module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 589,
"column": 57
} | {
"line": 589,
"column": 68
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 12
} | [
{
"pp": "case coe\nV : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nn : ℕ\nhG : G.chromaticNumber ≤ ↑n\nhH : H.chromaticNumber ≤ ↑n\n⊢ (G ⊕g H).chromaticNumber ≤ ↑n",
"usedConstants": [
"Iff.mpr",
"ENat.instNatCast",
"SimpleGraph.chromaticNumber_le_iff_colorable",
"S... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 591,
"column": 58
} | {
"line": 591,
"column": 69
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 595,
"column": 8
} | {
"line": 595,
"column": 19
} | [
{
"pp": "case refine_1.inr.inl.refine_3\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCove... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 596,
"column": 6
} | {
"line": 596,
"column": 17
} | [
{
"pp": "case refine_1.inr.inr\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 244,
"column": 86
} | {
"line": 247,
"column": 7
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nh : G.IsAcyclic\nu v w : V\np : G.Walk u v\nhp : p.IsPath\nhadj : G.Adj v w\nhsupp : w ∈ p.support\n⊢ w = p.penultimate",
"usedConstants": [
"List.mem_reverse._simp_1",
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk.support",
"Membership.mem",... | by
rw [← snd_reverse]
apply h.eq_snd_of_adj_start hp.reverse hadj
simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 603,
"column": 4
} | {
"line": 603,
"column": 28
} | [
{
"pp": "case refine_2\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ : ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right\nleft ri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 316,
"column": 6
} | {
"line": 316,
"column": 73
} | [
{
"pp": "case cons.inr\nV : Type u_1\nG : SimpleGraph V\nhG : G.IsAcyclic\nv w u' v' w✝ : V\nhead : G.Adj u' v'\ntail : G.Walk v' w✝\nih : List.IsChain (fun x1 x2 ↦ x1 ≠ x2) tail.edges → tail.IsPath\nh : List.IsChain (fun x1 x2 ↦ x1 ≠ x2) (cons head tail).edges\nhcc : (∀ y ∈ tail.edges.head?, s(u', v') ≠ y) ∧ L... | have := IsPath.mk' this |>.eq_snd_of_mem_edges (Sym2.eq_swap ▸ hhh) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 335,
"column": 41
} | {
"line": 335,
"column": 52
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 344,
"column": 54
} | {
"line": 344,
"column": 65
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 348,
"column": 49
} | {
"line": 348,
"column": 60
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 356,
"column": 4
} | {
"line": 357,
"column": 45
} | [
{
"pp": "case inr\nV : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝¹ : Nonempty V\ninhabited_h : Inhabited V\nthis✝ : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x... | · rw [Sym2.eq_swap]
exact this y x h.symm (le_of_not_ge h') | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 361,
"column": 56
} | {
"line": 361,
"column": 67
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 414,
"column": 75
} | {
"line": 414,
"column": 85
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : ¬u = v\nhadj : ¬G.Adj u v\nhacyc : (G ⊔ edge u v).IsAcyclic\nhreach : G.Reachable u v\n⊢ s(u, v) ∈ (G ⊔ edge u v).edgeSet",
"usedConstants": [
"False",
"SimpleGraph.edge",
"eq_false",
"Sym2.mk",
"congrArg",
"and_sel... | simp [huv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 414,
"column": 75
} | {
"line": 414,
"column": 85
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : ¬u = v\nhadj : ¬G.Adj u v\nhacyc : (G ⊔ edge u v).IsAcyclic\nhreach : G.Reachable u v\n⊢ s(u, v) ∈ (G ⊔ edge u v).edgeSet",
