module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Data.Nat.BitIndices | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 49
} | [
{
"pp": "case bit\nb : Bool\nn : ℕ\nhs : n.bitIndices.SortedLT\n⊢ List.Pairwise (fun a b ↦ a < b) n.bitIndices",
"usedConstants": [
"Preorder.toLT",
"List.SortedLT.pairwise",
"Nat.bitIndices",
"imp_self._simp_1",
"Nat.instPreorder",
"Nat",
"LT.lt",
"True",
... | exact List.Pairwise.imp (by simp) hs.pairwise | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Nat.BitIndices | {
"line": 115,
"column": 30
} | {
"line": 115,
"column": 68
} | [
{
"pp": "a : ℕ\nL : List ℕ\nhL✝ : (a :: L).SortedLT\nhL : List.Pairwise (fun x1 x2 ↦ x1 < x2) L\nhaL : ∀ (a' : ℕ), a' ∈ L → a + 1 ≤ a'\nx y : ℕ\nhx : x ∈ L\nx✝ : y ∈ L\n⊢ x - (a + 1) < y - (a + 1) ↔ x < y",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",... | rw [tsub_lt_tsub_iff_right (haL _ hx)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.BitIndices | {
"line": 115,
"column": 30
} | {
"line": 115,
"column": 68
} | [
{
"pp": "a : ℕ\nL : List ℕ\nhL✝ : (a :: L).SortedLT\nhL : List.Pairwise (fun x1 x2 ↦ x1 < x2) L\nhaL : ∀ (a' : ℕ), a' ∈ L → a + 1 ≤ a'\nx y : ℕ\nhx : x ∈ L\nx✝ : y ∈ L\n⊢ x - (a + 1) < y - (a + 1) ↔ x < y",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",... | rw [tsub_lt_tsub_iff_right (haL _ hx)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.BitIndices | {
"line": 115,
"column": 30
} | {
"line": 115,
"column": 68
} | [
{
"pp": "a : ℕ\nL : List ℕ\nhL✝ : (a :: L).SortedLT\nhL : List.Pairwise (fun x1 x2 ↦ x1 < x2) L\nhaL : ∀ (a' : ℕ), a' ∈ L → a + 1 ≤ a'\nx y : ℕ\nhx : x ∈ L\nx✝ : y ∈ L\n⊢ x - (a + 1) < y - (a + 1) ↔ x < y",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",... | rw [tsub_lt_tsub_iff_right (haL _ hx)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Compactness | {
"line": 110,
"column": 15
} | {
"line": 110,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Set α) → s.Finite → (a : α) → β a\nχ : (a : α) → β a\nhχ : ∀ (s : Finset α), ∃ t, s ⊆ t ∧ ∀ x ∈ s, χ x = g ↑t ⋯ x\ns : Set α\nhs : s.Finite\nt : Finset α\nht : hs.toFinset ⊆ t\nht' : ∀ x ∈ hs.toFinset, χ x = g ↑t ⋯ x\n⊢ ∃ (ht : (... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Compactness | {
"line": 110,
"column": 15
} | {
"line": 110,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Set α) → s.Finite → (a : α) → β a\nχ : (a : α) → β a\nhχ : ∀ (s : Finset α), ∃ t, s ⊆ t ∧ ∀ x ∈ s, χ x = g ↑t ⋯ x\ns : Set α\nhs : s.Finite\nt : Finset α\nht : hs.toFinset ⊆ t\nht' : ∀ x ∈ hs.toFinset, χ x = g ↑t ⋯ x\n⊢ ∃ (ht : (... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Compactness | {
"line": 110,
"column": 15
} | {
"line": 110,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Set α) → s.Finite → (a : α) → β a\nχ : (a : α) → β a\nhχ : ∀ (s : Finset α), ∃ t, s ⊆ t ∧ ∀ x ∈ s, χ x = g ↑t ⋯ x\ns : Set α\nhs : s.Finite\nt : Finset α\nht : hs.toFinset ⊆ t\nht' : ∀ x ∈ hs.toFinset, χ x = g ↑t ⋯ x\n⊢ ∃ (ht : (... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Compactness | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Set α) → s.Finite → (a : ↑s) → β ↑a\nχ : (a : α) → β a\nhχ : ∀ (s : Finset α), ∃ t, ∃ (hst : s ⊆ t), ∀ (x : ↥s), χ ↑x = g ↑t ⋯ (inclusion hst x)\ns : Set α\nhs : s.Finite\nt : Finset α\nht✝ : hs.toFinset ⊆ t\nhst : ∀ (x : ↥hs.toF... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Compactness | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Set α) → s.Finite → (a : ↑s) → β ↑a\nχ : (a : α) → β a\nhχ : ∀ (s : Finset α), ∃ t, ∃ (hst : s ⊆ t), ∀ (x : ↥s), χ ↑x = g ↑t ⋯ (inclusion hst x)\ns : Set α\nhs : s.Finite\nt : Finset α\nht✝ : hs.toFinset ⊆ t\nhst : ∀ (x : ↥hs.toF... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Compactness | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Set α) → s.Finite → (a : ↑s) → β ↑a\nχ : (a : α) → β a\nhχ : ∀ (s : Finset α), ∃ t, ∃ (hst : s ⊆ t), ∀ (x : ↥s), χ ↑x = g ↑t ⋯ (inclusion hst x)\ns : Set α\nhs : s.Finite\nt : Finset α\nht✝ : hs.toFinset ⊆ t\nhst : ∀ (x : ↥hs.toF... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Colex | {
"line": 116,
"column": 8
} | {
"line": 116,
"column": 32
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\nf : α → β\n𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)\ns✝ t✝ u✝ : Finset α\na✝ b✝ : α\ns t u : Colex (Finset α)\nhst : s ≤ t\nhtu : t ≤ u\na : α\nhas : a ∈ ofColex s\nhau : a ∉ ofColex u\nhat : a ∈ ofColex t\nb : α\nhbu : b ∈ of... | exact ⟨b, hbu, hbs, hab⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Colex | {
"line": 116,
"column": 8
} | {
"line": 116,
"column": 32
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\nf : α → β\n𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)\ns✝ t✝ u✝ : Finset α\na✝ b✝ : α\ns t u : Colex (Finset α)\nhst : s ≤ t\nhtu : t ≤ u\na : α\nhas : a ∈ ofColex s\nhau : a ∉ ofColex u\nhat : a ∈ ofColex t\nb : α\nhbu : b ∈ of... | exact ⟨b, hbu, hbs, hab⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Colex | {
"line": 116,
"column": 8
} | {
"line": 116,
"column": 32
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\nf : α → β\n𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)\ns✝ t✝ u✝ : Finset α\na✝ b✝ : α\ns t u : Colex (Finset α)\nhst : s ≤ t\nhtu : t ≤ u\na : α\nhas : a ∈ ofColex s\nhau : a ∉ ofColex u\nhat : a ∈ ofColex t\nb : α\nhbu : b ∈ of... | exact ⟨b, hbu, hbs, hab⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 69
