module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.UniformSpace.ProdApproximation | {
"line": 110,
"column": 27
} | {
"line": 110,
"column": 50
} | {
"line": 110,
"column": 51
} | [
{
"pp": "X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformS... | [
"X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformSpace R := Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 94,
"column": 42
} | {
"line": 94,
"column": 53
} | {
"line": 94,
"column": 54
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nncs : CauSeq ℤ_[p] norm\nF : Polynomial R\nhnorm : Tendsto (fun i ↦ ‖(Polynomial.aeval (↑ncs i)) F‖) atTop (𝓝 0)\n⊢ Tendsto (fun e ↦ ‖(Polynomial.aeval (↑ncs e)) F - 0‖) atTop (𝓝 0)",
"ppTerm": "?m... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nncs : CauSeq ℤ_[p] norm\nF : Polynomial R\nhnorm : Tendsto (fun i ↦ ‖(Polynomial.aeval (↑ncs i)) F‖) atTop (𝓝 0)\n⊢ Tendsto (fun e ↦ ‖(Polynomial.aeval (↑ncs e)) F‖) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 116,
"column": 47
} | {
"line": 116,
"column": 58
} | {
"line": 116,
"column": 59
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nha : (Polynomial.aeval a) F = 0\nz' : ℤ_[p]\nhz' : (Polynomial.aeval z') F = 0\nhnormz' : ‖z' - a‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq ... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nha : (Polynomial.aeval a) F = 0\nz' : ℤ_[p]\nhz' : (Polynomial.aeval z') F = 0\nhnormz' : ‖z' - a‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 138,
"column": 72
} | {
"line": 139,
"column": 14
} | {
"line": 141,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\n⊢ T_gen p F a = ‖(Polynomial.aeval a) F‖ / ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"P... | [] | by
simp [T_gen] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 15
} | {
"line": 220,
"column": 16
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval z) F) / ↑((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Pell | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 71
} | {
"line": 393,
"column": 72
} | [
{
"pp": "x y : ℤ\nhy : y ≠ 0\na : ℤ\nh₀ : 0 < a * a\nhxy : (x + a * y) * (x - a * y) = 1\n⊢ False",
"ppTerm": "?m.146",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ\nhy : y ≠ 0\na : ℤ\nh₀ : 0 < a * a\nhxy : (x + a * y) * (x - a * y) = 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 227,
"column": 24
} | {
"line": 227,
"column": 73
} | {
"line": 227,
"column": 73
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval... | [] | by simp only [PadicInt.coe_neg, PadicInt.coe_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Pell | {
"line": 565,
"column": 35
} | {
"line": 565,
"column": 50
} | {
"line": 565,
"column": 50
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ d * a.y * (a.y * a₁.x) = (a.x ^ 2 - 1) * a₁.x",
"ppTerm": "?m.130",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Pell.Solution₁.x",
"HSub... | [
"d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ d * a.y * (a.y * a₁.x) = d * a.y ^ 2 * a₁.x"
] | rw [← a.prop_y] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 229,
"column": 15
} | {
"line": 229,
"column": 74
} | {
"line": 229,
"column": 75
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nz z' z1 : ℤ_[p]\nhz' : z' = z - z1\nn : ℕ\nhz : ih_gen n z\nh1 : ‖↑((Polynomial.aeval z) F) / ↑((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 15
} | {
"line": 389,
"column": 16
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nhnsol : (Polynomial.aeval a) F ≠ 0\nn : ℕ\nthis : (fun x ↦ 2 ^ x) 1 ≤ (fun x ↦ 2 ^ x)... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nhnsol : (Polynomial.aeval a) F ≠ 0\nn : ℕ\nthis : (fun x ↦ 2 ^ x) 1 ≤ (fun x ↦ 2 ^ x) (n + 1)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 50,
"column": 36
} | {
"line": 57,
"column": 34
} | {
"line": 59,
"column": 0
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : DecidableEq K⟮X⟯\ninst✝ : IsTrivialOn K v\nhlt : 1 < v X\n⊢ v.IsEquiv (inftyValuation K)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"WithZero.instNont... | [] | by
refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_
rcases eq_or_ne f 0 with rfl | hf
· simp
· have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]
rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,
valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]
g... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Rayleigh | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 25
} | {
"line": 70,
"column": 2
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ ∀ ⦃a : ℤ⦄, a ∈ {x | ∃ k, beattySeq r k = x} → a ∉ {x | ∃ k, beattySeq' s k = x}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.Rayleigh.0.Beatty.no_collision.match_1_3",
"False",
"Set.ofPred",
... | [
"r s : ℝ\nhrs : r.HolderConjugate s\nj k : ℤ\nh₁ : beattySeq r k = j\nm : ℤ\nh₂ : beattySeq' s m = j\n⊢ False"
] | intro j ⟨k, h₁⟩ ⟨m, h₂⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 15
} | {
"line": 88,
"column": 16
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝¹ : Field K\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\na b : K[X]\nhab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0\nhb : ¬IsUnit b\nπᵥ : K[X] := ⋯.min {p | v ↑p < 1 ∧ p ≠ 0} hne\nhπᵥ : πᵥ = a * b\nhbpos : 0 < ↑b.natDegre... | [
"K : Type u_1\nΓ : Type u_2\ninst✝¹ : Field K\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\na b : K[X]\nhab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0\nhb : ¬IsUnit b\nπᵥ : K[X] := ⋯.min {p | v ↑p < 1 ∧ p ≠ 0} hne\nhπᵥ : πᵥ = a * b\nhbpos : 0 < ↑b.natDegree\n⊢ 0 < b.n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 136,
"column": 15
} | {
"line": 136,
"column": 26
} | {