"usedConstants": [
"False",
"SimpleGraph.edge",
"eq_false",
"Sym2.mk",
"congrArg",
"and_sel... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 414,
"column": 75
} | {
"line": 414,
"column": 85
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : ¬u = v\nhadj : ¬G.Adj u v\nhacyc : (G ⊔ edge u v).IsAcyclic\nhreach : G.Reachable u v\n⊢ s(u, v) ∈ (G ⊔ edge u v).edgeSet",
"usedConstants": [
"False",
"SimpleGraph.edge",
"eq_false",
"Sym2.mk",
"congrArg",
"and_sel... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Hasse | {
"line": 133,
"column": 22
} | {
"line": 135,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nV : Type u_3\ninst✝ : DecidableEq V\nG : SimpleGraph V\nu v : V\nw : G.Walk u v\na b : Fin (w.length + 1)\nh : (pathGraph (w.length + 1)).Adj a b\n⊢ w.toSubgraph.coe.Adj ⟨w.support[a], ⋯⟩ ⟨w.support[b], ⋯⟩",
"usedConstants": [
"_private.Mathlib.Combinatorics.Simple... | by
grind [support_getElem_eq_getVert, Subgraph.coe_adj, pathGraph_adj, toSubgraph_adj_getVert,
Subgraph.Adj.symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 53,
"column": 14
} | {
"line": 53,
"column": 25
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝ : ∃ G, p G\nG : SimpleGraph V\nhp : p G\n⊢ G ∈ {G | p G}",
"usedConstants": [
"instFintypeSimpleGraphOfDecidableEq",
"Eq.mpr",
"Finset.mem_filter._simp_1",
"Finset.univ",
"congrArg",
"Finset",
"C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 55,
"column": 12
} | {
"line": 55,
"column": 23
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝ : ∃ G, p G\nG : SimpleGraph V\nhp : p G\nG' : SimpleGraph V\nhp' : G' ∈ {G | p G}\nh : ∀ x' ∈ {G | p G}, #x'.edgeFinset ≤ #G'.edgeFinset\n⊢ p G'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 55,
"column": 62
} | {
"line": 55,
"column": 73
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\np : SimpleGraph V → Prop\nx✝² : ∃ G, p G\nG : SimpleGraph V\nhp✝ : p G\nG' : SimpleGraph V\nhp' : G' ∈ {G | p G}\nh : ∀ x' ∈ {G | p G}, #x'.edgeFinset ≤ #G'.edgeFinset\nx✝¹ : SimpleGraph V\nx✝ : DecidableRel x✝¹.Adj\nhp : p x✝¹\n⊢ ?m.148 ∈ {G | p G}",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 15
} | [
{
"pp": "n : ℕ\nV : Type u_1\nW : Type u_2\nH : SimpleGraph W\ninst✝ : Fintype V\nhc : Fintype.card V = n\ne : Fin n ≃ V\nG : SimpleGraph (Fin n)\nh : G ∈ {G | H.Free G}\nG' : SimpleGraph V := SimpleGraph.map (⇑e.toEmbedding) G\n⊢ G' ∈ univ ∧ H.Free G",
"usedConstants": [
"instFintypeSimpleGraphOfDeci... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 15
} | [
{
"pp": "n : ℕ\nV : Type u_1\nW : Type u_2\nH : SimpleGraph W\ninst✝ : Fintype V\nhc : Fintype.card V = n\ne : V ≃ Fin n := Fintype.equivFinOfCardEq hc\nG : SimpleGraph V\nh : G ∈ {G | H.Free G}\nG' : SimpleGraph (Fin n) := SimpleGraph.map (⇑e.toEmbedding) G\n⊢ G' ∈ univ ∧ H.Free G",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 446,
"column": 57
} | {
"line": 446,
"column": 68
} | [
{
"pp": "V : Type u_1\nG F : SimpleGraph V\nhle : F ≤ G\nhF : F.IsAcyclic\nh : F.Reachable = G.Reachable\nthis : ¬Maximal (fun F ↦ F ≤ G ∧ F.IsAcyclic) F\nH : SimpleGraph V\nhFH : F < H\nhHG : H ≤ G\nhH : H.IsAcyclic\ne : Sym2 V\nheH : e ∈ H.edgeSet\nheF : e ∉ F.edgeSet\nh_bridge : (F ⊔ fromEdgeSet {e}).IsBridg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 107,
"column": 64
} | {
"line": 107,
"column": 75
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nh : H.Free G\n⊢ G ∈ {G | H.Free G}",
"usedConstants": [
"SimpleGraph.Free",
"Eq.mpr",
"Finset.mem_filter._simp_1",
"Finset.univ",
"congrArg",
"Finset... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 513,
"column": 35