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nH : Submodule K V\nh : finrank K ↥((Equiv.refl (Submodule K V)) H) = 1\n⊢ H ∈ Set.range Projectivization.submodule",
"usedConstants": [
"IsNoetherianRing.strongRankCondition",
"Submodule",
... | rcases finrank_eq_one_iff'.1 h with ⟨v : H, hv₀, hv : ∀ w : H, _⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 74
} | [
{
"pp": "case h\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nH : Submodule K V\nh : finrank K ↥((Equiv.refl (Submodule K V)) H) = 1\nv : ↥H\nhv₀ : v ≠ 0\nhv : ∀ (w : ↥H), ∃ c, c • v = w\n⊢ (mk K ↑v ⋯).submodule = H",
"usedConstants": [
"Projectiviz... | rw [submodule_mk, SetLike.ext'_iff, Submodule.span_singleton_eq_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Projectivization.Constructions | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 77
} | [
{
"pp": "case h.h\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nv : Fin 3 → F\nhv : v ≠ 0\nw : Fin 3 → F\nhw : w ≠ 0\nh : mk F v hv ≠ mk F w hw\n⊢ ((mk F v hv).cross (mk F w hw)).orthogonal (mk F v hv)",
"usedConstants": [
"Projectivization.mk",
"dot_self_cross",
"Iff.mpr",
... | rw [cross_mk_of_ne hv hw h, orthogonal_mk, dotProduct_comm, dot_self_cross] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Projectivization.Constructions | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 77
} | [
{
"pp": "case h.h\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nv : Fin 3 → F\nhv : v ≠ 0\nw : Fin 3 → F\nhw : w ≠ 0\nh : mk F v hv ≠ mk F w hw\n⊢ ((mk F v hv).cross (mk F w hw)).orthogonal (mk F v hv)",
"usedConstants": [
"Projectivization.mk",
"dot_self_cross",
"Iff.mpr",
... | rw [cross_mk_of_ne hv hw h, orthogonal_mk, dotProduct_comm, dot_self_cross] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Constructions | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 77
} | [
{
"pp": "case h.h\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nv : Fin 3 → F\nhv : v ≠ 0\nw : Fin 3 → F\nhw : w ≠ 0\nh : mk F v hv ≠ mk F w hw\n⊢ ((mk F v hv).cross (mk F w hw)).orthogonal (mk F v hv)",
"usedConstants": [
"Projectivization.mk",
"dot_self_cross",
"Iff.mpr",
... | rw [cross_mk_of_ne hv hw h, orthogonal_mk, dotProduct_comm, dot_self_cross] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Colex | {
"line": 520,
"column": 7
} | {
"line": 520,
"column": 23
} | [
{
"pp": "n : ℕ\nhn : 2 ≤ n\na₁✝ a₂✝ : Finset ℕ\nh : (fun s ↦ ∑ i ∈ s, n ^ i) a₁✝ = (fun s ↦ ∑ i ∈ s, n ^ i) a₂✝\n⊢ a₁✝ = a₂✝",
"usedConstants": [
"congrArg",
"Finset",
"Nat.instMonoid",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Eq.mp",
"LE.le",
"Monoid.toPo... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Configuration | {
"line": 269,
"column": 25
} | {
"line": 269,
"column": 27
} | [
{
"pp": "case h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : p ∉ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := ... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Configuration | {
"line": 269,
"column": 25
} | {
"line": 269,
"column": 27
} | [
{
"pp": "case h\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : p ∉ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := ... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Enumerative.Catalan.Basic | {
"line": 93,
"column": 10
} | {
"line": 93,
"column": 13
} | [
{
"pp": "n i : ℕ\nh : i ≤ n\nl₁ : ↑i + 1 ≠ 0\nl₂ : ↑n - ↑i + 1 ≠ 0\nh₁ : ↑(i + 1).centralBinom = (↑i + 1) * ↑(i + 1).centralBinom / (↑i + 1)\nh₂ : ↑(n - i + 1).centralBinom = (↑n - ↑i + 1) * ↑(n - i + 1).centralBinom / (↑n - ↑i + 1)\nh₃ : (↑i + 1) * ↑(i + 1).centralBinom = 2 * (2 * ↑i + 1) * ↑i.centralBinom\nh₄... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Catalan.Basic | {
"line": 103,
"column": 2
} | {
"line": 105,
"column": 23
} | [
{
"pp": "n : ℕ\n⊢ catalan n = n.centralBinom / (n + 1)",
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Int.cast_natCast",
"catalan",
"Int.instDiv",
"Dvd.dvd",
"instHDiv",
"Int.cast_div_ofNat_charZero._simp_1",
"congrArg",
"Nat.cast_add._simp_1",... | suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
have h := Nat.succ_dvd_centralBinom n
exact mod_cast this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 138,
"column": 4
} | {
"line": 139,
"column": 28
} | [
{
"pp": "case neg\nm n : ℕ\nhm : ¬m = 0\nhn : ¬n = 0\n⊢ (multinomial {n} fun k ↦ k * count k (replicate m n)) *\n ∏ x ∈ {n}.erase 0, ∏ j ∈ Finset.range (count x (replicate m n)), (j * x + x - 1).choose (x - 1) =\n ∏ p ∈ Finset.range m, (p * n + n - 1).choose (n - 1)",
"usedConstants": [
"Eq.mp... | · rw [show ({n} : Finset ℕ).erase 0 = {n} by simp [Ne.symm hn]]
simp [count_replicate] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 52,
"column": 2
} | {
"line": 72,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nhA : #(A * A) ≤ #A\nha : a ∈ A\n⊢ a •> ↑(stabilizer G A) = ↑A ∧ ↑(stabilizer G A) <• a = ↑A",
"usedConstants": [
"smul_left_cancel_iff._simp_2",
"Set.ext",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
... | have smul_A {a} (ha : a ∈ A) : a •> A = A * A :=
eq_of_subset_of_card_le (smul_finset_subset_mul ha) (by simpa)
have A_smul {a} (ha : a ∈ A) : A <• a = A * A :=
eq_of_subset_of_card_le (op_smul_finset_subset_mul ha) (by simpa)
have smul_A_eq_A_smul {a} (ha : a ∈ A) : a •> A = A <• a := by rw [smul_A ha, A_s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 52,
"column": 2
} | {
