module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Colex | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 97
} | {
"line": 335,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : ∃ a ∈ ofColex (toColex t), a ∉ ofColex (toColex s) ∧ ∀ b ∈ ofColex (toColex s), b ∉ ofColex (toColex t) → b < a\nu : Finset α := {w ∈ t \\ s | ∀ a ∈ s, a ∉ t → a < w}\n⊢ ∃ w ∈ t, w ∉ s ∧ ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)",
"pp... | [
"case refine_1\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : ∃ a ∈ ofColex (toColex t), a ∉ ofColex (toColex s) ∧ ∀ b ∈ ofColex (toColex s), b ∉ ofColex (toColex t) → b < a\nu : Finset α := {w ∈ t \\ s | ∀ a ∈ s, a ∉ t → a < w}\nmem_u : ∀ {w : α}, w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w\n⊢ ∃ w ∈ ... | have mem_u {w : α} : w ∈ u ↔ w ∈ t ∧ w ∉ s ∧ ∀ a ∈ s, a ∉ t → a < w := by simp [u, and_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Derangements.Basic | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | {
"line": 126,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n⊢ ∀ (x : Perm α), x ∉ fiber none",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Option.none",
"Equiv.Perm",
"Set.instMembership",
"derangements.Equiv.RemoveNone.fiber",
"Set"
],
"us... | [
"α : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nhyp : f ∈ fiber none\n⊢ False"
] | intro f hyp | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 145,
"column": 30
} | {
"line": 145,
"column": 43
} | {
"line": 145,
"column": 44
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\n⊢ ↑(#(A * A)) < ↑(2 * #A)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"HMul.hMul"... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\n⊢ ↑(#(A * A)) < ↑2 * ↑(#A)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 225,
"column": 30
} | {
"line": 225,
"column": 43
} | {
"line": 225,
"column": 44
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₀ : A.Nonempty\nh₁ : ∀ a ∈ A⁻¹ * A, 1 / 2 * ↑(#A) < ↑(#({xy ∈ A ×ˢ A | xy.1 * xy.2⁻¹ = a}))\nh₂ :\n ∀ x ∈ A ×ˢ A,\n (fun x ↦\n match x with\n | (x, y) => x * y⁻¹)\n x ∈\n ... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₀ : A.Nonempty\nh₁ : ∀ a ∈ A⁻¹ * A, 1 / 2 * ↑(#A) < ↑(#({xy ∈ A ×ˢ A | xy.1 * xy.2⁻¹ = a}))\nh₂ :\n ∀ x ∈ A ×ˢ A,\n (fun x ↦\n match x with\n | (x, y) => x * y⁻¹)\n x ∈\n A⁻¹ * A... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 230,
"column": 2
} | {
"line": 231,
"column": 73
} | {
"line": 233,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nha : a ∈ A\n⊢ A ⊆ a •> (A⁻¹ * A)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"DivInvOneMonoid.to... | [] | rw [← smul_mul_assoc]
exact subset_mul_right _ (by simp [← inv_smul_mem_iff, inv_mem_inv ha]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 230,
"column": 2
} | {
"line": 231,
"column": 73
} | {
"line": 233,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nha : a ∈ A\n⊢ A ⊆ a •> (A⁻¹ * A)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"DivInvOneMonoid.to... | [] | rw [← smul_mul_assoc]
exact subset_mul_right _ (by simp [← inv_smul_mem_iff, inv_mem_inv ha]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Catalan.Basic | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 54
} | {
"line": 110,
"column": 4
} | [
{
"pp": "case ind\nd : ℕ\nhd : ∀ m ≤ d, ↑(catalan m) = ↑m.centralBinom / (↑m + 1)\n⊢ ↑(catalan d.succ) = ↑d.succ.centralBinom / (↑d.succ + 1)",
"ppTerm": "?ind",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"catala... | [
"case ind\nd : ℕ\nhd : ∀ m ≤ d, ↑(catalan m) = ↑m.centralBinom / (↑m + 1)\n⊢ ∑ x, ↑(catalan ↑x) * ↑(catalan (d - ↑x)) = ↑d.succ.centralBinom / (↑d.succ + 1)"
] | simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 274,
"column": 61
} | {
"line": 277,
"column": 38
} | {
"line": 278,
"column": 2
} | [
{
"pp": "n✝ n : ℕ\nih : ∀ m < n + 1, m.bell = ∑ p, p.parts.bell\n⊢ ∑ i ∈ Finset.Iic n, ∑ 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.151",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq... | [] | by
rw [← Nat.range_succ_eq_Iic, Finset.sum_range, ← Fintype.sum_sigma']
refine Fintype.sum_equiv (sigmaPartitionWithPartEquiv n) _ _ ?_
simp [sigmaPartitionWithPartEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 18
} | {
"line": 396,
"column": 2
} | [
{
"pp": "case inr\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))\nA... | [
"case inr\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))\nA_nonempty : ... | let S := A * A⁻¹ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 333,
"column": 8
} | {
"line": 333,
"column": 34
} | {
"line": 333,
"column": 34
} | [
{
"pp": "p q : DyckWord\nh✝ : p ≠ 0\nh : ¬p = 0\n⊢ p.take (p.firstReturn + 1) ⋯ ≠ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"DyckWord.toList_ne_nil",
"DyckWord.count_take_firstReturn_add_one",
"DyckWord",
"id",
"Ne",
... | [
"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 := ⋯ } ≠ []"
] | rw [← toList_ne_nil, take] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 149,
"column": 2
} | {
"line": 154,
"column": 45
} | {
"line": 155,
"column": 2
} | [
{
"pp": "case hg\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nn : ℕ\np : n.Partition\ns : Finset ℕ\nhs : Icc 1 n ⊆ s\nhs0 : 0 ∉ s\n⊢ ∀ x ∈ s \\ p.parts.toFinset, (coeff (p.toFinsuppAntidiag x)) (1 + ∑' (j : ℕ), f x (j + 1) • X ^ (x * (j + 1))) = 1",
... | [
"case hfg\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nn : ℕ\np : n.Partition\ns : Finset ℕ\nhs : Icc 1 n ⊆ s\nhs0 : 0 ∉ s\n⊢ ∀ x ∈ p.parts.toFinset,\n f x (Multiset.count x p.parts) = (coeff (p.toFinsuppAntidiag x)) (1 + ∑' (j : ℕ), f x (j + 1) • X ^ (x ... | · intro x hx
rw [mem_sdiff, Multiset.mem_toFinset] at hx
have hx0 : x ≠ 0 := fun h ↦ hs0 (h ▸ hx.1)
have hsum := (summable_genFun_term' f hx0).map_tsum _
(WithPiTopology.continuous_constantCoeff R)
simp [toFinsuppAntidiag, hsum, hx.2, hx0] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 752,
"column": 4
} | {
"line": 760,
"column": 55
} | {
"line": 762,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nS : Finset G\nε : ℝ\nhε₀ : 0 < ε\nhε₁ : ε ≤ 1\nhS : S.Nonempty\nK : ℝ := 1 - ε / 2\nhK : K < 1\nex : Finset G → ℝ := expansion K S\nκ : ℝ := connectivity K S\nH : Subgroup G\nw✝ : Fintype ↥H\nhH : IsAtom K S (↑H).toFinset\nZ : Finset G\nhZS : Z ⊆ S... | [] | rw [← mul_le_mul_iff_right₀ (a := ε / 2) (by positivity)]
calc
ε / 2 * (Fintype.card H)
_ = ε / 2 * #(H : Set G).toFinset := by
simp only [Set.toFinset_card, SetLike.coe_sort_coe]
_ = (1 - K) * #(H : Set G).toFinset := by ring
_ ≤ ex (Set.toFinset H) := mu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 752,
"column": 4
} | {
"line": 760,
"column": 55
} | {
