module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 43
} | {
"line": 273,
"column": 43
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\nlp : 0 < (↑p).length\n⊢ p.firstReturn < (↑p).length",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.length_range",
"id",
"List.range",
"Nat",
"LT.lt",
"DyckWord.firstReturn",
... | [
"p : DyckWord\nh : p ≠ 0\nlp : 0 < (↑p).length\n⊢ p.firstReturn < (range (↑p).length).length"
] | ← length_range (n := p.toList.length) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 512,
"column": 4
} | {
"line": 512,
"column": 8
} | {
"line": 513,
"column": 4
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\nthis✝ : DecidableLE α := Classical.decRel LE.le\nmud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b ↦ (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }\nthis : ... | [
"𝕜 : Type u_2\nα : Type u_5\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\nthis✝ : DecidableLE α := ⋯\nmud : IncidenceAlgebra 𝕜 αᵒᵈ := ⋯\nthis : mud * zeta 𝕜 * mu 𝕜 = mu 𝕜 * zeta 𝕜 * mu 𝕜\n⊢ mud = mu 𝕜"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 540,
"column": 2
} | {
"line": 540,
"column": 6
} | {
"line": 541,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder α\ninst✝² : OrderTop α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\nf g : α → 𝕜\nh : ∀ (x : α), g x = ∑ y ∈ Ici x, f y\nx : α\nthis : DecidableLE α := Classical.decRel LE.le\n⊢ f x = ∑ y ∈ Ici x, (mu 𝕜) x y * g y",
"pp... | [
"𝕜 : Type u_2\nα : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder α\ninst✝² : OrderTop α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\nf g : α → 𝕜\nh : ∀ (x : α), g x = ∑ y ∈ Ici x, f y\nx : α\nthis : DecidableLE α := ⋯\n⊢ ∑ y ∈ Ici x, (mu 𝕜) x y * g y = f x"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 131,
"column": 12
} | {
"line": 131,
"column": 24
} | {
"line": 131,
"column": 24
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : TopologicalSpace R\ninst✝³ : T2Space R\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : IsTopologicalRing R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\ni : ℕ\nhi : m ∣ i + 1\nj : ℕ\nhj : i + 1 = m * j\n⊢ j * m = m * j",
"ppTerm": "?refine_2",
"assigned":... | [
"case refine_2\nR : Type u_1\ninst✝⁴ : TopologicalSpace R\ninst✝³ : T2Space R\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : IsTopologicalRing R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\ni : ℕ\nhi : m ∣ i + 1\nj : ℕ\nhj : i + 1 = m * j\n⊢ j * m = j * m"
] | mul_comm m j | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 752,
"column": 4
} | {
"line": 752,
"column": 61
} | {
"line": 753,
"column": 4
} | [
{
"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... | [
"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\nhHZS : ↑H ... | rw [← mul_le_mul_iff_right₀ (a := ε / 2) (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | {
"line": 55,
"column": 78
} | {
"line": 57,
"column": 29
} | {
"line": 59,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ ruzsaSzemerediNumber α ≤ (Fintype.card α).choose 3",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"instFintypeSimpleGraphOfDecidableEq",
"Nat.choose",
"Finset",
"SimpleGraph.Adj",
"DecidableRe... | [] | by
classical
exact Nat.findGreatest_le _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 8
} | {
"line": 83,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq β\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nf : α ↪ β\nG : SimpleGraph α\nw✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\n⊢ 3 ≠ 1",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 29
} | {
"line": 157,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx : α\ne : β\nG H : Graph α β\nhHG : H ≤ G\ny : α\nhxy : y ≠ x\nhe : H.IsLink e x y\n⊢ G.IsNonloopAt e x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Graph.IsLink.mono",
"Ne",
"And",
"Graph.IsLink",
"And.intro",
"... | [] | exact ⟨y, hxy, he.mono hHG⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 290,
"column": 15
} | {
"line": 290,
"column": 19
} | {
"line": 291,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤ G\n⊢ H ≤i G ↔ ∀ (e : β) (x y : α), G.IsLink e x y → x ∈ V(H) → y ∈ V(H) → e ∈ E(H)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Graph.IsInducedSubgraph",
"Membership.mem",
"Graph.IsLink",
"Graph.edg... | [
"α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤ G\n⊢ (∀ (e : β) (x y : α), G.IsLink e x y → x ∈ V(H) → y ∈ V(H) → e ∈ E(H)) ↔ H ≤i G"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 358,
"column": 15
} | {
"line": 358,
"column": 19
} | {
"line": 359,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤i G\n⊢ H ≤c G ↔ ∀ (x y : α), G.Adj x y → x ∈ V(H) → y ∈ V(H)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Graph.IsClosedSubgraph",
"Membership.mem",
"Graph.Adj",
"Graph.vertexSet",
"Set.instMem... | [
"α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤i G\n⊢ (∀ (x y : α), G.Adj x y → x ∈ V(H) → y ∈ V(H)) ↔ H ≤c G"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Combinatorics.Graph.Basic | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 24
} | {
"line": 246,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx y z : α\ne : β\nG : Graph α β\nhx : G.Inc e x\nhy : G.Inc e y\nhz : G.Inc e z\nhxy : x ≠ y\nhxz : x ≠ z\nhyz : y ≠ z\n⊢ False",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"False",
"Graph.Inc",
"Graph.IsLink",
"Exists.casesOn... | [
"α : Type u_1\nβ : Type u_2\nx y z : α\ne : β\nG : Graph α β\nhy : G.Inc e y\nhz : G.Inc e z\nhxy : x ≠ y\nhxz : x ≠ z\nhyz : y ≠ z\nx' : α\nhx' : G.IsLink e x x'\n⊢ False"
] | obtain ⟨x', hx'⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Graph.Basic | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 40
} | {
"line": 350,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Graph α β\nhE : ∀ (e : β), e ∈ E(G) ↔ ∃ x y, G.IsLink e x y\n⊢ { vertexSet := V(G), IsLink := G.IsLink, edgeSet := E(G), isLink_symm := ⋯, eq_or_eq_of_isLink_of_isLink := ⋯,\n edge_mem_iff_exists_isLink := hE, left_mem_of_isLink := ⋯ } =\n G",
"ppTerm": "?m.6... | [] | cases G with | _ _ _ _ _ _ h _ => simp | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Topology.Algebra.Semigroup | {
"line": 55,
"column": 14
} | {
"line": 55,
"column": 31
} | {
"line": 55,
"column": 32
} | [
{
"pp": "case ht.refine_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r\nS : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}\nN : Set M\nhN... | [
"case ht.refine_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r\nS : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}\nN : Set M\nhN : Minimal (... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.HalesJewett | {
