ↑b ⋯ → f a = f b\nx✝ : FiniteMulArchimedeanClass M\nA : MulArchimedeanClass M\nhA : A ≠ ⊤\na b : M\nh' : MulArchimedeanClass... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 774,
"column": 4
} | {
"line": 774,
"column": 15
} | [
{
"pp": "case refine_2.mk\nM : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na✝ b : M\nα : Type u_2\nf : { a // a ≠ 1 } → α\nh : ∀ (a b : { a // a ≠ 1 }), mk ↑a ⋯ = mk ↑b ⋯ → f a = f b\nx✝ : FiniteMulArchimedeanClass M\na : M\nhA : MulArchimedeanClass.mk a ≠ ⊤\n⊢ MulArchimed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 793,
"column": 31
} | {
"line": 793,
"column": 40
} | [
{
"pp": "case mk.mk\nM : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na✝ b✝ : M\nα : Type u_2\ninst✝ : PartialOrder α\nf : { a // a ≠ 1 } → α\nh : ∀ (a b : { a // a ≠ 1 }), mk ↑a ⋯ ≤ mk ↑b ⋯ → f a ≤ f b\na : M\nha : a ≠ 1\nb : M\nhb : b ≠ 1\nhAB : mk a ha ≤ mk b hb\n⊢ lift... | | mk b hb
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 794,
"column": 4
} | {
"line": 794,
"column": 15
} | [
{
"pp": "case mk.mk\nM : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na✝ b✝ : M\nα : Type u_2\ninst✝ : PartialOrder α\nf : { a // a ≠ 1 } → α\nh : ∀ (a b : { a // a ≠ 1 }), mk ↑a ⋯ ≤ mk ↑b ⋯ → f a ≤ f b\na : M\nha : a ≠ 1\nb : M\nhb : b ≠ 1\nhAB : mk a ha ≤ mk b hb\n⊢ lift... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Div | {
"line": 185,
"column": 25
} | {
"line": 185,
"column": 47
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\na : ℕ\nha : 0 < a\n⊢ GaloisConnection (fun x ↦ a • x) fun x ↦ x / a",
"usedConstants": [
"instHSMul",
"instHDiv",
"instDistribSMul",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"Nat.instAddMonoid",
"AddZe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Div | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 62
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMulZeroClass α β\ninst✝ : FloorDiv α β\nf✝ : ι →₀ β\na _a : α\nha : 0 < _a\nf _g : ι →₀ β\ni : ι\n⊢ ((fun x ↦ _a • x) f) i ≤ _g i ↔ f i ≤ ((fun x ↦ m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Div | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 62
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMulZeroClass α β\ninst✝ : CeilDiv α β\nf✝ : ι →₀ β\na _a : α\nha : 0 < _a\nf _g : ι →₀ β\ni : ι\n⊢ ((fun x ↦ mapRange (fun x ↦ x ⌈/⌉ _a) ⋯ x) f) i ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 13
} | [
{
"pp": "r : ℝ≥0∞\nn : ℕ∞\nhn₀ : n ≠ 0\nhn : n ≠ ⊤\n⊢ ⌈r⌉ₑ < n ↔ r ≤ ↑n - 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 104,
"column": 19
} | {
"line": 104,
"column": 30
} | [
{
"pp": "r s : ℝ≥0∞\nhrs : r ≤ s\n⊢ ⌊r⌋ₑ ≤ ⌊s⌋ₑ",
"usedConstants": [
"Eq.mpr",
"Preorder.toLE",
"instPreorderENat",
"id",
"ENat.toENNReal",
"LE.le",
"ENat.floor",
"ENat",
"ENNReal.instLE",
"ENNReal",
"Eq",
"ENat.le_floor._simp_1"
]
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 106,
"column": 66
} | {
"line": 106,
"column": 77
} | [
{
"pp": "r s : ℝ≥0∞\nhrs : r ≤ s\n⊢ ⌈r⌉ₑ ≤ ⌈s⌉ₑ",
"usedConstants": [
"Eq.mpr",
"Preorder.toLE",
"instPreorderENat",
"id",
"ENat.ceil",
"ENat.toENNReal",
"ENat.ceil_le._simp_1",
"LE.le",
"ENat",
"ENNReal.instLE",
"ENNReal",
"Eq"
]
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 113,
"column": 42
} | {
"line": 113,
"column": 53
} | [
{
"pp": "⊢ ⌊0⌋ₑ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 114,
"column": 41
} | {
"line": 114,
"column": 52
} | [
{
"pp": "⊢ ⌈0⌉ₑ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 115,
"column": 41
} | {
"line": 115,
"column": 52
} | [
{