"line": 762,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nS : Finset G\nε : ℝ\nhε₀ : 0 < ε\nhε₁ : ε ≤ 1\nhS : S.Nonempty\nK : ℝ := 1 - ε / 2\nhK : K < 1\nex : Finset G → ℝ := expansion K S\nκ : ℝ := connectivity K S\nH : Subgroup G\nw✝ : Fintype ↥H\nhH : IsAtom K S (↑H).toFinset\nZ : Finset G\nhZS : Z ⊆ S... | [] | rw [← mul_le_mul_iff_right₀ (a := ε / 2) (by positivity)]
calc
ε / 2 * (Fintype.card H)
_ = ε / 2 * #(H : Set G).toFinset := by
simp only [Set.toFinset_card, SetLike.coe_sort_coe]
_ = (1 - K) * #(H : Set G).toFinset := by ring
_ ≤ ex (Set.toFinset H) := mu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 50
} | {
"line": 341,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx y : α\ne : β\nG H : Graph α β\nhe : G.IsLink e x y\nhHG : H ≤c G\n⊢ [x ∈ V(H), y ∈ V(H), e ∈ E(H)].TFAE",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Iff.mp",
"Graph.vertexSet",
"Set.instMembership",
... | [
"α : Type u_1\nβ : Type u_2\nx y : α\ne : β\nG H : Graph α β\nhe : G.IsLink e x y\nhHG : H ≤c G\ntfae_1_to_2 : x ∈ V(H) → y ∈ V(H)\n⊢ [x ∈ V(H), y ∈ V(H), e ∈ E(H)].TFAE"
] | tfae_have 1 → 2 := (hHG.mem_iff_of_isLink he).mp | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.Combinatorics.Graph.Maps | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 56
} | {
"line": 50,
"column": 4
} | [
{
"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\nh : e ∈ E(G)\n⊢ ∃ x y, Relation.Map (G.IsLink e) f f x y",
"ppTerm": "?m.167",
"assigned": true,
"usedConstants": [
"Graph.exists_is... | [
"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\nh : e ∈ E(G)\nx y : α\nhxy : G.IsLink e x y\n⊢ ∃ x y, Relation.Map (G.IsLink e) f f x y"
] | obtain ⟨x, y, hxy⟩ := exists_isLink_of_mem_edgeSet h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 160,
"column": 2
} | {
"line": 161,
"column": 46
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI D X : Set α\n⊢ (M ＼ D).IsBasis I X ↔ M.IsBasis I X ∧ Disjoint X D",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"and_iff_left_of_imp",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCom... | [] | rw [← restrict_compl, isBasis_restrict_iff, subset_sdiff, ← and_assoc,
and_iff_left_of_imp IsBasis.subset_ground] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 160,
"column": 2
} | {
"line": 161,
"column": 46
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI D X : Set α\n⊢ (M ＼ D).IsBasis I X ↔ M.IsBasis I X ∧ Disjoint X D",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"and_iff_left_of_imp",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCom... | [] | rw [← restrict_compl, isBasis_restrict_iff, subset_sdiff, ← and_assoc,
and_iff_left_of_imp IsBasis.subset_ground] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 160,
"column": 2
} | {
"line": 161,
"column": 46
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI D X : Set α\n⊢ (M ＼ D).IsBasis I X ↔ M.IsBasis I X ∧ Disjoint X D",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"and_iff_left_of_imp",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCom... | [] | rw [← restrict_compl, isBasis_restrict_iff, subset_sdiff, ← and_assoc,
and_iff_left_of_imp IsBasis.subset_ground] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 28
} | {
"line": 182,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nB D : Set α\nhD : M.Coindep D\nh : M.IsBasis B (M.E \\ D)\nhss : B ⊆ M.E \\ D\n⊢ M.IsBase B ∧ Disjoint B D",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"congrArg",
"Matroid.E",
"Disjoint",
"SemilatticeInf.t... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nB D : Set α\nhD : M.Coindep D\nh : M.IsBasis B (M.E \\ D)\nhss : B ⊆ M.E ∧ Disjoint B D\n⊢ M.IsBase B ∧ Disjoint B D"
] | rw [subset_sdiff] at hss | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 54
} | {
"line": 240,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nD X : Set α\n⊢ M.closure (X \\ D) \\ D ⊆ M.E",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.E",
"Matroid.closure_subset_ground",
"PartialOrder.toPreorder",
"Chai... | [] | exact sdiff_subset.trans (M.closure_subset_ground _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 85,
"column": 83
} | {
"line": 86,
"column": 19
} | {
"line": 88,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC₁ C₂ : Set α\n⊢ M ／ C₁ ／ C₂ = M ／ C₂ ／ C₁",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.instUnion",
"Set.union_comm",
"congr",
"True",
"eq_self",
"of_eq_true",
"Matroid.contract_contrac... | [] | by
simp [union_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 57
} | {
"line": 181,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.IsBasis I X\nJ : Set α\nhJ : J ⊆ (M✶ ＼ I ＼ (X \\ I)).E\ne : α\nhe : e ∈ X \\ I\n⊢ e ∈ M.closure X \\ I",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"Set.sdiff_subset_sdiff_left",
"Matroid.IsBasis.subset_ground",
... | [] | exact sdiff_subset_sdiff_left (M.subset_closure X) he | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 61
} | {
"line": 366,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.loops\n⊢ M ／ X = M ＼ X",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"LE.le",
"Matroid.contract_empty",
"SDiff.sdiff",
"Set.instLE",
"Set.sdiff_empty",
"congr... | [] | simp [(empty_isBasis_iff.2 hX).contract_eq_contract_delete] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 61
} | {
"line": 366,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.loops\n⊢ M ／ X = M ＼ X",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"LE.le",
"Matroid.contract_empty",
"SDiff.sdiff",
"Set.instLE",
"Set.sdiff_empty",
"congr... | [] | simp [(empty_isBasis_iff.2 hX).contract_eq_contract_delete] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 61
} | {
"line": 366,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.loops\n⊢ M ／ X = M ＼ X",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"LE.le",
"Matroid.contract_empty",
"SDiff.sdiff",
"Set.instLE",
"Set.sdiff_empty",
"congr... | [] | simp [(empty_isBasis_iff.2 hX).contract_eq_contract_delete] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 85,
"column": 4
} | {
"line": 86,
"column": 31
} | {
"line": 87,
"column": 4
} | [
{
"pp": "case inr\nι : 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).I... | [
"case inr\nι : 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 (inser... | rw [preimage_singleton_eq_empty.2 (by simpa), preimage_singleton_eq_empty.2 (by simpa),
sdiff_empty, union_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 52
} | {
"line": 488,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nK : Set α\nhK : M.IsCircuit K\nC : Set α\nhKC : ¬K ⊆ C\nh' : (M ／ (C ∩ K)).IsCircuit (K \\ C)\nhwin : (M ／ (C ∩ K) ／ (C \\ K)).Dep (K \\ C)\n⊢ (M ／ C).Dep (K \\ C)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Matroid.Dep",