"line": 433,
"column": 10
} | {
"line": 433,
"column": 25
} | {
"line": 433,
"column": 26
} | [
{
"pp": "case neg.refine_2\nα : Type u\ninst✝¹ : Fintype α\nκ : Type (max v u)\ninst✝ : Finite κ\nval✝ : Fintype κ\nh✝ : Nonempty α\nr : ℕ\nι : Type\n_inst✝ : Fintype ι\nι' : Type\n_inst : Fintype ι'\nC : (ι ⊕ ι' → Option α) → κ\nl' : Line α ι'\nC' : (ι → Option α) → κ\nhl' : ∀ (x : α), (fun v' v ↦ C (Sum.elim ... | [
"case neg.refine_2\nα : Type u\ninst✝¹ : Fintype α\nκ : Type (max v u)\ninst✝ : Finite κ\nval✝ : Fintype κ\nh✝ : Nonempty α\nr : ℕ\nι : Type\n_inst✝ : Fintype ι\nι' : Type\n_inst : Fintype ι'\nC : (ι ⊕ ι' → Option α) → κ\nl' : Line α ι'\nC' : (ι → Option α) → κ\nhl' : ∀ (x : α), (fun v' v ↦ C (Sum.elim v (some ∘ v'... | vertical_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.HalesJewett | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 49
} | {
"line": 440,
"column": 4
} | [
{
"pp": "case neg.refine_3.inr\nα : Type u\ninst✝¹ : Fintype α\nκ : Type (max v u)\ninst✝ : Finite κ\nval✝ : Fintype κ\nh✝ : Nonempty α\nr : ℕ\nι : Type\n_inst✝ : Fintype ι\nι' : Type\n_inst : Fintype ι'\nC : (ι ⊕ ι' → Option α) → κ\nl' : Line α ι'\nC' : (ι → Option α) → κ\nhl' : ∀ (x : α), (fun v' v ↦ C (Sum.e... | [] | · simp only [prod_apply, s.is_focused q hq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Hindman | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 25
} | {
"line": 169,
"column": 26
} | [
{
"pp": "case refine_4\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)}\nU : Ultrafilter M\nhU : ∀ (i : ℕ), U ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : ∀ (i : ℕ), V ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈... | [
"case refine_4\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)}\nU : Ultrafilter M\nhU : ∀ (i : ℕ), U ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : ∀ (i : ℕ), V ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Hindman | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 14
} | {
"line": 195,
"column": 4
} | [
{
"pp": "M : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns₀ : Set M\nsU : s₀ ∈ U\nexists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty\nelem : { s // s ∈ U } → M := fun p ↦ ⋯.some\nsucc : { s // s ∈ U } → { s // s ∈ U } := fun p ↦ ⟨↑p ∩ {m | elem p * ... | [
"M : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns₀ : Set M\nsU : s₀ ∈ U\nexists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty\nelem : { s // s ∈ U } → M := fun p ↦ ⋯.some\nsucc : { s // s ∈ U } → { s // s ∈ U } := fun p ↦ ⟨↑p ∩ {m | elem p * m ∈ ↑p}, ⋯⟩\... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 135,
"column": 72
} | {
"line": 135,
"column": 90
} | {
"line": 136,
"column": 4
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI B : Set α\nhI : M.Indep I\n⊢ (M✶ ＼ I).IsBase ((M.E \\ I) \\ B) ∧ B ⊆ M.E \\ I ↔ M.IsBase (B ∪ I) ∧ Disjoint B I",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBo... | [
"α : Type u_1\nM : Matroid α\nI B : Set α\nhI : M.Indep I\n⊢ M✶.IsBasis ((M.E \\ I) \\ B) (M✶.E \\ I) ∧ B ⊆ M.E \\ I ↔ M.IsBase (B ∪ I) ∧ Disjoint B I"
] | delete_isBase_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 141,
"column": 22
} | {
"line": 141,
"column": 51
} | {
"line": 141,
"column": 52
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI B : Set α\nhI : M.Indep I\n⊢ M✶.Spanning (M✶.E \\ I)",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.dual",
"id",
"LE.le",
"SDiff.sdiff",
"Matroid.Indep.subs... | [
"α : Type u_1\nM : Matroid α\nI B : Set α\nhI : M.Indep I\n⊢ M✶.Coindep I"
] | ← coindep_iff_compl_spanning, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 47
} | {
"line": 402,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\nC : Set α\nh : e ∈ M.E \\ M.closure C\nheC : e ∈ C\n⊢ e ∈ M.closure (C ∩ M.E)",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Set.inter_subset_right._simp_1",
"Matroid.E",
"Membership.mem",
"Matroid.subset_closure",
... | [] | exact (M.subset_closure (C ∩ M.E)) ⟨heC, h.1⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 35
} | {
"line": 123,
"column": 36
} | [
{
"pp": "U : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type u_3\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nhφψ : (φ ⋙q ψ).IsCovering\nφsur : Surjective φ.obj\n⊢ ψ.IsCovering",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Prefunctor.IsCover... | [
"case refine_1\nU : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type u_3\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nhφψ : (φ ⋙q ψ).IsCovering\nφsur : Surjective φ.obj\nv : V\n⊢ Bijective (ψ.star v)",
"case refine_2\nU : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Q... | refine ⟨fun v => ?_, fun v => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 89
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case refine_1\nU : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : φ.IsCovering\nu : Symmetrify U\n⊢ Bijective (φ.symmetrify.star u)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Quiver.symmetrifyStar",
"Equiv.instEquivLike",
... | [] | simp [φ.symmetrifyStar, φ.symmetrifyCostar, hφ.star_bijective u, hφ.costar_bijective u] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 89
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case refine_2\nU : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : φ.IsCovering\nu : Symmetrify U\n⊢ Bijective (φ.symmetrify.costar u)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"Sum.map",
"Prefunctor.... | [] | simp [φ.symmetrifyStar, φ.symmetrifyCostar, hφ.star_bijective u, hφ.costar_bijective u] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Quiver.Path.Decomposition | {
"line": 39,
"column": 6
} | {
"line": 40,
"column": 42
} | {
"line": 41,
"column": 6
} | [
{
"pp": "case pos\nV : 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 :... | [
"case pos\nV : 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 e)... | obtain ⟨u, hu_not_S, v, hv_S, e_uv, p₁, p₂, hp'⟩ :=
ih p' S ha_not_in_S hc_in_S p'_len | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 24
} | {
"line": 105,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nh : p.vertices ≠ []\n⊢ p.vertices.getLast h = b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"List.getLast_append_of_ne_nil",
"List.getLast",
"False",
"Quiver.Hom",
"Quiver.Path.nil",
"co... | [] | induction p with
| nil => simp only [vertices_nil, getLast_singleton]
| cons p' e ih => simp | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 24
} | {
"line": 105,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nh : p.vertices ≠ []\n⊢ p.vertices.getLast h = b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"List.getLast_append_of_ne_nil",