"pp": "⊢ ⌊1⌋ₑ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 116,
"column": 40
} | {
"line": 116,
"column": 51
} | [
{
"pp": "⊢ ⌈1⌉ₑ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 124,
"column": 52
} | {
"line": 124,
"column": 63
} | [
{
"pp": "r : ℝ≥0∞\n⊢ ⌈r⌉ₑ = 0 ↔ r = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 130,
"column": 20
} | {
"line": 130,
"column": 31
} | [
{
"pp": "r : ℝ≥0\n⊢ ⌈↑r⌉ₑ ≤ ⌊↑r⌋ₑ + 1",
"usedConstants": [
"ENNReal.ofNNReal",
"instAddMonoidWithOneENat",
"instAddENat",
"id",
"ENat.ceil",
"LE.le",
"instLEENat",
"AddMonoidWithOne.toOne",
"instHAdd",
"ENat.floor",
"HAdd.hAdd",
"ENat",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 205,
"column": 26
} | {
"line": 205,
"column": 37
} | [
{
"pp": "r : ℝ≥0\n⊢ ↑⌈↑r⌉ₑ < ↑r + 1",
"usedConstants": [
"ENNReal.instAdd",
"ENNReal.ofNNReal",
"Preorder.toLT",
"PartialOrder.toPreorder",
"id",
"ENat.ceil",
"ENat.toENNReal",
"instHAdd",
"HAdd.hAdd",
"LT.lt",
"ENNReal",
"One.toOfNat1"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 80,
"column": 38
} | {
"line": 80,
"column": 49
} | [
{
"pp": "H : Type u_1\ninst✝² : CommGroup H\ninst✝¹ : PartialOrder H\ninst✝ : IsOrderedMonoid H\na✝ a : H\n⊢ a ∈ (Submonoid.oneLE H).carrier → a⁻¹ ∈ (Submonoid.oneLE H).carrier → a = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 92,
"column": 23
} | {
"line": 92,
"column": 34
} | [
{
"pp": "H✝ : Type u_1\ninst✝⁵ : CommGroup H✝\ninst✝⁴ : PartialOrder H✝\ninst✝³ : IsOrderedMonoid H✝\na : H✝\nH : Type u_2\ninst✝² : CommGroup H\ninst✝¹ : LinearOrder H\ninst✝ : IsOrderedMonoid H\n⊢ ∀ (a : H), a ∈ oneLE H ∨ a⁻¹ ∈ oneLE H",
"usedConstants": [
"Eq.mpr",
"GroupCone.mem_oneLE._simp_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 103,
"column": 31
} | {
"line": 103,
"column": 42
} | [
{
"pp": "S : Type u_1\nG : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : SetLike S G\nC : S\ninst✝ : GroupConeClass S G\na b c : G\nnab : b / a ∈ C\nnbc : c / b ∈ C\n⊢ c / a ∈ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 37
} | [
{
"pp": "S : Type u_1\nG : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : SetLike S G\nC : S\ninst✝ : GroupConeClass S G\na b : G\nnab : b / a ∈ C\nnba : a / b ∈ C\n⊢ a = b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 105,
"column": 71
} | {
"line": 105,
"column": 82
} | [
{
"pp": "S : Type u_1\nG : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : SetLike S G\nC : S\ninst✝ : GroupConeClass S G\na b : G\nnab : b / a ∈ C\nnba : a / b ∈ C\n⊢ (b / a)⁻¹ ∈ C",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 117,
"column": 21
} | {
"line": 117,
"column": 32
} | [
{
"pp": "S : Type u_1\nG : Type u_2\ninst✝⁴ : CommGroup G\ninst✝³ : SetLike S G\nC : S\ninst✝² : GroupConeClass S G\ninst✝¹ : HasMemOrInvMem C\ninst✝ : DecidablePred fun x ↦ x ∈ C\na b : G\n⊢ a ≤ b ∨ b ≤ a",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"PartialOrder.toPreorde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 127,
"column": 42
} | {
"line": 127,
"column": 61
} | [
{
"pp": "S : Type u_1\nG : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : SetLike S G\nC : S\ninst✝ : GroupConeClass S G\nx✝ : PartialOrder G := PartialOrder.mkOfGroupCone C\na b : G\nnab : a ≤ b\nc : G\n⊢ a * c ≤ b * c",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"Monoid.toMulO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Ideal | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 22
