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.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 238,
"column": 52
} | {
"line": 238,
"column": 63
} | {
"line": 238,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\ninst✝⁶ : DecidableEq α\ninst✝⁵ : DecidableEq β\ninst✝⁴ : DecidableEq γ\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype γ\ninst✝ : Expli... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\ninst✝⁶ : DecidableEq α\ninst✝⁵ : DecidableEq β\ninst✝⁴ : DecidableEq γ\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype γ\ninst✝ : ExplicitDisjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 239,
"column": 38
} | {
"line": 239,
"column": 61
} | {
"line": 239,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\ninst✝⁶ : DecidableEq α\ninst✝⁵ : DecidableEq β\ninst✝⁴ : DecidableEq γ\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype γ\ninst✝ : Expli... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\ninst✝⁶ : DecidableEq α\ninst✝⁵ : DecidableEq β\ninst✝⁴ : DecidableEq γ\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype γ\ninst✝ : ExplicitDisjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 152,
"column": 4
} | {
"line": 153,
"column": 11
} | {
"line": 153,
"column": 12
} | [
{
"pp": "G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound ε ≤ Fintype.card G\nA : Finset G\nhAε : ε * ↑(Fintype.card G) ≤ ↑(#A)\nhA : ThreeAPFree ↑A\nB : Finset (G × G) :=\n {x |\n match x with\n | (x, y) => y - x ∈ A}\nthis : ε * ↑(Fintype.card G) ^ 2... | [
"G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound ε ≤ Fintype.card G\nA : Finset G\nhAε : ε * ↑(Fintype.card G) ≤ ↑(#A)\nhA : ThreeAPFree ↑A\nB : Finset (G × G) :=\n {x |\n match x with\n | (x, y) => y - x ∈ A}\nthis : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#B)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 157,
"column": 20
} | {
"line": 157,
"column": 31
} | {
"line": 157,
"column": 32
} | [
{
"pp": "G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound ε ≤ Fintype.card G\nA : Finset G\nhAε : ε * ↑(Fintype.card G) ≤ ↑(#A)\nhA : ThreeAPFree ↑A\nB : Finset (G × G) :=\n {x |\n match x with\n | (x, y) => y - x ∈ A}\nthis✝ : ε * ↑(Fintype.card G) ^ ... | [
"G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound ε ≤ Fintype.card G\nA : Finset G\nhAε : ε * ↑(Fintype.card G) ≤ ↑(#A)\nhA : ThreeAPFree ↑A\nB : Finset (G × G) :=\n {x |\n match x with\n | (x, y) => y - x ∈ A}\nthis✝ : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#B)\nx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Energy | {
"line": 148,
"column": 8
} | {
"line": 148,
"column": 19
} | {
"line": 148,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns t u : Finset α\n⊢ (∑ c ∈ u, #({xy ∈ s ×ˢ t | xy.1 * xy.2 = c})) ^ 2 ≤ #u * ∑ c ∈ u, #({xy ∈ s ×ˢ t | xy.1 * xy.2 = c}) ^ 2",
"ppTerm": "?m.141",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns t u : Finset α\n⊢ (∑ c ∈ u, #({xy ∈ s ×ˢ t | xy.1 * xy.2 = c})) ^ 2 ≤ #u * ∑ c ∈ u, #({xy ∈ s ×ˢ t | xy.1 * xy.2 = c}) ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 178,
"column": 8
} | {
"line": 178,
"column": 49
} | {
"line": 178,
"column": 50
} | [
{
"pp": "n : ℕ\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound (ε / 3) ≤ n\nA : Finset ℕ\nhAn : ↑A ⊆ Set.Iio n\nhAε : ε * ↑n ≤ ↑(#A)\nhA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)\nthis✝ : ↑A = Fin.val '' Nat.cast '' ↑A\nthis : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val\nx : ℕ\nhx : x ∈ Set.Iio n\n⊢ x < n",... | [
"n : ℕ\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound (ε / 3) ≤ n\nA : Finset ℕ\nhAn : ↑A ⊆ Set.Iio n\nhAε : ε * ↑n ≤ ↑(#A)\nhA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)\nthis✝ : ↑A = Fin.val '' Nat.cast '' ↑A\nthis : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val\nx : ℕ\nhx : x ∈ Set.Iio n\n⊢ x < n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 132,
"column": 8
} | {
"line": 132,
"column": 55
} | {
"line": 132,
"column": 56
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\ns : Finset ι\nf : ι → MvPolynomial σ K\nh : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ\nhq : 0 < q - 1\nS : Finset (σ → K) := {x... | [
"K : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\ns : Finset ι\nf : ι → MvPolynomial σ K\nh : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ\nhq : 0 < q - 1\nS : Finset (σ → K) := {x | ∀ i ∈ s, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 13
} | {
"line": 169,
"column": 14
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : Fintype K\ninst✝⁵ : Field K\ninst✝⁴ : Fintype σ\ninst✝³ : DecidableEq σ\ninst✝² : DecidableEq K\np : ℕ\ninst✝¹ : CharP K p\ninst✝ : Fintype ι\nf : ι → MvPolynomial σ K\nh : ∑ i, (f i).totalDegree < Fintype.card σ\n⊢ p ∣ Fintype.card { x // ∀ (i : ι), (... | [
"K : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : Fintype K\ninst✝⁵ : Field K\ninst✝⁴ : Fintype σ\ninst✝³ : DecidableEq σ\ninst✝² : DecidableEq K\np : ℕ\ninst✝¹ : CharP K p\ninst✝ : Fintype ι\nf : ι → MvPolynomial σ K\nh : ∑ i, (f i).totalDegree < Fintype.card σ\n⊢ p ∣ Fintype.card { x // ∀ (i : ι), (eval x) (f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 495,
"column": 8
} | {
"line": 495,
"column": 19
} | {
"line": 495,
"column": 20
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤... | [
"case refine_1\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 33
} | {
"line": 68,
"column": 34
} | [
{
"pp": "M : Type u_1\ninst✝³ : DecidableEq M\ninst✝² : AddCommMonoid M\nA : Finset M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nA_nonneg : ∀ x ∈ A, 0 ≤ x\n⊢ ∀ x ∈ A.subsetSum, 0 ≤ x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.subsetSum",... | [
"M : Type u_1\ninst✝³ : DecidableEq M\ninst✝² : AddCommMonoid M\nA : Finset M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nA_nonneg : ∀ x ∈ A, 0 ≤ x\n⊢ ∀ a ⊆ A, 0 ≤ ∑ b ∈ a, b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 60
} | {
"line": 79,
"column": 6
} | [
{
"pp": "M : Type u_1\ninst✝³ : DecidableEq M\ninst✝² : AddCommMonoid M\nA : Finset M\na : M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nhA : ∀ x ∈ A, 0 < x\nhAa : ∀ x ∈ A, x < a\nha : 0 < a\nthis : ∀ x ∈ A.subsetSum, 0 ≤ x\n⊢ Disjoint (insert 0 A) (a +ᵥ A.subsetSum)",