"line": 136,
"column": 27
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"ppTerm": "?m.62",
"... | [
"K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Rayleigh | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 46
} | {
"line": 135,
"column": 0
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ {x | ∃ k, beattySeq' r k = x}ᶜ = {x | ∃ k, beattySeq s k = x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Set.ofPred",
"Compl.compl",
"Real.HolderConjugate.symm",
... | [] | rw [← compl_beattySeq hrs.symm, compl_compl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Rayleigh | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 46
} | {
"line": 135,
"column": 0
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ {x | ∃ k, beattySeq' r k = x}ᶜ = {x | ∃ k, beattySeq s k = x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Set.ofPred",
"Compl.compl",
"Real.HolderConjugate.symm",
... | [] | rw [← compl_beattySeq hrs.symm, compl_compl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Rayleigh | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 46
} | {
"line": 135,
"column": 0
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ {x | ∃ k, beattySeq' r k = x}ᶜ = {x | ∃ k, beattySeq s k = x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Set.ofPred",
"Compl.compl",
"Real.HolderConjugate.symm",
... | [] | rw [← compl_beattySeq hrs.symm, compl_compl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SelbergSieve | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 60
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case h_ne\ns : BoundingSieve\nd : ℕ\nhdP : d ∣ s.prodPrimes\nhd_ne_one : d ≠ 1\nhd_sq : Squarefree d\nthis : d ≠ 0\n⊢ d.primeFactors.Nonempty",
"ppTerm": "?h_ne",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"LT.lt",
"True",
"Finset.Nonempty",
... | [] | simp only [nonempty_primeFactors, show 1 < d by lia] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.SelbergSieve | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 60
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case h_ne\ns : BoundingSieve\nd : ℕ\nhdP : d ∣ s.prodPrimes\nhd_ne_one : d ≠ 1\nhd_sq : Squarefree d\nthis : d ≠ 0\n⊢ d.primeFactors.Nonempty",
"ppTerm": "?h_ne",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"LT.lt",
"True",
"Finset.Nonempty",
... | [] | simp only [nonempty_primeFactors, show 1 < d by lia] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.SelbergSieve | {
"line": 156,
"column": 8
} | {
"line": 156,
"column": 60
} | {
"line": 157,
"column": 4
} | [
{
"pp": "case h_ne\ns : BoundingSieve\nd : ℕ\nhdP : d ∣ s.prodPrimes\nhd_ne_one : d ≠ 1\nhd_sq : Squarefree d\nthis : d ≠ 0\n⊢ d.primeFactors.Nonempty",
"ppTerm": "?h_ne",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"LT.lt",
"True",
"Finset.Nonempty",
... | [] | simp only [nonempty_primeFactors, show 1 < d by lia] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 173,
"column": 4
} | {
"line": 174,
"column": 11
} | {
"line": 174,
"column": 12
} | [
{
"pp": "case right\nK : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\np : K[X]\nhp : p ≠ 0\nπ : K[X] := ⋯\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\nhπirr : Irreducible π\nk : ℕ\nq : K[X]\... | [
"case right\nK : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\np : K[X]\nhp : p ≠ 0\nπ : K[X] := πᵥ\nhne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty\nhπirr : Irreducible π\nk : ℕ\nq : K[X]\nhnq : ¬q %... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 178,
"column": 27
} | {
"line": 178,
"column": 38
} | {
"line": 178,
"column": 39
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\nf : K⟮X⟯\nhf : f ≠ 0\n⊢ v ↑πᵥ ≠ 0",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\nf : K⟮X⟯\nhf : f ≠ 0\n⊢ ¬πᵥ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 190,
"column": 27
} | {
"line": 190,
"column": 38
} | {
"line": 190,
"column": 39
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\nhv : v.IsRankOneDiscrete\n⊢ v ↑πᵥ ≠ 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\nhv : v.IsRankOneDiscrete\n⊢ ¬πᵥ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SelbergSieve | {
"line": 307,
"column": 40
} | {
"line": 307,
"column": 51
} | {
"line": 307,
"column": 52
} | [
{
"pp": "s : BoundingSieve\nl : ℕ\nhl : Squarefree l\nhnu_nonzero : s.nu l ≠ 0\nd e : ℕ\nhd : (d, e) ∈ l.divisorsAntidiagonal\n⊢ d * e = l ∧ ?m.92",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : BoundingSieve\nl : ℕ\nhl : Squarefree l\nhnu_nonzero : s.nu l ≠ 0\nd e : ℕ\nhd : (d, e) ∈ l.divisorsAntidiagonal\n⊢ d * e = l ∧ ?m.92"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SelbergSieve | {
"line": 308,
"column": 42
} | {
"line": 308,
"column": 79
} | {
"line": 308,
"column": 80
} | [
{
"pp": "s : BoundingSieve\nd e : ℕ\nhl : Squarefree (d * e)\nhnu_nonzero : s.nu (d * e) ≠ 0\nhd : (d, e) ∈ (d * e).divisorsAntidiagonal\n⊢ d.Coprime e ∧ ?m.123",
"ppTerm": "?m.125",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : BoundingSieve\nd e : ℕ\nhl : Squarefree (d * e)\nhnu_nonzero : s.nu (d * e) ≠ 0\nhd : (d, e) ∈ (d * e).divisorsAntidiagonal\n⊢ d.Coprime e ∧ ?m.123"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 194,
"column": 56
} | {
"line": 194,
"column": 67
} | {
"line": 194,
"column": 68
} | [
{
"pp": "x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).re - ↑(round (toComplex x / toComplex y).re)| ≤ 2⁻¹",
"ppTerm": "?m.149",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).re - ↑(round (toComplex x / toComplex y).re)| ≤ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 195,
"column": 56
} | {
"line": 195,
"column": 67
} | {
"line": 195,
"column": 68
} | [
{