"line": 72,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nhA : #(A * A) ≤ #A\nha : a ∈ A\n⊢ a •> ↑(stabilizer G A) = ↑A ∧ ↑(stabilizer G A) <• a = ↑A",
"usedConstants": [
"smul_left_cancel_iff._simp_2",
"Set.ext",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
... | have smul_A {a} (ha : a ∈ A) : a •> A = A * A :=
eq_of_subset_of_card_le (smul_finset_subset_mul ha) (by simpa)
have A_smul {a} (ha : a ∈ A) : A <• a = A * A :=
eq_of_subset_of_card_le (op_smul_finset_subset_mul ha) (by simpa)
have smul_A_eq_A_smul {a} (ha : a ∈ A) : a •> A = A <• a := by rw [smul_A ha, A_s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 10
} | [
{
"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 : G\nhh₁ : h₁ ∈ H\nhz₁ : z₁ ∈ Z\nhh₂ : h₂ ∈ H\nh : h₁ * z₁ = h₂ * z₁\nhz₂ : z₁ ∈ Z\n⊢ (h₁, z₁) = (h₂, z₁)",
"usedConstants": [
"CancelMonoid.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 367,
"column": 49
} | {
"line": 367,
"column": 65
} | [
{
"pp": "p : DyckWord\n⊢ (p.nest.take (p.nest.firstReturn + 1) ⋯).denest ⋯ = p",
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"False",
"DyckStep.U",
"congrArg",
"DyckWord.firstReturn_nest",
"DyckWord.count_take_firstReturn_add_one",
"DyckWord",
... | firstReturn_nest | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 373,
"column": 50
} | {
"line": 373,
"column": 66
} | [
{
"pp": "p : DyckWord\n⊢ p.nest.drop (p.nest.firstReturn + 1) ⋯ = 0",
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"False",
"DyckStep.U",
"congrArg",
"DyckWord.firstReturn_nest",
"DyckWord.count_take_firstReturn_add_one",
"DyckWord",
"id",
... | firstReturn_nest | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 29
} | [
{
"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\nhg' : g ∉ Set.range toFinsuppAntidiag\ni : ℕ\nhi : i ∈ s\nhi' : (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1)... | apply prod_eq_zero hi hi' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 452,
"column": 12
} | {
"line": 452,
"column": 20
} | [
{
"pp": "case h₂.h\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA : Finset G\nhK₁ : 1 < K\nhKφ : K < φ\nhA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)\nhA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)\nK_pos hK₀ : 0 < K\nhKφ' : 0 < φ - K\nhKψ' : 0 < K - ψ\nhK₂' : 0 < 2 - K\nconst_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 452,
"column": 12
} | {
"line": 452,
"column": 20
} | [
{
"pp": "case h₂.h\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA : Finset G\nhK₁ : 1 < K\nhKφ : K < φ\nhA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)\nhA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)\nK_pos hK₀ : 0 < K\nhKφ' : 0 < φ - K\nhKψ' : 0 < K - ψ\nhK₂' : 0 < 2 - K\nconst_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 452,
"column": 12
} | {
"line": 452,
"column": 20
} | [
{
"pp": "case h₂.h\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA : Finset G\nhK₁ : 1 < K\nhKφ : K < φ\nhA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)\nhA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)\nK_pos hK₀ : 0 < K\nhKφ' : 0 < φ - K\nhKψ' : 0 < K - ψ\nhK₂' : 0 < 2 - K\nconst_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 65
} | [
{
"pp": "case neg\n𝕜 : Type u_2\nα : Type u_5\ninst✝¹ : Zero 𝕜\ninst✝ : LE α\nf g : IncidenceAlgebra 𝕜 α\nh : ∀ (a b : α), a ≤ b → f a b = g a b\na b : α\nhab : ¬a ≤ b\n⊢ f a b = g a b",
"usedConstants": [
"Eq.mpr",
"congrArg",
"IncidenceAlgebra.apply_eq_zero_of_not_le",
"id",
... | rw [apply_eq_zero_of_not_le hab, apply_eq_zero_of_not_le hab] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 65
} | [
{
"pp": "case neg\n𝕜 : Type u_2\nα : Type u_5\ninst✝¹ : Zero 𝕜\ninst✝ : LE α\nf g : IncidenceAlgebra 𝕜 α\nh : ∀ (a b : α), a ≤ b → f a b = g a b\na b : α\nhab : ¬a ≤ b\n⊢ f a b = g a b",
"usedConstants": [
"Eq.mpr",
"congrArg",
"IncidenceAlgebra.apply_eq_zero_of_not_le",
"id",
... | rw [apply_eq_zero_of_not_le hab, apply_eq_zero_of_not_le hab] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 65
} | [
{
"pp": "case neg\n𝕜 : Type u_2\nα : Type u_5\ninst✝¹ : Zero 𝕜\ninst✝ : LE α\nf g : IncidenceAlgebra 𝕜 α\nh : ∀ (a b : α), a ≤ b → f a b = g a b\na b : α\nhab : ¬a ≤ b\n⊢ f a b = g a b",
"usedConstants": [
"Eq.mpr",
"congrArg",
"IncidenceAlgebra.apply_eq_zero_of_not_le",
"id",
... | rw [apply_eq_zero_of_not_le hab, apply_eq_zero_of_not_le hab] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 235,
"column": 4
} | {
"line": 238,
"column": 54
} | [
{
"pp": "F : Type u_1\n𝕜 : Type u_2\n𝕝 : Type u_3\n𝕞 : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : Preorder α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : DecidableEq α\ninst✝ : Semiring 𝕜\nf g h : IncidenceAlgebra 𝕜 α\n⊢ f * g * h = f * (g * h)",
"usedConstants": [
"Eq.mpr",
"Finset.mul_sum... | ext a b
simp only [mul_apply, sum_mul, mul_sum, sum_sigma']
apply sum_nbij' (fun ⟨a, b⟩ ↦ ⟨b, a⟩) (fun ⟨a, b⟩ ↦ ⟨b, a⟩) <;>
aesop (add simp mul_assoc) (add unsafe le_trans) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 235,
"column": 4
} | {
"line": 238,
"column": 54
} | [
{
"pp": "F : Type u_1\n𝕜 : Type u_2\n𝕝 : Type u_3\n𝕞 : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : Preorder α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : DecidableEq α\ninst✝ : Semiring 𝕜\nf g h : IncidenceAlgebra 𝕜 α\n⊢ f * g * h = f * (g * h)",
"usedConstants": [
"Eq.mpr",
"Finset.mul_sum... | ext a b