"congrArg",
... | [] | rwa [contract_contract, inter_union_sdiff] at hwin | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 540,
"column": 23
} | {
"line": 540,
"column": 46
} | {
"line": 540,
"column": 46
} | [
{
"pp": "α : Type u_1\nR C : Set α\nM : Matroid α\nh : Disjoint C R\n⊢ (M ／ C ↾ R).E = ((M ↾ (R ∪ C)) ／ C).E",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.union_sdiff_right",
"Disjoint.sdiff_eq_right",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congr... | [] | simp [h.sdiff_eq_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 540,
"column": 23
} | {
"line": 540,
"column": 46
} | {
"line": 540,
"column": 46
} | [
{
"pp": "α : Type u_1\nR C : Set α\nM : Matroid α\nh : Disjoint C R\n⊢ (M ／ C ↾ R).E = ((M ↾ (R ∪ C)) ／ C).E",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.union_sdiff_right",
"Disjoint.sdiff_eq_right",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congr... | [] | simp [h.sdiff_eq_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 540,
"column": 23
} | {
"line": 540,
"column": 46
} | {
"line": 540,
"column": 46
} | [
{
"pp": "α : Type u_1\nR C : Set α\nM : Matroid α\nh : Disjoint C R\n⊢ (M ／ C ↾ R).E = ((M ↾ (R ∪ C)) ／ C).E",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.union_sdiff_right",
"Disjoint.sdiff_eq_right",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congr... | [] | simp [h.sdiff_eq_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 146,
"column": 8
} | {
"line": 146,
"column": 61
} | {
"line": 147,
"column": 8
} | [
{
"pp": "case pos\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nb : ι → MvPolynomial σ R\nhb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))\nf : MvPolynomial σ R\ni : ι\nhb0 : m.degree (b i) = 0\nj : ι\nhj : j = i\n⊢ m.toSyn (m.degree (b j) + m.degree ((Finsupp.single i (⋯.unit... | [
"case pos\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nb : ι → MvPolynomial σ R\nhb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))\nf : MvPolynomial σ R\ni : ι\nhb0 : m.degree (b i) = 0\nj : ι\nhj : j = i\n⊢ m.toSyn (m.degree (⋯.unit⁻¹ • f)) ≤ m.toSyn (m.degree f)"
] | simp only [hj, hb0, Finsupp.single_eq_same, zero_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 109,
"column": 6
} | {
"line": 125,
"column": 23
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case hcard\nR : Type u_1\ninst✝³ : CommRing R\nσ✝ : Type u_2\ninst✝² : Finite σ✝\ninst✝¹ : IsDomain R\nσ : Type u_2\ninst✝ : Fintype σ\nh :\n ∀ (P : MvPolynomial σ R) (S : σ → Finset R),\n (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0\nP : MvP... | [] | · apply lt_of_le_of_lt _ (Hdeg none)
rw [Polynomial.natDegree_le_iff_coeff_eq_zero]
intro d hd
simp only [hQ]
rw [MvPolynomial.coeff_eval_eq_eval_coeff]
convert! map_zero (MvPolynomial.eval x)
ext m
simp only [coeff_zero]
set n := (embDomain Function.Embed... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 215,
"column": 15
} | {
"line": 215,
"column": 79
} | {
"line": 215,
"column": 79
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nσ : Type u_2\ninst✝¹ : Finite σ\ninst✝ : IsDomain R\nS : σ → Finset R\nSne : ∀ (i : σ), (S i).Nonempty\nf : MvPolynomial σ R\nHeval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0\nthis : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel\ni : σ\n⊢ IsUn... | [] | by simp only [(Alon.monic_P ..).leadingCoeff_eq_one, isUnit_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 222,
"column": 47
} | {
"line": 222,
"column": 59
} | {
"line": 222,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nσ : Type u_2\ninst✝¹ : Finite σ\ninst✝ : IsDomain R\nS : σ → Finset R\nSne : ∀ (i : σ), (S i).Nonempty\nf : MvPolynomial σ R\nHeval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0\nthis : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel\nh : σ →₀ MvPo... | [] | simp [Sne i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 222,
"column": 47
} | {
"line": 222,
"column": 59
} | {
"line": 222,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nσ : Type u_2\ninst✝¹ : Finite σ\ninst✝ : IsDomain R\nS : σ → Finset R\nSne : ∀ (i : σ), (S i).Nonempty\nf : MvPolynomial σ R\nHeval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0\nthis : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel\nh : σ →₀ MvPo... | [] | simp [Sne i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 222,
"column": 47
} | {
"line": 222,
"column": 59
} | {
"line": 222,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nσ : Type u_2\ninst✝¹ : Finite σ\ninst✝ : IsDomain R\nS : σ → Finset R\nSne : ∀ (i : σ), (S i).Nonempty\nf : MvPolynomial σ R\nHeval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) f = 0\nthis : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel\nh : σ →₀ MvPo... | [] | simp [Sne i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path.Decomposition | {
"line": 29,
"column": 40
} | {
"line": 44,
"column": 32
} | {
"line": 46,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nS : Set V\nha_not_in_S : ¬a ∈ S\nhb_in_S : b ∈ S\n⊢ ∃ u, ¬u ∈ S ∧ ∃ v, v ∈ S ∧ ∃ e p₁ p₂, p = p₁.comp (e.toPath.comp p₂)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMon... | [] | by
induction h_len : p.length generalizing a b S ha_not_in_S hb_in_S with
| zero =>
obtain rfl := eq_of_length_zero p h_len
exact (ha_not_in_S hb_in_S).elim
| succ n ih =>
have h_pos : 0 < p.length := by simp [h_len]
obtain ⟨c, p', e, rfl⟩ := (length_ne_zero_iff_eq_cons p).mp h_pos.ne'
by_case... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 188,
"column": 6
} | {
"line": 189,
"column": 48
} | {
"line": 190,
"column": 4
} | [
{
"pp": "case cons.inl\nV : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nb✝ c : V\np' : Path a b✝\ne : b✝ ⟶ c\nih : ∀ {n : ℕ}, n ≤ p'.length → ∃ v p₁ p₂, p' = p₁.comp p₂ ∧ p₁.length = n\nn : ℕ\nhn : n ≤ p'.length + 1\nh : n ≤ p'.length\n⊢ ∃ v p₁ p₂, p'.cons e = p₁.comp p₂ ∧ p₁.length = n",
"ppTerm": "... | [] | obtain ⟨d, p₁, p₂, hp, hl⟩ := ih h
exact ⟨d, p₁, p₂.cons e, by simp [hp], hl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 188,
"column": 6
} | {
"line": 189,
"column": 48
} | {
"line": 190,
"column": 4
} | [
{
"pp": "case cons.inl\nV : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nb✝ c : V\np' : Path a b✝\ne : b✝ ⟶ c\nih : ∀ {n : ℕ}, n ≤ p'.length → ∃ v p₁ p₂, p' = p₁.comp p₂ ∧ p₁.length = n\nn : ℕ\nhn : n ≤ p'.length + 1\nh : n ≤ p'.length\n⊢ ∃ v p₁ p₂, p'.cons e = p₁.comp p₂ ∧ p₁.length = n",
"ppTerm": "... | [] | obtain ⟨d, p₁, p₂, hp, hl⟩ := ih h
exact ⟨d, p₁, p₂.cons e, by simp [hp], hl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 92,
"column": 33
} | {
"line": 92,
"column": 93
} | {
