module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 7
} | {
"line": 732,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nB : Set (MvPolynomial σ R)\n⊢ m.leadingTerm '' insert 0 B = insert 0 (m.leadingTerm '' B)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Nat.instMulZeroClass",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 7
} | {
"line": 732,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nB : Set (MvPolynomial σ R)\n⊢ m.leadingTerm '' insert 0 B = insert 0 (m.leadingTerm '' B)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Nat.instMulZeroClass",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 7
} | {
"line": 732,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nB : Set (MvPolynomial σ R)\n⊢ m.leadingTerm '' insert 0 B = insert 0 (m.leadingTerm '' B)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Nat.instMulZeroClass",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Squarefree | {
"line": 92,
"column": 26
} | {
"line": 92,
"column": 49
} | {
"line": 92,
"column": 49
} | [
{
"pp": "n k : ℕ\nhn : n ≠ 1\nhk : k ≠ 0\n⊢ Squarefree n ∧ k = 1 → Squarefree (n ^ k)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.instMonoid",
"id",
"Ne",
"instOfNatNat",
"NPow.toPow",
"And.casesOn",
"And"... | [] | rintro ⟨hn, rfl⟩; simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 92,
"column": 26
} | {
"line": 92,
"column": 49
} | {
"line": 92,
"column": 49
} | [
{
"pp": "n k : ℕ\nhn : n ≠ 1\nhk : k ≠ 0\n⊢ Squarefree n ∧ k = 1 → Squarefree (n ^ k)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.instMonoid",
"id",
"Ne",
"instOfNatNat",
"NPow.toPow",
"And.casesOn",
"And"... | [] | rintro ⟨hn, rfl⟩; simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 811,
"column": 2
} | {
"line": 811,
"column": 7
} | {
"line": 813,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\n⊢ m.degree p = m.degree q ∧ m.leadingCoeff p = m.leadingCoeff q ∨ m.leadingCoeff p = 0 ∧ m.leadingCoeff q = 0 ↔\n m.leadingCoeff p = m.leadingCoeff q ∧ m.degree p = m.degree q",
"ppTerm": "?m.44",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Nat.Squarefree | {
"line": 141,
"column": 2
} | {
"line": 146,
"column": 70
} | {
"line": 147,
"column": 2
} | [
{
"pp": "n k : ℕ\npk : Prime k\ndk : k ∣ n\ndkk : ¬k * k ∣ n\no : Option ℕ\nH : (n / k).MinSqFacProp o\n⊢ n.MinSqFacProp o",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.Coprime",
"False",
"Nat.Prime",
"Dvd.dvd",
"HMul.hMul",
"Nat.... | [
"n k : ℕ\npk : Prime k\ndk : k ∣ n\ndkk : ¬k * k ∣ n\no : Option ℕ\nH : (n / k).MinSqFacProp o\nthis : ∀ (p : ℕ), Prime p → p * p ∣ n → k * (p * p) ∣ n\n⊢ n.MinSqFacProp o"
] | have : ∀ p, Prime p → p * p ∣ n → k * (p * p) ∣ n := fun p pp dp =>
have :=
(coprime_primes pk pp).2 fun e => by
subst e
contradiction
(coprime_mul_iff_right.2 ⟨this, this⟩).mul_dvd_of_dvd_of_dvd dk dp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.MvPolynomial.SchwartzZippel | {
"line": 146,
"column": 8
} | {
"line": 147,
"column": 95
} | {
"line": 148,
"column": 8
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.n... | [
"case h\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.natDegree\npₖ... | have hpₓdeg : pₓ.natDegree = k := by
rw [hpₓ, hk, Polynomial.natDegree_map_of_leadingCoeff_ne_zero _ (mem_filter.1 hxₜ).2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 92
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case mpr.inl\np : ℕ\nhp : Nat.Prime p\nhq : ∀ (y : ℕ), (fun p ↦ Nat.Prime p ∧ p ∣ 0) y → y = p\nhn : p ∣ 0\n⊢ ∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = 0",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"dvd_zero",
"Nat.Prime",
"Dvd.dvd",
"Nat.instSemigroupWit... | [
"case mpr.inr\nn p : ℕ\nhq : ∀ (y : ℕ), (fun p ↦ Nat.Prime p ∧ p ∣ n) y → y = p\nhp : Nat.Prime p\nhn : p ∣ n\nhn₀ : n ≠ 0\n⊢ ∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n"
] | · cases (hq 2 ⟨Nat.prime_two, dvd_zero 2⟩).trans (hq 3 ⟨Nat.prime_three, dvd_zero 3⟩).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 9
} | {
"line": 161,
"column": 0
} | [
{
"pp": "case inr\nk n : ℕ\nhn : n ≠ 0\n⊢ ∑ d ∈ n.divisors, d ^ k = 0 ↔ n = 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instPowNat",
"Nat.mem_divisors._simp_1",
"Eq.mpr",
"False",
"Dvd.dvd",
"Finset.sum_eq_zero_iff._simp_1",
"eq_false",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 7
} | {
"line": 201,
"column": 0
} | [
{
"pp": "k x✝ : ℕ\n⊢ ∀ x ∈ x✝.divisors, (pow k) x = x ^ k",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Nat.mem_divisors._simp_1",
"Eq.mpr",
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"Dvd.dvd",
"ite_eq_right_iff._simp_1",
"Arithmeti... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 9
} | {
"line": 244,
"column": 2
} | [
{
"pp": "case pos\nk n : ℕ\nhn0 : n = 0\n⊢ (σ k) n = 1 ↔ n = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"Nat.instOne",
"congrArg",
"instOfNatNat",