simp only [mul_apply, sum_mul, mul_sum, sum_sigma']
apply sum_nbij' (fun ⟨a, b⟩ ↦ ⟨b, a⟩) (fun ⟨a, b⟩ ↦ ⟨b, a⟩) <;>
aesop (add simp mul_assoc) (add unsafe le_trans) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 172,
"column": 15
} | {
"line": 172,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\n⊢ Filter.Tendsto (fun s ↦ ∏ b ∈ s, (1 + ∑' (j : ℕ), f (b + 1) (j + 1) • X ^ ((b + 1) * (j + 1))))\n (SummationFilter.unconditional ℕ).filter (nhds (genFun f))",
"usedConstants": [
"Eq.mpr"... | WithPiTopology.tendsto_iff_coeff_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Stirling | {
"line": 150,
"column": 19
} | {
"line": 150,
"column": 41
} | [
{
"pp": "n✝ : ℕ\nx✝ : 0 < n✝ + 1\n⊢ stirlingSecond 0 (n✝ + 1) = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat",
"Eq.refl",
"OfNat.ofNat",
"Nat.succ",
"Eq",
"Nat.stir... | by rw [stirlingSecond] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 172,
"column": 2
} | {
"line": 187,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\n⊢ HasProd (fun i ↦ 1 + ∑' (j : ℕ), f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1))) (genFun f)",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"not_exists._simp_1",
"Set.i... | rw [HasProd, WithPiTopology.tendsto_iff_coeff_tendsto]
refine fun d ↦ tendsto_atTop_of_eventually_const (fun s (hs : s ≥ range d) ↦ ?_)
have : ∏ i ∈ s, ((1 : R⟦X⟧) + ∑' j, f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1)))
= ∏ i ∈ s.map (addRightEmbedding 1), (1 + ∑' j, f i (j + 1) • X ^ (i * (j + 1))) := by simp
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 172,
"column": 2
} | {
"line": 187,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\n⊢ HasProd (fun i ↦ 1 + ∑' (j : ℕ), f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1))) (genFun f)",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"not_exists._simp_1",
"Set.i... | rw [HasProd, WithPiTopology.tendsto_iff_coeff_tendsto]
refine fun d ↦ tendsto_atTop_of_eventually_const (fun s (hs : s ≥ range d) ↦ ?_)
have : ∏ i ∈ s, ((1 : R⟦X⟧) + ∑' j, f (i + 1) (j + 1) • X ^ ((i + 1) * (j + 1)))
= ∏ i ∈ s.map (addRightEmbedding 1), (1 + ∑' j, f i (j + 1) • X ^ (i * (j + 1))) := by simp
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 12
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : CommRing α\ns : Finset α\nhs : ThreeAPFree ↑s\nd a : α\nha : a ∈ s\nd' c : α\nhc : c ∈ s\nh₂ : d' + c = d + a\nhb : a ∈ s\nh₁ : d' + 2 * a = d + 2 * a\nthis : a + c = a + a\n⊢ d' = d ∨ d + a = d' + a ∨ d + 2 * a = d' + 2 * c",
"usedConstants": [
"HMul... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 180,
"column": 94
} | {
"line": 184,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhGH : G < H\n⊢ V(G) ⊂ V(H) ∨ E(G) ⊂ E(H)",
"usedConstants": [
"lt_iff_le_and_ne",
"Eq.mpr",
"instDecidableNot",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"HasSSubset.SSubset",
"Preorder.toLE",... | by
rw [lt_iff_le_and_ne] at hGH
simp only [ssubset_iff_subset_ne, hGH.1.vertexSet_mono, ne_eq, true_and, hGH.1.edgeSet_mono]
by_contra! heq
exact hGH.2 <| hGH.1.compatible.ext heq.1 heq.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Graph α β\nhe : V(G) = ∅\ne : β\nx y : α\n⊢ ¬G.IsLink e x y",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"False.elim",
"Membership.mem",
"Eq.mp",
"Graph.IsLink.left_mem",
"Graph.IsLink",
... | exact fun h ↦ by simpa [he] using h.left_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Graph.Maps | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 43
} | [
{
"pp": "α : Type u_1\nα' : Type u_2\nα'' : Type u_3\nβ : Type u_4\nG✝ H : Graph α β\nf✝ g : α → α'\nu v : α\ne✝ : β\nx✝ y✝ : α'\nf : α → α'\nG : Graph α β\ne : β\nx y : α\nh : G.IsLink e x y\n⊢ f x ∈ f '' V(G)",
"usedConstants": [
"Graph.IsLink.left_mem",
"Set.mem_image_of_mem",
"Graph.ve... | exact Set.mem_image_of_mem _ h.left_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Graph.Basic | {
"line": 394,
"column": 10
} | {
"line": 394,
"column": 18
} | [
{
"pp": "case hV.h\nα : Type u_1\nβ : Type u_2\nG : Graph α β\nV : Set α\nE : Set β\nIsLink : β → α → α → Prop\nhV : V(G) = V\nhE : E(G) = E\nh_isLink : ∀ (e : β) (x y : α), G.IsLink e x y ↔ IsLink e x y\nx✝ : α\n⊢ x✝ ∈ V(G.copy hV hE h_isLink) ↔ x✝ ∈ V(G)",
"usedConstants": [
"congrArg",
"Membe... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Graph.Basic | {
"line": 394,
"column": 10
} | {
"line": 394,
"column": 18
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nG : Graph α β\nV : Set α\nE : Set β\nIsLink : β → α → α → Prop\nhV : V(G) = V\nhE : E(G) = E\nh_isLink : ∀ (e : β) (x y : α), G.IsLink e x y ↔ IsLink e x y\ne✝ : β\nx✝ y✝ : α\n⊢ (G.copy hV hE h_isLink).IsLink e✝ x✝ y✝ ↔ G.IsLink e✝ x✝ y✝",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.HalesJewett | {
"line": 228,
"column": 42
} | {
"line": 229,
"column": 81
} | [
{
"pp": "η : Type u_1\nα : Type u_2\nι : Type u_3\nl : Line (η → α) ι\na : η → α\nie : ι × η\n⊢ ↑l.toSubspace a ie = ↑l a ie.1 ie.2",
"usedConstants": [
"Combinatorics.Subspace.idxFun",
"Combinatorics.Line.toSubspace._proof_1",
"congrArg",
"Combinatorics.Line.toFun",
"Combinato... | by
cases h : l.idxFun ie.1 <;> simp [toSubspace, h, coe_apply, Subspace.coe_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Hindman | {
"line": 151,
"column": 2
} | {
"line": 160,
"column": 38
} | [
{
"pp": "case refine_2\nM : Type u_1\ninst✝ : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}\n⊢ S.Nonempty",