"pp": "x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).im - ↑(round (toComplex x / toComplex y).im)| ≤ 2⁻¹",
"ppTerm": "?m.171",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(toComplex x / toComplex y).im - ↑(round (toComplex x / toComplex y).im)| ≤ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumFourSquares | {
"line": 130,
"column": 20
} | {
"line": 130,
"column": 31
} | {
"line": 130,
"column": 32
} | [
{
"pp": "case h\np : ℕ\nhp : Prime p\nthis : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\na b c d : ℕ\nhmin : ∀ m < 1, ¬(m ... | [
"case h\np : ℕ\nhp : Prime p\nthis : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\na b c d : ℕ\nhmin : ∀ m < 1, ¬(m < p ∧ 0 < m ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 58
} | {
"line": 250,
"column": 59
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\n⊢ ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b",
"ppTerm": "?m.84",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\n⊢ ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 253,
"column": 91
} | {
"line": 254,
"column": 85
} | {
"line": 254,
"column": 85
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\nhab : ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b\na b : ℤ[i]\nhpab : ↑p = a * b\nhau : ¬IsUnit a\nhbu : ¬IsUnit b\n⊢ (norm a).natAbs * (norm b).natAbs = p ^ 2",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
... | [] | by
rw [← Int.natCast_inj, Int.natCast_pow, sq, ← @norm_natCast (-1), hpab]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 255,
"column": 32
} | {
"line": 255,
"column": 64
} | {
"line": 255,
"column": 65
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\nhab : ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b\na b : ℤ[i]\nhpab : ↑p = a * b\nhau : ¬IsUnit a\nhbu : ¬IsUnit b\nhnap : (norm a).natAbs = p\n⊢ a.re.natAbs ^ 2 + a.im.natAbs ^ 2 = p",
"ppTerm": "?m.127",
"assigned": true,
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : ¬Irreducible ↑p\nhpu : ¬IsUnit ↑p\nhab : ∃ a b, ↑p = a * b ∧ ¬IsUnit a ∧ ¬IsUnit b\na b : ℤ[i]\nhpab : ↑p = a * b\nhau : ¬IsUnit a\nhbu : ¬IsUnit b\nhnap : (norm a).natAbs = p\n⊢ a.re.natAbs * a.re.natAbs + a.im.natAbs * a.im.natAbs = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 42
} | {
"line": 92,
"column": 43
} | [
{
"pp": "m n : ℕ\nhc : m.Coprime n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\nthis : IsSquare (-1)\n⊢ IsSquare (-1)",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m n : ℕ\nhc : m.Coprime n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\nthis : IsSquare (-1)\n⊢ IsSquare (-1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 90
} | {
"line": 230,
"column": 0
} | [
{
"pp": "case inr.refine_2\nn : ℕ\nhn₀ : n > 0\nH : ∀ q ∈ n.primeFactors, q % 4 = 3 → Even (padicValNat q n)\nb a : ℕ\nhb₀ : 0 < b\nha₀ : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\nq : ℕ\nhq : q ∈ b.primeFactors\nhq4 : q % 4 = 3\nthis✝¹ : Fact (Prime q)\nthis✝ : n ≠ 0 → b.primeFactors ⊆ n.primeFactors\nthis... | [] | grind [factorization_def, prime_of_mem_primeFactors, padicValNat.mul, padicValNat.pow] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.SumFourSquares | {
"line": 161,
"column": 63
} | {
"line": 161,
"column": 86
} | {
"line": 161,
"column": 87
} | [
{
"pp": "p : ℕ\nhp : Prime p\nthis✝ : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 ... | [
"p : ℕ\nhp : Prime p\nthis✝ : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumFourSquares | {
"line": 191,
"column": 8
} | {
"line": 192,
"column": 30
} | {
"line": 192,
"column": 31
} | [
{
"pp": "p : ℕ\nhp : Prime p\nthis✝² : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0... | [
"p : ℕ\nhp : Prime p\nthis✝² : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 26
} | {
"line": 179,
"column": 26
} | [
{
"pp": "case h.right.refine_1\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^... | [
"case h.right.refine_1\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - ... | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleNumber | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 41
} | {
"line": 189,
"column": 42
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\nhpos : 0 < remainder (↑m) n\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n",
"ppTerm": "?m.78",
"ass... | [
"m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\nhpos : 0 < remainder (↑m) n\n⊢ remainder (↑m) n < ((↑m ^ n !) ^ n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Irrational | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 35
} | {
"line": 97,
"column": 36
} | [
{
"pp": "x : ℝ\nhx : Irrational x\nn : ℕ\nε : ℝ\nH : ∀ k ≤ n, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k)\nr : ℚ\nhr : r.den ≤ n\n⊢ ε ≤ dist x ↑r",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"Rat.num",
"instHDiv",
... | [
"x : ℝ\nhx : Irrational x\nn : ℕ\nε : ℝ\nH : ∀ k ≤ n, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k)\nr : ℚ\nhr : r.den ≤ n\n⊢ ε ≤ dist x (↑r.num / ↑r.den)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 39
} | {
"line": 84,
"column": 2
} | [
{
"pp": "p x C : ℝ\nhC : ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\nn : ℕ\nhle : 1 ≤ n\nm : ℤ\nhne : x ≠ ↑m / ↑n\nhlt : |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < max C 1 / ↑n ^ p",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
... | [
"p x C : ℝ\nhC : ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\nn : ℕ\nhle : 1 ≤ n\nm : ℤ\nhne : x ≠ ↑m / ↑n\nhlt : |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ C / ↑n ^ p ≤ max C 1 / ↑n ^ p"
] | refine ⟨hle, m, hne, hlt.trans_le ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 48
} | {
"line": 102,
"column": 6
} | [