"iff_self",
"zero_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 9
} | {
"line": 244,
"column": 2
} | [
{
"pp": "case pos\nk n : ℕ\nhn0 : n = 0\n⊢ (σ k) n = 1 ↔ n = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"Nat.instOne",
"congrArg",
"instOfNatNat",
"iff_self",
"zero_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 9
} | {
"line": 244,
"column": 2
} | [
{
"pp": "case pos\nk n : ℕ\nhn0 : n = 0\n⊢ (σ k) n = 1 ↔ n = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"Nat.instOne",
"congrArg",
"instOfNatNat",
"iff_self",
"zero_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 308,
"column": 16
} | {
"line": 308,
"column": 21
} | {
"line": 309,
"column": 2
} | [
{
"pp": "case pos.zero\nm : ℕ\nhm : m = 0\n⊢ Ω (m ^ 0) = 0 * Ω m",
"ppTerm": "?pos.zero✝",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Nat.instMulZeroClass",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"Monoid.toMulOneClass",
"congrArg",
"Nat... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 308,
"column": 16
} | {
"line": 308,
"column": 21
} | {
"line": 309,
"column": 2
} | [
{
"pp": "case pos.succ\nm : ℕ\nhm : m = 0\nn✝ : ℕ\n⊢ Ω (m ^ (n✝ + 1)) = (n✝ + 1) * Ω m",
"ppTerm": "?pos.succ✝",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"Nat.instOne",
"congrArg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Archimedean.IndicatorCard | {
"line": 58,
"column": 9
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 14
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : AddCommMonoid R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : Archimedean R\nr : R\nh : 0 < r\ns : Set ℕ\nh_mono : Monotone fun n ↦ ∑ k ∈ Finset.range n, s.indicator (fun x ↦ r) k\nhs : s.Infinite\nn : R\nn' : ℕ\nhn' : n < ... | [
"case h\nR : Type u_1\ninst✝⁴ : AddCommMonoid R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : Archimedean R\nr : R\nh : 0 < r\ns : Set ℕ\nh_mono : Monotone fun n ↦ ∑ k ∈ Finset.range n, s.indicator (fun x ↦ r) k\nhs : s.Infinite\nn : R\nn' : ℕ\nhn' : n < n' • r\nt : ... | hn', | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 621,
"column": 27
} | {
"line": 621,
"column": 32
} | {
"line": 621,
"column": 32
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : MonoidWithZero R\nm n : ℕ\nhm : m ≠ 0\nhn : n ≠ 0\nhmn : m.Coprime n\nh : m = 1\n⊢ 1 (m * n) = 1 m * 1 n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"MulOne.toOne",
"HMul.hMul",
"ArithmeticFunction.instFunL... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 621,
"column": 27
} | {
"line": 621,
"column": 32
} | {
"line": 621,
"column": 32
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : MonoidWithZero R\nm n : ℕ\nhm : m ≠ 0\nhn : n ≠ 0\nhmn : m.Coprime n\nh : ¬m = 1\n⊢ 1 (m * n) = 1 m * 1 n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNa... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 11
} | {
"line": 490,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\na b : M\nh✝ : mk a < mk b\nhneg : a ≤ 1\nh : a < b ∧ b < a⁻¹\n⊢ a < b",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"DivisionCommMonoid.toDivisionMonoid",
"D... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 21
} | {
"line": 638,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝² : CommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedMonoid M\n⊢ StrictAntiOn subgroup (Set.Iio ⊤)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"UpperSet",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Member... | [
"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⊢ subgroup t < subgroup s"
] | intro s hs t ht hst | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Order.Group.Cyclic | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 22
} | {
"line": 42,
"column": 2
} | [
{
"pp": "case inl\nG : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\nH : Subgroup G\ninst✝ : Nontrivial ↥H\nhH : IsCyclic ↥H\na : G\nha : Subgroup.zpowers a = H\nha1 : a < 1\n⊢ ∃ a < 1, Subgroup.zpowers a = H",
"ppTerm": "?inl",
"assigned": true,
"usedConstants"... | [
"case inr.inl\nG : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\nH : Subgroup G\ninst✝ : Nontrivial ↥H\nhH : IsCyclic ↥H\nha : Subgroup.zpowers 1 = H\n⊢ ∃ a < 1, Subgroup.zpowers a = H",
"case inr.inr\nG : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrd... | · exact ⟨a, ha1, ha⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Group.Ideal | {
"line": 41,
"column": 4
} | {
"line": 45,
"column": 26
} | {
"line": 46,
"column": 2
} | [
{
"pp": "case mp\nM : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : WellQuasiOrderedLE M\ninst✝ : CanonicallyOrderedMul M\nI : SemigroupIdeal M\nhpwo : {x | x ∈ I}.IsPWO\nx : M\n⊢ x ∈ I → ∃ y z, Minimal (fun x ↦ x ∈ I) z ∧ y * z = x",
"ppTerm": "?mp",
"assigned": true,
"usedCons... | [] | intro hx
rcases hpwo.exists_le_minimal hx with ⟨z, hz, hz'⟩
rw [le_iff_exists_mul'] at hz
rcases hz with ⟨y, rfl⟩