"ppTerm": "?m.71",
... | [
"M : Type u_1\ninst✝³ : DecidableEq M\ninst✝² : AddCommMonoid M\nA : Finset M\na : M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nhA : ∀ x ∈ A, 0 < x\nhAa : ∀ x ∈ A, x < a\nha : 0 < a\nthis : ∀ x ∈ A.subsetSum, 0 ≤ x\n⊢ (∀ x ∈ A.subsetSum, ¬a + x = 0) ∧ ∀ a_1 ∈ A, ∀ x ∈ A.subsetSum, ¬a + x = a_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 76
} | {
"line": 84,
"column": 77
} | [
{
"pp": "case refine_2.refine_2\nι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ZMod p\nhs : #s = 2 * p - 1\nthis : NeZero p\nN : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }\nzero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩\nhN₀ : 0... | [
"case refine_2.refine_2\nι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ZMod p\nhs : #s = 2 * p - 1\nthis : NeZero p\nN : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }\nzero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩\nhN₀ : 0 < N\nhs' : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 62
} | {
"line": 89,
"column": 63
} | [
{
"pp": "case refine_3\nι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ZMod p\nhs : #s = 2 * p - 1\nthis : NeZero p\nN : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }\nzero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩\nhN₀ : 0 < N\nhs'... | [
"case refine_3\nι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ZMod p\nhs : #s = 2 * p - 1\nthis : NeZero p\nN : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }\nzero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩\nhN₀ : 0 < N\nhs' : 2 * p - 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 9
} | {
"line": 98,
"column": 10
} | [
{
"pp": "ι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ℤ\nhs : #s = 2 * p - 1\n⊢ ∃ t ⊆ s, #t = p ∧ ↑p ∣ ∑ i ∈ t, a i",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ℤ\nhs : #s = 2 * p - 1\n⊢ ∃ t ⊆ s, #t = p ∧ ↑p ∣ ∑ i ∈ t, a i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 118,
"column": 11
} | {
"line": 118,
"column": 22
} | {
"line": 118,
"column": 23
} | [
{
"pp": "case one\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * 1 - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = 1 ∧ ↑1 ∣ ∑ i ∈ t, a i",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"MulOne.toOne",
"Dvd.dvd",
"and_true",
"Monoid.toMulOne... | [
"case one\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * 1 - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 508,
"column": 2
} | {
"line": 513,
"column": 54
} | {
"line": 515,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | [] | apply (edgeDensity_chunk_aux (hP := hP) hPα hPε hU hV).trans
have key : (16 : ℝ) ^ #P.parts = #((chunk hP G ε hU).parts ×ˢ (chunk hP G ε hV).parts) := by
rw [card_product, cast_mul, card_chunk (m_pos hPα).ne', card_chunk (m_pos hPα).ne', ←
cast_mul, ← mul_pow]; norm_cast
simp_rw [key]
convert! sum_div_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 508,
"column": 2
} | {
"line": 513,
"column": 54
} | {
"line": 515,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | [] | apply (edgeDensity_chunk_aux (hP := hP) hPα hPε hU hV).trans
have key : (16 : ℝ) ^ #P.parts = #((chunk hP G ε hU).parts ×ˢ (chunk hP G ε hV).parts) := by
rw [card_product, cast_mul, card_chunk (m_pos hPα).ne', card_chunk (m_pos hPα).ne', ←
cast_mul, ← mul_pow]; norm_cast
simp_rw [key]
convert! sum_div_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.BitIndices | {
"line": 132,
"column": 28
} | {
"line": 132,
"column": 39
} | {
"line": 132,
"column": 40
} | [
{
"pp": "a n : ℕ\nha : a ∈ n.bitIndices\n⊢ n.testBit a = true",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a n : ℕ\nha : a ∈ n.bitIndices\n⊢ n.testBit a = true"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Compactness | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 54
} | {
"line": 74,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : Finset α → (a : α) → β a\ninstTop : (a : α) → TopologicalSpace (β a) := fun a ↦ ⊥\ninstDiscr : ∀ (a : α), DiscreteTopology (β a)\ne : Finset α → Set ((a : α) → β a) := fun s ↦ {f | ∃ t, s ⊆ t ∧ ∀ x ∈ s, f x = g t x}\nthis : ∀ (s : Fin... | [
"α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : Finset α → (a : α) → β a\ninstTop : (a : α) → TopologicalSpace (β a) := fun a ↦ ⊥\ninstDiscr : ∀ (a : α), DiscreteTopology (β a)\ne : Finset α → Set ((a : α) → β a) := fun s ↦ {f | ∃ t, s ⊆ t ∧ ∀ x ∈ s, f x = g t x}\nthis : ∀ (s : Finset α), s.re... | have he' (s : Finset α) : IsClosed (e s) := by
rw [← this]
exact (isClosed_discrete _).preimage (by fun_prop) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Compactness | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 18
} | {
"line": 95,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Finset α) → (a : ↥s) → β ↑a\nthis : ∀ (a : α), Nonempty (β a)\ng' : Finset α → (a : α) → β a := fun s a ↦ if ha : a ∈ s then g s ⟨a, ha⟩ else Classical.arbitrary (β a)\nhg : ∀ (s : Finset α) (x : ↥s), g s x = g' s ↑x\n⊢ ∃ χ, ∀ (s... | [
"α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Finset α) → (a : ↥s) → β ↑a\nthis : ∀ (a : α), Nonempty (β a)\ng' : Finset α → (a : α) → β a := fun s a ↦ if ha : a ∈ s then g s ⟨a, ha⟩ else Classical.arbitrary (β a)\nhg : ∀ (s : Finset α) (x : ↥s), g s x = g' s ↑x\n⊢ ∃ χ, ∀ (s : Finset α)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 134,
"column": 6
} | {
"line": 144,
"column": 29
} | {
"line": 146,
"column": 4
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n)... | [] | obtain ⟨𝒜, h𝒜card, h𝒜disj, h𝒜⟩ := this _ le_rfl
-- By induction hypothesis on `m`, find a subfamily `ℬ` of size `m` such that the sum over
-- `t ∈ ℬ` of `(∑ i ∈ t, a i) / n` is divisible by `m`.
obtain ⟨ℬ, hℬ𝒜, hℬcard, hℬ⟩ := ihm (fun t ↦ (∑ i ∈ t, a i) / n) h𝒜card.ge
-- We are done.
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 134,
"column": 6
} | {
"line": 144,
"column": 29
} | {
"line": 146,
"column": 4
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n)... | [] | obtain ⟨𝒜, h𝒜card, h𝒜disj, h𝒜⟩ := this _ le_rfl
-- By induction hypothesis on `m`, find a subfamily `ℬ` of size `m` such that the sum over
-- `t ∈ ℬ` of `(∑ i ∈ t, a i) / n` is divisible by `m`.
obtain ⟨ℬ, hℬ𝒜, hℬcard, hℬ⟩ := ihm (fun t ↦ (∑ i ∈ t, a i) / n) h𝒜card.ge
-- We are done.