} | {
"line": 513,
"column": 59
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nthis : Fintype V\nh : G.IsTree\n⊢ Nat.card ↑G.edgeSet + 1 = Nat.card V",
"usedConstants": [
"Eq.mpr",
"Fintype.card_ofFinset",
"SimpleGraph.decidableMemEdgeSet",
"Finset.univ",
"congrArg",
"SimpleGraph.Adj",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 45
} | [
{
"pp": "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",
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"SimpleGraph.IsAcyclic",
"Simple... | simp_rw [isAcyclic_iff_forall_adj_isBridge] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 100,
"column": 4
} | {
"line": 101,
"column": 42
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nK : Finset V\nleft✝ : K ⊆ univ\nhn : #K ≤ Fintype.card V\nhG : G.IsTuranMaximal #K\nh : G.CliqueFree #K\n⊢ ∃ a ∈ K, ∃ b ∈ K, a ≠ b ∧ ¬G.Adj a b",
"usedConstants": [
"Finset",
"SimpleGraph.Adj",
"Membe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 60
} | [
{
"pp": "V : Type u_1\ninst✝ : Fintype V\nr : ℕ\nhr : 0 < r\n⊢ ∃ H x, H.IsTuranMaximal r",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Adj",
"DecidableRel",
"Exists",
"id",
"instOfNatNat",
"_private.Mathlib.Combinatorics.SimpleGraph.Extremal.Turan.0.SimpleGraph.exists... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 214,
"column": 31
} | {
"line": 214,
"column": 42
} | [
{
"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\nlarge : Finset V\nhl : large ∈ fp.parts\nsmall : Finset V\nhs : small ∈ fp.parts\nineq : #small + 1 < #large\nw : V\nhw : w ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.EdgeLabeling | {
"line": 202,
"column": 56
} | {
"line": 202,
"column": 79
} | [
{
"pp": "case Adj.h.h.a\nV : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nx y : V\n⊢ (EdgeLabeling.labelGraph G.toTopEdgeLabeling 1).Adj x y ↔ G.Adj x y",
"usedConstants": [
"Eq.mpr",
"False",
"Iff.of_eq",
"congrArg",
"SimpleGraph.Adj",
"Exists",
"Simple... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 278,
"column": 2
} | {
"line": 279,
"column": 60
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn r : ℕ\nf : G ≃g turanGraph n r\nhr : 0 < r\nJ : SimpleGraph V\nw✝ : DecidableRel J.Adj\nj : J.IsTuranMaximal r\ng : J ≃g turanGraph n r\n⊢ G.IsTuranMaximal r",
"usedConstants": [
"SimpleGraph.turanGraph",
... | use (turanGraph_cliqueFree (n := n) hr).comap f.isContained,
fun H _ cf ↦ (f.symm.comp g).card_edgeFinset_eq ▸ j.2 cf | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 33
} | [
{
"pp": "α : Type u\nG : SimpleGraph α\ninst✝ : Fintype α\nn : ℕ\nC : G.Coloring (Fin n)\nt : ℕ\nh : ∀ (c : Fin n), Fintype.card ↑(C.colorClass c) ≤ t\nthis : ∀ (c : Fin n), Nonempty (↑(C.colorClass c) ↪ Fin t)\nF : (c : Fin n) → ↑(C.colorClass c) ↪ Fin t\nc₁ c₂ : Fin n\nv₁ : ↑(C.colorClass c₁)\nv₂ : ↑(C.colorC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 381,
"column": 18
} | {
"line": 381,
"column": 34
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\n⊢ #(K.parts.disjiUnion id ⋯) = r * t",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"SimpleGraph.CompleteEquipartiteSubgraph.disjoint",
"Finset",
"Finset.card_disjiUnion",
... | card_disjiUnion, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite | {
"line": 78,
"column": 28
} | {
"line": 78,
"column": 39
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝⁴ : DecidableEq V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\nG' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\ninst✝¹ : Fintype ↑c'.supp\ninst✝ : DecidablePred fun c ↦ c.supp ⊆ c'.supp\nx : G.ConnectedComponent\nx✝¹ : x ∈ ↑{c | c.supp ⊆ c'.supp}\ny :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite | {
"line": 80,
"column": 29
} | {
"line": 80,
"column": 40
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝⁴ : DecidableEq V\ninst✝³ : Fintype V\ninst✝² : DecidableRel G.Adj\nG' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\ninst✝¹ : Fintype ↑c'.supp\ninst✝ : DecidablePred fun c ↦ c.supp ⊆ c'.supp\n⊢ ↑({c | c.supp ⊆ c'.supp}.disjiUnion (fun c ↦ c.supp.toFinset) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 42,
"column": 4
} | {
"line": 44,
"column": 9
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nC : Set G.ConnectedComponent\n⊢ Set.InjOn G.connectedComponentMk (Quot.out '' C)",
"usedConstants": [
"SimpleGraph.connectedComponentMk",
"Quot.out",
"congrArg",
"Quot.out_eq",
"Membership.mem",
"Eq.mp",
"id",
"SimpleGra... | rintro x ⟨c, ⟨hc, rfl⟩⟩ y ⟨d, ⟨hd, rfl⟩⟩ hxy
simp only [connectedComponentMk] at hxy
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 42,
"column": 4
} | {
"line": 44,
"column": 9
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nC : Set G.ConnectedComponent\n⊢ Set.InjOn G.connectedComponentMk (Quot.out '' C)",
"usedConstants": [
"SimpleGraph.connectedComponentMk",
"Quot.out",
"congrArg",
"Quot.out_eq",
"Membership.mem",
"Eq.mp",
"id",
"SimpleGra... | rintro x ⟨c, ⟨hc, rfl⟩⟩ y ⟨d, ⟨hd, rfl⟩⟩ hxy
simp only [connectedComponentMk] at hxy
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 419,
"column": 6
} | {
"line": 419,
"column": 17
} | [
{
"pp": "case neg.refine_1\nα : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nf : (completeEquipartiteGraph r t).Copy G\nht : ¬t = 0\ni : Fin r\nx✝¹ x✝ : Fin t\nh : (fun j ↦ f (i, j)) x✝¹ = (fun j ↦ f (i, j)) x✝\n⊢ x✝¹ = x✝",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 422,
"column": 57
} | {
"line": 422,
"column": 68
} | [
{
"pp": "α : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nf : (completeEquipartiteGraph r t).Copy G\nht : ¬t = 0\ni₁ i₂ : Fin r\nh :\n ∀ (a : V),\n a ∈ map { toFun := fun j ↦ f (i₁, j), inj' := ⋯ } univ ↔ a ∈ map { toFun := fun j ↦ f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 87,
"column": 10
} | {
"line": 87,
"column": 76
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : Even K.supp.ncard\n⊢ ?m.31 ∉ G.oddComponents",
"usedConstants": [
"SimpleGraph.oddComponents",
"Membership.mem",
"id",
"SimpleGraph.ConnectedComponent",
"Set.in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 86,
"column": 4
} | {
"line": 87,
"column": 86
} | [
{
"pp": "case pos\nV : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : Even K.supp.ncard\n⊢ Even (K.supp \\ s).ncard",
"usedConstants": [
"Eq.mpr",
"congrArg",
"_private.Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents.0.Sim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite | {
"line": 118,
"column": 4
} | {
"line": 119,
"column": 11
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Finite V\nG' : SimpleGraph V\nh : G ≤ G'\nc : G'.ConnectedComponent\nhc : Odd c.supp.ncard\nh' : {c' | Odd c'.supp.ncard ∧ c'.supp ⊆ c.supp}.ncard = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 30
} | [
{
"pp": "case neg\nV : Type u\nG : SimpleGraph V\ns : Set V\nK : G.ConnectedComponent\nhrep : Represents s G.oddComponents\nh : ¬Even K.supp.ncard\nthis : K.supp.ncard ≠ 0\n⊢ Even K.supp.ncard ↔ Even 1",
"usedConstants": [
"Nat.not_even_iff_odd._simp_1",
"Eq.mpr",
"False",
"Nat.not_e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 41
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Subsingleton α\nu : α\n⊢ G.eccent u = 0",
"usedConstants": [