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Iff.mpr",
"Eq.mpr",
"ultrafilter_compact",
... | · apply IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed
· intro n U hU
filter_upwards [hU]
rw [← Stream'.drop_drop, ← Stream'.tail_eq_drop]
exact FP.tail _
· intro n
exact ⟨pure _, mem_pure.mpr <| FP.head _⟩
· exact (ultrafilter_isClosed_basic _).isCompact
· int... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.HalesJewett | {
"line": 453,
"column": 32
} | {
"line": 453,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝¹ : Finite α\nκ : Type v\ninst✝ : Finite κ\nι : Type\nιfin : Fintype ι\nhι : ∀ (C : (ι → α) → ULift.{u, v} κ), ∃ l, IsMono C l\nC : (ι → α) → κ\nl : Line α ι\nc : ULift.{u, v} κ\nhc : ∀ (x : α), (ULift.up ∘ C) (↑l x) = c\nx : α\n⊢ C (↑l x) = c.down",
"usedConstants": [
"Eq.mp... | ← hc x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 215,
"column": 4
} | {
"line": 218,
"column": 29
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nI J X Y : Set α\nhIX : M.IsBasis I X\nhJY : M.IsBasis J Y\nhIJ : I ⊆ J\n⊢ J \\ I ⊆ Y \\ X",
"usedConstants": [
"Eq.mpr",
"Set.diff_subset",
"Set.diff_self_inter",
"congrArg",
"Disjoint",
"SemilatticeInf.toPartialOrder",... | rw [subset_diff, and_iff_right (diff_subset.trans hJY.subset),
hIX.eq_of_subset_indep (hJY.indep.inter_right X) (subset_inter hIJ hIX.subset)
inter_subset_right, diff_self_inter]
exact disjoint_sdiff_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 215,
"column": 4
} | {
"line": 218,
"column": 29
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nI J X Y : Set α\nhIX : M.IsBasis I X\nhJY : M.IsBasis J Y\nhIJ : I ⊆ J\n⊢ J \\ I ⊆ Y \\ X",
"usedConstants": [
"Eq.mpr",
"Set.diff_subset",
"Set.diff_self_inter",
"congrArg",
"Disjoint",
"SemilatticeInf.toPartialOrder",... | rw [subset_diff, and_iff_right (diff_subset.trans hJY.subset),
hIX.eq_of_subset_indep (hJY.indep.inter_right X) (subset_inter hIJ hIX.subset)
inter_subset_right, diff_self_inter]
exact disjoint_sdiff_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 38
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nM✝ M : (i : ι) → Matroid (α i)\nB₁ B₂ : Set ((i : ι) × α i)\nh₁ : (fun B ↦ ∀ (i : ι), (M i).IsBase (Sigma.mk i ⁻¹' B)) B₁\nh₂ : (fun B ↦ ∀ (i : ι), (M i).IsBase (Sigma.mk i ⁻¹' B)) B₂\ni : ι\ne : α i\nhe₁ : ⟨i, e⟩ ∈ B₁\nhe₂ : ⟨i, e⟩ ∉ B₂\nf : α i\nhfB : (M i).IsBase (ins... | obtain (rfl | hne) := eq_or_ne i j | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Quiver.ConnectedComponent | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 46
} | [
{
"pp": "V : Type u_2\ninst✝ : Quiver V\nh : IsSStronglyConnected V\n⊢ IsStronglyConnected V",
"usedConstants": [
"Exists",
"instOfNatNat",
"Nonempty.intro",
"Quiver.Path",
"Exists.casesOn",
"Nat",
"Quiver.Path.length",
"LT.lt",
"Nonempty",
"instLT... | intro i j; obtain ⟨p, _⟩ := h i j; exact ⟨p⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.ConnectedComponent | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 46
} | [
{
"pp": "V : Type u_2\ninst✝ : Quiver V\nh : IsSStronglyConnected V\n⊢ IsStronglyConnected V",
"usedConstants": [
"Exists",
"instOfNatNat",
"Nonempty.intro",
"Quiver.Path",
"Exists.casesOn",
"Nat",
"Quiver.Path.length",
"LT.lt",
"Nonempty",
"instLT... | intro i j; obtain ⟨p, _⟩ := h i j; exact ⟨p⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path.Decomposition | {
"line": 38,
"column": 40
} | {
"line": 38,
"column": 48
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\nn : ℕ\nih :\n ∀ {a b : V} (p : Path a b) (S : Set V),\n ¬a ∈ S → b ∈ S → p.length = n → ∃ u, ¬u ∈ S ∧ ∃ v, v ∈ S ∧ ∃ e p₁ p₂, p = p₁.comp (e.toPath.comp p₂)\na b : V\nS : Set V\nha_not_in_S : ¬a ∈ S\nhb_in_S : b ∈ S\nc : V\np' : Path a c\ne : c ⟶ b\nh_len : (p'.cons ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 21
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nh₁ : p.vertices.getLast ⋯ = b\nh₂ : p.vertices.getLast ⋯ ∈ p.vertices\n⊢ b ∈ p.vertices",
"usedConstants": [
"List.getLast",
"congrArg",
"Quiver.Path.vertices_ne_nil",
"Membership.mem",
"Eq.mp",
"List",
... | simpa [h₁] using h₂ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Schnirelmann | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 22
} | [
{
"pp": "A : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\n⊢ schnirelmannDensity A = 1 ↔ {0}ᶜ ⊆ A",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"schnirelmannDensity",
"Preorder.toLE",
"Set... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 161,
"column": 85
} | {
"line": 161,
"column": 93
} | [
{
"pp": "case e_s.h.e_a.e_a.e_s.h\nA : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\nB : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ B\nh : ∀ n > 0, n ∈ A ↔ n ∈ B\nn : ℕ\nhn : 0 < n\nx : ℕ\n⊢ x ∈ {a ∈ Ioc 0 ↑⟨n, hn⟩ | a ∈ A} ↔ x ∈ {a ∈ Ioc 0 ↑⟨n, hn⟩ | a ∈ B}",
"usedConstants": [
"Finset.mem_filter._simp_... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Schnirelmann | {
"line": 175,
"column": 33
} | {
"line": 175,
"column": 41
} | [
{
"pp": "A : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\nB : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ B\nh : A = B\n⊢ ∀ n > 0, n ∈ A ↔ n ∈ B",
"usedConstants": [
"congrArg",
"Membership.mem",
"instOfNatNat",
"iff_self",
"GT.gt",
"Iff",
"Nat",
"LT.lt",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Schnirelmann | {
"line": 175,
"column": 33
} | {
"line": 175,