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 170,
"column": 8
} | {
"line": 170,
"column": 45
} | {
"line": 170,
"column": 46
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n)... | [
"m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n) - 1 ≤ #s\nk... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 13
} | {
"line": 225,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) ≤ toColex (t \\ s) ↔ toColex s ≤ toColex t",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) ≤ toColex (t \\ s) ↔ toColex s ≤ toColex t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 13
} | {
"line": 229,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) < toColex (t \\ s) ↔ toColex s < toColex t",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) < toColex (t \\ s) ↔ toColex s < toColex t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 303,
"column": 6
} | {
"line": 303,
"column": 40
} | {
"line": 303,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : toColex s ≤ toColex t\nhst : s ≠ t\nm : α := (s ∆ t).max' ⋯\nhmt : m ∉ t\n⊢ m ∈ s",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"id",
"Finset.instSetLike",
"SetLike... | [
"α : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : toColex s ≤ toColex t\nhst : s ≠ t\nm : α := (s ∆ t).max' ⋯\nhmt : m ∉ t\n⊢ (s ∆ t).max' ⋯ ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 37
} | {
"line": 309,
"column": 38
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : ∀ (hst : s ≠ t), (s ∆ t).max' ⋯ ∈ t\na : α\nhas : a ∈ ofColex (toColex s)\nhat : a ∉ ofColex (toColex t)\nhst : s ≠ t\n⊢ (s ∆ t).max' ⋯ ∉ ofColex (toColex s)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [... | [
"case refine_2\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : ∀ (hst : s ≠ t), (s ∆ t).max' ⋯ ∈ t\na : α\nhas : a ∈ ofColex (toColex s)\nhat : a ∉ ofColex (toColex t)\nhst : s ≠ t\n⊢ (s ∆ t).max' ⋯ ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 51
} | {
"line": 184,
"column": 52
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ns : Finset ι\na : ι → ZMod n\nhs : 2 * n - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = n ∧ ∑ i ∈ t, a i = 0",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nn : ℕ\ns : Finset ι\na : ι → ZMod n\nhs : 2 * n - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = n ∧ ∑ i ∈ t, a i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 192,
"column": 83
} | {
"line": 192,
"column": 94
} | {
"line": 192,
"column": 95
} | [
{
"pp": "n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * ?m.31 - 1 ≤ #s.toEnumFinset",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"Int.instDecidableEq",
"id",
"ins... | [
"n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * ?m.31 ≤ s.card + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 193,
"column": 78
} | {
"line": 193,
"column": 89
} | {
"line": 193,
"column": 90
} | [
{
"pp": "n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ℤ × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ↑n ∣ ∑ i ∈ t, i.1\n⊢ (Multiset.map Prod.fst t.val).card = n ∧ ↑n ∣ (Multiset.map Prod.fst t.val).sum",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Multiset.sum",
... | [
"n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ℤ × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ↑n ∣ ∑ i ∈ t, i.1\n⊢ #t = n ∧ ↑n ∣ ∑ a ∈ t, a.1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 201,
"column": 84
} | {
"line": 201,
"column": 95
} | {
"line": 201,
"column": 96
} | [
{
"pp": "n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * n - 1 ≤ #s.toEnumFinset",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"HMul.hMul",
"congrArg",
"ZMod.decidableEq",
"HSub.hSub",
"id",
"ins... | [
"n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * n ≤ s.card + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 202,
"column": 78
} | {
"line": 202,
"column": 89
} | {
"line": 202,
"column": 90
} | [
{
"pp": "n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ZMod n × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ∑ i ∈ t, i.1 = 0\n⊢ (Multiset.map Prod.fst t.val).card = n ∧ (Multiset.map Prod.fst t.val).sum = 0",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Multise... | [
"n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ZMod n × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ∑ i ∈ t, i.1 = 0\n⊢ #t = n ∧ ∑ a ∈ t, a.1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 68
} | {
"line": 356,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\na : α\nha : a ∈ s\nhst : toColex s ≤ toColex s\nhcard : #s ≤ #s\nht : s.Nonempty\nm : α := s.min' ht\n⊢ toColex (s.erase a) ≤ toColex (s.erase m)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Finset.min'",
... | [] | exact (erase_le_erase ha <| min'_mem _ _).2 <| min'_le _ _ <| ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Colex | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 68
} | {
"line": 356,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\na : α\nha : a ∈ s\nhst : toColex s ≤ toColex s\nhcard : #s ≤ #s\nht : s.Nonempty\nm : α := s.min' ht\n⊢ toColex (s.erase a) ≤ toColex (s.erase m)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Finset.min'",
... | [] | exact (erase_le_erase ha <| min'_mem _ _).2 <| min'_le _ _ <| ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Colex | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 68
} | {
"line": 356,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\na : α\nha : a ∈ s\nhst : toColex s ≤ toColex s\nhcard : #s ≤ #s\nht : s.Nonempty\nm : α := s.min' ht\n⊢ toColex (s.erase a) ≤ toColex (s.erase m)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Finset.min'",
... | [] | exact (erase_le_erase ha <| min'_mem _ _).2 <| min'_le _ _ <| ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 19
} | {
"line": 244,
"column": 20
} | [
{
"pp": "case pos\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : D = D'\n⊢ LinearIndepOn K id {D.rep, D'.rep}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instIsTorsionFreeOfIsDomainOfNoZeroSMul... | [
"case pos\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : D = D'\n⊢ ¬D'.rep = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 15
} | {
"line": 247,
"column": 16
} | [
{