"List.getLast",
"False",
"Quiver.Hom",
"Quiver.Path.nil",
"co... | [] | induction p with
| nil => simp only [vertices_nil, getLast_singleton]
| cons p' e ih => simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 24
} | {
"line": 105,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nh : p.vertices ≠ []\n⊢ p.vertices.getLast h = b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"List.getLast_append_of_ne_nil",
"List.getLast",
"False",
"Quiver.Hom",
"Quiver.Path.nil",
"co... | [] | induction p with
| nil => simp only [vertices_nil, getLast_singleton]
| cons p' e ih => simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 216,
"column": 2
} | {
"line": 243,
"column": 44
} | {
"line": 245,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ p.vertices\n⊢ ∃ p₁ p₂, p = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"False",
"Quiver.Hom",
"Quiver.Path.nil",
"congrArg",
"Quiver.Path.rec... | [] | induction p with
| nil =>
have hxa : v = a := by
simpa [vertices_nil, List.mem_singleton] using hv
subst hxa
exact ⟨Path.nil, Path.nil, by simp only [comp_nil],
by simp only [vertices_nil, tail_cons, not_mem_nil, not_false_eq_true]⟩
| cons pPrev e ih =>
have hv' : v ∈ pPrev.vertices ∨ v ... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 216,
"column": 2
} | {
"line": 243,
"column": 44
} | {
"line": 245,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ p.vertices\n⊢ ∃ p₁ p₂, p = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"False",
"Quiver.Hom",
"Quiver.Path.nil",
"congrArg",
"Quiver.Path.rec... | [] | induction p with
| nil =>
have hxa : v = a := by
simpa [vertices_nil, List.mem_singleton] using hv
subst hxa
exact ⟨Path.nil, Path.nil, by simp only [comp_nil],
by simp only [vertices_nil, tail_cons, not_mem_nil, not_false_eq_true]⟩
| cons pPrev e ih =>
have hv' : v ∈ pPrev.vertices ∨ v ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 216,
"column": 2
} | {
"line": 243,
"column": 44
} | {
"line": 245,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ p.vertices\n⊢ ∃ p₁ p₂, p = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"False",
"Quiver.Hom",
"Quiver.Path.nil",
"congrArg",
"Quiver.Path.rec... | [] | induction p with
| nil =>
have hxa : v = a := by
simpa [vertices_nil, List.mem_singleton] using hv
subst hxa
exact ⟨Path.nil, Path.nil, by simp only [comp_nil],
by simp only [vertices_nil, tail_cons, not_mem_nil, not_false_eq_true]⟩
| cons pPrev e ih =>
have hv' : v ∈ pPrev.vertices ∨ v ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 135,
"column": 98
} | {
"line": 136,
"column": 35
} | {
"line": 138,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\na : α\n⊢ nonMemberSubfamily a (image (insert a) 𝒜) = ∅",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"congrArg",
"true_or",
"Finset",
"Finset.mem_image._simp_1",
"Finset.nonMemberSubfamil... | [] | by
simp [eq_empty_iff_forall_notMem] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 82,
"column": 51
} | {
"line": 100,
"column": 65
} | {
"line": 102,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\n⊢ ∑ s, ↑(card α) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) = ↑(card α) * ∑ k ∈ range (card α), (↑k)⁻¹ + 1",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"mul_div",
"Iff.mpr",
... | [] | by
rw [← powerset_univ, powerset_card_disjiUnion, sum_disjiUnion]
have : ∀ {x : ℕ}, ∀ s ∈ powersetCard x (univ : Finset α),
(card α / ((card α - #s) * (card α).choose #s) : ℚ) =
card α / ((card α - x) * (card α).choose x) := by
intro n s hs
rw [mem_powersetCard_univ.1 hs]
simp_rw [Finset.sum_con... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Birkhoff | {
"line": 229,
"column": 97
} | {
"line": 230,
"column": 81
} | {
"line": 232,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : DistribLattice α\ninst✝¹ : Fintype α\ninst✝ : DecidablePred SupIrred\na b : α\n⊢ birkhoffSet (a ⊔ b) = birkhoffSet a ∪ birkhoffSet b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"OrderEmbedding.birkhoffSet._proof_1",
"Eq.mpr",
"RelEmbedd... | [] | by
unfold OrderEmbedding.birkhoffSet; split <;> simp [eq_iff_true_of_subsingleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Birkhoff | {
"line": 232,
"column": 97
} | {
"line": 233,
"column": 81
} | {
"line": 235,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : DistribLattice α\ninst✝¹ : Fintype α\ninst✝ : DecidablePred SupIrred\na b : α\n⊢ birkhoffSet (a ⊓ b) = birkhoffSet a ∩ birkhoffSet b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"OrderEmbedding.birkhoffSet._proof_1",
"Eq.mpr",
"RelEmbedd... | [] | by
unfold OrderEmbedding.birkhoffSet; split <;> simp [eq_iff_true_of_subsingleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 22
} | {
"line": 254,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v a : α\ninst✝ : DecidableEq α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : a ∉ s\nb : α\nhb : b ∈ s\nh : compress u v b = a\n⊢ a ∈ s",
"ppTerm": "?inr",
"assigned... | [
"case inr\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v a : α\ninst✝ : DecidableEq α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : a ∉ s\nb : α\nhb : b ∈ s\nh : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \\ v else b) = a\n⊢ a ∈ s"
] | unfold compress at h | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 255,
"column": 4
} | {
"line": 256,
"column": 50
} | {
"line": 257,
"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 a : α\ninst✝ : DecidableEq α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : a ∉ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nh : (b ⊔ u) \\ v = a\n⊢ a ∈ s",
"ppTer... | [] | rw [← h, le_sdiff_right] at hva
rwa [← h, hvu hva, hva, sup_bot_eq, sdiff_bot] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 255,
"column": 4
} | {
"line": 256,
"column": 50
} | {
"line": 257,
"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 a : α\ninst✝ : DecidableEq α\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : a ∉ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nh : (b ⊔ u) \\ v = a\n⊢ a ∈ s",
"ppTer... | [] | rw [← h, le_sdiff_right] at hva
rwa [← h, hvu hva, hva, sup_bot_eq, sdiff_bot] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 52
} | {
"line": 171,
"column": 6
} | [
{
"pp": "case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nhs : (𝓓 a 𝒜).Shatters s\nu : Finset α\nht : s ∩ u ⊆ s\nhu : u ∉ 𝒜 ∧ insert a u ∈ 𝒜\nha✝ : a ∈ s\nv : Finset α\nhsv : s ∩ v = insert a (s ∩ u)\nhv : v ∈ 𝒜 ∧ v.erase a ∈ 𝒜\nha : a ∈ s ∩ u\n⊢ False",
... | [
"case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nhs : (𝓓 a 𝒜).Shatters s\nu : Finset α\nht : s ∩ u ⊆ s\nhu : u ∉ 𝒜 ∧ u ∈ 𝒜\nha✝ : a ∈ s\nv : Finset α\nhsv : s ∩ v = insert a (s ∩ u)\nhv : v ∈ 𝒜 ∧ v.erase a ∈ 𝒜\nha : a ∈ s ∩ u\n⊢ False"