"column": 41
} | [
{
"pp": "A : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\nB : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ B\nh : A = B\n⊢ ∀ n > 0, n ∈ A ↔ n ∈ B",
"usedConstants": [
"congrArg",
"Membership.mem",
"instOfNatNat",
"iff_self",
"GT.gt",
"Iff",
"Nat",
"LT.lt",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Schnirelmann | {
"line": 175,
"column": 33
} | {
"line": 175,
"column": 41
} | [
{
"pp": "A : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\nB : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ B\nh : A = B\n⊢ ∀ n > 0, n ∈ A ↔ n ∈ B",
"usedConstants": [
"congrArg",
"Membership.mem",
"instOfNatNat",
"iff_self",
"GT.gt",
"Iff",
"Nat",
"LT.lt",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 20
} | [
{
"pp": "case ind.inr.succ.h𝒜₁\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset α\n𝒜 : Finset (Finset α)\nhs : s ∉ 𝒜\nh𝒜₁ : (insert s 𝒜).Nonempty\nh𝒜₂ : univ ∉ insert s 𝒜\nh𝒜₃ : (insert s 𝒜).Nontrivial\nih :\n ∀ (𝒜_1 : Finset (Finset α)),\n #𝒜_1 < #𝒜 + 1 ... | · exact h𝒜.image _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 375,
"column": 6
} | {
"line": 375,
"column": 97
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v : Finset α\nhuv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\ns : Finset α\nhs𝒜' : s ∈ ∂ 𝒜'\nhs𝒜 : s ∉ ∂ 𝒜\nm : ∀ y ∉ s, insert y s ∉ 𝒜\nx : α\nleft✝ : x ∉ s\nright✝ : insert x s ∈... | refine sup_sdiff_mem_of_mem_compression (by rwa [hxy.eq]) ((erase_subset _ _).trans ‹_›) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 41
} | [
{
"pp": "case intro\nα : Type u_1\ninst✝¹ : LinearOrder α\n𝒜 : Finset (Finset α)\nr : ℕ\ninst✝ : Finite α\nh₁ : IsInitSeg 𝒜 r\nval✝ : Fintype α\n⊢ IsInitSeg (∂ 𝒜) (r - 1)",
"usedConstants": [
"Nat.eq_zero_or_pos"
]
}
] | obtain rfl | hr := Nat.eq_zero_or_pos r | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 133,
"column": 2
} | {
"line": 139,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns U V : Finset α\nhU : U.Nonempty\nhV : V.Nonempty\nh : U.max' hU < V.max' hV\nhA : compress U V s ≠ s\n⊢ toColex (compress U V s) < toColex s",
"usedConstants": [
"Eq.mpr",
"False",
"Finset.instGeneralizedBooleanAlgebra",
"Preorder.toLT"... | rw [compress, ite_ne_right_iff] at hA
rw [compress, if_pos hA.1, lt_iff_exists_filter_lt]
simp_rw [mem_sdiff (s := s), filter_inj, and_assoc]
refine ⟨_, hA.1.2 <| max'_mem _ hV, notMem_sdiff_of_mem_right <| max'_mem _ _, fun a ha ↦ ?_⟩
have : a ∉ V := fun H ↦ ha.not_ge (le_max' _ _ H)
have : a ∉ U := fun H ↦ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 133,
"column": 2
} | {
"line": 139,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns U V : Finset α\nhU : U.Nonempty\nhV : V.Nonempty\nh : U.max' hU < V.max' hV\nhA : compress U V s ≠ s\n⊢ toColex (compress U V s) < toColex s",
"usedConstants": [
"Eq.mpr",
"False",
"Finset.instGeneralizedBooleanAlgebra",
"Preorder.toLT"... | rw [compress, ite_ne_right_iff] at hA
rw [compress, if_pos hA.1, lt_iff_exists_filter_lt]
simp_rw [mem_sdiff (s := s), filter_inj, and_assoc]
refine ⟨_, hA.1.2 <| max'_mem _ hV, notMem_sdiff_of_mem_right <| max'_mem _ _, fun a ha ↦ ?_⟩
have : a ∉ V := fun H ↦ ha.not_ge (le_max' _ _ H)
have : a ∉ U := fun H ↦ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 70
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nH : G.Subgraph\nhr : G.Reachable u v\nh : ∀ v ∈ H.verts, ∀ (w : V), G.Adj v w → H.Adj v w\nhu : u ∈ H.verts\nv' : V\nhv' : v' ∈ H.verts\np : G.Walk v' v\n⊢ v ∈ H.verts",
"usedConstants": [
"SimpleGraph.Subgraph.edge_vert",
"SimpleGraph.Subgraph.Ad... | by_cases hnp : p.Nil
· exact hnp.eq ▸ hv'
exact aux (H.edge_vert (h _ hv' _ (Walk.adj_snd hnp)).symm) p.tail | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 70
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nH : G.Subgraph\nhr : G.Reachable u v\nh : ∀ v ∈ H.verts, ∀ (w : V), G.Adj v w → H.Adj v w\nhu : u ∈ H.verts\nv' : V\nhv' : v' ∈ H.verts\np : G.Walk v' v\n⊢ v ∈ H.verts",
"usedConstants": [
"SimpleGraph.Subgraph.edge_vert",
"SimpleGraph.Subgraph.Ad... | by_cases hnp : p.Nil
· exact hnp.eq ▸ hv'
exact aux (H.edge_vert (h _ hv' _ (Walk.adj_snd hnp)).symm) p.tail | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 193,
"column": 2
} | {
"line": 211,
"column": 69
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, ... | · rw [add_zero, collapse_eq hat, mul_add]
split_ifs
· refine (add_le_add (h ‹_› ‹_›) <| h ‹_› ‹_›).trans ?_
rw [collapse_of_mem ‹_› (union_mem_sups ‹_› ‹_›) (union_mem_sups ‹_› ‹_›) rfl
(union_insert _ _ _), inter_insert_of_notMem ‹_›, ← mul_add]
gcongr
· exact add_nonneg (h₄ _) (h₄ _)... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 891,
"column": 66
} | {
"line": 891,
"column": 74
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ne : Sym2 V\nh : ¬G.Reachable u v\nh'✝ : G.IsBridge e\nx✝ : V\np : (G ⊔ edge u v).Walk x✝ x✝\nhp : p.IsCycle\nhpe : e ∈ p.edges\ne' : Sym2 V\nhe' : e' ∈ p.edges\nh' : e' ∈ (edge u v).edgeSet\n⊢ p.IsCycle ∧ s(u, v) ∈ p.edges",
"usedConstants": [
"Eq.mpr",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 891,
"column": 66
} | {
"line": 891,
"column": 74
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ne : Sym2 V\nh : ¬G.Reachable u v\nh'✝ : G.IsBridge e\nx✝ : V\np : (G ⊔ edge u v).Walk x✝ x✝\nhp : p.IsCycle\nhpe : e ∈ p.edges\ne' : Sym2 V\nhe' : e' ∈ p.edges\nh' : e' ∈ (edge u v).edgeSet\n⊢ p.IsCycle ∧ s(u, v) ∈ p.edges",