"pp": "case neg\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : LinearIndepOn K id (Set.range ![D'.rep, D.rep])\n⊢ LinearIndepOn K id {D.rep, D'.rep}",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"case neg\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : LinearIndepOn K id (Set.range ![D'.rep, D.rep])\n⊢ LinearIndepOn K id {D.rep, D'.rep}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 69
} | {
"line": 248,
"column": 70
} | [
{
"pp": "case neg\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : LinearIndependent K ![D'.rep, D.rep]\n⊢ Function.Injective ![D'.rep, D.rep]",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Projecti... | [
"case neg\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : LinearIndependent K ![D'.rep, D.rep]\n⊢ ¬D'.rep = D.rep"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 462,
"column": 19
} | {
"line": 462,
"column": 30
} | {
"line": 462,
"column": 31
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\n𝒜 : Finset (Finset α)\nr : ℕ\ninst✝ : Fintype α\nh𝒜 : IsInitSeg 𝒜 r\nh𝒜₀ : 𝒜.Nonempty\na : Finset α\nha : a ∈ 𝒜\n⊢ toColex a ∈ ⇑ofColex ⁻¹' ↑𝒜",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\n𝒜 : Finset (Finset α)\nr : ℕ\ninst✝ : Fintype α\nh𝒜 : IsInitSeg 𝒜 r\nh𝒜₀ : 𝒜.Nonempty\na : Finset α\nha : a ∈ 𝒜\n⊢ a ∈ 𝒜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Derangements.Basic | {
"line": 69,
"column": 15
} | {
"line": 69,
"column": 48
} | {
"line": 69,
"column": 49
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\nf : Perm (Subtype p)\nhf : ∀ a ∈ fixedPoints ⇑↑((Perm.subtypeEquivSubtypePerm p) f), ¬p a\na : α\nha : p a\nhfa : f ⟨a, ha⟩ = ⟨a, ha⟩\n⊢ (Perm.ofSubtype f) a = a",
"ppTerm": "?refine_2",
"assigned": true,
"use... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\nf : Perm (Subtype p)\nhf : ∀ a ∈ fixedPoints ⇑↑((Perm.subtypeEquivSubtypePerm p) f), ¬p a\na : α\nha : p a\nhfa : f ⟨a, ha⟩ = ⟨a, ha⟩\n⊢ ↑(f ⟨a, ha⟩) = a"
] | Perm.ofSubtype_apply_of_mem _ ha, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 36
} | {
"line": 108,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA B : Finset G\nhA : ↑(#(B * A)) ≤ K * ↑(#A)\na : G\nha : a ∈ B\nb : G\nhb : b ∈ B\n⊢ (2 - K) * ↑(#A) ≤ ↑(A.convolution A⁻¹ (a⁻¹ * b))",
"ppTerm": "?m.106",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA B : Finset G\nhA : ↑(#(B * A)) ≤ K * ↑(#A)\na : G\nha : a ∈ B\nb : G\nhb : b ∈ B\n⊢ (2 - K) * ↑(#A) ≤ ↑(A.convolution A⁻¹ (a⁻¹ * b))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 36
} | {
"line": 115,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA B : Finset G\nhA : ↑(#(B * A)) < K * ↑(#A)\na : G\nha : a ∈ B\nb : G\nhb : b ∈ B\n⊢ (2 - K) * ↑(#A) < ↑(A.convolution A⁻¹ (a⁻¹ * b))",
"ppTerm": "?m.106",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA B : Finset G\nhA : ↑(#(B * A)) < K * ↑(#A)\na : G\nha : a ∈ B\nb : G\nhb : b ∈ B\n⊢ (2 - K) * ↑(#A) < ↑(A.convolution A⁻¹ (a⁻¹ * b))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 15
} | {
"line": 124,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : #(A * A) < 2 * #A\nx : G\nhx : x ∈ A\ny : G\nhy : y ∈ A\n⊢ (x •> A ∩ y •> A).Nonempty",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : #(A * A) < 2 * #A\nx : G\nhx : x ∈ A\ny : G\nhy : y ∈ A\n⊢ (x •> A ∩ y •> A).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 13
} | {
"line": 142,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : #(A * A) < 2 * #A\n⊢ A * A⁻¹ ⊆ A⁻¹ * A",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : #(A * A) < 2 * #A\n⊢ A * A⁻¹ ⊆ A⁻¹ * A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 142,
"column": 61
} | {
"line": 142,
"column": 88
} | {
"line": 142,
"column": 89
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : #(A * A) < 2 * #A\n⊢ #(A⁻¹ * A⁻¹) < 2 * #A⁻¹",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Finset.card_inv",
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toIn... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : #(A * A) < 2 * #A\n⊢ #(A * A) < 2 * #A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 169,
"column": 43
} | {
"line": 169,
"column": 54
} | {
"line": 169,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na b c d x✝ y✝ : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nx : G\nhx : x ∈ A\ny : G\nhy : y ∈ A\n⊢ ↑(#(A * ?m.78)) < ?m.77 * ↑(#?m.78)",
"ppTerm": "?m.82",
"assigned": false,
"usedConstants": [],
"us... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na b c d x✝ y✝ : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nx : G\nhx : x ∈ A\ny : G\nhy : y ∈ A\n⊢ ↑(#(A * ?m.78)) < ?m.77 * ↑(#?m.78)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Digraph.Basic | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 51
} | {
"line": 64,
"column": 52
} | [
{
"pp": "V : Type u_1\nadj adj' : V → V → Bool\nh : (fun v w ↦ adj v w = true) = fun v w ↦ adj' v w = true\nv w : V\n⊢ adj v w = adj' v w",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nadj adj' : V → V → Bool\nh : (fun v w ↦ adj v w = true) = fun v w ↦ adj' v w = true\nv w : V\n⊢ adj v w = adj' v w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Configuration | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 59
} | {
"line": 428,
"column": 60
} | [
{
"pp": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Finite P\ninst✝ : Finite L\np : P\n⊢ 2 < lineCount L p",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Configuration.lineCount",
"id",
"i... | [
"P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Finite P\ninst✝ : Finite L\np : P\n⊢ 1 < order P L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Configuration | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 60
} | {
"line": 433,
"column": 61
} | [
{
"pp": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Finite P\ninst✝ : Finite L\nl : L\n⊢ 2 < pointCount P l",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"_private.Mathlib.Combinatorics.Configuration.... | [
"P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Finite P\ninst✝ : Finite L\nl : L\n⊢ 1 < order P L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 165,
"column": 19
} | {
"line": 165,
"column": 30
} | {
"line": 165,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na✝ b✝ c d x y : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\na : G\nha : a⁻¹ ∈ A\nb : G\nhb : b ∈ A\n⊢ b⁻¹⁻¹ ∈ A",
"ppTerm": "?m.581",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvOneM... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na✝ b✝ c d x y : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\na : G\nha : a⁻¹ ∈ A\nb : G\nhb : b ∈ A\n⊢ b ∈ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 213,