] | rw [insert_eq_self.2 (mem_inter.1 ha).2] at hu | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 167,
"column": 4
} | {
"line": 171,
"column": 21
} | {
"line": 172,
"column": 4
} | [
{
"pp": "case pos.inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nhs : (𝓓 a 𝒜).Shatters s\nu : Finset α\nht : s ∩ u ⊆ s\nhu : u ∉ 𝒜 ∧ insert a u ∈ 𝒜\nha : a ∈ s\nv : Finset α\nhsv : s ∩ v = insert a (s ∩ u)\nhv : v ∈ 𝒜 ∧ v.erase a ∈ 𝒜\n⊢ ∃ u_1 ∈ 𝒜, s ∩ u_1 = s ∩ u",... | [
"case pos.inr\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nhs : (𝓓 a 𝒜).Shatters s\nu : Finset α\nht : s ∩ u ⊆ s\nhu : u ∉ 𝒜 ∧ insert a u ∈ 𝒜\nha : a ∈ s\nv : Finset α\nhsv : s ∩ v = insert a (s ∩ u)\nhv : v ∉ 𝒜 ∧ insert a v ∈ 𝒜\n⊢ ∃ u_1 ∈ 𝒜, s ∩ u_1 = s ∩ u"
] | · refine ⟨erase v a, hv.2, ?_⟩
rw [inter_erase, hsv, erase_insert]
rintro ha
rw [insert_eq_self.2 (mem_inter.1 ha).2] at hu
exact hu.1 hu.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Setoid.Partition | {
"line": 280,
"column": 43
} | {
"line": 280,
"column": 64
} | {
"line": 280,
"column": 64
} | [
{
"pp": "α : Type u_1\nr s : Setoid α\n| mkClasses r.classes ⋯ ≤ s",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Setoid.mkClasses",
"congrArg",
"Setoid.mkClasses_classes",
"Setoid.classes_eqv_classes",
"Setoid.classes",
"Setoid",
"LE.le",
... | [
"α : Type u_1\nr s : Setoid α\n| mkClasses r.classes ⋯ ≤ mkClasses s.classes ⋯"
] | ← mkClasses_classes s | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 292,
"column": 4
} | {
"line": 293,
"column": 22
} | {
"line": 295,
"column": 0
} | [
{
"pp": "case succ\nn n✝ : ℕ\nih : ∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset (Fin n))}, Set.Sized r ↑𝒜 → #𝒞 ≤ #𝒜 → IsInitSeg 𝒞 r → #(∂^[n✝] 𝒞) ≤ #(∂^[n✝] 𝒜)\nr : ℕ\n𝒜 𝒞 : Finset (Finset (Fin n))\nh₁ : Set.Sized r ↑𝒜\nh₂ : #𝒞 ≤ #𝒜\nh₃ : IsInitSeg 𝒞 r\n⊢ #(∂^[n✝ + 1] 𝒞) ≤ #(∂^[n✝ + 1] 𝒜)",
"ppTerm": "?s... | [] | refine ih h₁.shadow (kruskal_katona h₁ h₂ h₃) ?_
convert! h₃.shadow | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 292,
"column": 4
} | {
"line": 293,
"column": 22
} | {
"line": 295,
"column": 0
} | [
{
"pp": "case succ\nn n✝ : ℕ\nih : ∀ {r : ℕ} {𝒜 𝒞 : Finset (Finset (Fin n))}, Set.Sized r ↑𝒜 → #𝒞 ≤ #𝒜 → IsInitSeg 𝒞 r → #(∂^[n✝] 𝒞) ≤ #(∂^[n✝] 𝒜)\nr : ℕ\n𝒜 𝒞 : Finset (Finset (Fin n))\nh₁ : Set.Sized r ↑𝒜\nh₂ : #𝒞 ≤ #𝒜\nh₃ : IsInitSeg 𝒞 r\n⊢ #(∂^[n✝ + 1] 𝒞) ≤ #(∂^[n✝ + 1] 𝒜)",
"ppTerm": "?s... | [] | refine ih h₁.shadow (kruskal_katona h₁ h₂ h₃) ?_
convert! h₃.shadow | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 195,
"column": 4
} | {
"line": 200,
"column": 69
} | {
"line": 201,
"column": 4
} | [
{
"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 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⦄, s ⊆ insert a... | · 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 _)
· exact le_collapse_of_mem ‹_› h₃ rfl <| inter_mem_in... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 12
} | {
"line": 381,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Nonempty V\nn : ℕ\nhc : G.Colorable n\n⊢ ∀ b ∈ {n | G.Colorable n}, Nat.succ 0 ≤ b",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"setOf",
"Membership.mem",
"Nat",
"SimpleGraph.Colorable",
"Set.instMembership",
... | [
"V : Type u\nG : SimpleGraph V\ninst✝ : Nonempty V\nn : ℕ\nhc : G.Colorable n\nm : ℕ\nhm : m ∈ {n | G.Colorable n}\n⊢ Nat.succ 0 ≤ m"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 829,
"column": 2
} | {
"line": 848,
"column": 54
} | {
"line": 850,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\n⊢ G.Adj v w ∧ (G.deleteEdges {s(v, w)}).Reachable v w ↔ ∃ u p, p.IsCycle ∧ s(v, w) ∈ p.edges",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"Mathlib.Tactic.Push.not_exists._simp_1",
"List.mem_rev... | [] | classical
rw [reachable_deleteEdges_iff_exists_walk]
constructor
· rintro ⟨h, p, hp⟩
refine ⟨w, Walk.cons h.symm p.toPath, ?_, ?_⟩
· apply Path.cons_isCycle
rw [Sym2.eq_swap]
intro h
cases hp (Walk.edges_toPath_subset_edges p h)
· simp
· rintro ⟨u, c, hc, he⟩
refine ⟨c.adj_of_m... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 829,
"column": 2
} | {
"line": 848,
"column": 54
} | {
"line": 850,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\n⊢ G.Adj v w ∧ (G.deleteEdges {s(v, w)}).Reachable v w ↔ ∃ u p, p.IsCycle ∧ s(v, w) ∈ p.edges",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"Mathlib.Tactic.Push.not_exists._simp_1",
"List.mem_rev... | [] | classical
rw [reachable_deleteEdges_iff_exists_walk]
constructor
· rintro ⟨h, p, hp⟩
refine ⟨w, Walk.cons h.symm p.toPath, ?_, ?_⟩
· apply Path.cons_isCycle
rw [Sym2.eq_swap]
intro h
cases hp (Walk.edges_toPath_subset_edges p h)
· simp
· rintro ⟨u, c, hc, he⟩
refine ⟨c.adj_of_m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 829,
"column": 2
} | {
"line": 848,
"column": 54
} | {
"line": 850,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\n⊢ G.Adj v w ∧ (G.deleteEdges {s(v, w)}).Reachable v w ↔ ∃ u p, p.IsCycle ∧ s(v, w) ∈ p.edges",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"Mathlib.Tactic.Push.not_exists._simp_1",
"List.mem_rev... | [] | classical
rw [reachable_deleteEdges_iff_exists_walk]
constructor
· rintro ⟨h, p, hp⟩
refine ⟨w, Walk.cons h.symm p.toPath, ?_, ?_⟩
· apply Path.cons_isCycle
rw [Sym2.eq_swap]
intro h
cases hp (Walk.edges_toPath_subset_edges p h)
· simp
· rintro ⟨u, c, hc, he⟩
refine ⟨c.adj_of_m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 109,
"column": 23
} | {
"line": 109,
"column": 40
} | {
"line": 109,
"column": 41
} | [
{
"pp": "V : Type u_1\nv : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhv : v ∈ s\nw : V\n⊢ G.Adj v w ↔ w ∈ {w | w ∈ t ∧ G.Adj v w}",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Adj",
"setOf",
"Membership.... | [
"V : Type u_1\nv : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhv : v ∈ s\nw : V\n⊢ G.Adj v w ↔ w ∈ t ∧ G.Adj v w"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 135,
"column": 33
} | {
"line": 135,
"column": 50
} | {
"line": 135,
"column": 51
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhw : w ∈ t\nv : V\n⊢ G.Adj v w ↔ v ∈ {v | v ∈ s ∧ G.Adj v w}",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Adj",
"setOf",
"Membership.... | [
"V : Type u_1\nw : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhw : w ∈ t\nv : V\n⊢ G.Adj v w ↔ v ∈ s ∧ G.Adj v w"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 34
} | {
"line": 296,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nc : V → Fin 2\nhc : ∀ {a b : V}, G.Adj a b → (completeGraph (Fin 2)).Adj (c a) (c b)\nv w : V\nhvw : (completeGraph (Fin 2)).Adj (c v) (c w)\n⊢ v ∈ {v | c v = 0} ∧ w ∈ {v | c v = 1} ∨ v ∈ {v | c v = 1} ∧ w ∈ {v | c v = 0}",