exact ⟨y, z, hz', rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Ideal | {
"line": 41,
"column": 4
} | {
"line": 45,
"column": 26
} | {
"line": 46,
"column": 2
} | [
{
"pp": "case mp\nM : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : WellQuasiOrderedLE M\ninst✝ : CanonicallyOrderedMul M\nI : SemigroupIdeal M\nhpwo : {x | x ∈ I}.IsPWO\nx : M\n⊢ x ∈ I → ∃ y z, Minimal (fun x ↦ x ∈ I) z ∧ y * z = x",
"ppTerm": "?mp",
"assigned": true,
"usedCons... | [] | intro hx
rcases hpwo.exists_le_minimal hx with ⟨z, hz, hz'⟩
rw [le_iff_exists_mul'] at hz
rcases hz with ⟨y, rfl⟩
exact ⟨y, z, hz', rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 35
} | {
"line": 43,
"column": 35
} | [
{
"pp": "case h₂\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nr : Finset ℤ := Ioc (c - ↑(#s)) c\nx : ℤ\nmx : x ∈ r \\ s\n⊢ c - ↑(#s) ≤ x",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"HSub.hSub",
... | [
"case h₂\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nr : Finset ℤ := Ioc (c - ↑(#s)) c\nx : ℤ\nmx : (c - ↑(#s) < x ∧ x ≤ c) ∧ x ∉ s\n⊢ c - ↑(#s) ≤ x"
] | rw [mem_sdiff, mem_Ioc] at mx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 52,
"column": 36
} | {
"line": 52,
"column": 50
} | {
"line": 52,
"column": 51
} | [
{
"pp": "case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℤ\nmx : x ∈ ↑(Ioc (c - ↑(#s)) c)\n⊢ x ∈ (fun x ↦ c - ↑x) '' Set.Iio #s",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HSub.hSub",
"Membership.mem",
"Exists",
"_private.Mathli... | [
"case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, x ≤ c\nx : ℤ\nmx : x ∈ ↑(Ioc (c - ↑(#s)) c)\n⊢ ∃ x_1 ∈ Set.Iio #s, c - ↑x_1 = x"
] | Set.mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.GroupWithZero.Bounds | {
"line": 27,
"column": 28
} | {
"line": 27,
"column": 58
} | {
"line": 27,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : Nonempty α\ninst✝² : Preorder β\ninst✝¹ : Zero β\ninst✝ : Preorder γ\nf : α → β\ng : β → γ\nhf : BddAbove (range f)\nhf0 : 0 ≤ f\nhg : MonotoneOn g {x | 0 ≤ x}\n⊢ range f ⊆ {x | 0 ≤ x}",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants":... | [] | rintro x ⟨a, rfl⟩; exact hf0 a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.GroupWithZero.Bounds | {
"line": 27,
"column": 28
} | {
"line": 27,
"column": 58
} | {
"line": 27,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : Nonempty α\ninst✝² : Preorder β\ninst✝¹ : Zero β\ninst✝ : Preorder γ\nf : α → β\ng : β → γ\nhf : BddAbove (range f)\nhf0 : 0 ≤ f\nhg : MonotoneOn g {x | 0 ≤ x}\n⊢ range f ⊆ {x | 0 ≤ x}",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants":... | [] | rintro x ⟨a, rfl⟩; exact hf0 a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 63,
"column": 21
} | {
"line": 63,
"column": 28
} | {
"line": 63,
"column": 28
} | [
{
"pp": "s : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nr : Finset ℤ := Ico c (c + ↑(#s))\nx : ℤ\nmx : x ∈ r ∧ x ∉ s\n⊢ x ≤ c + ↑(#s)",
"ppTerm": "?m.175",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",... | [
"s : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nr : Finset ℤ := Ico c (c + ↑(#s))\nx : ℤ\nmx : (c ≤ x ∧ x < c + ↑(#s)) ∧ x ∉ s\n⊢ x ≤ c + ↑(#s)"
] | mem_Ico | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 76,
"column": 20
} | {
"line": 76,
"column": 27
} | {
"line": 76,
"column": 27
} | [
{
"pp": "case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℕ\nmx : x ∈ range #s\n⊢ c + ↑x ∈ Ico c (c + ↑(#s))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preo... | [
"case refine_1\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℕ\nmx : x ∈ range #s\n⊢ c ≤ c + ↑x ∧ c + ↑x < c + ↑(#s)"
] | mem_Ico | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 78,
"column": 36
} | {
"line": 78,
"column": 50
} | {
"line": 78,
"column": 51
} | [
{
"pp": "case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℤ\nmx : x ∈ ↑(Ico c (c + ↑(#s)))\n⊢ x ∈ (fun x ↦ c + ↑x) '' Set.Iio #s",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Algebra.Order.Group.Int.Sum.0.Finset.sum_range_le_sum._si... | [
"case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℤ\nmx : x ∈ ↑(Ico c (c + ↑(#s)))\n⊢ ∃ x_1 ∈ Set.Iio #s, c + ↑x_1 = x"
] | Set.mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Group.Int.Sum | {
"line": 79,
"column": 17
} | {
"line": 79,
"column": 24
} | {
"line": 79,
"column": 24
} | [
{
"pp": "case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℤ\nmx : x ∈ Ico c (c + ↑(#s))\n⊢ ∃ x_1 < #s, c + ↑x_1 = x",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE... | [
"case refine_3\ns : Finset ℤ\nc : ℤ\nhs : ∀ x ∈ s, c ≤ x\nx : ℤ\nmx : c ≤ x ∧ x < c + ↑(#s)\n⊢ ∃ x_1 < #s, c + ↑x_1 = x"
] | mem_Ico | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Archimedean | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 11
} | {
"line": 58,
"column": 4