"SimpleGraph.edist_eq_zero_iff._simp_1",
"Eq.mpr",
"instCompleteLinearOrderENat",
"CommSemiring.toSemiring",
"iSup",
"id",
"ConditionallyCompleteLinearOrde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 17
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : SimpleGraph α\nh : G.radius = 0\n⊢ Nonempty α",
"usedConstants": [
"Eq.mpr",
"False",
"iInf",
"instCompleteLinearOrderENat",
"congrArg",
"CommSemiring.toSemiring",
"ENat.iInf_eq_zero._simp_1",
"Exists",
"id",
... | · contrapose! h
simp [radius] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 24
} | [
{
"pp": "case h\nα : Type u_1\nG : SimpleGraph α\nx✝ : Nonempty α ∧ Subsingleton α\nleft✝ : Nonempty α\nright✝ : Subsingleton α\n⊢ G.eccent Classical.ofNonempty = 0",
"usedConstants": [
"SimpleGraph.edist_eq_zero_iff._simp_1",
"Eq.mpr",
"Classical.ofNonempty",
"instCompleteLinearOrde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 73,
"column": 4
} | {
"line": 74,
"column": 11
} | [
{
"pp": "case hf.hm\nW : Type u_1\nH : SimpleGraph W\nn : ℕ\nhn : n ≥ 2\nG : SimpleGraph (Fin (n + 1))\ninst✝ : DecidableRel G.Adj\nh : H.Free G\nv : Fin (n + 1)\nhv : v ∈ univ\n⊢ ↑(#(bipartiteAbove (fun v e ↦ v ∉ e) G.edgeFinset v)) ≤ extremalNumber n H",
"usedConstants": [
"instDecidableNot",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 47
} | [
{
"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)\nhz : ∀ᶠ (x : ℕ) in atTop, H.turanDensity * ↑(x.choose 2) ≠ 0\n⊢ (fun n ↦ ↑(extremalNumber n H)) ~[atTop] fun n ↦ H.turanDensity * ↑(n.choose 2)",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 45
} | [
{
"pp": "W : Type u_1\nH : SimpleGraph W\nε : ℝ\nhε_pos : 0 < ε\nh : ∀ (a : ℕ), ∃ b ≥ a, ∃ G inst, ↑(#G.edgeFinset) ≥ (H.turanDensity + ε) * ↑(b.choose 2) ∧ IsEmpty (H.Copy G)\n⊢ H.turanDensity + ε ≤ sInf {x | ∃ n ∈ Set.Ici 2, ↑(extremalNumber n H) / ↑(n.choose 2) = x}",
"usedConstants": [
"Real",
... | refine le_csInf ?_ (fun x ⟨m, hm, hx⟩ ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Finsubgraph | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 12
} | [
{
"pp": "V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\ninst✝ : Finite W\nh : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F\nval✝ : Fintype W\nthis : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')\n⊢ (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F).obj G'... | intro G' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.SimpleGraph.Finsubgraph | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 12
} | [
{
"pp": "V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\ninst✝ : Finite W\nh : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F\nval✝ : Fintype W\nthis : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')\n⊢ (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F).obj G'... | intro G' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 59
} | [
{
"pp": "case a\nW : Type u_1\nH : SimpleGraph W\nε : ℝ\nhε_pos : 0 < ε\nV : Type u_2\ninst✝¹ : Fintype V\nh_verts : Fintype.card V ≥ H.turanDensityConst ε\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\n⊢ Fintype.card V ≥ Nat.find ⋯",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Girth | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\n⊢ 3 ≤ G.egirth",
"usedConstants": [
"SimpleGraph.le_egirth._simp_1",
"Eq.mpr",
"instCompleteLinearOrderENat",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ChainCompletePartialOrder.instOfCompleteLattice",
"ENat.instNatCa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 18