"ppTerm": "?m.55",
"assigned": true,
... | [
"V : Type u_1\nG : SimpleGraph V\nc : V → Fin 2\nhc : ∀ {a b : V}, G.Adj a b → (completeGraph (Fin 2)).Adj (c a) (c b)\nv w : V\nhvw : ¬c v = c w\n⊢ c v = 0 ∧ c w = 1 ∨ c v = 1 ∧ c w = 0"
] | simp [Set.mem_setOf_eq] at hvw ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 93
} | {
"line": 108,
"column": 0
} | [
{
"pp": "case inr\nV : Type u\nG : SimpleGraph V\nH K : G.Subgraph\nhH : H.Preconnected\nhK : K.Preconnected\nu : V\nhu : u ∈ H.verts\nhu' : u ∈ K.verts\nv : V\nhv : v ∈ K.verts\n⊢ (H ⊔ K).coe.Reachable ⟨u, ⋯⟩ ⟨v, ⋯⟩",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatti... | [] | · exact Reachable.map (Subgraph.inclusion (le_sup_right : K ≤ H ⊔ K)) (hK ⟨u, hu'⟩ ⟨v, hv⟩) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 465,
"column": 4
} | {
"line": 465,
"column": 86
} | {
"line": 465,
"column": 86
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\nh_bipartite : (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ\n⊢ #(G.neighborFinset w) ≤ #((between (↑s) (↑s)ᶜ G).neighborFinset w) + #sᶜ",
"ppTerm": "?m.52",
... | [
"V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\nh_bipartite : (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ\n⊢ #(G.neighborFinset w) ≤ #((between (↑s) (↑s)ᶜ G).neighborFinset w ∪ sᶜ)"
] | ← card_union_of_disjoint (isBipartiteWith_neighborFinset_disjoint' h_bipartite hw) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 8
} | {
"line": 312,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\n⊢ 1 ≠ 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidableE... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 380,
"column": 90
} | {
"line": 397,
"column": 63
} | {
"line": 399,
"column": 0
} | [
{
"pp": "V : 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\n⊢ ∃ x a b, G.Adj x a ∧ G.Adj a b ∧ ¬G.Adj x b ∧ x ≠ b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Metric.0.SimpleGraph.W... | [] | by
use v, p.getVert 1, p.getVert 2
have hnp : ¬p.Nil := by grind [Nil.length_eq_zero]
have : p.tail.tail.length < p.tail.length := by
rw [← p.tail.length_tail_add_one (by
simp only [not_nil_iff_lt_length, ← p.length_tail_add_one hnp] at hp ⊢
lia)]
lia
have : p.tail.length < p.length := by rw... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 25
} | {
"line": 452,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nc v : V\nr : ℕ∞\n⊢ v ∈ G.ball c r ↔ c ∈ G.ball v r",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimpleGraph.ball",
"congrArg",
"setOf",
"Membership.mem",
"iff_self",
"funext",
"Iff",
"SimpleGraph.e... | [] | simp [ball, edist_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 25
} | {
"line": 452,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nc v : V\nr : ℕ∞\n⊢ v ∈ G.ball c r ↔ c ∈ G.ball v r",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimpleGraph.ball",
"congrArg",
"setOf",
"Membership.mem",
"iff_self",
"funext",
"Iff",
"SimpleGraph.e... | [] | simp [ball, edist_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 25
} | {
"line": 452,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nc v : V\nr : ℕ∞\n⊢ v ∈ G.ball c r ↔ c ∈ G.ball v r",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimpleGraph.ball",
"congrArg",
"setOf",
"Membership.mem",
"iff_self",
"funext",
"Iff",
"SimpleGraph.e... | [] | simp [ball, edist_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Prod | {
"line": 284,
"column": 39
} | {
"line": 284,
"column": 62
} | {
"line": 285,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nH : SimpleGraph β\nx y : α × β\ntop_case : (G □ H).edist x y = ⊤ ↔ G.edist x.1 y.1 = ⊤ ∨ H.edist x.2 y.2 = ⊤\nh : ¬(G □ H).edist x y = ⊤\nrGH : G.edist x.1 y.1 ≠ ⊤ ∧ H.edist x.2 y.2 ≠ ⊤\nwG : G.Walk x.1 y.1\nhwG : ↑wG.length = G.edist x.1 y.1\nwH : H.Walk ... | [] | by exact_mod_cast w_len | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Prod | {
"line": 285,
"column": 54
} | {
"line": 285,
"column": 74
} | {
"line": 286,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nH : SimpleGraph β\nx y : α × β\ntop_case : (G □ H).edist x y = ⊤ ↔ G.edist x.1 y.1 = ⊤ ∨ H.edist x.2 y.2 = ⊤\nh : ¬(G □ H).edist x y = ⊤\nrGH : G.edist x.1 y.1 ≠ ⊤ ∧ H.edist x.2 y.2 ≠ ⊤\nwG : G.Walk x.1 y.1\nhwG : ↑wG.length = G.edist x.1 y.1\nwH : H.Walk ... | [] | simp only [hwG, hwH] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Prod | {
"line": 285,
"column": 54
} | {
"line": 285,
"column": 74
} | {
"line": 286,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nH : SimpleGraph β\nx y : α × β\ntop_case : (G □ H).edist x y = ⊤ ↔ G.edist x.1 y.1 = ⊤ ∨ H.edist x.2 y.2 = ⊤\nh : ¬(G □ H).edist x y = ⊤\nrGH : G.edist x.1 y.1 ≠ ⊤ ∧ H.edist x.2 y.2 ≠ ⊤\nwG : G.Walk x.1 y.1\nhwG : ↑wG.length = G.edist x.1 y.1\nwH : H.Walk ... | [] | simp only [hwG, hwH] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Prod | {
"line": 285,
"column": 54
} | {
"line": 285,
"column": 74
} | {
"line": 286,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nH : SimpleGraph β\nx y : α × β\ntop_case : (G □ H).edist x y = ⊤ ↔ G.edist x.1 y.1 = ⊤ ∨ H.edist x.2 y.2 = ⊤\nh : ¬(G □ H).edist x y = ⊤\nrGH : G.edist x.1 y.1 ≠ ⊤ ∧ H.edist x.2 y.2 ≠ ⊤\nwG : G.Walk x.1 y.1\nhwG : ↑wG.length = G.edist x.1 y.1\nwH : H.Walk ... | [] | simp only [hwG, hwH] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 8
} | {
"line": 50,
"column": 0
} | [
{
"pp": "⊢ ∀ (x x_1 : Fin 2), (cycleGraph 2).Adj x x_1 = ⊤.Adj x x_1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"SimpleGraph.Top.adjDecidable",
"SimpleGraph.instDecidableRelFinAdjCycleGraph",
"SimpleGraph.Adj",
"instDecidableEqFin",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 8
} | {
"line": 54,
"column": 0
} | [
{
"pp": "⊢ ∀ (x x_1 : Fin 3), (cycleGraph 3).Adj x x_1 = ⊤.Adj x x_1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"SimpleGraph.Top.adjDecidable",
"SimpleGraph.instDecidableRelFinAdjCycleGraph",
"SimpleGraph.Adj",
"instDecidableEqFin",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 37
} | {
"line": 142,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (have hadj := ⋯;\n cons hadj (cycleCons n (Fin.last (n + 2)))).length =\n n + 3",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"congrArg",
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"SimpleGraph.cycleGraph",... | [] | simp [cycleGraph.length_cycle_cons] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 31
} | {
"line": 579,
"column": 32
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ns : Set V\n⊢ (induce s G).Preconnected ↔ (⊤.induce s).Preconnected",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.induce_eq_coe_induce_top",
"congrArg",