"column": 36
} | {
"line": 213,
"column": 47
} | {
"line": 213,
"column": 48
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₀ : A.Nonempty\na : G\nha : a ∈ A⁻¹ * A\n⊢ ↑(#(A * ?m.103)) < ?m.102 * ↑(#?m.103)",
"ppTerm": "?m.106",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₀ : A.Nonempty\na : G\nha : a ∈ A⁻¹ * A\n⊢ ↑(#(A * ?m.103)) < ?m.102 * ↑(#?m.103)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Catalan.Tree | {
"line": 36,
"column": 73
} | {
"line": 36,
"column": 84
} | {
"line": 36,
"column": 85
} | [
{
"pp": "a b : Finset (BinaryTree Unit)\nx✝¹ x✝ : BinaryTree Unit × BinaryTree Unit\nx₁ x₂ y₁ y₂ : BinaryTree Unit\nh : (fun x ↦ node () x.1 x.2) (x₁, x₂) = (fun x ↦ node () x.1 x.2) (y₁, y₂)\n⊢ (x₁, x₂) = (y₁, y₂)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Bina... | [
"a b : Finset (BinaryTree Unit)\nx✝¹ x✝ : BinaryTree Unit × BinaryTree Unit\nx₁ x₂ y₁ y₂ : BinaryTree Unit\nh : (fun x ↦ node () x.1 x.2) (x₁, x₂) = (fun x ↦ node () x.1 x.2) (y₁, y₂)\n⊢ x₁ = y₁ ∧ x₂ = y₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 78
} | {
"line": 81,
"column": 78
} | [
{
"pp": "p q : DyckWord\n⊢ ∀ (i : ℕ), count D (take i (↑p ++ ↑q)) ≤ count U (take i (↑p ++ ↑q))",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"Nat.instIsOrderedAddMonoid",
"DyckStep.U",
"congrArg",
"covariant_swap_a... | [] | simp only [take_append, count_append]
exact fun _ ↦ add_le_add (p.count_D_le_count_U _) (q.count_D_le_count_U _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 78
} | {
"line": 81,
"column": 78
} | [
{
"pp": "p q : DyckWord\n⊢ ∀ (i : ℕ), count D (take i (↑p ++ ↑q)) ≤ count U (take i (↑p ++ ↑q))",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"Nat.instIsOrderedAddMonoid",
"DyckStep.U",
"congrArg",
"covariant_swap_a... | [] | simp only [take_append, count_append]
exact fun _ ↦ add_le_add (p.count_D_le_count_U _) (q.count_D_le_count_U _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 42
} | {
"line": 119,
"column": 43
} | [
{
"pp": "case cons\ns : DyckStep\ntail✝ : List DyckStep\ncount_U_eq_count_D✝ : count U (s :: tail✝) = count D (s :: tail✝)\nnonneg : ∀ (i : ℕ), count D (take i (s :: tail✝)) ≤ count U (take i (s :: tail✝))\nh : ↑{ toList := s :: tail✝, count_U_eq_count_D := count_U_eq_count_D✝, count_D_le_count_U := nonneg } ≠ ... | [
"case cons\ns : DyckStep\ntail✝ : List DyckStep\ncount_U_eq_count_D✝ : count U (s :: tail✝) = count D (s :: tail✝)\nnonneg : ∀ (i : ℕ), count D (take i (s :: tail✝)) ≤ count U (take i (s :: tail✝))\nh : ↑{ toList := s :: tail✝, count_U_eq_count_D := count_U_eq_count_D✝, count_D_le_count_U := nonneg } ≠ []\nf : ¬s =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 39
} | {
"line": 98,
"column": 4
} | [
{
"pp": "case e_a.a\nm : Multiset ℕ\nx : ℕ\nhx : x ∈ m.toFinset.erase 0\n⊢ x ! ^ count x m * ((count x m)! * ∏ j ∈ Finset.range (count x m), (j * x + x - 1).choose (x - 1)) = (x * count x m)!",
"ppTerm": "?e_a.a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"... | [
"case e_a.a.hx\nm : Multiset ℕ\nx : ℕ\nhx : x ∈ m.toFinset.erase 0\n⊢ x ≠ 0"
] | rw [← mul_assoc, bell_mul_eq_lemma] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 296,
"column": 2
} | {
"line": 313,
"column": 88
} | {
"line": 315,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nha : a ∈ A\n⊢ a •> (A⁻¹ * A) = (A⁻¹ * A) <• a",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Finset.mul_inv_eq_inv_mul_of_doubling_lt_two",
"Eq.mpr",
"l... | [] | refine subset_antisymm ?_ ?_
· rw [subset_smul_finset_iff, ← op_inv]
calc
a •> (A⁻¹ * A) <• a⁻¹ ⊆ a •> (A⁻¹ * A) * A⁻¹ := op_smul_finset_subset_mul (by simpa)
_ ⊆ A * (A⁻¹ * A) * A⁻¹ := by grw [smul_finset_subset_mul (by simpa)]
_ = A⁻¹ * A := by
simp_rw [← coe_inj, coe_mul]
rw [... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 296,
"column": 2
} | {
"line": 313,
"column": 88
} | {
"line": 315,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nha : a ∈ A\n⊢ a •> (A⁻¹ * A) = (A⁻¹ * A) <• a",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Finset.mul_inv_eq_inv_mul_of_doubling_lt_two",
"Eq.mpr",
"l... | [] | refine subset_antisymm ?_ ?_
· rw [subset_smul_finset_iff, ← op_inv]
calc
a •> (A⁻¹ * A) <• a⁻¹ ⊆ a •> (A⁻¹ * A) * A⁻¹ := op_smul_finset_subset_mul (by simpa)
_ ⊆ A * (A⁻¹ * A) * A⁻¹ := by grw [smul_finset_subset_mul (by simpa)]
_ = A⁻¹ * A := by
simp_rw [← coe_inj, coe_mul]
rw [... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 334,
"column": 4
} | {
"line": 335,
"column": 11
} | {
"line": 335,
"column": 12
} | [
{
"pp": "case refine_3\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nH : Subgroup G := A.invMulSubgroup h\na : G\nha : a ∈ A\n⊢ a •> ↑H = ↑H <• a",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
... | [
"case refine_3\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nH : Subgroup G := A.invMulSubgroup h\na : G\nha : a ∈ A\n⊢ a •> (A⁻¹ * A) = (A⁻¹ * A) <• a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 56
} | {
"line": 362,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nA Z : Finset G\nhZA : ↑Z ⊆ ↑A\nhZinj : Set.InjOn (fun x ↦ ↑H <• x) ↑Z\nhHZA : (fun x ↦ ↑H <• x) '' ↑Z = (fun x ↦ ↑H <• x) '' ↑A\n⊢ ↑H * ↑Z = ↑H * ↑A",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nA Z : Finset G\nhZA : ↑Z ⊆ ↑A\nhZinj : Set.InjOn (fun x ↦ ↑H <• x) ↑Z\nhHZA : (fun x ↦ ↑H <• x) '' ↑Z = (fun x ↦ ↑H <• x) '' ↑A\n⊢ ↑H * ↑Z = ↑H * ↑A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 173,
"column": 75
} | {
"line": 173,
"column": 91
} | {
"line": 173,
"column": 91
} | [
{
"pp": "case inr\np q : DyckWord\nh : p ≠ 0\ni : ℕ\nhi : i > 0\n⊢ count D (List.take (i - ([U] ++ ↑p).length) [D]) + count D [U] ≤\n count U (List.take (i - ([U] ++ ↑p).length) [D]) + count U [U]",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckS... | [
"case inr\np q : DyckWord\nh : p ≠ 0\ni : ℕ\nhi : i > 0\n⊢ (count D (List.take (i - ([U] ++ ↑p).length) [D]) + if U = D then 1 else 0) ≤\n count U (List.take (i - ([U] ++ ↑p).length) [D]) + count U [U]"
] | count_singleton' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 54
} | {
"line": 372,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : DecidableEq G\nH : Subgroup G\ninst✝ : Fintype ↥H\nZ : Finset G\nhZ : Set.InjOn (fun x ↦ ↑H <• x) ↑Z\nh₁ z₁ h₂ z₂ : G\nh : h₁ * z₁ = h₂ * z₂\nhh₁ : h₁ ∈ H\nhz₁ : z₁ ∈ Z\nhh₂ : h₂ ∈ H\nhz₂ : z₂ ∈ Z\n⊢ z₂ * z₁⁻¹ ∈ H",
"ppTerm": "?m.93",
"assigned": true,