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\nK : Type u_2\ninst✝⁵ : Ring K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsOrderedRing K\ninst✝² : Archimedean K\ninst✝¹ : Module K M\ninst✝ : PosSMulMono K M\nk : K\na : M\n__spread✝⁻⁰ : AddSubgroup M := add... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Module.Archimedean | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 11
} | {
"line": 58,
"column": 4
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\nK : Type u_2\ninst✝⁵ : Ring K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsOrderedRing K\ninst✝² : Archimedean K\ninst✝¹ : Module K M\ninst✝ : PosSMulMono K M\nk : K\na : M\n__spread✝⁻⁰ : AddSubgroup M := add... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Module.Archimedean | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 11
} | {
"line": 58,
"column": 4
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\nK : Type u_2\ninst✝⁵ : Ring K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsOrderedRing K\ninst✝² : Archimedean K\ninst✝¹ : Module K M\ninst✝ : PosSMulMono K M\nk : K\na : M\n__spread✝⁻⁰ : AddSubgroup M := add... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Basic | {
"line": 527,
"column": 50
} | {
"line": 527,
"column": 86
} | {
"line": 528,
"column": 6
} | [
{
"pp": "case coe.coe\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na✝¹ a✝ : NonemptyInterval α\n⊢ ↑(a✝¹ * a✝) = 1 ↔ ∃ a b, ↑a✝¹ = pure a ∧ ↑a✝ = pure b ∧ a * b = 1",
"ppTerm": "?coe.coe",
"assigned": true,
"usedConstants": [
"Interval.pure",
"I... | [
"case coe.coe\nα : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na✝¹ a✝ : NonemptyInterval α\n⊢ ↑(a✝¹ * a✝) = ↑1 ↔ ∃ a b, ↑a✝¹ = pure a ∧ ↑a✝ = pure b ∧ a * b = 1"
] | ← NonemptyInterval.coe_one_interval, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 113,
"column": 30
} | {
"line": 113,
"column": 45
} | {
"line": 113,
"column": 45
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Subsingleton R\nx✝¹ x✝ : R⟦Γ⟧\n⊢ x✝¹.coeff = x✝.coeff",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pi.instSubsingleton",
"HahnSeries.coeff",
"Subsinglet... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 71
} | {
"line": 420,
"column": 0
} | [
{
"pp": "case neg\nΓ : Type u_1\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : Zero Γ\nx : R⟦Γ⟧\nh : ¬x = 0\n⊢ x.leadingCoeff = x.coeff x.order",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"HahnSeries.support",
"Iff.mpr",
"HahnSeries.orderTop_ne_top",... | [] | · simp [leadingCoeff_of_ne_zero, orderTop_of_ne_zero, order_of_ne, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 11
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case mpr\nM : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c x : M\n⊢ (∃ a_1, (a ≤ a_1 ∧ a_1 < b) ∧ a_1 * c = x) → a * c ≤ x ∧ x < b * c",
"ppTerm": "?mpr",
"assigned": true,
"usedConst... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 11
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case mpr\nM : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c x : M\n⊢ (∃ a_1, (a ≤ a_1 ∧ a_1 < b) ∧ a_1 * c = x) → a * c ≤ x ∧ x < b * c",
"ppTerm": "?mpr",
"assigned": true,
"usedConst... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 11
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case mpr\nM : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c x : M\n⊢ (∃ a_1, (a ≤ a_1 ∧ a_1 < b) ∧ a_1 * c = x) → a * c ≤ x ∧ x < b * c",
"ppTerm": "?mpr",
"assigned": true,
"usedConst... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 30
} | {
"line": 88,
"column": 6
} | [
{
"pp": "case hg\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero R\ninst✝ : LinearOrder R\nx : Lex R⟦Γ⟧\ni : Γ\nhj : ∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j\nhi : (ofLex 0).coeff i < (ofLex x).coeff i\n⊢ i ∈ (ofLex x).support",
"ppTerm": "?hg",
"assigned": true,
"usedConsta... | [
"case hx\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero R\ninst✝ : LinearOrder R\nx : Lex R⟦Γ⟧\ni : Γ\nhj : ∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j\nhi : (ofLex 0).coeff i < (ofLex x).coeff i\n⊢ ∀ g' ∈ (ofLex x).support, i ≤ g'"
] | · simpa using hi.ne.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 204,
"column": 63
} | {
"line": 204,
"column": 68
} | {
"line": 206,
"column": 0
} | [
{
"pp": "M : Type u_1\nG✝ : Type u_2\ninst✝⁹ : AddCancelCommMonoid M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : LocallyFiniteOrder M\ninst✝⁵ : AddCommGroup G✝\ninst✝⁴ : LinearOrder G✝\ninst✝³ : IsOrderedAddMonoid G✝\ninst✝² : LocallyFiniteOrder G✝\nG : Type u_3\ninst✝¹ : LinearOrderedCommG... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 37
} | {
"line": 180,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nh : (ofLex x).orderTop = (ofLex y).orderTop\n⊢ ArchimedeanClass.mk x ≤ ArchimedeanClass.mk y ↔\n ArchimedeanClass.mk (ofLex x).leadingCoeff ≤ Archimedean... | [
"Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nh : (ofLex x).orderTop = (ofLex y).orderTop\n⊢ (∃ n, |y| ≤ n • |x|) ↔ ∃ n, |(ofLex y).leadingCoeff| ≤ n • |(ofLex x).leadingCoeff|"
] | simp_rw [ArchimedeanClass.mk_le_mk] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Order.PUnit | {
"line": 21,
"column": 37
} | {
"line": 21,
"column": 52
} | {
"line": 21,
"column": 52
} | [
{
"pp": "x✝² x✝¹ : PUnit\nx✝ : x✝² ≤ x✝¹\n⊢ x✝¹ = x✝² + unit",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instSubsingletonPUnit",
"instHAdd",
"HAdd.hAdd",
"PUnit",
"PUnit.unit",
"Subsingleton.elim",
"PUnit.instAdd_mathlib"
],
"usedFVars... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 37
} | {
"line": 240,
"column": 2
} | [
{
"pp": "case inr.inr\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nhgt : (ofLex y).orderTop < (ofLex x).orderTop\n⊢ ArchimedeanClass.mk x ≤ ArchimedeanClass.mk y ↔\n (ofLex x).orderTop < (ofLex y).orderTop ... | [
"case inr.inr\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx y : Lex R⟦Γ⟧\nhgt : (ofLex y).orderTop < (ofLex x).orderTop\n⊢ (∃ n, |y| ≤ n • |x|) ↔\n (ofLex x).orderTop < (ofLex y).orderTop ∨\n (ofLex x).orderTop = (ofLex ... | simp_rw [ArchimedeanClass.mk_le_mk] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 553,
"column": 37
} | {
"line": 553,
"column": 42
} | {
"line": 554,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : R⟦Γ⟧\nh : x.leadingCoeff * y.leadingCoeff ≠ 0\n⊢ x.leadingCoeff ≠ 0",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 554,
"column": 37
} | {
"line": 554,
"column": 42
} | {
"line": 555,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\nx y : R⟦Γ⟧\nh : x.leadingCoeff * y.leadingCoeff ≠ 0\nhx : x.leadingCoeff ≠ 0\n⊢ y.leadingCoeff ≠ 0",
"ppTerm": "?m.64",
"assigned": true,
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 229,
"column": 2
} | {
"line": 231,
"column": 16
} | {
"line": 233,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : CommRing R\ninst✝ : IsStrictOrderedRing R\nx : R\nhx : 0 ≤ mk x\n⊢ ∃ n, ↑n < x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"I... | [] | obtain ⟨n, hn⟩ := exists_nat_gt_of_mk_nonneg (mk_neg x ▸ hx)
use -n
simpa [neg_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 229,
"column": 2
} | {
"line": 231,
"column": 16
} | {
"line": 233,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : CommRing R\ninst✝ : IsStrictOrderedRing R\nx : R\nhx : 0 ≤ mk x\n⊢ ∃ n, ↑n < x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"I... | [] | obtain ⟨n, hn⟩ := exists_nat_gt_of_mk_nonneg (mk_neg x ▸ hx)
use -n
simpa [neg_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 86
} | {
"line": 78,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx : R\n⊢ x * x ∈ P",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"CommSemiring.toSemiring",
"RingPreordering.mem_of_isSquare",
"instDistribOfSemiring",
"CommRing.toCommSemiring",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 86
} | {
"line": 78,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx : R\n⊢ x * x ∈ P",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"CommSemiring.toSemiring",
"RingPreordering.mem_of_isSquare",
"instDistribOfSemiring",
"CommRing.toCommSemiring",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 86
} | {
"line": 78,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx : R\n⊢ x * x ∈ P",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"CommSemiring.toSemiring",
"RingPreordering.mem_of_isSquare",
"instDistribOfSemiring",
"CommRing.toCommSemiring",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 79,
"column": 80
} | {
"line": 79,
"column": 85
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx : R\n⊢ x ^ 2 ∈ P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"IsSquare.sq",
"instOfNatNat",
"RingPreordering.mem_of_isSquare",
"NPow.toPow",
"CommRing.toCo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 79,
"column": 80
} | {
"line": 79,
"column": 85
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx : R\n⊢ x ^ 2 ∈ P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"IsSquare.sq",
"instOfNatNat",
"RingPreordering.mem_of_isSquare",
"NPow.toPow",
"CommRing.toCo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 79,
"column": 80
} | {
"line": 79,
"column": 85
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx : R\n⊢ x ^ 2 ∈ P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"IsSquare.sq",
"instOfNatNat",
"RingPreordering.mem_of_isSquare",
"NPow.toPow",
"CommRing.toCo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 118,
"column": 23
} | {
"line": 118,
"column": 28
} | {
"line": 120,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\na✝ b✝ : R\nha : a✝ ∈ S\nhb : b✝ ∈ S\n⊢ a✝ * b✝ ∈ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"HMul.hMul",
"CommSemiring.toSemiring",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 118,
"column": 23
} | {
"line": 118,
"column": 28
} | {