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\np : Set V\ninst✝ : DecidablePred fun x ↦ x ∈ p\nf : ↑p → V\nh₁ : ∀ (x : ↑p), f x ∉ p\nh₂ : ∀ (x : ↑p), G.Adj (↑x) (f x)\nv w : V\nh : if h : v ∈ p then f ⟨v, h⟩ = w else if h : w ∈ p then f ⟨w, h⟩ = v else False\n⊢ G.Adj v w",
"usedConstants": [
"SimpleGraph.A... | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 51
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\n⊢ (⋃ i ∈ t, s i).ncard ≤ ∑ᶠ (i : ι) (_ : i ∈ t), (s i).ncard",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 51
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\n⊢ (⋃ i ∈ t, s i).encard ≤ ∑ᶠ (i : ι) (_ : i ∈ t), (s i).encard",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 13
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Fintype ι\ns : ι → Set α\n⊢ (⋃ i, s i).ncard ≤ ∑ i, (s i).ncard",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 13
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Fintype ι\ns : ι → Set α\n⊢ (⋃ i, s i).encard ≤ ∑ i, (s i).encard",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Finite ι\ns : ι → Set α\n⊢ (⋃ i, s i).ncard ≤ ∑ᶠ (i : ι), (s i).ncard",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 13
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Finite ι\ns : ι → Set α\n⊢ (⋃ i, s i).encard ≤ ∑ᶠ (i : ι), (s i).encard",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 143,
"column": 11
} | {
"line": 143,
"column": 26
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na b : α\np : G.Walk a b\ninst✝ : Fintype α\nh✝ : Nonempty α\nx✝ : p.IsPath ∧ p.length = Fintype.card α - 1\nhp : p.IsPath\nh : p.length = Fintype.card α - 1\nthis : Injective fun x ↦ p.support.get x\n⊢ Fintype.card α = p.support.length"... | length_support, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 15
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\nβ : Type u_2\ninst✝ : DecidableEq β\nH : SimpleGraph β\na : α\nf : G →g H\nhf : Bijective ⇑f\nx v✝ : α\ny : G.Adj a v✝\np : G.Walk v✝ a\nhp : (cons y p).IsHamiltonianCycle\n__IsCycle✝ : (Walk.map f (cons y p)).IsCycle := IsCycle.map (B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 191,
"column": 2
} | {
"line": 193,
"column": 11
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na : α\np : G.Walk a a\ninst✝ : Fintype α\nhp : p.IsHamiltonianCycle\n⊢ p.length = Fintype.card α",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Walk.length_tail_add_one",
"SimpleGraph.Walk.IsHamiltonianCycle.isHamiltonian_tail"... | rw [← length_tail_add_one hp.not_nil, hp.isHamiltonian_tail.length_eq, Nat.sub_add_cancel]
rw [Nat.succ_le_iff, Fintype.card_pos_iff]
exact ⟨a⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 191,
"column": 2
} | {
"line": 193,
"column": 11
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na : α\np : G.Walk a a\ninst✝ : Fintype α\nhp : p.IsHamiltonianCycle\n⊢ p.length = Fintype.card α",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Walk.length_tail_add_one",
"SimpleGraph.Walk.IsHamiltonianCycle.isHamiltonian_tail"... | rw [← length_tail_add_one hp.not_nil, hp.isHamiltonian_tail.length_eq, Nat.sub_add_cancel]
rw [Nat.succ_le_iff, Fintype.card_pos_iff]
exact ⟨a⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | [
{
"pp": "case h\nV : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nh : M.IsMatching\nhGG' : G ≤ G'\nv✝ : V\nhv✝ : v✝ ∈ (Subgraph.map (Hom.ofLE hGG') M).verts\nw✝ : V\nhv : w✝ ∈ M.verts\nhv' : (Hom.ofLE hGG') w✝ = v✝\nw : V\nhw : (fun w ↦ M.Adj w✝ w) w ∧ ∀ (y : V), (fun w ↦ M.Adj w✝ w) y → y = w\n⊢ (fun w ↦ (S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 77
} | [
{