{
"pp": "p q x : ℝ\nh : LiouvilleWith p x\nhlt : q < p\nC : ℝ\n_hC₀ : 0 < C\nhC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ ∀ᶠ (n : ℕ) in atTop, C < ↑n ^ (p - q)",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"p q x : ℝ\nh : LiouvilleWith p x\nhlt : q < p\nC : ℝ\n_hC₀ : 0 < C\nhC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p\n⊢ ∀ᶠ (n : ℕ) in atTop, C < ↑n ^ (p - q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 44
} | {
"line": 204,
"column": 45
} | [
{
"pp": "f : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval ... | [
"f : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 35
} | {
"line": 206,
"column": 36
} | [
{
"pp": "case h.right.refine_2\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^... | [
"case h.right.refine_2\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 91
} | {
"line": 132,
"column": 6
} | [
{
"pp": "p x : ℝ\nr : ℚ\nhr : r ≠ 0\nh : LiouvilleWith p (x * ↑r)\n⊢ LiouvilleWith p x",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p x : ℝ\nr : ℚ\nhr : r ≠ 0\nh : LiouvilleWith p (x * ↑r)\n⊢ LiouvilleWith p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 54,
"column": 30
} | {
"line": 54,
"column": 40
} | {
"line": 54,
"column": 41
} | [
{
"pp": "p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\nhb : 1 ≤ b\na : ℤ\nhlt : |x - ↑a / ↑b| < (↑b ^ (2 + 1 / (↑n + 1)))⁻¹\n⊢ ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))",
"ppTerm": "?m.308",
"assigned": true,
... | [
"p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\nhb : 1 ≤ b\na : ℤ\nhlt : |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / (↑n + 1))\n⊢ ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 16
} | [
{
"pp": "p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\na : ℤ\nhlt : |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n.succ)\nhb : 1 ≤ ↑b\nhb0 : 0 < ↑b\n⊢ 2 ≤ 2 + 1 / ↑(n + 1)",
"ppTerm": "?m.445",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\na : ℤ\nhlt : |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n.succ)\nhb : 1 ≤ ↑b\nhb0 : 0 < ↑b\n⊢ 0 ≤ ↑n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Wilson | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 58
} | {
"line": 66,
"column": 59
} | [
{
"pp": "case refine_3.refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : 0 < p\nb : ℕ\nhb : b ≠ 0 ∧ b < p\nh : (↑b).val = val 0\n⊢ b = 0",
"ppTerm": "?refine_3.refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_3.refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : 0 < p\nb : ℕ\nhb : b ≠ 0 ∧ b < p\nh : (↑b).val = val 0\n⊢ b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Wilson | {
"line": 44,
"column": 2
} | {
"line": 68,
"column": 48
} | {
"line": 70,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ ↑(p - 1)! = -1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine
calc
((p - 1)! : ZMod p) = ∏ x ∈ Ico 1 (succ (p - 1)), (x : ZMod p) := by
rw [← Finset.prod_Ico_id_eq_factorial, prod_natCast]
_ = ∏ x : (ZMod p)ˣ, (x : ZMod p) := ?_
_ = -1 := by
simp_rw [← Units.coeHom_apply, ← map_prod (Units.coeHom (ZMod p)),
prod_univ_units_id... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Wilson | {
"line": 44,
"column": 2
} | {
"line": 68,
"column": 48
} | {
"line": 70,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ ↑(p - 1)! = -1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine
calc
((p - 1)! : ZMod p) = ∏ x ∈ Ico 1 (succ (p - 1)), (x : ZMod p) := by
rw [← Finset.prod_Ico_id_eq_factorial, prod_natCast]
_ = ∏ x : (ZMod p)ˣ, (x : ZMod p) := ?_
_ = -1 := by
simp_rw [← Units.coeHom_apply, ← map_prod (Units.coeHom (ZMod p)),
prod_univ_units_id... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Bounds.Lattice | {
"line": 31,
"column": 2
} | {
"line": 32,
"column": 9
} | {
"line": 32,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\n⊢ GaloisConnection (⇑OrderDual.toDual ∘ upperBounds) (lowerBounds ∘ ⇑OrderDual.ofDual)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"OrderDual.instLE",
"OrderDual.toDual",
"Eq.mpr",
"Equiv.instEquivLike",
"OrderDua... | [
"α : Type u_1\ninst✝ : Preorder α\n⊢ ∀ (a a_1 : Set α), (∀ x ∈ a_1, ∀ x_1 ∈ a, x_1 ≤ x) ↔ ∀ x ∈ a, ∀ x_1 ∈ a_1, x ≤ x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 183,
"column": 15
} | {
"line": 183,
"column": 26
} | {
"line": 183,
"column": 27
} | [
{
"pp": "p x : ℝ\nr : ℚ\nh : LiouvilleWith p (x + ↑r)\n⊢ LiouvilleWith p x",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p x : ℝ\nr : ℚ\nh : LiouvilleWith p (x + ↑r)\n⊢ LiouvilleWith p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 275,
"column": 20
} | {
"line": 275,
"column": 30
} | {
"line": 275,
"column": 31
} | [
{
"pp": "p : ℝ\nhp : 1 < p\nM : ℤ\nh : LiouvilleWith p ↑M\nn : ℕ\nhn : 0 < n\nm : ℤ\nhne : ↑M ≠ ↑m / ↑n\nhlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)\nhn' : 0 < ↑n\n⊢ (↑n)⁻¹ ≤ |↑M - ↑m / ↑n|",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
... | [
"p : ℝ\nhp : 1 < p\nM : ℤ\nh : LiouvilleWith p ↑M\nn : ℕ\nhn : 0 < n\nm : ℤ\nhne : ↑M ≠ ↑m / ↑n\nhlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)\nhn' : 0 < ↑n\n⊢ 1 / ↑n ≤ |↑M - ↑m / ↑n|"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 73
} | {
"line": 117,
"column": 4
} | [
{
"pp": "⊢ ∀ᵐ (x : ℝ), ∀ p > 2, ¬LiouvilleWith p x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"Iff.of_eq",
"congrArg",
"Set.ofPred",
"_private.Mathlib.NumberTheory.Transcende... | [
"⊢ volume (⋃ i, ⋃ (_ : i > 2), {x | LiouvilleWith i x}) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 46
} | {
"line": 125,
"column": 47
} | [
{
"pp": "⊢ volume {x | Liouville x} = 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ volume {x | Liouville x} = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 340,