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : DecidableEq G\nH : Subgroup G\ninst✝ : Fintype ↥H\nZ : Finset G\nhZ : Set.InjOn (fun x ↦ ↑H <• x) ↑Z\nh₁ z₁ h₂ z₂ : G\nh : h₁ * z₁ = h₂ * z₂\nhh₁ : h₁ ∈ H\nhz₁ : z₁ ∈ Z\nhh₂ : h₂ ∈ H\nhz₂ : z₂ ∈ Z\n⊢ h₂⁻¹ * h₁ ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 38
} | {
"line": 128,
"column": 39
} | [
{
"pp": "m : Multiset ℕ\na : ℕ\nha : a ≠ 0\nrest : ℕ := ∏ j ∈ (m.toFinset.erase 0).erase a, (count j m)!\nhrest : rest = ∏ j ∈ ((a ::ₘ m).toFinset.erase 0).erase a, (count j (a ::ₘ m))!\nc : ℕ := (map (fun x ↦ x !) m).prod * (count a m)! * rest\nhsplit : (count a m)! * rest = ∏ j ∈ m.toFinset.erase 0, (count j ... | [
"m : Multiset ℕ\na : ℕ\nha : a ≠ 0\nrest : ℕ := ∏ j ∈ (m.toFinset.erase 0).erase a, (count j m)!\nhrest : rest = ∏ j ∈ ((a ::ₘ m).toFinset.erase 0).erase a, (count j (a ::ₘ m))!\nc : ℕ := (map (fun x ↦ x !) m).prod * (count a m)! * rest\nhsplit : (count a m)! * rest = ∏ j ∈ m.toFinset.erase 0, (count j m)!\n⊢ m.bel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 15
} | {
"line": 201,
"column": 16
} | [
{
"pp": "p q : DyckWord\nh : p ≠ 0\nhn : p.IsNested\nthis :\n (count U (↑p).dropLast.tail + if (U == U) = true then 1 else 0) + count U [D] =\n count D (U :: (↑p).dropLast.tail ++ [D])\n⊢ count U (↑p).dropLast.tail = count D (↑p).dropLast.tail",
"ppTerm": "?m.38",
"assigned": false,
"usedConstan... | [
"p q : DyckWord\nh : p ≠ 0\nhn : p.IsNested\nthis :\n (count U (↑p).dropLast.tail + if (U == U) = true then 1 else 0) + count U [D] =\n count D (U :: (↑p).dropLast.tail ++ [D])\n⊢ count U (↑p).dropLast.tail = count D (↑p).dropLast.tail"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 77
} | {
"line": 137,
"column": 78
} | [
{
"pp": "m : Multiset ℕ\na : ℕ\nha : a ≠ 0\nrest : ℕ := ∏ j ∈ (m.toFinset.erase 0).erase a, (count j m)!\nhrest : rest = ∏ j ∈ ((a ::ₘ m).toFinset.erase 0).erase a, (count j (a ::ₘ m))!\nc : ℕ := (map (fun x ↦ x !) m).prod * (count a m)! * rest\nhm0 : m.bell * c = m.sum !\nhm : m.sum ! * a ! = m.bell * a ! * c\... | [
"m : Multiset ℕ\na : ℕ\nha : a ≠ 0\nrest : ℕ := ∏ j ∈ (m.toFinset.erase 0).erase a, (count j m)!\nhrest : rest = ∏ j ∈ ((a ::ₘ m).toFinset.erase 0).erase a, (count j (a ::ₘ m))!\nc : ℕ := (map (fun x ↦ x !) m).prod * (count a m)! * rest\nhm0 : m.bell * c = m.sum !\nhm : m.sum ! * a ! = m.bell * a ! * c\nhc : 0 < a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 35
} | {
"line": 275,
"column": 2
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\nlp : 0 < (↑p).length\n⊢ p.firstReturn < (range (↑p).length).length",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableEqDyckStep",
"DyckStep.U",
"List.findIdx_lt_length_of_exists",
"instOfNatNat",
"List.range",
... | [
"p : DyckWord\nh : p ≠ 0\nlp : 0 < (↑p).length\n⊢ ∃ x ∈ range (↑p).length, decide (count U (List.take (x + 1) ↑p) = count D (List.take (x + 1) ↑p)) = true"
] | apply findIdx_lt_length_of_exists | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 33
} | {
"line": 194,
"column": 2
} | [
{
"pp": "m n : ℕ\nhn : n ≠ 0\n⊢ (m * n)! / (n ! ^ m * m !) = m.uniformBell n",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Nat.instMonoid",
"instMulNat",
"NPow.toPow",
"Nat.div_eq_of_eq_mul_left",
"HPow.hPow",
"Nat.factorial",
... | [
"case H1\nm n : ℕ\nhn : n ≠ 0\n⊢ 0 < n ! ^ m * m !",
"case H2\nm n : ℕ\nhn : n ≠ 0\n⊢ (m * n)! = m.uniformBell n * (n ! ^ m * m !)"
] | apply Nat.div_eq_of_eq_mul_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 42
} | {
"line": 236,
"column": 43
} | [
{
"pp": "n : ℕ\np : (n + 1).Partition\n⊢ n + 1 = ∑ a ∈ p.parts.toFinset, count a p.parts * a",
"ppTerm": "?m.139",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\np : (n + 1).Partition\n⊢ n + 1 = ∑ a ∈ p.parts.toFinset, count a p.parts * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 242,
"column": 6
} | {
"line": 242,
"column": 41
} | {
"line": 242,
"column": 42
} | [
{
"pp": "n : ℕ\np : (n + 1).Partition\na : ℕ\nha : a ∈ p.parts.toFinset\nha0 : a ≠ 0\n⊢ (p.parts.erase a).sum + a = n + 1",
"ppTerm": "?m.195",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
"congrArg",
"Nat.Partition.parts",
"id",
"instOfNatNat",
... | [
"n : ℕ\np : (n + 1).Partition\na : ℕ\nha : a ∈ p.parts.toFinset\nha0 : a ≠ 0\n⊢ a + (p.parts.erase a).sum = n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 249,
"column": 40
} | {
"line": 249,
"column": 51
} | {
"line": 249,
"column": 52
} | [
{
"pp": "n : ℕ\nx : (i : Fin n.succ) × { p // ↑i + 1 ∈ p.parts }\n⊢ ↑x.fst + 1 ∈ (↑x.snd).parts.toFinset",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Finset",
"Nat.Partition.parts",
"Membership.mem",
"Multiset",
"... | [
"n : ℕ\nx : (i : Fin n.succ) × { p // ↑i + 1 ∈ p.parts }\n⊢ ↑x.fst + 1 ∈ (↑x.snd).parts"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 17
} | {
"line": 307,
"column": 18
} | [
{
"pp": "case neg.right\np q : DyckWord\nh : ¬p = 0\nu : ↑(p + q) = ↑p ++ ↑q\nv : p.firstReturn < (↑p).length\nj : ℕ\nhj : j < p.firstReturn\n⊢ decide (count U (List.take (j + 1) ↑p) = count D (List.take (j + 1) ↑p)) = false",
"ppTerm": "?neg.right✝",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case neg.right\np q : DyckWord\nh : ¬p = 0\nu : ↑(p + q) = ↑p ++ ↑q\nv : p.firstReturn < (↑p).length\nj : ℕ\nhj : j < p.firstReturn\n⊢ ¬count U (List.take (j + 1) ↑p) = count D (List.take (j + 1) ↑p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 275,
"column": 51
} | {
"line": 275,
"column": 71
} | {
"line": 275,
"column": 71
} | [
{
"pp": "n✝ n : ℕ\nih : ∀ m < n + 1, m.bell = ∑ p, p.parts.bell\n⊢ ∑ i, ∑ p, n.choose ↑i * ((↑p).parts.erase (↑i + 1)).bell =\n ∑ x, n.choose (↑x.snd - 1) * (x.fst.parts.erase ↑x.snd).bell",
"ppTerm": "?m.357",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
... | [
"n✝ n : ℕ\nih : ∀ m < n + 1, m.bell = ∑ p, p.parts.bell\n⊢ ∑ x, n.choose ↑x.fst * ((↑x.snd).parts.erase (↑x.fst + 1)).bell =\n ∑ x, n.choose (↑x.snd - 1) * (x.fst.parts.erase ↑x.snd).bell"
] | ← Fintype.sum_sigma' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 319,
"column": 28
} | {
"line": 319,
"column": 43
} | {
"line": 319,
"column": 44
} | [
{
"pp": "case right\np : DyckWord\nu : ↑p.nest = U :: ↑p ++ [D]\nj : ℕ\nhj : j < (↑p).length + 1\n⊢ decide (count U (List.take (j + 1) (U :: (↑p ++ [D]))) = count D (List.take (j + 1) (U :: (↑p ++ [D])))) = false",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"instDecidableEqDyckSt... | [