"usedConstants": [
"Eq.mpr",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 891,
"column": 66
} | {
"line": 891,
"column": 74
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ne : Sym2 V\nh : ¬G.Reachable u v\nh'✝ : G.IsBridge e\nx✝ : V\np : (G ⊔ edge u v).Walk x✝ x✝\nhp : p.IsCycle\nhpe : e ∈ p.edges\ne' : Sym2 V\nhe' : e' ∈ p.edges\nh' : e' ∈ (edge u v).edgeSet\n⊢ p.IsCycle ∧ s(u, v) ∈ p.edges",
"usedConstants": [
"Eq.mpr",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 940,
"column": 65
} | {
"line": 940,
"column": 73
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu x : V\nhxu : x ≠ u\nhx : (G.neighborSet x).Subsingleton\nw : V\nhxv : x ≠ u\nhw : Walk.nil.IsTrail\nhxw : x ∈ Walk.nil.support\n⊢ False",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"False.elim",
"SimpleGraph.Walk.support",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 43
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ns t : Finset V\ninst✝ : Fintype ↑(G.neighborSet w)\nh : G.IsBipartiteWith ↑s ↑t\nhw : w ∈ t\n⊢ G.neighborFinset w ⊆ s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.Adj",
"SimpleGraph.neighborFinset",
"Cl... | rw [isBipartiteWith_neighborFinset' h hw] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 85
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\na : V\nha : a ∈ (G.subgraphOfAdj hvw).verts\nb : V\nhb : b ∈ (G.subgraphOfAdj hvw).verts\n⊢ (G.subgraphOfAdj hvw).coe.Reachable ⟨a, ha⟩ ⟨b, hb⟩",
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"Set.instSingl... | simp only [subgraphOfAdj_verts, Set.mem_insert_iff, Set.mem_singleton_iff] at ha hb | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 478,
"column": 28
} | {
"line": 478,
"column": 36
} | [
{
"pp": "case refine_1.inl.inl\nW₁ : Type u_2\nW₂ : Type u_3\nval✝¹ val✝ : W₁\nh : s(Sum.inl val✝¹, Sum.inl val✝) ∈ (completeBipartiteGraph W₁ W₂).edgeSet\n⊢ s(Sum.inl val✝¹, Sum.inl val✝) ∈ Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)",
"usedConstants": [
"Sum.isRight",
"False",
"and_tru... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 478,
"column": 28
} | {
"line": 478,
"column": 36
} | [
{
"pp": "case refine_1.inl.inr\nW₁ : Type u_2\nW₂ : Type u_3\nval✝¹ : W₁\nval✝ : W₂\nh : s(Sum.inl val✝¹, Sum.inr val✝) ∈ (completeBipartiteGraph W₁ W₂).edgeSet\n⊢ s(Sum.inl val✝¹, Sum.inr val✝) ∈ Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)",
"usedConstants": [
"False",
"Sym2.Rel",
"Sum.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 478,
"column": 28
} | {
"line": 478,
"column": 36
} | [
{
"pp": "case refine_1.inr.inl\nW₁ : Type u_2\nW₂ : Type u_3\nval✝¹ : W₂\nval✝ : W₁\nh : s(Sum.inr val✝¹, Sum.inl val✝) ∈ (completeBipartiteGraph W₁ W₂).edgeSet\n⊢ s(Sum.inr val✝¹, Sum.inl val✝) ∈ Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)",
"usedConstants": [
"False",
"Sym2.Rel",
"Sum.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 478,
"column": 28
} | {
"line": 478,
"column": 36
} | [
{
"pp": "case refine_1.inr.inr\nW₁ : Type u_2\nW₂ : Type u_3\nval✝¹ val✝ : W₂\nh : s(Sum.inr val✝¹, Sum.inr val✝) ∈ (completeBipartiteGraph W₁ W₂).edgeSet\n⊢ s(Sum.inr val✝¹, Sum.inr val✝) ∈ Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)",
"usedConstants": [
"Sum.isRight",
"False",
"and_tru... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 495,
"column": 4
} | {
"line": 495,
"column": 20
} | [
{
"pp": "V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nW₁ : Type u_2\nW₂ : Type u_3\ninst✝ : DecidableRel fun x1 x2 ↦ x1 ∈ x2\nhG : G.IsBipartiteWith s t\n⊢ Function.Injective fun x ↦\n match x with\n | ⟨e, he⟩ =>\n Sym2.hrec (motive := fun x ↦ x ∈ G.edgeSet → ↑(completeBipartiteGraph ↑s ↑t)... | rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 199,
"column": 14
} | {
"line": 199,
"column": 22
} | [
{
"pp": "case nil\nV : Type u\nG : SimpleGraph V\nw u x : V\nhadj : nil.toSubgraph.Adj w x\n⊢ ¬nil.Nil",
"usedConstants": [
"False.elim",
"SimpleGraph.Walk.nil",
"SimpleGraph.Walk.Nil",
"Not"
]
}
] | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 199,
"column": 14
} | {
"line": 199,
"column": 22
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nw u v x v✝ : V\nh✝ : G.Adj u v✝\np✝ : G.Walk v✝ v\nhadj : (cons h✝ p✝).toSubgraph.Adj w x\n⊢ ¬(cons h✝ p✝).Nil",
"usedConstants": [
"False",
"congrArg",
"SimpleGraph.Walk.not_nil_cons._simp_1",
"SimpleGraph.Walk.cons",
"True",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 10
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv u : V\ni : ℕ\np : G.Walk u v\nhp : p.IsPath\nh : i ≠ 0\nh' : i < p.length\nthis : p.getVert (i - 1) ≠ p.getVert (i + 1)\n⊢ {p.getVert (i - 1), p.getVert (i + 1)}.ncard = 2",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"HSub.hSub",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 12
} | [
{
"pp": "case refine_4\nV : Type u_1\nG : SimpleGraph V\nu v : V\nh : u ≠ v ∧ ¬G.Adj u v ∧ (G.commonNeighbors u v).Nonempty\nw : V\nhw : G.Adj u w ∧ G.Adj v w\nthis : G.edist u v ≤ ↑(Walk.cons ⋯ (Walk.cons ⋯ Walk.nil)).length\n⊢ G.edist u v ≤ 2",
"usedConstants": [
"SimpleGraph.Adj.symm",
"ENat.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 161,
"column": 76
} | {
"line": 161,
"column": 84
} | [
{
"pp": "case refine_5.inl\nV : Type u_1\nG : SimpleGraph V\nu v : V\nh : u ≠ v ∧ ¬G.Adj u v ∧ (G.commonNeighbors u v).Nonempty\nhc : G.edist u v < 2\nh✝ : G.edist u v = 0\n⊢ False",
"usedConstants": [