"line": 93,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nthis :\n ∀ {x : ℕ},\n ∀ s ∈ powersetCard x univ,\n ↑(card α) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) = ↑(card α) / ((↑(card α) - ↑x) * ↑((card α).choose x))\n⊢ ↑(card α) * ∑ i ∈ range (card α + 1), ↑((card α).choose i) / ((↑(card α)... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nthis :\n ∀ {x : ℕ},\n ∀ s ∈ powersetCard x univ,\n ↑(card α) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) = ↑(card α) / ((↑(card α) - ↑x) * ↑((card α).choose x))\n⊢ ↑(card α) * ∑ i ∈ range (card α + 1), ↑((card α).choose i) / ((↑(card α) - ↑i) * ↑((... | ← sum_insert (f := fun x : ℕ ↦ (x⁻¹ : ℚ)) notMem_range_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 106,
"column": 39
} | {
"line": 106,
"column": 79
} | {
"line": 106,
"column": 80
} | [
{
"pp": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableLE α\na b : α\n⊢ b \\ (b \\ a) ⊔ a \\ (b \\ a) = a",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"disjoint_sdiff_self_right",
"Lattice.toSemilatticeSu... | [
"α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableLE α\na b : α\n⊢ b \\ (b \\ a) ⊔ a = a"
] | disjoint_sdiff_self_right.sdiff_eq_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 140,
"column": 4
} | {
"line": 143,
"column": 28
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\na b : α\nhb : b ∈ s ∧ compress u v b ∉ s\nhas : Disjoint u a ∧ v ≤ a\nha : a ∈ s ∧ (a ⊔ u) \\ v ∉ s\nhab : (a ⊔ u) \\ v = compress u v b\n⊢... | [] | rw [compress] at hb hab
split_ifs at hb hab with hbs
· exact sup_sdiff_injOn u v has hbs hab
· exact (hb.2 hb.1).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 140,
"column": 4
} | {
"line": 143,
"column": 28
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\na b : α\nhb : b ∈ s ∧ compress u v b ∉ s\nhas : Disjoint u a ∧ v ≤ a\nha : a ∈ s ∧ (a ⊔ u) \\ v ∉ s\nhab : (a ⊔ u) \\ v = compress u v b\n⊢... | [] | rw [compress] at hb hab
split_ifs at hb hab with hbs
· exact sup_sdiff_injOn u v has hbs hab
· exact (hb.2 hb.1).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 140,
"column": 2
} | {
"line": 143,
"column": 28
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\na b : α\nhb : b ∈ s ∧ compress u v b ∉ s\nhas : Disjoint u a ∧ v ≤ a\nha : a ∈ s ∧ (a ⊔ u) \\ v ∉ s\nhab : (a ⊔ u) \\ v = compress u v b\n⊢... | [
"case neg\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\na b : α\nhb : b ∈ s ∧ compress u v b ∉ s\nhas : ¬(Disjoint u a ∧ v ≤ a)\nha : a ∈ s ∧ a ∉ s\nhab : a = compress u v b\n⊢ a = b"
] | · rw [compress] at hb hab
split_ifs at hb hab with hbs
· exact sup_sdiff_injOn u v has hbs hab
· exact (hb.2 hb.1).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 191,
"column": 71
} | {
"line": 194,
"column": 31
} | {
"line": 196,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ninst✝ : DecidableEq α\nu v : α\ns : Finset α\n⊢ #(𝓒 u v s) = #s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"UV.compress_injOn",
"Eq.mpr",
"instDecida... | [] | by
rw [compression, card_union_of_disjoint compress_disjoint, filter_image,
card_image_of_injOn compress_injOn, ← card_union_of_disjoint (disjoint_filter_filter_not s _ _),
filter_union_filter_not_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 394,
"column": 73
} | {
"line": 394,
"column": 92
} | {
"line": 395,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : Nonempty α\nhs : s ≠ univ\nthis :\n ∀ (t : Finset α),\n (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =\n if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card ... | [
"α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : Nonempty α\nhs : s ≠ univ\nthis :\n ∀ (t : Finset α),\n (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =\n if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t... | filter_subset_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 28
} | {
"line": 420,
"column": 2
} | [
{
"pp": "case ind.inr.zero\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\n𝒜 : Finset (Finset α)\nh𝒜₁ : 𝒜.Nonempty\nh𝒜₂ : univ ∉ 𝒜\nh𝒜₃ : 𝒜.Nontrivial\nhm : 0 = #𝒜\nih : ∀ (𝒜 : Finset (Finset α)), #𝒜 < 0 → 𝒜.Nonempty → univ ∉ 𝒜 → supSum 𝒜 = ↑(card α) * ∑ k ∈ range (ca... | [] | cases h𝒜₁.card_pos.ne hm | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 28
} | {
"line": 420,
"column": 2
} | [
{
"pp": "case ind.inr.zero\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\n𝒜 : Finset (Finset α)\nh𝒜₁ : 𝒜.Nonempty\nh𝒜₂ : univ ∉ 𝒜\nh𝒜₃ : 𝒜.Nontrivial\nhm : 0 = #𝒜\nih : ∀ (𝒜 : Finset (Finset α)), #𝒜 < 0 → 𝒜.Nonempty → univ ∉ 𝒜 → supSum 𝒜 = ↑(card α) * ∑ k ∈ range (ca... | [] | cases h𝒜₁.card_pos.ne hm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 28
} | {
"line": 420,
"column": 2
} | [
{
"pp": "case ind.inr.zero\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\n𝒜 : Finset (Finset α)\nh𝒜₁ : 𝒜.Nonempty\nh𝒜₂ : univ ∉ 𝒜\nh𝒜₃ : 𝒜.Nontrivial\nhm : 0 = #𝒜\nih : ∀ (𝒜 : Finset (Finset α)), #𝒜 < 0 → 𝒜.Nonempty → univ ∉ 𝒜 → supSum 𝒜 = ↑(card α) * ∑ k ∈ range (ca... | [] | cases h𝒜₁.card_pos.ne hm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 315,
"column": 6
} | {
"line": 317,
"column": 40
} | {
"line": 318,
"column": 6
} | [
{
"pp": "case pos\nα : 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\nH : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ ∂ 𝒜 ∧ (s ∪ v) \\ u ∉ ∂ 𝒜'\nt : Finset α\nHt :... | [
"case pos\nα : 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\nH : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ ∂ 𝒜 ∧ (s ∪ v) \\ u ∉ ∂ 𝒜'\nt : Finset α\nHt : t ∈ 𝒜'\na ... | have hsb :=
sup_sdiff_mem_of_mem_compression ht ((erase_subset _ _).trans hvt)
(disjoint_erase_comm.2 huvs.1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SetFamily.HarrisKleitman | {
"line": 124,
"column": 9
} | {
"line": 124,
"column": 31
} | {
"line": 124,
"column": 32
} | [
{
"pp": "case h₂\nα : Type u_1\ninst✝¹ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ninst✝ : Fintype α\nh𝒜 : IsLowerSet ↑𝒜ᶜ\nhℬ : IsUpperSet ↑ℬ\nthis : 2 ^ Fintype.card α * #(𝒜ᶜ ∩ ℬ) ≤ 2 ^ Fintype.card α * #ℬ - #𝒜 * #ℬ\n⊢ #𝒜 * #ℬ ≤ 2 ^ Fintype.card α * #ℬ",
"ppTerm": "?h₂✝",
"assigned": true,
"use... | [
"case h₂\nα : Type u_1\ninst✝¹ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ninst✝ : Fintype α\nh𝒜 : IsLowerSet ↑𝒜ᶜ\nhℬ : IsUpperSet ↑ℬ\nthis : 2 ^ Fintype.card α * #(𝒜ᶜ ∩ ℬ) ≤ 2 ^ Fintype.card α * #ℬ - #𝒜 * #ℬ\n⊢ #𝒜 * #ℬ ≤ Fintype.card (Finset α) * #ℬ"
] | ← Fintype.card_finset, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 85
} | {
"line": 84,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\ncards' : #(s.erase (s.min' hs)) = #t\nk : α\nhks : k ∈ s.erase (s.min' hs)\nhkt : k ∉ t\nz : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))\nj : α := tᶜ.min' ⋯\nhjk : j ≤ k\nthis : j ∉ t\n⊢ #s = ... | [] | rw [card_insert_of_notMem ‹j ∉ t›, ← ‹_ = #t›, card_erase_add_one (min'_mem _ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 33