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : WellQuasiOrderedLE M\ninst✝ : CanonicallyOrderedMul M\nf : ℕ →o SemigroupIdeal M\ns : Finset M\nhI : ⨆ i, f i = closure ↑s\nx : M\nhx : x ∈ ↑(closure ↑s)\n⊢ ∃ i, x ∈ f i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.DenselyOrdered | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝³ : Group α\ninst✝² : LinearOrder α\ninst✝¹ : MulLeftMono α\ninst✝ : DenselyOrdered α\na b : α\nh : ∀ (ε : α), 1 < ε → a / ε ≤ b\nε : α\nε1 : ε < 1\n⊢ a * ε ≤ b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.DenselyOrdered | {
"line": 105,
"column": 19
} | {
"line": 105,
"column": 41
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : LinearOrder M\ninst✝³ : DenselyOrdered M\ninst✝² : CommMonoid M\ninst✝¹ : ExistsMulOfLE M\ninst✝ : IsOrderedCancelMonoid M\ny : M\nhy : 1 < y\nz : M\nhz : 1 < z\nhx : 1 < y * z\nhyx : y < y * z\nhxy : y * z ≤ y ^ 2\n⊢ z ^ 2 ≤ y * z",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.DenselyOrdered | {
"line": 110,
"column": 12
} | {
"line": 110,
"column": 23
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : LinearOrder M\ninst✝³ : DenselyOrdered M\nx : M\ninst✝² : CommMonoid M\ninst✝¹ : ExistsMulOfLE M\ninst✝ : IsOrderedCancelMonoid M\nhx : 1 < x\n⊢ ∃ y, 1 < y ∧ y ^ 1 < x",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Preorder.toLT",
"Monoid.toMulOneClas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.DenselyOrdered | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 23
} | [
{
"pp": "case right\nM : Type u_2\ninst✝³ : LinearOrder M\ninst✝² : DenselyOrdered M\nx : M\ninst✝¹ : CommGroup M\ninst✝ : IsOrderedCancelMonoid M\nhx : x < 1\nn : ℕ\ny : M\nhy : 1 < y\nhy' : y ^ n < x⁻¹\n⊢ x < y⁻¹ ^ n",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"DivisionCommMonoid.toD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Lex | {
"line": 86,
"column": 18
} | {
"line": 86,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero α\ninst✝ : LinearOrderedCommGroupWithZero β\n⊢ Monotone (↑((WithZero.map' (toLexMulEquiv (αˣ × βˣ)).toMonoidHom).comp (MonoidWithZeroHom.inl α β))).toFun",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Lex | {
"line": 94,
"column": 18
} | {
"line": 94,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero α\ninst✝ : LinearOrderedCommGroupWithZero β\n⊢ Monotone (↑((WithZero.map' (toLexMulEquiv (αˣ × βˣ)).toMonoidHom).comp (MonoidWithZeroHom.inr α β))).toFun",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Lex | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 17
} | [
{
"pp": "case coe.coe\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero α\ninst✝ : LinearOrderedCommGroupWithZero β\na✝¹ a✝ : Lex (αˣ × βˣ)\n⊢ ↑a✝¹ ≤ ↑a✝ →\n (↑((MonoidWithZeroHom.fst α β).comp (WithZero.map' (toLexMulEquiv (αˣ × βˣ)).symm.toMonoidHom))).toFun ↑a✝¹ ≤\n (↑((MonoidWithZ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 31
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na b : α\n⊢ Ico (a + 1) b = Ioo a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 31
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 31
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\nb : α\nhb : ¬IsMax b\na : α\n⊢ Ico a (b + 1) = Icc a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 31
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\nb : α\nhb : ¬IsMax b\na : α\n⊢ Ioo a (b + 1) = Ioc a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.SuccPred | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 31
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : One α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\nb : α\nhb : ¬IsMax b\na : α\n⊢ Ico (a + 1) (b + 1) = Ioc a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.