"case right\np : DyckWord\nu : ↑p.nest = U :: ↑p ++ [D]\nj : ℕ\nhj : j < (↑p).length + 1\n⊢ decide (count U (U :: List.take j (↑p ++ [D])) = count D (U :: List.take j (↑p ++ [D]))) = false"
] | take_succ_cons, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 40
} | {
"line": 513,
"column": 41
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\nthis✝ : DecidableLE α := ⋯\nmud : IncidenceAlgebra 𝕜 αᵒᵈ := ⋯\nthis : mud * zeta 𝕜 * mu 𝕜 = mu 𝕜 * zeta 𝕜 * mu 𝕜\n⊢ mud = mu 𝕜",
"ppTerm": "?m.189",
"ass... | [
"𝕜 : Type u_2\nα : Type u_5\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\nthis✝ : DecidableLE α := ⋯\nmud : IncidenceAlgebra 𝕜 αᵒᵈ := ⋯\nthis : mud * zeta 𝕜 * mu 𝕜 = mu 𝕜 * zeta 𝕜 * mu 𝕜\n⊢ mud = mu 𝕜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 334,
"column": 36
} | {
"line": 334,
"column": 47
} | {
"line": 334,
"column": 48
} | [
{
"pp": "p q : DyckWord\nh✝ : p ≠ 0\nh : ¬p = 0\n⊢ ↑{ toList := List.take (p.firstReturn + 1) ↑p, count_U_eq_count_D := ⋯, count_D_le_count_U := ⋯ } ≠ []",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
... | [
"p q : DyckWord\nh✝ : p ≠ 0\nh : ¬p = 0\n⊢ ¬↑p = []"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 114,
"column": 60
} | {
"line": 114,
"column": 71
} | {
"line": 114,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nx : ℕ\nhx : g x ≠ 0\n⊢ x ∈ g.support",
... | [
"R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nx : ℕ\nhx : g x ≠ 0\n⊢ ¬g x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 388,
"column": 2
} | {
"line": 389,
"column": 19
} | {
"line": 391,
"column": 0
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\n⊢ p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DyckWord.semilength_add",
"DyckWord.nest_insidePart_add_outsidePart",
"instAddDyckWord",
"congr... | [] | rw [← congrArg semilength (nest_insidePart_add_outsidePart h), semilength_add, semilength_nest,
add_right_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 388,
"column": 2
} | {
"line": 389,
"column": 19
} | {
"line": 391,
"column": 0
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\n⊢ p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DyckWord.semilength_add",
"DyckWord.nest_insidePart_add_outsidePart",
"instAddDyckWord",
"congr... | [] | rw [← congrArg semilength (nest_insidePart_add_outsidePart h), semilength_add, semilength_nest,
add_right_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 388,
"column": 2
} | {
"line": 389,
"column": 19
} | {
"line": 391,
"column": 0
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\n⊢ p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DyckWord.semilength_add",
"DyckWord.nest_insidePart_add_outsidePart",
"instAddDyckWord",
"congr... | [] | rw [← congrArg semilength (nest_insidePart_add_outsidePart h), semilength_add, semilength_nest,
add_right_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 128,
"column": 60
} | {
"line": 128,
"column": 71
} | {
"line": 128,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nh : g 0 ≠ 0\n⊢ 0 ∈ g.support",
"ppTerm": "?m.253... | [
"R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nh : g 0 ≠ 0\n⊢ ¬g 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 15
} | {
"line": 131,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nhgne0 : ∀ (i : ℕ), g i ≠... | [
"case refine_1\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nhgne0 : ∀ (i : ℕ), g i ≠ 0 ↔ i ≠ 0 ∧... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 35
} | {
"line": 140,
"column": 36
} | [
{
"pp": "case refine_4\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nhgne0 : ∀ (i : ℕ), g i ≠... | [
"case refine_4\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nhgne0 : ∀ (i : ℕ), g i ≠ 0 ↔ i ≠ 0 ∧... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 139,
"column": 4
} | {
"line": 140,
"column": 97
} | {
"line": 142,
"column": 0
} | [
{
"pp": "case refine_4\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nhgne0 : ∀ (i : ℕ), g i ≠... | [] | ext x
simpa [toFinsuppAntidiag] using Nat.div_mul_cancel <| aux_dvd_of_coeff_ne_zero hs0 hg hprod x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 139,
"column": 4
} | {
"line": 140,
"column": 97
} | {
"line": 142,
"column": 0
} | [
{
"pp": "case refine_4\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nhgne0 : ∀ (i : ℕ), g i ≠... | [] | ext x
simpa [toFinsuppAntidiag] using Nat.div_mul_cancel <| aux_dvd_of_coeff_ne_zero hs0 hg hprod x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 86,
"column": 8
} | {
"line": 86,
"column": 34
} | {
"line": 86,
"column": 35
} | [
{
"pp": "case e'_5\nR : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\ni : ℕ\n⊢ ∑ j ∈ range m, X ^ ((i + 1) * j) = 1 + ∑' (j : ℕ), (if j + 1 < m then 1 else 0) • X ^ ((i + 1) * (j + 1))",
"ppTerm": "?e'_5",
"assigned": true,
"... | [
"case e'_5\nR : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\ni : ℕ\n⊢ X ^ ((i + 1) * 0) + ∑ x ∈ Ico 1 m, X ^ ((i + 1) * x) =\n 1 + ∑' (j : ℕ), (if j + 1 < m then 1 else 0) • X ^ ((i + 1) * (j + 1))"
] | sum_range_eq_add_Ico _ hm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 93,
"column": 62
} | {
"line": 93,
"column": 73
} | {
"line": 93,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\ni b : ℕ\nhb : b ∉ range (m - 1)\n⊢ (if b + 1 < m then 1 else 0) = 0",
"ppTerm": "?m.334",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring... | [
"R : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\ni b : ℕ\nhb : b ∉ range (m - 1)\n⊢ m ≤ b + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 85,
"column": 2
} | {
"line": 93,
"column": 86
} | {
"line": 94,
"column": 2
} | [
{
"pp": "case e'_5\nR : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\n⊢ (fun i ↦ ∑ j ∈ range m, X ^ ((i + 1) * j)) = fun i ↦\n 1 + ∑' (j : ℕ), (if j + 1 < m then 1 else 0) • X ^ ((i + 1) * (j + 1))",
"ppTerm": "?e'_5",
"assign... | [
"case e'_6\nR : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\n⊢ (PowerSeries.mk fun n ↦ ↑(#(countRestricted n m))) = genFun fun i c ↦ if c < m then 1 else 0"
] | · ext1 i
rw [sum_range_eq_add_Ico _ hm, sum_Ico_eq_sum_range]
congrm $(by simp) + ?_
trans ∑ k ∈ range (m - 1), (if k + 1 < m then (1 : R) else 0) • X ^ ((i + 1) * (k + 1))
· refine sum_congr rfl fun b hn ↦ ?_
rw [add_comm 1 b]
have : b + 1 < m := by grind
simp [this]
· exact (tsum... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 15
} | {
"line": 100,
"column": 16