"SimpleGraph.Subgraph",
"SimpleGraph.Precon... | [
"V : Type u\nG : SimpleGraph V\ns : Set V\n⊢ (⊤.induce s).coe.Preconnected ↔ (⊤.induce s).Preconnected"
] | induce_eq_coe_induce_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 584,
"column": 6
} | {
"line": 584,
"column": 31
} | {
"line": 584,
"column": 32
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ns : Set V\n⊢ (induce s G).Connected ↔ (⊤.induce s).Connected",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.induce_eq_coe_induce_top",
"congrArg",
"SimpleGraph.Subgraph",
"Set.Elem",
"id",
... | [
"V : Type u\nG : SimpleGraph V\ns : Set V\n⊢ (⊤.induce s).coe.Connected ↔ (⊤.induce s).Connected"
] | induce_eq_coe_induce_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 36
} | {
"line": 176,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nn : ℕ\nhn : 2 < n\n⊢ cycleGraph n ⊑ G ↔ ∃ v p, p.IsCycle ∧ p.length = n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsContained",
"SimpleGraph.Walk.length",
"SimpleGraph.cycleGraph",
"SimpleGraph.Walk.IsCycl... | [
"case refine_1\nV : Type u_1\nG : SimpleGraph V\nn : ℕ\nhn : 2 < n\nx✝ : cycleGraph n ⊑ G\nh : (cycleGraph n).Copy G\n⊢ ∃ v p, p.IsCycle ∧ p.length = n",
"case refine_2\nV : Type u_1\nG : SimpleGraph V\nn : ℕ\nhn : 2 < n\nh' : ∃ v p, p.IsCycle ∧ p.length = n\n⊢ cycleGraph n ⊑ G"
] | refine ⟨fun ⟨h⟩ ↦ ?_, fun h' ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 106,
"column": 67
} | {
"line": 110,
"column": 35
} | {
"line": 112,
"column": 0
} | [
{
"pp": "M : Type u_1\ns : Set M\ninst✝ : MulOneClass M\n⊢ mulCayley (s \\ {1}) = mulCayley s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"HMul.hMul",
"eq_false",
"and_true",
"congrArg",
"SimpleGraph.Adj... | [] | by
nth_rw 2 [← Set.sdiff_union_inter s {1}]
rw [mulCayley_union]
ext u v
simp +contextual [mulCayley_adj'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 84
} | {
"line": 120,
"column": 2
} | [
{
"pp": "V : Type u_1\nW : Type u_2\ninst✝ : Fintype V\nH : SimpleGraph W\nm : ℕ\n⊢ extremalNumber (Fintype.card V) H ≤ m ↔\n ∀ ⦃G : SimpleGraph V⦄ [inst : DecidableRel G.Adj], H.Free G → #G.edgeFinset ≤ m",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instFintypeSimpleGraphOfDe... | [
"V : Type u_1\nW : Type u_2\ninst✝ : Fintype V\nH : SimpleGraph W\nm : ℕ\n⊢ (∀ (b : SimpleGraph V), H.Free b → #b.edgeFinset ≤ m) ↔\n ∀ ⦃G : SimpleGraph V⦄ [inst : DecidableRel G.Adj], H.Free G → #G.edgeFinset ≤ m"
] | simp_rw [extremalNumber_of_fintypeCard_eq rfl, Finset.sup_le_iff, mem_filter_univ] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 159,
"column": 51
} | {
"line": 169,
"column": 29
} | {
"line": 171,
"column": 0
} | [
{
"pp": "n₁ n₂ : ℕ\nW₁ : Type u_4\nW₂ : Type u_5\nH₁ : SimpleGraph W₁\nH₂ : SimpleGraph W₂\nh : n₁ = n₂\ne : H₁ ≃g H₂\n⊢ extremalNumber n₁ H₁ = extremalNumber n₂ H₂",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SimpleGraph.Free",
"Eq.mpr",
"Fintype.card_fin",
"Si... | [] | by
rw [h, le_antisymm_iff]
and_intros
on_goal 2 =>
replace e := e.symm
all_goals
rw [← Fintype.card_fin n₂, extremalNumber_le_iff]
intro G _ h
apply card_edgeFinset_le_extremalNumber
contrapose h
exact h.trans' ⟨e.toCopy⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 16
} | {
"line": 234,
"column": 16
} | [
{
"pp": "V : Type u_1\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nr : ℕ\nh : G.IsTuranMaximal r\ninst✝ : DecidableEq V\nfp : Finpartition univ := h.finpartition\nl : #fp.parts < min (#univ) r\n⊢ False",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Preorder.... | [
"V : Type u_1\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nr : ℕ\nh : G.IsTuranMaximal r\ninst✝ : DecidableEq V\nfp : Finpartition univ := h.finpartition\nl : #fp.parts < #univ ∧ #fp.parts < r\n⊢ False"
] | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 484,
"column": 2
} | {
"line": 485,
"column": 28
} | {
"line": 486,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nthis : Fintype V\n⊢ G.IsTree ↔ G.Connected ∧ Nat.card ↑G.edgeSet + 1 = Nat.card V",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_ofFinset",
"SimpleGraph.decidableMemEdgeSet",
"Finse... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nthis : Fintype V\nx✝ : G.Connected ∧ Nat.card ↑G.edgeSet + 1 = Nat.card V\nh₁ : G.Connected\nh₂ : Nat.card ↑G.edgeSet + 1 = Nat.card V\n⊢ G.IsAcyclic"
] | refine ⟨fun h ↦ ⟨h.connected, by simpa [edgeFinset] using h.card_edgeFinset⟩,
fun ⟨h₁, h₂⟩ ↦ ⟨h₁, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 497,
"column": 2
} | {
"line": 503,
"column": 7
} | {
"line": 504,
"column": 2
} | [
{
"pp": "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",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Finset.card_univ",
"SimpleGraph.sum_degrees_eq_twice_... | [
"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.minDegree\n⊢ G.minDegree = 1"
] | · have := h.card_edgeFinset
have := G.sum_degrees_eq_twice_card_edges
have hle : ∑ v : V, 2 ≤ ∑ v, G.degree v := by
gcongr
exact le_trans q (G.minDegree_le_degree _)
rw [Finset.sum_const, Finset.card_univ, smul_eq_mul] at hle
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 526,
"column": 2
} | {
"line": 543,
"column": 5
} | {
"line": 545,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nhconn : G.Connected\nv : V\ninst✝ : Fintype ↑(G.neighborSet v)\nhdeg : G.degree v = 1\nu : V\nadj_vu : G.Adj v u\nhu : ∀ (y : V), (fun w ↦ G.Adj v w) y → y = u\nw x : ↑{v}ᶜ\npwu : G.Walk (↑w) u\nhpwu : pwu.IsPath\npux : G.Walk u ↑x\nhpux : pux.IsPath\n⊢ ∃ x, True",
... | [] | classical
use ((pwu.append pux).toPath.val.induce {v}ᶜ ?_).copy (SetCoe.ext rfl) (SetCoe.ext rfl)
/- Each path between vertex u and another vertex in G.induce {v}ᶜ
is contained in G.induce {v}ᶜ. -/
intro z hz
rw [Set.mem_compl_iff, Set.mem_singleton_iff]
obtain ⟨pwz, pzx, p_eq_pwzx⟩ := mem_support_iff_exist... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 64
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case inl\nr t : ℕ\nht_eq0 : t = 0\n⊢ (completeEquipartiteGraph r t).IsCompleteMultipartite",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Bot.bot",
"LE.le",
"instLENat",
"... | [
"case inl\nr t : ℕ\nht_eq0 : t = 0\n⊢ ⊥.IsCompleteMultipartite"
] | rw [completeEquipartiteGraph_eq_bot_iff.mpr (Or.inr ht_eq0)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 301,
"column": 71
} | {
"line": 309,
"column": 87
} | {
"line": 311,
"column": 0
} | [
{
"pp": "r t : ℕ\n⊢ #(completeEquipartiteGraph r t).edgeFinset = r.choose 2 * t ^ 2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Nat.even_mul_pred_self",
"Finset.card_univ",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SimpleGraph.sum_degrees_eq_t... | [] | by