} | {
"line": 109,
"column": 2
} | [
{
"pp": "case e'_4\n𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : #𝒜 * (r + 1) ≤ #(∂ 𝒜) * (Fintype.card α - r)\n⊢ (Finty... | [] | exact Or.inl (mul_comm _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 101,
"column": 2
} | {
"line": 108,
"column": 33
} | {
"line": 109,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : #𝒜 * r ≤ #(∂ 𝒜) * (Fintype.card α - r + 1)\n⊢ #𝒜 * (Fintype.ca... | [
"case inr.hb\n𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r ≠ 0\nhr' : r ≤ Fintype.card α\nh𝒜 : #𝒜 * r ≤ #(∂ 𝒜) * (Fintype.card α - r + 1)\n⊢ 0 < (Fintype.card α).choos... | · rcases r with - | r
· exact (hr rfl).elim
rw [tsub_add_eq_add_tsub hr', add_tsub_add_eq_tsub_right] at h𝒜
apply le_of_mul_le_mul_right _ (pos_iff_ne_zero.2 hr)
convert! Nat.mul_le_mul_right ((Fintype.card α).choose r) h𝒜 using 1
· simpa [mul_assoc, Nat.choose_succ_right_eq] using Or.inl (mul_com... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 168,
"column": 8
} | {
"line": 168,
"column": 23
} | {
"line": 168,
"column": 23
} | [
{
"pp": "case pos\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⦄, ... | [
"case pos\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⦄, s ⊆ insert a... | collapse_eq hat | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 500,
"column": 92
} | {
"line": 503,
"column": 53
} | {
"line": 505,
"column": 0
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nφ : G ≃g G'\nv : V\nC : G.ConnectedComponent\n⊢ G'.connectedComponentMk (φ v) = map (RelIso.toRelEmbedding φ).toRelHom C ↔ G.connectedComponentMk v = C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGra... | [] | by
refine C.ind fun u => ?_
simp only [Iso.reachable_iff, ConnectedComponent.map_mk, RelEmbedding.coe_toRelHom,
RelIso.coe_toRelEmbedding, ConnectedComponent.eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 507,
"column": 85
} | {
"line": 510,
"column": 57
} | {
"line": 512,
"column": 0
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nφ : G ≃g G'\nv' : V'\nC : G.ConnectedComponent\n⊢ G.connectedComponentMk (φ.symm v') = C ↔ G'.connectedComponentMk v' = map (RelIso.toRelEmbedding φ).toRelHom C",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"... | [] | by
refine C.ind fun u => ?_
simp only [Iso.symm_apply_reachable, ConnectedComponent.eq, ConnectedComponent.map_mk,
RelEmbedding.coe_toRelHom, RelIso.coe_toRelEmbedding] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Setoid.Partition | {
"line": 502,
"column": 2
} | {
"line": 503,
"column": 30
} | {
"line": 505,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs : IndexedPartition s\nβ : Type u_3\nf : ι → α → β\nt : Set β\nx : α\nx✝ : x ∈ ⋃ i, s i ∩ f i ⁻¹' t\na : Set α\ni : ι\nhi : (fun i ↦ s i ∩ f i ⁻¹' t) i = a\nha : x ∈ a\n⊢ x ∈ hs.piecewise f ⁻¹' t",
"ppTerm": "?refine_2",
"assigned": tr... | [] | · rw [← hi, ← (mem_iff_index_eq hs).mp ha.1] at ha
simp_all [piecewise_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 53
} | {
"line": 139,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nC : G.Coloring α\ninst✝¹ : Fintype V\ninst✝ : Fintype α\n⊢ Fintype (G.Coloring α)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"id",
"SimpleGraph.Coloring",
"Fintype"
],
"us... | [
"V : Type u\nG : SimpleGraph V\nn : ℕ\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nC : G.Coloring α\ninst✝¹ : Fintype V\ninst✝ : Fintype α\n⊢ Fintype (G.Adj →r (completeGraph α).Adj)"
] | change Fintype (RelHom G.Adj (completeGraph α).Adj) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 202,
"column": 6
} | {
"line": 204,
"column": 38
} | {
"line": 205,
"column": 4
} | [
{
"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⦄, ... | [] | exact (h ‹_› ‹_›).trans <| mul_le_mul (le_collapse_of_mem ‹_› h₃ rfl <|
inter_mem_infs ‹_› ‹_›) (le_collapse_of_mem ‹_› h₄ rfl <| union_mem_sups ‹_› ‹_›)
(h₄ _) <| collapse_nonneg h₃ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 23
} | {
"line": 361,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable (↑(Nat.find ?m.12)).toNat",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"setOf",
"Classical.propDecidable",
"Membership.mem",
"Nat",
"SimpleGraph.Colorable",
"Set.instMemb... | [] | apply Nat.find_spec | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 23
} | {
"line": 361,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable (↑(Nat.find ?m.12)).toNat",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"setOf",
"Classical.propDecidable",
"Membership.mem",
"Nat",
"SimpleGraph.Colorable",
"Set.instMemb... | [] | apply Nat.find_spec | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 23
} | {
"line": 361,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable (↑(Nat.find ?m.12)).toNat",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"setOf",
"Classical.propDecidable",
"Membership.mem",
"Nat",
"SimpleGraph.Colorable",
"Set.instMemb... | [] | apply Nat.find_spec | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 52
} | {
"line": 499,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nhadj : G.Adj u v\nh : G.chromaticNumber < 2\nc : G.Coloring (Fin 1)\n⊢ False",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"RelHom.instFunLike",
"SimpleGraph.Adj",
"SimpleGraph.completeGraph",
"instOfNatNat",
... | [] | exact c.valid hadj (Subsingleton.elim (c u) (c v)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 353,
"column": 4
} | {
"line": 357,
"column": 15
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case inl\nn : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\nb : r = 0\n⊢ #𝒜 ≤ (n - 1).choose (r - 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"Nat.choose",
"Finset.dis... | [] | convert! Nat.zero_le _
rw [Finset.card_eq_zero, eq_empty_iff_forall_notMem]
refine fun A HA ↦ h𝒜 HA HA ?_
rw [disjoint_self_iff_empty, ← Finset.card_eq_zero, ← b]
exact h₂ HA | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 353,
"column": 4
} | {
"line": 357,
"column": 15
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case inl\nn : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\nb : r = 0\n⊢ #𝒜 ≤ (n - 1).choose (r - 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"Nat.choose",
"Finset.dis... | [] | convert! Nat.zero_le _
rw [Finset.card_eq_zero, eq_empty_iff_forall_notMem]
refine fun A HA ↦ h𝒜 HA HA ?_
rw [disjoint_self_iff_empty, ← Finset.card_eq_zero, ← b]
exact h₂ HA | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 40