"line": 120,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\na✝ b✝ : R\nha : a✝ ∈ S\nhb : b✝ ∈ S\n⊢ a✝ * b✝ ∈ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"HMul.hMul",
"CommSemiring.toSemiring",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 118,
"column": 23
} | {
"line": 118,
"column": 28
} | {
"line": 120,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\na✝ b✝ : R\nha : a✝ ∈ S\nhb : b✝ ∈ S\n⊢ a✝ * b✝ ∈ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"HMul.hMul",
"CommSemiring.toSemiring",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 117,
"column": 17
} | {
"line": 117,
"column": 22
} | {
"line": 118,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\n⊢ 1 ∈ S",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"MulOne.toOne",
"CommSemiring.toSemiring",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 117,
"column": 17
} | {
"line": 117,
"column": 22
} | {
"line": 118,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\n⊢ 1 ∈ S",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"MulOne.toOne",
"CommSemiring.toSemiring",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 117,
"column": 17
} | {
"line": 117,
"column": 22
} | {
"line": 118,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\n⊢ 1 ∈ S",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"MulOne.toOne",
"CommSemiring.toSemiring",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 28
} | {
"line": 117,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\na✝ b✝ : R\nha : a✝ ∈ S\nhb : b✝ ∈ S\n⊢ a✝ + b✝ ∈ S",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 28
} | {
"line": 117,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\na✝ b✝ : R\nha : a✝ ∈ S\nhb : b✝ ∈ S\n⊢ a✝ + b✝ ∈ S",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 28
} | {
"line": 117,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\na✝ b✝ : R\nha : a✝ ∈ S\nhb : b✝ ∈ S\n⊢ a✝ + b✝ ∈ S",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 115,
"column": 18
} | {
"line": 115,
"column": 23
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\n⊢ 0 ∈ S",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"CommSemiring.toSemiring",
"SubsemiringClass.toAdd... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 115,
"column": 18
} | {
"line": 115,
"column": 23
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\n⊢ 0 ∈ S",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"CommSemiring.toSemiring",
"SubsemiringClass.toAdd... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 115,
"column": 18
} | {
"line": 115,
"column": 23
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nS : Set R\nhS : S = ↑P\n⊢ 0 ∈ S",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"CommSemiring.toSemiring",
"SubsemiringClass.toAdd... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 142,
"column": 17
} | {
"line": 142,
"column": 22
} | {
"line": 143,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ ∀ {a b : R}, a ∈ ↑P ∩ -↑P → b ∈ ↑P ∩ -↑P → a + b ∈ ↑P ∩ -↑P",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 142,
"column": 17
} | {
"line": 142,
"column": 22
} | {
"line": 143,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ ∀ {a b : R}, a ∈ ↑P ∩ -↑P → b ∈ ↑P ∩ -↑P → a + b ∈ ↑P ∩ -↑P",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 142,
"column": 17
} | {
"line": 142,
"column": 22
} | {
"line": 143,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ ∀ {a b : R}, a ∈ ↑P ∩ -↑P → b ∈ ↑P ∩ -↑P → a + b ∈ ↑P ∩ -↑P",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 141,
"column": 18
} | {
"line": 141,
"column": 23
} | {
"line": 142,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ 0 ∈ ↑P ∩ -↑P",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUni... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 141,
"column": 18
} | {
"line": 141,
"column": 23
} | {
"line": 142,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ 0 ∈ ↑P ∩ -↑P",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUni... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 141,
"column": 18
} | {
"line": 141,
"column": 23
} | {
"line": 142,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ 0 ∈ ↑P ∩ -↑P",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUni... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 143,
"column": 17
} | {
"line": 143,
"column": 22
} | {
"line": 145,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ ∀ {x : R}, x ∈ ↑P ∩ -↑P → -x ∈ ↑P ∩ -↑P",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnita... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 143,
"column": 17
} | {
"line": 143,
"column": 22
} | {
"line": 145,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ ∀ {x : R}, x ∈ ↑P ∩ -↑P → -x ∈ ↑P ∩ -↑P",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnita... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 143,
"column": 17
} | {
"line": 143,
"column": 22
} | {