rw [← mul_right_inj' two_ne_zero, ← sum_degrees_eq_twice_card_edges]
conv_lhs =>
rhs; intro v
rw [degree_completeEquipartiteGraph v]
rw [sum_const, smul_eq_mul, card_univ, card_prod, Fintype.card_fin, Fintype.card_fin]
conv_rhs =>
rw [← Nat.mul_assoc, Nat.choose_two_right, Nat.mul_div_cancel' r.e... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 17
} | {
"line": 59,
"column": 18
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nu : α\nh✝ : ∀ (v : α), ∃ w, ¬G.Reachable v w\nv : α\nh : ¬G.Reachable u v\n⊢ G.eccent u = ⊤",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"eq_top_iff",
"instTopENat",
"congrAr... | [
"α : Type u_1\nG : SimpleGraph α\nu : α\nh✝ : ∀ (v : α), ∃ w, ¬G.Reachable v w\nv : α\nh : ¬G.Reachable u v\n⊢ ⊤ ≤ G.eccent u"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 51
} | {
"line": 90,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Nontrivial α\nu v : α\n⊢ ⊤.edist u v ≤ 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"False",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"eq_false",
"instReflLe",
"congrArg",
"CommSemiring.toSemiri... | [] | cases eq_or_ne u v <;> simp_all [edist_top_of_ne] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 17
} | {
"line": 183,
"column": 18
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Nonempty α\nh : ∀ (v : α), ∃ w, ¬G.Reachable v w\nw✝ : α\nhw : ¬G.Reachable Classical.ofNonempty w✝\n⊢ G.ediam = ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"eq_top_iff",
... | [
"α : Type u_1\nG : SimpleGraph α\ninst✝ : Nonempty α\nh : ∀ (v : α), ∃ w, ¬G.Reachable v w\nw✝ : α\nhw : ¬G.Reachable Classical.ofNonempty w✝\n⊢ ⊤ ≤ G.ediam"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 180,
"column": 2
} | {
"line": 184,
"column": 22
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Nonempty α\nh : ¬G.Connected\n⊢ G.ediam = ⊤",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Classical.ofNonempty",
"instC... | [] | rw [connected_iff_exists_forall_reachable] at h
push Not at h
obtain ⟨_, hw⟩ := h Classical.ofNonempty
rw [eq_top_iff, ← edist_eq_top_of_not_reachable hw]
exact edist_le_ediam | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 180,
"column": 2
} | {
"line": 184,
"column": 22
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Nonempty α\nh : ¬G.Connected\n⊢ G.ediam = ⊤",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Classical.ofNonempty",
"instC... | [] | rw [connected_iff_exists_forall_reachable] at h
push Not at h
obtain ⟨_, hw⟩ := h Classical.ofNonempty
rw [eq_top_iff, ← edist_eq_top_of_not_reachable hw]
exact edist_le_ediam | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 46
} | {
"line": 311,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nh✝ : Subsingleton α\n⊢ ⊥.ediam = 0 ∨ ⊥.ediam = ⊤",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instTopENat",
"CommSemiring.toSemiring",
"SimpleGraph.ediam",
"Bot.bot",
"SimpleGraph.ediam_eq_zero_of_subsingleton",
"Simpl... | [] | exact Or.inl ediam_eq_zero_of_subsingleton | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 46
} | {
"line": 311,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nh✝ : Subsingleton α\n⊢ ⊥.ediam = 0 ∨ ⊥.ediam = ⊤",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instTopENat",
"CommSemiring.toSemiring",
"SimpleGraph.ediam",
"Bot.bot",
"SimpleGraph.ediam_eq_zero_of_subsingleton",
"Simpl... | [] | exact Or.inl ediam_eq_zero_of_subsingleton | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 46
} | {
"line": 311,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nh✝ : Subsingleton α\n⊢ ⊥.ediam = 0 ∨ ⊥.ediam = ⊤",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instTopENat",
"CommSemiring.toSemiring",
"SimpleGraph.ediam",
"Bot.bot",
"SimpleGraph.ediam_eq_zero_of_subsingleton",
"Simpl... | [] | exact Or.inl ediam_eq_zero_of_subsingleton | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Ends.Defs | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 6
} | {
"line": 145,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nK : Set V\nGc : G.Preconnected\nhK : K.Nonempty\nv : V\nvnK : v ∉ K\nC : G.ComponentCompl K := G.componentComplMk vnK\ndis : K ∩ ↑C ⊆ ∅ := Set.disjoint_iff.mp (ComponentCompl.disjoint_right C)\nh : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.... | [
"V : Type u\nG : SimpleGraph V\nK : Set V\nGc : G.Preconnected\nhK : K.Nonempty\nv : V\nvnK : v ∉ K\nC : G.ComponentCompl K := ⋯\ndis : K ∩ ↑C ⊆ ∅ := ⋯\nh : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.2\n⊢ ↑C = Set.univ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Combinatorics.SimpleGraph.Finsubgraph | {
"line": 110,
"column": 12
} | {
"line": 110,
"column": 21
} | {
"line": 110,
"column": 22
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Finite V\nι : Sort u_1\nf : ι → G.Finsubgraph\n⊢ ↑(sSup (range fun i ↦ f i)) = ⨆ i, ↑(f i)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SimpleGraph.Finsubgraph.coe_sSup",
"Eq.mpr",
"congrArg",
"iSup",
"Simple... | [
"V : Type u\nG : SimpleGraph V\ninst✝ : Finite V\nι : Sort u_1\nf : ι → G.Finsubgraph\n⊢ ⨆ G_1 ∈ range fun i ↦ f i, ↑G_1 = ⨆ i, ↑(f i)"
] | coe_sSup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 40
} | {
"line": 88,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ns : Set α\nhe : Even s.ncard\nhs : s.Finite\n⊢ ∃ t u, t ∪ u = s ∧ Disjoint t u ∧ #↑t = #↑u",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"Set.Finite.toFinset",
"Nat",
"Even",
"Finset.card",
"in... | [
"case inr\nα : Type u_1\ns : Set α\nhs : s.Finite\nhe : Even #hs.toFinset\n⊢ ∃ t u, t ∪ u = s ∧ Disjoint t u ∧ #↑t = #↑u"
] | rw [ncard_eq_toFinset_card s hs] at he | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 95,
"column": 2
} | {
"line": 112,
"column": 64
} | {
"line": 114,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\n⊢ ∃ h, Bijective h ∧ ∀ (a : ↑p₁), G.Adj ↑a ↑(h a)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.mem_u... | [] | classical
obtain ⟨f, hf₁, hf₂⟩ := Finset.all_card_le_biUnion_card_iff_exists_injective
(fun x ↦ G.neighborFinset x) |>.mp fun s ↦ by
have := h₂ s
simpa [← Set.ncard_coe_finset, neighborFinset_def]
have (x : V) (h : x ∈ p₁) : f x ∉ p₁ := h₁.disjoint |>.notMem_of_mem_right <|
isBipartiteWith_neighbo... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 95,
"column": 2
} | {
"line": 112,
"column": 64
} | {
"line": 114,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\n⊢ ∃ h, Bijective h ∧ ∀ (a : ↑p₁), G.Adj ↑a ↑(h a)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.mem_u... | [] | classical
obtain ⟨f, hf₁, hf₂⟩ := Finset.all_card_le_biUnion_card_iff_exists_injective
(fun x ↦ G.neighborFinset x) |>.mp fun s ↦ by
have := h₂ s