"column": 36
} | {
"line": 340,
"column": 58
} | {
"line": 340,
"column": 59
} | [
{
"pp": "x : ℝ\nH : ∀ (p : ℝ), LiouvilleWith p x\nn b : ℕ\nhb : 1 < b\na : ℤ\nhne : x ≠ ↑a / ↑b\nhlt : |x - ↑a / ↑b| < ↑b ^ (-↑n)\n⊢ |x - ↑a / ↑↑b| < 1 / ↑↑b ^ n",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Real",
... | [
"x : ℝ\nH : ∀ (p : ℝ), LiouvilleWith p x\nn b : ℕ\nhb : 1 < b\na : ℤ\nhne : x ≠ ↑a / ↑b\nhlt : |x - ↑a / ↑b| < ↑b ^ (-↑n)\n⊢ |x - ↑a / ↑b| < (↑b ^ n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.WellApproximable | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 66
} | {
"line": 154,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedCommGroup A\na : A\nn : ℕ\nδ : ℝ\nhn : 0 < n\nhan : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n\nf : ↑{b | orderOf b = n} → ↑{b | orderOf b = n} := fun b ↦ ⟨a * ↑b, ⋯⟩\nhf : Surjective f\n⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball ... | [
"A : Type u_1\ninst✝ : SeminormedCommGroup A\na : A\nn : ℕ\nδ : ℝ\nhn : 0 < n\nhan : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n\nf : ↑{b | orderOf b = n} → ↑{b | orderOf b = n} := fun b ↦ ⟨a * ↑b, ⋯⟩\nhf : Surjective f\n⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball x δ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Comparable | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 13
} | {
"line": 347,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ a < b ∨ a = b ∨ b < a ∨ IncompRel (fun x1 x2 ↦ x1 ≤ x2) a b",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ a < b ∨ a = b ∨ b < a ∨ IncompRel (fun x1 x2 ↦ x1 ≤ x2) a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompletePartialOrder | {
"line": 87,
"column": 2
} | {
"line": 89,
"column": 16
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CompletePartialOrder α\ninst✝ : Preorder β\nf : α → β\n⊢ ScottContinuous f ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → IsLUB (f '' d) (f (sSup d))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"con... | [] | refine ⟨fun h d hd₁ hd₂ ↦ h hd₁ hd₂ hd₂.isLUB_sSup, fun h d hne hd a hda ↦ ?_⟩
rw [hda.unique hd.isLUB_sSup]
exact h hne hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompletePartialOrder | {
"line": 87,
"column": 2
} | {
"line": 89,
"column": 16
} | {
"line": 91,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CompletePartialOrder α\ninst✝ : Preorder β\nf : α → β\n⊢ ScottContinuous f ↔ ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → IsLUB (f '' d) (f (sSup d))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"con... | [] | refine ⟨fun h d hd₁ hd₂ ↦ h hd₁ hd₂ hd₂.isLUB_sSup, fun h d hne hd a hda ↦ ?_⟩
rw [hda.unique hd.isLUB_sSup]
exact h hne hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompleteSublattice | {
"line": 104,
"column": 12
} | {
"line": 104,
"column": 22
} | {
"line": 104,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\nι : Sort u_3\nf : ι → ↥L\n⊢ ↑(sSup (range f)) = ⨆ i, ↑(f i)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteSublattice.instSupSet",
"congrArg",
"iSup",
"Membership.m... | [
"α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\nι : Sort u_3\nf : ι → ↥L\n⊢ ⨆ N ∈ range f, ↑N = ⨆ i, ↑(f i)"
] | coe_sSup', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompleteSublattice | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 62
} | {
"line": 152,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sSup s ∈ ⇑f ⁻¹' ↑L",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sSup (⇑f '' s) ∈ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompleteSublattice | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 62
} | {
"line": 155,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sInf s ∈ ⇑f ⁻¹' ↑L",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"sInfHomCla... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sInf (⇑f '' s) ∈ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Concept | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 95
} | {
"line": 419,
"column": 2
} | [
{
"pp": "α : Type u_2\nr' : α → α → Prop\nc' : Concept α α r'\ninst✝¹ : Std.Trichotomous r'\ninst✝ : IsTrans α r'\nx : α\nx✝ : x ∈ ⊤\nhx : x ∉ c'.extent\ny : α\nhy : y ∈ c'.extent\n⊢ r' y x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Concept.mem_extent_of_rel_extent",
"Not... | [
"α : Type u_2\nr' : α → α → Prop\nc' : Concept α α r'\ninst✝¹ : Std.Trichotomous r'\ninst✝ : IsTrans α r'\nx : α\nx✝ : x ∈ ⊤\nhx : x ∉ c'.extent\ny : α\nhy : y ∈ c'.extent\n⊢ ¬x = y"
] | apply Not.imp_symm <| Std.Trichotomous.trichotomous x y (hx <| mem_extent_of_rel_extent · hy) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.Completion | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 13
} | {
"line": 106,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ principal a ≤ principal b ↔ a ≤ b",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : Preorder α\na b : α\n⊢ principal a ≤ principal b ↔ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Completion | {
"line": 180,
"column": 2
} | {
"line": 184,
"column": 6
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : PartialOrder β\nf : β ↪o α\nx : β\n⊢ (factorEmbedding f) (principal x) = f x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"sSup_le_iff._simp_2",
"Eq.mpr",
"instReflLe",
"congrArg",
"Se... | [] | rw [factorEmbedding_apply]
apply le_antisymm (by simp)
rw [le_sSup_iff]
refine fun y hy ↦ hy ?_
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Completion | {
"line": 180,
"column": 2
} | {
"line": 184,
"column": 6
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : PartialOrder β\nf : β ↪o α\nx : β\n⊢ (factorEmbedding f) (principal x) = f x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"sSup_le_iff._simp_2",