"line": 145,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\n⊢ ∀ {x : R}, x ∈ ↑P ∩ -↑P → -x ∈ ↑P ∩ -↑P",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnita... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 246,
"column": 48
} | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\nF : Type u_2\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : FunLike F α R\ninst✝² : ZeroHomClass F α R\ninst✝¹ : NonnegHomClass F α R\ninst✝ : SubmultiplicativeHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℕ\na b : α\n⊢ f 0 = 0",
"ppT... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 246,
"column": 48
} | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\nF : Type u_2\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : FunLike F α R\ninst✝² : ZeroHomClass F α R\ninst✝¹ : NonnegHomClass F α R\ninst✝ : SubmultiplicativeHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℕ\na b : α\n⊢ f 0 = 0",
"ppT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 246,
"column": 48
} | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\nF : Type u_2\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : FunLike F α R\ninst✝² : ZeroHomClass F α R\ninst✝¹ : NonnegHomClass F α R\ninst✝ : SubmultiplicativeHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℕ\na b : α\n⊢ f 0 = 0",
"ppT... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 246,
"column": 59
} | {
"line": 246,
"column": 64
} | {
"line": 246,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\nF : Type u_2\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : FunLike F α R\ninst✝² : ZeroHomClass F α R\ninst✝¹ : NonnegHomClass F α R\ninst✝ : SubmultiplicativeHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℕ\na b : α\n⊢ ∀ (x : α), 0 ≤ f x... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 246,
"column": 59
} | {
"line": 246,
"column": 64
} | {
"line": 246,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\nF : Type u_2\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : FunLike F α R\ninst✝² : ZeroHomClass F α R\ninst✝¹ : NonnegHomClass F α R\ninst✝ : SubmultiplicativeHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℕ\na b : α\n⊢ ∀ (x : α), 0 ≤ f x... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.IsNonarchimedean | {
"line": 246,
"column": 59
} | {
"line": 246,
"column": 64
} | {
"line": 246,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder R\nF : Type u_2\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : FunLike F α R\ninst✝² : ZeroHomClass F α R\ninst✝¹ : NonnegHomClass F α R\ninst✝ : SubmultiplicativeHomClass F α R\nf : F\nhna : IsNonarchimedean ⇑f\nn : ℕ\na b : α\n⊢ ∀ (x : α), 0 ≤ f x... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Quotient | {
"line": 77,
"column": 2
} | {
"line": 86,
"column": 33
} | {
"line": 88,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Preorder α\ns : Setoid α\ninst✝ : Preorder β\nf : α → β\nhf : Monotone f\nH : ∀ (x₁ x₂ : α), x₁ ≈ x₂ → f x₁ = f x₂\n⊢ Monotone (Quotient.lift f H)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Quotient.instPreorder",
"Preorder.to... | [] | intro x y h
induction x using Quotient.inductionOn with | h x
induction y using Quotient.inductionOn with | h y
induction h
on_goal 2 => rename_i IH; apply IH.trans
all_goals
rename_i h
cases h with
| inl h => exact hf h
| inr h => exact (H _ _ h).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Quotient | {
"line": 77,
"column": 2
} | {
"line": 86,
"column": 33
} | {
"line": 88,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Preorder α\ns : Setoid α\ninst✝ : Preorder β\nf : α → β\nhf : Monotone f\nH : ∀ (x₁ x₂ : α), x₁ ≈ x₂ → f x₁ = f x₂\n⊢ Monotone (Quotient.lift f H)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Quotient.instPreorder",
"Preorder.to... | [] | intro x y h
induction x using Quotient.inductionOn with | h x
induction y using Quotient.inductionOn with | h y
induction h
on_goal 2 => rename_i IH; apply IH.trans
all_goals
rename_i h
cases h with
| inl h => exact hf h
| inr h => exact (H _ _ h).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 60,
"column": 39
} | {
"line": 60,
"column": 44
} | {
"line": 61,
"column": 2
} | [
{
"pp": "F : Type u_2\ninst✝ : Field F\nP : RingPreordering F\na : F\nha : a ∈ P\n⊢ a * (a⁻¹ * a⁻¹) ∈ P",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"RingPreordering.mul_self_mem._simp_1",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInv... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 66,
"column": 38
} | {
"line": 66,
"column": 43
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx : R\n⊢ 0 ∈ P",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"CommSemiring.toSemiring",
"SubsemiringClass.toAddSubmonoidClass",
"AddMonoid.toAdd... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 66,
"column": 38
} | {
"line": 66,
"column": 43
} | {
"line": 68,
"column": 0
} | [
{