simpa [← Set.ncard_coe_finset, neighborFinset_def]
have (x : V) (h : x ∈ p₁) : f x ∉ p₁ := h₁.disjoint |>.notMem_of_mem_right <|
isBipartiteWith_neighbo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 95,
"column": 2
} | {
"line": 112,
"column": 64
} | {
"line": 114,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\n⊢ ∃ h, Bijective h ∧ ∀ (a : ↑p₁), G.Adj ↑a ↑(h a)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.mem_u... | [] | classical
obtain ⟨f, hf₁, hf₂⟩ := Finset.all_card_le_biUnion_card_iff_exists_injective
(fun x ↦ G.neighborFinset x) |>.mp fun s ↦ by
have := h₂ s
simpa [← Set.ncard_coe_finset, neighborFinset_def]
have (x : V) (h : x ∈ p₁) : f x ∉ p₁ := h₁.disjoint |>.notMem_of_mem_right <|
isBipartiteWith_neighbo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 35
} | {
"line": 192,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\ns : Finset α\nG : SimpleGraph α\nr k : ℕ\ninst✝ : DecidableEq α\nv w₁ w₂ : α\nt : Finset α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nC : G.Coloring (Fin (r + 1))\nh : Set.SurjOn (⇑C) (insert w₁ ↑s) Set.univ\nthis : Set.SurjOn (⇑C) (insert w₂ ↑t) Set.univ\nx : α\nhcx : C x = C v\n... | [
"α : Type u_1\ns : Finset α\nG : SimpleGraph α\nr k : ℕ\ninst✝ : DecidableEq α\nv w₁ w₂ : α\nt : Finset α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nC : G.Coloring (Fin (r + 1))\nh : Set.SurjOn (⇑C) (insert w₁ ↑s) Set.univ\nthis : Set.SurjOn (⇑C) (insert w₂ ↑t) Set.univ\nx : α\nhcx : C x = C v\ny : α\nhy : y ∈ insert... | apply (C.valid _ hcx.symm).elim | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.SimpleGraph.IncMatrix | {
"line": 81,
"column": 49
} | {
"line": 84,
"column": 26
} | {
"line": 86,
"column": 0
} | [
{
"pp": "R : Type u_1\nα : Type u_2\nG : SimpleGraph α\ninst✝² : MulZeroOneClass R\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel G.Adj\na b : α\ne : Sym2 α\nhab : a ≠ b\nh : ¬G.Adj a b\n⊢ incMatrix R G a e * incMatrix R G b e = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | by
rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_notMem]
rw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab]
exact Set.notMem_empty e | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.IncMatrix | {
"line": 145,
"column": 4
} | {
"line": 148,
"column": 59
} | {
"line": 150,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nα : Type u_2\nG : SimpleGraph α\ninst✝⁴ : NonAssocSemiring R\ninst✝³ : DecidableEq α\ninst✝² : DecidableRel G.Adj\ne : Sym2 α\ninst✝¹ : Fintype α\ninst✝ : Decidable (e ∈ G.edgeSet)\nh : e ∉ G.edgeSet\n⊢ ↑(#{x | e ∈ G.incidenceSet x}) = 0",
"ppTerm": "?neg✝",
"assigned": ... | [] | revert h
refine e.ind ?_
intro v w h
simp [mk'_mem_incidenceSet_iff, G.mem_edgeSet.not.mp h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.IncMatrix | {
"line": 145,
"column": 4
} | {
"line": 148,
"column": 59
} | {
"line": 150,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nα : Type u_2\nG : SimpleGraph α\ninst✝⁴ : NonAssocSemiring R\ninst✝³ : DecidableEq α\ninst✝² : DecidableRel G.Adj\ne : Sym2 α\ninst✝¹ : Fintype α\ninst✝ : Decidable (e ∈ G.edgeSet)\nh : e ∉ G.edgeSet\n⊢ ↑(#{x | e ∈ G.incidenceSet x}) = 0",
"ppTerm": "?neg✝",
"assigned": ... | [] | revert h
refine e.ind ?_
intro v w h
simp [mk'_mem_incidenceSet_iff, G.mem_edgeSet.not.mp h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 62
} | {
"line": 315,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\ninst✝ : DecidableRel G.Adj\nc : G.ConnectedComponent\nhM : M.IsPerfectMatching\nthis✝ : DecidablePred fun x ↦ x ∈ (M.induce c.supp).verts := fun a ↦ G.instDecidableMemSupp c a\nthis : Even (M.induce c.supp).ver... | [
"V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\ninst✝ : DecidableRel G.Adj\nc : G.ConnectedComponent\nhM : M.IsPerfectMatching\nthis✝ : DecidablePred fun x ↦ x ∈ (M.induce c.supp).verts := fun a ↦ G.instDecidableMemSupp c a\nthis : Even (Fintype.card ↑c.supp)\n⊢ Even (F... | simp only [Subgraph.induce_verts, Set.toFinset_card] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 150,
"column": 27
} | {
"line": 150,
"column": 54
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case inl\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : DecidableEq V\nh : G.IsSRGWith n k ℓ μ\nv u : V\nha : v ≠ u ∧ ¬G.Adj v u\nhne : v ≠ u\n⊢ ¬G.Adj v u ∧ ¬G.Adj u u",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa [adj_comm] using ha.2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 150,
"column": 27
} | {
"line": 150,
"column": 54
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case inr\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : DecidableEq V\nh : G.IsSRGWith n k ℓ μ\nw u : V\nha : u ≠ w ∧ ¬G.Adj u w\nhne : u ≠ w\n⊢ ¬G.Adj u u ∧ ¬G.Adj w u",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa [adj_comm] using ha.2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 375,
"column": 2
} | {
"line": 381,
"column": 30
} | {
"line": 384,
"column": 2
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | [
"α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\nW : Finset α... | have xcle : ∀ x ∈ Xᶜ, 1 ≤ #{z ∈ s ∩ t | ¬ G.Adj x z} := by
intro x hx
apply card_pos.2
obtain ⟨_, hy⟩ : ∃ y ∈ s ∩ t, ¬ G.Adj x y := by
contrapose! hx
simpa [X] using hx
exact ⟨_, mem_filter.2 hy⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Tiling.Tile | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 26
} | {
"line": 219,
"column": 27
} | [
{
"pp": "G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\nindex : ιₚ\ngroupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries\n⊢ (Quotient.liftOn' groupElts (fun g ↦ g • ↑(↑ps index)) ⋯).Nonempty ↔ (↑(↑ps index)).... | [
"G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\nindex : ιₚ\ngroupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries\n⊢ ((Quotient.mk'' (Quotient.out groupElts)).liftOn' (fun g ↦ g • ↑(↑ps index)) ⋯).Nonempty ↔ (↑(↑p... | ← groupElts.out_eq', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Tiling.Tile | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 26
} | {
"line": 230,
"column": 27
} | [
{
"pp": "G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\nindex : ιₚ\ngroupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries\n⊢ (Quotient.liftOn' groupElts (fun g ↦ g • ↑(↑ps index)) ⋯).Finite ↔ (↑(↑ps index)).Fi... | [
"G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\nindex : ιₚ\ngroupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries\n⊢ ((Quotient.mk'' (Quotient.out groupElts)).liftOn' (fun g ↦ g • ↑(↑ps index)) ⋯).Finite ↔ (↑(↑ps ... | ← groupElts.out_eq', | Lean.Elab.Tactic.evalRewriteSeq | null |
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