"Eq.mpr",
"instReflLe",
"congrArg",
"Se... | [] | rw [factorEmbedding_apply]
apply le_antisymm (by simp)
rw [le_sSup_iff]
refine fun y hy ↦ hy ?_
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Concept | {
"line": 527,
"column": 2
} | {
"line": 527,
"column": 13
} | {
"line": 527,
"column": 14
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nt : Set β\nc : Concept α β r\nh : c.intent ⊆ t\n⊢ c.intent ⊆ (ofAttributes r t).intent",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Concept.ofAttributes",
"Concept.intent",
"id",
... | [
"α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nt : Set β\nc : Concept α β r\nh : c.intent ⊆ t\n⊢ c.intent ⊆ upperPolar r (lowerPolar r t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Completion | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 52
} | {
"line": 229,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : a.extent ⊆ b.extent ∧ ∃ x ∈ b.extent, x ∉ a.extent\n⊢ ∃ c, a < principal c ∧ principal c ≤ b",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.t... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : a.extent ⊆ b.extent ∧ ∃ x ∈ b.extent, x ∉ a.extent\n⊢ ∃ c ∉ a.left, c ∈ b.left"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Completion | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 42
} | {
"line": 236,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : b.intent ⊆ a.intent ∧ ∃ x ∈ a.intent, x ∉ b.intent\n⊢ ∃ c, a ≤ principal c ∧ principal c < b",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.t... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : b.intent ⊆ a.intent ∧ ∃ x ∈ a.intent, x ∉ b.intent\n⊢ ∃ c ∈ a.right, c ∉ b.right"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Completion | {
"line": 245,
"column": 4
} | {
"line": 249,
"column": 74
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : DenselyOrdered α\na b : DedekindCut α\nh : a < b\n⊢ ∃ a_1, a < a_1 ∧ a_1 < b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"DedekindCut.principal_lt_principal._simp_1",
"Eq.mpr",
"Preorder.toLT",
... | [] | obtain ⟨c, hac, hcb⟩ := lt_iff_exists.mp h
obtain ⟨d, had, hdc⟩ := lt_iff_exists'.mp hac
simp only [principal_lt_principal] at hdc
obtain ⟨u, _, _⟩ := DenselyOrdered.dense d c hdc
exact ⟨principal u, had.trans_lt (by simpa), hcb.trans_lt' (by simpa)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Completion | {
"line": 245,
"column": 4
} | {
"line": 249,
"column": 74
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : DenselyOrdered α\na b : DedekindCut α\nh : a < b\n⊢ ∃ a_1, a < a_1 ∧ a_1 < b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"DedekindCut.principal_lt_principal._simp_1",
"Eq.mpr",
"Preorder.toLT",
... | [] | obtain ⟨c, hac, hcb⟩ := lt_iff_exists.mp h
obtain ⟨d, had, hdc⟩ := lt_iff_exists'.mp hac
simp only [principal_lt_principal] at hdc
obtain ⟨u, _, _⟩ := DenselyOrdered.dense d c hdc
exact ⟨principal u, had.trans_lt (by simpa), hcb.trans_lt' (by simpa)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Rank | {
"line": 48,
"column": 2
} | {
"line": 56,
"column": 42
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u\na : α\nr : α → α → Prop\no : Ordinal.{u}\nha : Acc r a\nho : o ≤ ha.rank\n⊢ ∃ b, ∃ (hb : Acc r b), hb.rank = o",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Acc.rank_eq",
"LE.le.eq_or_lt",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Or... | [] | obtain rfl | ho := ho.eq_or_lt
· exact ⟨a, ha, rfl⟩
· revert ho
refine ha.recOn fun a ha IH ho ↦ ?_
rw [rank_eq, Ordinal.lt_iSup_iff] at ho
obtain ⟨⟨b, hb⟩, ho⟩ := ho
rw [Order.lt_succ_iff] at ho
obtain rfl | ho := ho.eq_or_lt
exacts [⟨b, ha b hb, rfl⟩, IH _ hb ho] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Rank | {
"line": 48,
"column": 2
} | {
"line": 56,
"column": 42
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u\na : α\nr : α → α → Prop\no : Ordinal.{u}\nha : Acc r a\nho : o ≤ ha.rank\n⊢ ∃ b, ∃ (hb : Acc r b), hb.rank = o",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Acc.rank_eq",
"LE.le.eq_or_lt",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Or... | [] | obtain rfl | ho := ho.eq_or_lt
· exact ⟨a, ha, rfl⟩
· revert ho
refine ha.recOn fun a ha IH ho ↦ ?_
rw [rank_eq, Ordinal.lt_iSup_iff] at ho
obtain ⟨⟨b, hb⟩, ho⟩ := ho
rw [Order.lt_succ_iff] at ho
obtain rfl | ho := ho.eq_or_lt
exacts [⟨b, ha b hb, rfl⟩, IH _ hb ho] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.CardinalInter | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 31
} | {
"line": 236,
"column": 32
} | [
{
"pp": "ι α β : Type u\nc : Cardinal.{u}\nl✝ : Filter α\ninst✝¹ : CardinalInterFilter l✝ c\nl : Filter β\ninst✝ : CardinalInterFilter l c\nf : α → β\nS : Set (Set α)\nhSc : #↑S < c\nt : Set α → Set β\nhtl : ∀ s ∈ S, t s ∈ l\nht : ∀ s ∈ S, f ⁻¹' t s ⊆ s\n⊢ f ⁻¹' ⋂ i ∈ S, t i ⊆ ⋂₀ S",
"ppTerm": "?m.45",
... | [
"ι α β : Type u\nc : Cardinal.{u}\nl✝ : Filter α\ninst✝¹ : CardinalInterFilter l✝ c\nl : Filter β\ninst✝ : CardinalInterFilter l c\nf : α → β\nS : Set (Set α)\nhSc : #↑S < c\nt : Set α → Set β\nhtl : ∀ s ∈ S, t s ∈ l\nht : ∀ s ∈ S, f ⁻¹' t s ⊆ s\n⊢ ∀ t' ∈ S, ⋂ i ∈ S, f ⁻¹' t i ⊆ t'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CardinalInter | {
"line": 312,
"column": 6
} | {
"line": 312,
"column": 31
} | {
"line": 312,
"column": 32
} | [
{
"pp": "case mp.basic\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns✝ : Set α\nhreg : c.IsRegular\ns : Set α\nhs : s ∈ g\n⊢ #↑{s} < c ∧ ⋂₀ {s} ⊆ s",
"ppTerm": "?mp.basic",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.fintypeSing... | [