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\n⊢ Multipliable fun i ↦ ∑ j ∈ range 0, X ^ ((i + 1) * j)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"MvPowerSeries.instCommSemiring",
"CommSemiring.to... | [
"case inl\nR : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\n⊢ Multipliable fun i ↦ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 517,
"column": 6
} | {
"line": 517,
"column": 18
} | {
"line": 517,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA S : Finset G\nhS : S.Nonempty\n⊢ (1 - K) * ↑(#A) ≤ expansion K S A",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"AddGroupWithOne.toAddGrou... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA S : Finset G\nhS : S.Nonempty\n⊢ ↑(#A) - K * ↑(#A) ≤ expansion K S A"
] | one_sub_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Pentagonal.Ring | {
"line": 101,
"column": 32
} | {
"line": 101,
"column": 62
} | {
"line": 101,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T2Space R\nx : R\nhsum✝ : ∀ (k : ℕ), Summable fun x_1 ↦ powMulProdOneSubPow k x_1 x\nh : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)\nhsum : Summable fun x_1 ↦ x ^ x_1 * ∏ x_2 ∈ Finset.range (x... | [
"R : Type u_1\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T2Space R\nx : R\nhsum✝ : ∀ (k : ℕ), Summable fun x_1 ↦ powMulProdOneSubPow k x_1 x\nh : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)\nhsum : Summable fun x_1 ↦ x ^ x_1 * ∏ x_2 ∈ Finset.range (x_1 + 1), (1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Pentagonal.Ring | {
"line": 101,
"column": 74
} | {
"line": 101,
"column": 85
} | {
"line": 101,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T2Space R\nx : R\nhsum✝ : ∀ (k : ℕ), Summable fun x_1 ↦ powMulProdOneSubPow k x_1 x\nh : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)\nhsum : Summable fun x_1 ↦ x ^ x_1 * ∏ x_2 ∈ Finset.range (x... | [
"R : Type u_1\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T2Space R\nx : R\nhsum✝ : ∀ (k : ℕ), Summable fun x_1 ↦ powMulProdOneSubPow k x_1 x\nh : ∀ (k : ℕ), Multipliable fun n ↦ 1 - x ^ (n + k + 1)\nhsum : Summable fun x_1 ↦ x ^ x_1 * ∏ x_2 ∈ Finset.range (x_1 + 1), (1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 14
} | [
{
"pp": "n m : ℕ\nhm : 0 < m\n⊢ #(restricted n fun x ↦ ¬m ∣ x) = #(countRestricted n m)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n m : ℕ\nhm : 0 < m\n⊢ #(restricted n fun x ↦ ¬m ∣ x) = #(countRestricted n m)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 186,
"column": 83
} | {
"line": 186,
"column": 94
} | {
"line": 186,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs✝ : s ⊇ range d\nthis :\n ∏ i ∈ s, (1 + ∑' (j : ℕ), f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1))) =\n ∏ i ∈ Finset.map (addRightEmbedding 1) s, (1 + ∑' (j : ℕ), f i (j + 1) • ... | [
"R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs✝ : s ⊇ range d\nthis :\n ∏ i ∈ s, (1 + ∑' (j : ℕ), f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1))) =\n ∏ i ∈ Finset.map (addRightEmbedding 1) s, (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 636,
"column": 31
} | {
"line": 636,
"column": 51
} | {
"line": 637,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nS : Finset G\nhK : K < 1\nhS : S.Nonempty\nH : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A\nex : Finset G → ℝ := expansion K S\nκ : ℝ := connectivity K S\nκ_add_one_pos : 0 < κ + 1\none_sub_K_pos : 0 < 1 - K\nt : ℕ := ⌊(κ + 1) / (1 - K)... | [] | by norm_cast; gcongr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 17
} | {
"line": 72,
"column": 18
} | [
{
"pp": "case zero\nn : ℕ\nx✝ : n + 2 ≠ 0\nhk : 0 ∈ Iic (n + 1)\n⊢ Even (largeSchroder 0 * (n + 1 - 0).largeSchroder)",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.largeSchroder",
"Nat.instOrderedSub",
"HMul.hMul",
"congrArg",
"AddMonoid... | [
"case zero\nn : ℕ\nx✝ : n + 2 ≠ 0\nhk : 0 ∈ Iic (n + 1)\n⊢ Even (n + 1).largeSchroder"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 30
} | {
"line": 69,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : TopologicalSpace R\nk : ℕ\n⊢ Multipliable fun n ↦ 1 - X ^ (n + k + 1)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : TopologicalSpace R\nk : ℕ\n⊢ Multipliable fun n ↦ 1 + -X ^ (n + k + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries | {
"line": 86,
"column": 54
} | {
"line": 86,
"column": 65
} | {
"line": 86,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nh : n ∉ Set.range pentagonal\n⊢ ¬∃ k, pentagonal k = (Finsupp.single () n) ()",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Unit.unit",
"not_exists._simp_1",
"Nat.instMulZero... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nh : n ∉ Set.range pentagonal\n⊢ ∀ (x : ℤ), ¬pentagonal x = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.Pentagonal.PowerSeries | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 39
} | {
"line": 109,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : TopologicalSpace R\n⊢ HasSum (fun n ↦ C ((coeff n) (pentagonalSeries R)) * X ^ n) (pentagonalSeries R)",
"ppTerm": "?m.154",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : TopologicalSpace R\n⊢ HasSum (fun n ↦ C ((coeff n) (pentagonalSeries R)) * X ^ n) (pentagonalSeries R)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 73
} | {
"line": 158,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Fintype α\ninst✝¹ : CommRing α\ns : Finset α\nx : α × α × α\ninst✝ : Fact (IsUnit 2)\na : α\nha : a ∈ s\ny b : α\nhb : b ∈ s\nh : y + 2 * b = y + 2 * a\n⊢ y + a = y + b",
"ppTerm": "?m.173",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Add... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : Fintype α\ninst✝¹ : CommRing α\ns : Finset α\nx : α × α × α\ninst✝ : Fact (IsUnit 2)\na : α\nha : a ∈ s\ny b : α\nhb : b ∈ s\nh : y + 2 * b = y + 2 * a\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 73
} | {
"line": 162,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Fintype α\ninst✝¹ : CommRing α\ns : Finset α\nx : α × α × α\ninst✝ : Fact (IsUnit 2)\na : α\nha : a ∈ s\ny b : α\nhb : b ∈ s\nh : y + b = y + a\n⊢ y + 2 * a = y + 2 * b",
"ppTerm": "?m.247",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMu... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : Fintype α\ninst✝¹ : CommRing α\ns : Finset α\nx : α × α × α\ninst✝ : Fact (IsUnit 2)\na : α\nha : a ∈ s\ny b : α\nhb : b ∈ s\nh : y + b = y + a\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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