"SimpleGraph.edist_eq_zero_iff._simp_1",
"False",
"eq_false",
"CommSemiring.toSemi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 161,
"column": 76
} | {
"line": 161,
"column": 84
} | [
{
"pp": "case refine_5.inr\nV : Type u_1\nG : SimpleGraph V\nu v : V\nh : u ≠ v ∧ ¬G.Adj u v ∧ (G.commonNeighbors u v).Nonempty\nhc : G.edist u v < 2\nh✝ : G.edist u v = 1\n⊢ False",
"usedConstants": [
"False",
"instAddMonoidWithOneENat",
"eq_false",
"SimpleGraph.Adj",
"False.e... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 454,
"column": 4
} | {
"line": 454,
"column": 12
} | [
{
"pp": "case h.inl\nV : Type u\nG : SimpleGraph V\nu : V\ni : ℕ\np : G.Walk u u\nhpc : p.IsCycle\nv : V\nh : ¬i = 0\nh' : i < p.length\nhadj1 :\n ∃ i_1,\n (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨\n p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 63
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : p.length = G.dist v w\nhl : 1 < G.dist v w\nhnp : ¬p.Nil\nthis✝ : p.tail.tail.length < p.tail.length\nthis : p.tail.length < p.length\nhv : ¬v = p.getVert 2\nhadj : ¬G.Adj v (p.getVert 2)\n⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVer... | exact ⟨p.adj_snd hnp, p.adj_getVert_succ (hp ▸ hl), hadj, hv⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 449,
"column": 59
} | {
"line": 450,
"column": 25
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nc v : V\nr : ℕ∞\n⊢ v ∈ G.ball c r ↔ c ∈ G.ball v r",
"usedConstants": [
"SimpleGraph.ball",
"congrArg",
"setOf",
"Membership.mem",
"iff_self",
"funext",
"Iff",
"SimpleGraph.edist",
"ENat",
"congr",
"L... | by
simp [ball, edist_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 620,
"column": 2
} | {
"line": 622,
"column": 66
} | [
{
"pp": "case hv\nV : Type u\nG : SimpleGraph V\nv w : V\ns t : Set V\nsconn : (⊤.induce s).Preconnected\ntconn : (⊤.induce t).Preconnected\nhv : v ∈ s\nhw : w ∈ t\nha : G.Adj v w\n⊢ (⊤.induce {v, w} ⊔ ⊤.induce s ⊔ ⊤.induce t).verts = (⊤.induce (s ∪ t)).verts",
"usedConstants": [
"Eq.mpr",
"cong... | · simp only [Subgraph.verts_sup, Subgraph.induce_verts]
rw [Set.union_assoc]
simp [Set.insert_subset_iff, Set.singleton_subset_iff, hv, hw] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 693,
"column": 68
} | {
"line": 693,
"column": 76
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG'✝ : SimpleGraph V'\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nu v : ↑G''.verts\nhreachable : G''.coe.Reachable u v\n⊢ ↑↑u ∈ (Subgraph.coeSubgraph G'').verts",
"usedConstants": [
"Iff.mpr",
"RelHom.instFunLike",
"Iff.of_eq",
"congrA... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 693,
"column": 68
} | {
"line": 693,
"column": 76
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG'✝ : SimpleGraph V'\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nu v : ↑G''.verts\nhreachable : G''.coe.Reachable u v\n⊢ ↑↑u ∈ (Subgraph.coeSubgraph G'').verts",
"usedConstants": [
"Iff.mpr",
"RelHom.instFunLike",
"Iff.of_eq",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 693,
"column": 68
} | {
"line": 693,
"column": 76
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG'✝ : SimpleGraph V'\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nu v : ↑G''.verts\nhreachable : G''.coe.Reachable u v\n⊢ ↑↑u ∈ (Subgraph.coeSubgraph G'').verts",
"usedConstants": [
"Iff.mpr",
"RelHom.instFunLike",
"Iff.of_eq",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 694,
"column": 29
} | {
"line": 694,
"column": 37
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG'✝ : SimpleGraph V'\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nu v : ↑G''.verts\nhreachable : G''.coe.Reachable u v\n⊢ ↑↑v ∈ (Subgraph.coeSubgraph G'').verts",
"usedConstants": [
"Iff.mpr",
"RelHom.instFunLike",
"Iff.of_eq",
"congrA... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 694,
"column": 29
} | {
"line": 694,
"column": 37
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG'✝ : SimpleGraph V'\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nu v : ↑G''.verts\nhreachable : G''.coe.Reachable u v\n⊢ ↑↑v ∈ (Subgraph.coeSubgraph G'').verts",
"usedConstants": [
"Iff.mpr",
"RelHom.instFunLike",
"Iff.of_eq",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 694,
"column": 29
} | {
"line": 694,
"column": 37
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG'✝ : SimpleGraph V'\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nu v : ↑G''.verts\nhreachable : G''.coe.Reachable u v\n⊢ ↑↑v ∈ (Subgraph.coeSubgraph G'').verts",
"usedConstants": [
"Iff.mpr",
"RelHom.instFunLike",
"Iff.of_eq",
"congrA... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 49
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nh : ENat.card V ≤ 2\nv : V\np : G.Walk v v\nhp : p.IsCycle\nthis✝ : 3 ≤ p.length\nthis : p.support.tail.length ≤ 2\n⊢ False",
"usedConstants": [
"congrArg",
"HSub.hSub",
"SimpleGraph.Walk.support",
"Eq.mp",
"instSubNat",
"instOfNa... | rw [List.length_tail, p.length_support] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 10
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nh : G.Adj u v\nw : ↑(G.subgraphOfAdj h).verts\np : (G.subgraphOfAdj h).coe.Walk w w\nhp : p.IsCycle\nthis✝ : s(u, v) = s(↑w, ↑p.snd)\nthis : s(u, v) = s(↑p.penultimate, ↑w)\n⊢ False",
"usedConstants": [
"Sym2.Rel",
"Sym2.eq._simp_1",
"and_... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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