"case mp.basic\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns✝ : Set α\nhreg : c.IsRegular\ns : Set α\nhs : s ∈ g\n⊢ 1 < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CardinalInter | {
"line": 315,
"column": 25
} | {
"line": 315,
"column": 55
} | {
"line": 316,
"column": 4
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"ppTerm": "?mp.superset",
"assigned": true,
"usedConstants": [
... | [] | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.CardinalInter | {
"line": 315,
"column": 25
} | {
"line": 315,
"column": 55
} | {
"line": 316,
"column": 4
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"ppTerm": "?mp.superset",
"assigned": true,
"usedConstants": [
... | [] | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CardinalInter | {
"line": 315,
"column": 25
} | {
"line": 315,
"column": 55
} | {
"line": 316,
"column": 4
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"ppTerm": "?mp.superset",
"assigned": true,
"usedConstants": [
... | [] | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Partial | {
"line": 131,
"column": 4
} | {
"line": 132,
"column": 11
} | {
"line": 132,
"column": 12
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂.sets, r.core s ∈ l₁) → l₁ ≤ rcomap r l₂",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.mem_sets._simp_1",
"SetRel",
"c... | [
"case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂, r.core s ∈ l₁) → ∀ (x : Set α), ∀ x_1 ∈ l₂, r.core x_1 ⊆ x → x ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Partial | {
"line": 133,
"column": 4
} | {
"line": 134,
"column": 11
} | {
"line": 134,
"column": 12
} | [
{
"pp": "case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤ rcomap r l₂ → ∀ s ∈ l₂.sets, r.core s ∈ l₁",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.mem_sets._simp_1",
"SetRel",
"c... | [
"case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α), ∀ x_1 ∈ l₂, r.core x_1 ⊆ x → x ∈ l₁) → ∀ s ∈ l₂, r.core s ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Partial | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 57
} | {
"line": 165,
"column": 4
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nr : SetRel α β\ns : SetRel β γ\nl : Filter γ\nt : Set α\nu : Set β\nh : r.preimage u ⊆ t\nv : Set γ\nvsets : v ∈ l\nhv : s.preimage v ⊆ u\n⊢ ∃ t_1 ∈ l, r.preimage (s.preimage t_1) ⊆ t",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
... | [] | exact ⟨v, vsets, (SetRel.preimage_mono hv).trans h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.Partial | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 42
} | {
"line": 183,
"column": 43
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤\n { sets := SetRel.image {(s, t) | r.preimage s ⊆ t} l₂.sets, univ_sets := ⋯, sets_of_superset := ⋯,\n inter_sets := ⋯ } →\n ∀ s ∈ l₂, r.preimage s ∈ l₁",
"ppTerm": "?mp",
"assigned": true,
... | [
"case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α), ∀ x_1 ∈ l₂, r.preimage x_1 ⊆ x → x ∈ l₁) → ∀ s ∈ l₂, r.preimage s ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Partial | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 42
} | {
"line": 184,
"column": 43
} | [
{
"pp": "case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂, r.preimage s ∈ l₁) →\n l₁ ≤\n { sets := SetRel.image {(s, t) | r.preimage s ⊆ t} l₂.sets, univ_sets := ⋯, sets_of_superset := ⋯,\n inter_sets := ⋯ }",
"ppTerm": "?mpr",
"assigned": true,
... | [
"case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂, r.preimage s ∈ l₁) → ∀ (x : Set α), ∀ x_1 ∈ l₂, r.preimage x_1 ⊆ x → x ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Fin.InsertNth | {
"line": 28,
"column": 4
} | {
"line": 28,
"column": 15
} | {
"line": 28,
"column": 16
} | [
{
"pp": "case inr\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : Monotone f\nx : α\nhx : x ≤ f 0\ni : Fin n\n⊢ insertNth 0 x f i.succ.castSucc ≤ insertNth 0 x f i.succ.succ",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
... | [
"case inr\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : Monotone f\nx : α\nhx : x ≤ f 0\ni : Fin n\n⊢ f i.castSucc ≤ f i.succ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Fin.InsertNth | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 15
} | {
"line": 37,
"column": 16
} | [
{
"pp": "case inr\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : ∀ (i : Fin n), f i.castSucc < f i.succ\nx : α\nhx : x < f 0\ni : Fin n\n⊢ insertNth 0 x f i.succ.castSucc < insertNth 0 x f i.succ.succ",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case inr\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : ∀ (i : Fin n), f i.castSucc < f i.succ\nx : α\nhx : x < f 0\ni : Fin n\n⊢ f i.castSucc < f i.succ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 95
} | {
"line": 69,
"column": 95
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nh : s.chainHeight r ≠ ⊤\nthis : Nonempty { t // t ⊆ s ∧ IsChain r t }\n⊢ ⨆ i, ?m.28 i < ⊤",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.chainHeight",
"Eq.mpr",
"Set.encard",
"Preorder.toLT",
"instCompl... | [] | by rwa [← chainHeight_eq_iSup, lt_top_iff_ne_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Height | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 15
} | {
"line": 109,
"column": 16
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Set α\nr : α → α → Prop\nh : ∀ a ⊆ s, IsChain r a → a = ∅\nx : α\n⊢ x ∈ s ↔ x ∈ ∅",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
... | [
"case refine_1\nα : Type u_1\ns : Set α\nr : α → α → Prop\nh : ∀ a ⊆ s, IsChain r a → a = ∅\nx : α\n⊢ x ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 140,
"column": 29
} | {
"line": 140,
"column": 56
} | {
"line": 140,
"column": 57