"pp": "case sq_add\nR : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx x✝ s✝ : R\nhx✝ : IsSquare x✝\nhs✝ : IsSumSq s✝\na✝ : s✝ ∈ P\n⊢ x✝ + s✝ ∈ P",
"ppTerm": "?sq_add",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddSubmonoidClass.toAddMemCl... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 175,
"column": 51
} | {
"line": 175,
"column": 56
} | {
"line": 175,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx✝¹ : P.IsOrdering\na b : R\nx✝ : -(a * b) ∈ P\nthis : ¬(a ∈ P ∨ b ∈ P)\n⊢ -a ∈ P",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"eq_false",
"congrArg",
"Membersh... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 175,
"column": 51
} | {
"line": 175,
"column": 56
} | {
"line": 175,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx✝¹ : P.IsOrdering\na b : R\nx✝ : -(a * b) ∈ P\nthis : ¬(a ∈ P ∨ b ∈ P)\n⊢ -a ∈ P",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"eq_false",
"congrArg",
"Membersh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 175,
"column": 51
} | {
"line": 175,
"column": 56
} | {
"line": 175,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx✝¹ : P.IsOrdering\na b : R\nx✝ : -(a * b) ∈ P\nthis : ¬(a ∈ P ∨ b ∈ P)\n⊢ -a ∈ P",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"eq_false",
"congrArg",
"Membersh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 175,
"column": 71
} | {
"line": 175,
"column": 76
} | {
"line": 175,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx✝¹ : P.IsOrdering\na b : R\nx✝ : -(a * b) ∈ P\nthis : ¬(a ∈ P ∨ b ∈ P)\n⊢ -b ∈ P",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"eq_false",
"congrArg",
"Membersh... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 175,
"column": 71
} | {
"line": 175,
"column": 76
} | {
"line": 175,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx✝¹ : P.IsOrdering\na b : R\nx✝ : -(a * b) ∈ P\nthis : ¬(a ∈ P ∨ b ∈ P)\n⊢ -b ∈ P",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"eq_false",
"congrArg",
"Membersh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 175,
"column": 71
} | {
"line": 175,
"column": 76
} | {
"line": 175,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nx✝¹ : P.IsOrdering\na b : R\nx✝ : -(a * b) ∈ P\nthis : ¬(a ∈ P ∨ b ∈ P)\n⊢ -b ∈ P",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"eq_false",
"congrArg",
"Membersh... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 186,
"column": 14
} | {
"line": 186,
"column": 19
} | {
"line": 186,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nh : ∀ (a b : R), -(a * b) ∈ P → a ∈ P ∨ b ∈ P\nthis✝⁴ : HasMemOrNegMem P\nx y : R\nx✝ : x * y ∈ P.support\nthis✝³ : ¬(x ∈ P.support ∨ y ∈ P.support)\nthis✝² : -(-x * y) ∈ P → -x ∈ P ∨ y ∈ P\nthis✝¹ : -(-x * -y) ∈ P → -x ∈ P ∨ -y ∈ P\nthis✝ : -(x ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 186,
"column": 14
} | {
"line": 186,
"column": 19
} | {
"line": 186,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nh : ∀ (a b : R), -(a * b) ∈ P → a ∈ P ∨ b ∈ P\nthis✝⁴ : HasMemOrNegMem P\nx y : R\nx✝ : x * y ∈ P.support\nthis✝³ : ¬(x ∈ P.support ∨ y ∈ P.support)\nthis✝² : -(-x * y) ∈ P → -x ∈ P ∨ y ∈ P\nthis✝¹ : -(-x * -y) ∈ P → -x ∈ P ∨ -y ∈ P\nthis✝ : -(x ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 186,
"column": 14
} | {
"line": 186,
"column": 19
} | {
"line": 186,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nh : ∀ (a b : R), -(a * b) ∈ P → a ∈ P ∨ b ∈ P\nthis✝⁴ : HasMemOrNegMem P\nx y : R\nx✝ : x * y ∈ P.support\nthis✝³ : ¬(x ∈ P.support ∨ y ∈ P.support)\nthis✝² : -(-x * y) ∈ P → -x ∈ P ∨ y ∈ P\nthis✝¹ : -(-x * -y) ∈ P → -x ∈ P ∨ -y ∈ P\nthis✝ : -(x ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 9
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case refine_2.inl\nK : Type u_1\ninst✝¹¹ : DivisionRing K\ninst✝¹⁰ : LinearOrder K\ninst✝⁹ : IsOrderedRing K\ninst✝⁸ : Archimedean K\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : LinearOrder M\ninst✝⁵ : IsOrderedAddMonoid M\ninst✝⁴ : Module K M\ninst✝³ : IsOrderedModule K M\nR : Type u_3\ninst✝² : A... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 9
} | {
"line": 426,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹² : DivisionRing K\ninst✝¹¹ : LinearOrder K\ninst✝¹⁰ : IsOrderedRing K\ninst✝⁹ : Archimedean K\nM : Type u_2\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\ninst✝⁵ : Module K M\ninst✝⁴ : IsOrderedModule K M\nR : Type u_3\ninst✝³ : AddCommGroup R\nins... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 34
} | {
"line": 217,
"column": 34
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\n⊢ a ∈ IsLocalRing.maximalIdeal ↥A ↔ A.valuation ↑a < 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Preorder.toLT",
... | [
"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\n⊢ a ∈ nonunits ↥A ↔ A.valuation ↑a < 1"
] | IsLocalRing.mem_maximalIdeal | Lean.Elab.Tactic.evalRewriteSeq | null |
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