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case refine_2.inl\nV : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\nhk : k ≠ 0\nh : ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeReachable k u v\nhs : ∅.encard < ↑(k + 1)\n⊢ (G.deleteEdges ∅).Reachable u v",
"ppTerm": "?refine_2.inl",
"assigned": true,
"usedConstants": [
"SimpleGraph.del... | [
"case refine_2.inr\nV : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\nhk : k ≠ 0\nh : ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeReachable k u v\ns : Set (Sym2 V)\nhs : s.encard < ↑(k + 1)\ne : Sym2 V\nhe : e ∈ s\n⊢ (G.deleteEdges s).Reachable u v"
] | · simpa using (h s(u, u)).reachable hk | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 230,
"column": 12
} | {
"line": 230,
"column": 36
} | {
"line": 232,
"column": 0
} | [
{
"pp": "case cons.e_a.verts\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.reverse.toSubgraph = p✝.toSubgraph\nx✝ : V\n⊢ x✝ ∈ (G.subgraphOfAdj h✝ ⊔ G.singletonSubgraph u✝).verts ↔ x✝ ∈ (G.subgraphOfAdj h✝).verts",
"ppTerm": "?cons.e_a.verts",
"assigned... | [] | simp [-Set.bot_eq_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 230,
"column": 12
} | {
"line": 230,
"column": 36
} | {
"line": 232,
"column": 0
} | [
{
"pp": "case cons.e_a.Adj\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.reverse.toSubgraph = p✝.toSubgraph\nx✝¹ x✝ : V\n⊢ (G.subgraphOfAdj h✝ ⊔ G.singletonSubgraph u✝).Adj x✝¹ x✝ ↔ (G.subgraphOfAdj h✝).Adj x✝¹ x✝",
"ppTerm": "?cons.e_a.Adj",
"assigned... | [] | simp [-Set.bot_eq_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 167,
"column": 38
} | {
"line": 167,
"column": 85
} | {
"line": 168,
"column": 4
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nh : 2 < G.edist u v\nhn : u ≠ v\nhc : G.Adj u v\n⊢ False",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.not_ofNat_lt_one._simp_1",
"False",
"instCompleteLinearOrderENat",
"instCharZeroENat",
... | [] | simp +decide [edist_eq_one_iff_adj.mpr hc] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 167,
"column": 38
} | {
"line": 167,
"column": 85
} | {
"line": 168,
"column": 4
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nh : 2 < G.edist u v\nhn : u ≠ v\nhc : G.Adj u v\n⊢ False",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.not_ofNat_lt_one._simp_1",
"False",
"instCompleteLinearOrderENat",
"instCharZeroENat",
... | [] | simp +decide [edist_eq_one_iff_adj.mpr hc] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 167,
"column": 38
} | {
"line": 167,
"column": 85
} | {
"line": 168,
"column": 4
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nh : 2 < G.edist u v\nhn : u ≠ v\nhc : G.Adj u v\n⊢ False",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.not_ofNat_lt_one._simp_1",
"False",
"instCompleteLinearOrderENat",
"instCharZeroENat",
... | [] | simp +decide [edist_eq_one_iff_adj.mpr hc] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 332,
"column": 4
} | {
"line": 334,
"column": 45
} | {
"line": 335,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nhp : p.length = G.dist u v\n⊢ p.length ≤ p.bypass.length",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Trans.trans",
"SimpleGraph.dist",
"SimpleGraph.Walk.length",
"Classical.propDecidable",
... | [] | calc p.length
_ = G.dist u v := hp
_ ≤ p.bypass.length := dist_le p.bypass | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 360,
"column": 2
} | {
"line": 367,
"column": 5
} | {
"line": 369,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v u' v' : V\np₁ : G.Walk u v\np₂ : G.Walk u' v'\nh₁ : p₁.length = G.dist u v\nh₂ : p₂.IsSubwalk p₁\n⊢ p₂.length = G.dist u' v'",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"SimpleGraph.dist",
"... | [] | refine (dist_le _).eq_of_not_lt' fun hh ↦ ?_
obtain ⟨ru, rv, h⟩ := h₂
obtain ⟨s, _⟩ := p₂.reachable.exists_path_of_dist
let r := ru.append s |>.append rv
have : p₁.length = ru.length + p₂.length + rv.length := by simp [h]
have : r.length = ru.length + s.length + rv.length := by simp [r]
have := dist_le r
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 360,
"column": 2
} | {
"line": 367,
"column": 5
} | {
"line": 369,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v u' v' : V\np₁ : G.Walk u v\np₂ : G.Walk u' v'\nh₁ : p₁.length = G.dist u v\nh₂ : p₂.IsSubwalk p₁\n⊢ p₂.length = G.dist u' v'",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"SimpleGraph.dist",
"... | [] | refine (dist_le _).eq_of_not_lt' fun hh ↦ ?_
obtain ⟨ru, rv, h⟩ := h₂
obtain ⟨s, _⟩ := p₂.reachable.exists_path_of_dist
let r := ru.append s |>.append rv
have : p₁.length = ru.length + p₂.length + rv.length := by simp [h]
have : r.length = ru.length + s.length + rv.length := by simp [r]
have := dist_le r
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 263,
"column": 21
} | {
"line": 263,
"column": 73
} | {
"line": 263,
"column": 74
} | [
{
"pp": "case inl\nU : Type u_1\nU' : Type u_2\nV : Type u_3\nV' : Type u_4\nW : Type u_5\nW' : Type u_6\nγ : Type u_7\nG : SimpleGraph V\nH : SimpleGraph W\nI : SimpleGraph U\nG' : SimpleGraph V'\nH' : SimpleGraph W'\nI' : SimpleGraph U'\nv✝ v' : V\nw w' : W\ninst✝³ : DecidableEq V\ninst✝² : DecidableEq W\nins... | [
"case inl\nU : Type u_1\nU' : Type u_2\nV : Type u_3\nV' : Type u_4\nW : Type u_5\nW' : Type u_6\nγ : Type u_7\nG : SimpleGraph V\nH : SimpleGraph W\nI : SimpleGraph U\nG' : SimpleGraph V'\nH' : SimpleGraph W'\nI' : SimpleGraph U'\nv✝ v' : V\nw w' : W\ninst✝³ : DecidableEq V\ninst✝² : DecidableEq W\ninst✝¹ : G.Loca... | simp only [neighborSet_sum_inl, neighborSet_sum_inr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 263,
"column": 21
} | {
"line": 263,
"column": 73
} | {
"line": 263,
"column": 74
} | [
{
"pp": "case inr\nU : Type u_1\nU' : Type u_2\nV : Type u_3\nV' : Type u_4\nW : Type u_5\nW' : Type u_6\nγ : Type u_7\nG : SimpleGraph V\nH : SimpleGraph W\nI : SimpleGraph U\nG' : SimpleGraph V'\nH' : SimpleGraph W'\nI' : SimpleGraph U'\nv v' : V\nw✝ w' : W\ninst✝³ : DecidableEq V\ninst✝² : DecidableEq W\nins... | [
"case inr\nU : Type u_1\nU' : Type u_2\nV : Type u_3\nV' : Type u_4\nW : Type u_5\nW' : Type u_6\nγ : Type u_7\nG : SimpleGraph V\nH : SimpleGraph W\nI : SimpleGraph U\nG' : SimpleGraph V'\nH' : SimpleGraph W'\nI' : SimpleGraph U'\nv v' : V\nw✝ w' : W\ninst✝³ : DecidableEq V\ninst✝² : DecidableEq W\ninst✝¹ : G.Loca... | simp only [neighborSet_sum_inl, neighborSet_sum_inr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Hasse | {
"line": 82,
"column": 65
} | {
"line": 82,
"column": 80
} | {
"line": 82,
"column": 81
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\nx y : α × β\n⊢ (x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ x.2 ⋖ y.2 ∧ x.1 = y.1) ∨ y.1 ⋖ x.1 ∧ y.2 = x.2 ∨ y.2 ⋖ x.2 ∧ y.1 = x.1 ↔\n (x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ y.1 ⋖ x.1 ∧ x.2 = y.2) ∨ x.2 ⋖ y.2 ∧ x.1 = y.1 ∨ y.2 ⋖ x.2 ∧ x.1 = y.1",