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight (flip r)\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain (flip r) a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ (fun x y ↦ r x y ∨ r y x) x✝¹ x✝",
"ppTerm": "?m.58",
"assigned": true,
"u... | [
"α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight (flip r)\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain (flip r) a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ r x✝¹ x✝ ∨ r x✝ x✝¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 140,
"column": 29
} | {
"line": 140,
"column": 56
} | {
"line": 140,
"column": 57
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight r\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain r a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ (fun x y ↦ flip r x y ∨ flip r y x) x✝¹ x✝",
"ppTerm": "?m.109",
"assigned": true,
"used... | [
"α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight r\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain r a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ r x✝¹ x✝ ∨ r x✝ x✝¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Fin.InsertNth | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 11
} | {
"line": 72,
"column": 12
} | [
{
"pp": "case inl\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : Monotone f\nx : α\nhx : f (last n) ≤ x\ni : Fin n\n⊢ (last (n + 1)).insertNth x f i.castSucc.castSucc ≤ (last (n + 1)).insertNth x f i.castSucc.succ",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"E... | [
"case inl\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : Monotone f\nx : α\nhx : f (last n) ≤ x\ni : Fin n\n⊢ f i.castSucc ≤ f i.succ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 18
} | {
"line": 167,
"column": 19
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nhc :\n ((⇑(Subtype.relEmbedding (fun x1 x2 ↦ r x1 x2) fun x ↦ x ∈ s) '' univ).chainHeight fun x1 x2 ↦ r x1 x2) =\n univ.chainHeight (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2)\nhs : (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2) = fun x y ↦ r ↑x ↑y\n⊢ (univ.chainHeight... | [
"α : Type u_1\ns : Set α\nr : α → α → Prop\nhc :\n ((⇑(Subtype.relEmbedding (fun x1 x2 ↦ r x1 x2) fun x ↦ x ∈ s) '' univ).chainHeight fun x1 x2 ↦ r x1 x2) =\n univ.chainHeight (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2)\nhs : (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2) = fun x y ↦ r ↑x ↑y\n⊢ (univ.chainHeight fun x1 x2 ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 13
} | {
"line": 172,
"column": 14
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LE α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 ≤ x2) = s.chainHeight fun x1 x2 ↦ x1 ≤ x2",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns : Set α\ninst✝ : LE α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 ≤ x2) = s.chainHeight fun x1 x2 ↦ x1 ≤ x2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 13
} | {
"line": 177,
"column": 14
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LT α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 < x2) = s.chainHeight fun x1 x2 ↦ x1 < x2",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns : Set α\ninst✝ : LT α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 < x2) = s.chainHeight fun x1 x2 ↦ x1 < x2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Cocardinal | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 38
} | {
"line": 100,
"column": 39
} | [
{
"pp": "α : Type u\nc : Cardinal.{u}\nhreg : c.IsRegular\nx : α\n⊢ ∀ᶠ (a : α) in cocardinal α hreg, a ≠ x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.fintypeSingleton",
"Preorder.toLT",
"Classical.n... | [
"α : Type u\nc : Cardinal.{u}\nhreg : c.IsRegular\nx : α\n⊢ 1 < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Fin.InsertNth | {
"line": 81,
"column": 4
} | {
"line": 82,
"column": 11
} | {
"line": 82,
"column": 12
} | [
{
"pp": "case inl\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : ∀ (i : Fin n), f i.castSucc < f i.succ\nx : α\nhx : f (last n) < x\ni : Fin n\n⊢ (last (n + 1)).insertNth x f i.castSucc.castSucc < (last (n + 1)).insertNth x f i.castSucc.succ",
"ppTerm": "?inl",
"assigned": true,
... | [
"case inl\nn : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : ∀ (i : Fin n), f i.castSucc < f i.succ\nx : α\nhx : f (last n) < x\ni : Fin n\n⊢ f i.castSucc < f i.succ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 23,
"column": 2
} | {
"line": 23,
"column": 38
} | {
"line": 23,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Icc a b).ncard = b + 1 - a",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Icc a b).ncard = b + 1 - a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 38
} | {
"line": 26,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Ico a b).ncard = b - a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Ico a b).ncard = b - a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 38
} | {
"line": 29,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Ioc a b).ncard = b - a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Ioc a b).ncard = b - a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 38
} | {
"line": 32,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 38
} | {
"line": 35,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (uIcc a b).ncard = (↑b - ↑a).natAbs + 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (uIcc a b).ncard = (↑b - ↑a).natAbs + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 38
} | {
"line": 38,
"column": 39
} | [
{
"pp": "b : ℕ\n⊢ (Iic b).ncard = b + 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b : ℕ\n⊢ (Iic b).ncard = b + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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