"ppTer... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : PartialOrder β\nx y : α × β\n⊢ (x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ x.2 ⋖ y.2 ∧ x.1 = y.1) ∨ y.1 ⋖ x.1 ∧ y.2 = x.2 ∨ y.2 ⋖ x.2 ∧ x.1 = y.1 ↔\n (x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ y.1 ⋖ x.1 ∧ x.2 = y.2) ∨ x.2 ⋖ y.2 ∧ x.1 = y.1 ∨ y.2 ⋖ x.2 ∧ x.1 = y.1"
] | @eq_comm _ y.1, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 88,
"column": 2
} | {
"line": 96,
"column": 88
} | {
"line": 98,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ pathGraph n ≤ cycleGraph n",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Fin.instSub",
"congrArg",
"SimpleGraph.pathGraph_adj",
"SimpleGraph.cycleGraph_adj'",
"true_or",
"SimpleGraph.Adj",
"Parti... | [] | match n with
| 0 | 1 => simp
| n + 2 =>
intro u v h
rw [pathGraph_adj] at h
rw [cycleGraph_adj']
cases h with
| inl h | inr h =>
simp [Fin.coe_sub_iff_le.mpr (Nat.lt_of_succ_le h.le).le, Nat.eq_sub_of_add_eq' h] | Lean.Elab.Tactic.evalMatch | Lean.Parser.Tactic.match |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 88,
"column": 2
} | {
"line": 96,
"column": 88
} | {
"line": 98,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ pathGraph n ≤ cycleGraph n",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Fin.instSub",
"congrArg",
"SimpleGraph.pathGraph_adj",
"SimpleGraph.cycleGraph_adj'",
"true_or",
"SimpleGraph.Adj",
"Parti... | [] | match n with
| 0 | 1 => simp
| n + 2 =>
intro u v h
rw [pathGraph_adj] at h
rw [cycleGraph_adj']
cases h with
| inl h | inr h =>
simp [Fin.coe_sub_iff_le.mpr (Nat.lt_of_succ_le h.le).le, Nat.eq_sub_of_add_eq' h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 88,
"column": 2
} | {
"line": 96,
"column": 88
} | {
"line": 98,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ pathGraph n ≤ cycleGraph n",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Fin.instSub",
"congrArg",
"SimpleGraph.pathGraph_adj",
"SimpleGraph.cycleGraph_adj'",
"true_or",
"SimpleGraph.Adj",
"Parti... | [] | match n with
| 0 | 1 => simp
| n + 2 =>
intro u v h
rw [pathGraph_adj] at h
rw [cycleGraph_adj']
cases h with
| inl h | inr h =>
simp [Fin.coe_sub_iff_le.mpr (Nat.lt_of_succ_le h.le).le, Nat.eq_sub_of_add_eq' h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Circulant | {
"line": 43,
"column": 2
} | {
"line": 46,
"column": 38
} | {
"line": 48,
"column": 0
} | [
{
"pp": "case mpr\nG : Type u_1\ninst✝ : AddGroup G\ns : Set G\nu v : G\nh : u ≠ v\n⊢ u - v ∈ s \\ {0} ∨ v - u ∈ s \\ {0} → u - v ∈ s ∨ v - u ∈ s",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"HSub.hSub",
"Membership.mem",
"Set.instS... | [] | · intro h1
cases h1 with
| inl h1 => exact Or.inl h1.left
| inr h1 => exact Or.inr h1.left | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 409,
"column": 47
} | {
"line": 409,
"column": 73
} | {
"line": 410,
"column": 2
} | [
{
"pp": "case zero.inl\nα : Type u_1\nV : Type u_2\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\ninst✝ : Semiring α\nu : V\n⊢ (adjMatrix α G ^ 0) u u = ↑(#(G.finsetWalkLength 0 u u))",
"ppTerm": "?zero.inl",
"assigned": true,
"usedConstants": [
"d... | [] | simp [finsetWalkLength, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 409,
"column": 47
} | {
"line": 409,
"column": 73
} | {
"line": 410,
"column": 2
} | [
{
"pp": "case zero.inr\nα : Type u_1\nV : Type u_2\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\ninst✝ : Semiring α\nu v : V\nh : u ≠ v\n⊢ (adjMatrix α G ^ 0) u v = ↑(#(G.finsetWalkLength 0 u v))",
"ppTerm": "?zero.inr",
"assigned": true,
"usedConstants... | [] | simp [finsetWalkLength, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions | {
"line": 173,
"column": 14
} | {
"line": 173,
"column": 16
} | {
"line": 174,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\nG : SimpleGraph α\nh : G.Colorable 2\nu✝ : α\nw : G.Walk u✝ u✝\n⊢ ¬Odd w.length",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.length",
"Odd",
"Nat",
"Nat.instSemiring"
],
"usedFVars": [
"α",
"G",
... | [
"case mp\nα : Type u_1\nG : SimpleGraph α\nh : G.Colorable 2\nu✝ : α\nw : G.Walk u✝ u✝\nho : Odd w.length\n⊢ False"
] | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 30
} | {
"line": 497,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nh : G.IsTree\ninst✝² : Fintype V\ninst✝¹ : Nontrivial V\ninst✝ : DecidableRel G.Adj\n⊢ G.minDegree = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"SimpleGraph.minDegree",
"instOfNatNat",
"LE.le",
"instLENat",
"dit... | [
"case pos\nV : Type u_1\nG : SimpleGraph V\nh : G.IsTree\ninst✝² : Fintype V\ninst✝¹ : Nontrivial V\ninst✝ : DecidableRel G.Adj\nq : 2 ≤ G.minDegree\n⊢ G.minDegree = 1",
"case neg\nV : Type u_1\nG : SimpleGraph V\nh : G.IsTree\ninst✝² : Fintype V\ninst✝¹ : Nontrivial V\ninst✝ : DecidableRel G.Adj\nq : ¬2 ≤ G.minD... | by_cases q : 2 ≤ G.minDegree | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 32
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case inr.a\nn r c : ℕ\nhr : r > 0\nw x : ℕ\nmw : w < r ∧ c ≡ n + w [MOD r]\nmx : x < r ∧ c ≡ n + x [MOD r]\nthis : w ≡ x [MOD r]\n⊢ w = x",
"ppTerm": "?inr.a",
"assigned": true,
"usedConstants": [
"id",
"Nat.instMod",
"instHMod",
"HMod.hMod",
"Nat.ModEq",
... | [
"case inr.a\nn r c : ℕ\nhr : r > 0\nw x : ℕ\nmw : w < r ∧ c ≡ n + w [MOD r]\nmx : x < r ∧ c ≡ n + x [MOD r]\nthis : w % r = x % r\n⊢ w = x"
] | change w % r = x % r at this | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 59
} | {
"line": 389,
"column": 4
} | [
{
"pp": "case pos\nα : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nht : t = 0\n⊢ (completeEquipartiteGraph r t).Copy G",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrAr... | [
"case pos\nα : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nht : t = 0\n⊢ ⊥.Copy G"
] | rw [completeEquipartiteGraph_eq_bot_iff.mpr <| .inr ht] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 68
} | {
"line": 329,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Nontrivial α\n⊢ ¬G.diam = 0 ↔ ¬G.ediam = ⊤",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SimpleGraph.diam_ne_zero_of_ediam_ne_top",
"instTopENat",
"SimpleGraph.ediam",
"instOfNatNat",
"Nat",
"ENat",
... | [] | exact ⟨ediam_ne_top_of_diam_ne_zero, diam_ne_zero_of_ediam_ne_top⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 358,
"column": 75
} | {
"line": 359,
"column": 49
} | {
"line": 361,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nh : ¬G.Connected\n⊢ G.radius = ⊤",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"iInf",
"instCompleteLinearOrderENat",
"SimpleGraph.eccent_eq_top_of_not_connected",
"CompleteLattice.toLattice",
"instTopENat",
"co... | [] | by
simp [radius, eccent_eq_top_of_not_connected h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 490,
"column": 17
} | {
"line": 490,
"column": 26
} | {
"line": 490,
"column": 27
} | [
{
"pp": "case neg.refine_2.refine_3\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nht : ¬t = 0\nx✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s\nK : G.CompleteEquipartiteSubgraph r t\ns : Finset V\nhs : #s = t\nhadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s\n⊢ ∀ {p : Finset V}, p ∈ cons s K.parts ⋯ →... | [
"case neg.refine_2.refine_3\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nht : ¬t = 0\nx✝ : ∃ K s, #s = t ∧ ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s\nK : G.CompleteEquipartiteSubgraph r t\ns : Finset V\nhs : #s = t\nhadj : ∀ p ∈ K.parts, G.IsCompleteBetween ↑p ↑s\n⊢ ∀ {p : Finset V}, p = s ∨ p ∈ K.parts → #p = t"
] | mem_cons, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Ends.Defs | {
"line": 204,
"column": 2
} | {
"line": 215,
"column": 73
} | {
"line": 217,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nK : Finset V\nC : G.ComponentCompl ↑K\n⊢ C.supp.Infinite ↔ ∀ (L : Finset V) (h : K ⊆ L), ∃ D, hom h D = C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.ComponentCompl.setLike",
"False",
"SimpleGraph.ComponentCompl.dis... | [] | classical
constructor
· rintro Cinf L h
obtain ⟨v, ⟨vK, rfl⟩, vL⟩ := Set.Infinite.nonempty (Set.Infinite.sdiff Cinf L.finite_toSet)
exact ⟨componentComplMk _ vL, rfl⟩
· rintro h Cfin
obtain ⟨D, e⟩ := h (K ∪ Cfin.toFinset) Finset.subset_union_left
obtain ⟨v, vD⟩ := D.nonempty
le... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.Ends.Defs | {
"line": 204,
"column": 2
} | {
"line": 215,
"column": 73
} | {
"line": 217,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nK : Finset V\nC : G.ComponentCompl ↑K\n⊢ C.supp.Infinite ↔ ∀ (L : Finset V) (h : K ⊆ L), ∃ D, hom h D = C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.ComponentCompl.setLike",
"False",
"SimpleGraph.ComponentCompl.dis... | [] | classical
constructor
· rintro Cinf L h
obtain ⟨v, ⟨vK, rfl⟩, vL⟩ := Set.Infinite.nonempty (Set.Infinite.sdiff Cinf L.finite_toSet)
exact ⟨componentComplMk _ vL, rfl⟩
· rintro h Cfin
obtain ⟨D, e⟩ := h (K ∪ Cfin.toFinset) Finset.subset_union_left
obtain ⟨v, vD⟩ := D.nonempty
le... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Ends.Defs | {
"line": 204,
"column": 2
} | {
"line": 215,
"column": 73
} | {
"line": 217,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nK : Finset V\nC : G.ComponentCompl ↑K\n⊢ C.supp.Infinite ↔ ∀ (L : Finset V) (h : K ⊆ L), ∃ D, hom h D = C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.ComponentCompl.setLike",
"False",
"SimpleGraph.ComponentCompl.dis... | [] | classical
constructor
· rintro Cinf L h
obtain ⟨v, ⟨vK, rfl⟩, vL⟩ := Set.Infinite.nonempty (Set.Infinite.sdiff Cinf L.finite_toSet)
exact ⟨componentComplMk _ vL, rfl⟩
· rintro h Cfin
obtain ⟨D, e⟩ := h (K ∪ Cfin.toFinset) Finset.subset_union_left
obtain ⟨v, vD⟩ := D.nonempty
le... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 29
} | {
"line": 117,
"column": 4
} | [
{
"pp": "W : Type u_1\nH : SimpleGraph W\nh : H.turanDensity ≠ 0\nhπ : Tendsto (fun x ↦ ↑(extremalNumber x H) / (H.turanDensity * ↑(x.choose 2))) atTop (𝓝 1)\n⊢ ∃ a, ∀ (b : ℕ), a ≤ b → H.turanDensity * ↑(b.choose 2) ≠ 0",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Real",
"... | [
"W : Type u_1\nH : SimpleGraph W\nh : H.turanDensity ≠ 0\nhπ : Tendsto (fun x ↦ ↑(extremalNumber x H) / (H.turanDensity * ↑(x.choose 2))) atTop (𝓝 1)\nn : ℕ\nhn : 2 ≤ n\n⊢ H.turanDensity * ↑(n.choose 2) ≠ 0"
] | refine ⟨2, fun n hn ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 72
} | {
"line": 73,
"column": 72
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nv : Fin n\nht'_pos : 0 < t'\nhv : v ∈ K.vertsᶜ\np : Finset (Fin n)\nhp : p ∈ K.parts\nhs : ∀ (x : Finset (Fin n)), x ∉ powersetCard t p ∨ ∃ x_1 ∈ x, ¬G.Adj v x_1\n⊢ #((between (↑K.verts) (↑K.... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nv : Fin n\nht'_pos : 0 < t'\nhv : v ∈ K.vertsᶜ\np : Finset (Fin n)\nhp : p ∈ K.parts\nhs : ∀ (x : Finset (Fin n)), x ∉ powersetCard t p ∨ ∃ x_1 ∈ x, ¬G.Adj v x_1\n⊢ #({v_1 ∈ K.verts | (between (↑K.verts)... | isBipartiteWith_neighborFinset' (between_verts_isBipartiteWith K) hv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 136,
"column": 45
} | {
"line": 136,
"column": 64
} | {
"line": 136,
"column": 65
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑n * (↑(#K.verts) + (↑(#K.verts) * ε... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑n * (↑(#K.verts) * ε - ↑(#K.verts) * (1 / ↑r) +... | add_sub_sub_cancel, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 136,
"column": 79
} | {
"line": 136,
"column": 92
} | {
"line": 137,
"column": 10
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑n * (↑(r * t') * ε - ↑(r * t') * (1... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑n * (↑r * ↑t' * ε - ↑r * ↑t' * (1 / ↑r) + (↑t' ... | Nat.cast_mul, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 81
} | {
"line": 201,
"column": 2
} | [
{
"pp": "ε : ℝ\nhε : 0 < ε\nr t : ℕ\n⊢ ∀ᶠ (n : ℕ) in atTop,\n ∀ {G : SimpleGraph (Fin n)} [inst : DecidableRel G.Adj],\n ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n → completeEquipartiteGraph (r + 1) t ⊑ G",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
... | [
"case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nh0 : r = 0 ∨ t = 0\n⊢ ∀ᶠ (n : ℕ) in atTop,\n ∀ {G : SimpleGraph (Fin n)} [inst : DecidableRel G.Adj],\n ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n → completeEquipartiteGraph (r + 1) t ⊑ G",
"case inr\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nhr_pos : r ≠ 0\nht_pos : t ≠ 0\n⊢ ∀ᶠ (n : ℕ)... | rcases show (r = 0 ∨ t = 0) ∨ r ≠ 0 ∧ t ≠ 0 by tauto with h0 | ⟨hr_pos, ht_pos⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 46
} | {
"line": 112,
"column": 2
} | [
{
"pp": "α : Type u_1\na b c : α\ns : Finset α\nG : SimpleGraph α\nr : ℕ\ninst✝ : DecidableEq α\nhs : G.IsNClique r (insert a s)\nhc : c ∈ s\nha : a ∉ s\nhd : ∀ w ∈ insert a s, w ≠ c → G.Adj w b\n⊢ G.IsNClique r (insert b ((insert a s).erase c))",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants"... | [
"α : Type u_1\na b c : α\ns : Finset α\nG : SimpleGraph α\nr : ℕ\ninst✝ : DecidableEq α\nhs : G.IsNClique r (insert a s)\nhc : c ∈ s\nha : a ∉ s\nhd : ∀ w ∈ insert a s, w ∉ {c} → G.Adj b w\n⊢ G.IsNClique r (insert b ((insert a s).erase c))"
] | simp_rw [adj_comm, ← notMem_singleton] at hd | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
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