"pp": "case h.inr\nV : Type u_1\nG : SimpleGraph V\nM M' : G.Subgraph\nhM : M.IsMatching\nhM' : M'.IsMatching\nhd✝ : Disjoint M.support M'.support\nv : V\nhv : v ∈ (M ⊔ M').verts\nN N' : G.Subgraph\nhN : N.IsMatching\nhd : ∀ ⦃a : V⦄, a ∈ N.support → a ∉ N'.support\nhmN : v ∈ N.verts\nw : V\nhw : (fun w ↦ N.Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 77
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\nι : Type u_3\nf : ι → G.Subgraph\nhM : ∀ (i : ι), (f i).IsMatching\nhd : Pairwise fun i j ↦ Disjoint (f i).support (f j).support\nv : V\nhv : v ∈ (⨆ i, f i).verts\ni : ι\nhi : v ∈ (f i).verts\nw : V\nhw : (fun w ↦ (f i).Adj v w) w ∧ ∀ (y : V), (fun w ↦ (f i).A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 212,
"column": 13
} | {
"line": 212,
"column": 37
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nG' : SimpleGraph W\nM : G.Subgraph\nf : G ≃g G'\nh : (Subgraph.map f.toHom M).IsMatching\n⊢ M.IsMatching",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 35
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Fintype V\nh : M.IsPerfectMatching\n⊢ Even (Fintype.card V)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 77
} | [
{
"pp": "case h\nV : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\nh : M.IsMatching\nv✝ : V\nhv : v✝ ∈ M.verts\nw : V\nhvw : M.Adj v✝ w\nhw : ∀ (y : V), (fun w ↦ M.Adj v✝ w) y → y = w\n⊢ (fun w ↦ (M.induce (M.verts ∩ (G.connectedComponentMk v✝).supp)).Adj v✝ w) w ∧\n ∀ (y : V), (fun w ↦ (M.induce (M.verts ∩ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\nh : M.IsPerfectMatching\nc : G.ConnectedComponent\n⊢ (M.induce c.supp).IsMatching",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 290,
"column": 4
} | {
"line": 291,
"column": 32
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\nhMr : M.IsMatching\nhc : G.IsClique M.verts\nhu : M.verts.Finite\n⊢ Even M.verts.ncard",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Set.Finite.toFinset",
"Nat",
"Even",
"Finset.card",
"instAddNat",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 46
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Finite V\nu : Set V\nhM : M.IsPerfectMatching\nc : ↑(⊤.deleteVerts u).coe.oddComponents\nh : ∀ w ∈ u, ∀ (v : ↑(⊤.deleteVerts u).verts), M.Adj (↑v) w → v ∉ (↑c).supp\nhMmatch : (M.induce (Subtype.val '' (↑c).supp)).IsMatching\nthis✝ : Fintype ↑(M.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 80,
"column": 4
} | {
"line": 84,
"column": 50
} | [
{
"pp": "case a\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : Nontrivial V\nh : G.IsSRGWith n k ℓ μ\nht : G ≠ ⊤\nhm : μ ≠ 0\nhc : G.ediam < 2\n⊢ False",
"usedConstants": [
"False",
"instAddMonoidWithOneENat",
"instLinearOrderENat",
... | cases ENat.le_one_iff_eq_zero_or_eq_one.mp (Order.le_of_lt_succ hc) with
| inl hc =>
rw [ediam_eq_zero_iff_subsingleton] at hc
exact false_of_nontrivial_of_subsingleton V
| inr hc => exact ht.elim (ediam_eq_one.mp hc) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 13
} | [
{
"pp": "V : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nv w : V\nh : G.Adj v w\n⊢ ((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \\ ({w} ∪ {v}) = (G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ",
"usedConstants": [
"Finset.disjoint_singleton_right... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 154,
"column": 27
} | {
"line": 154,
"column": 49
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"False",
"and_true",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 154,
"column": 27
} | {
"line": 154,
"column": 49
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 215,
"column": 2
} | {
"line": 216,
"column": 31
} | [
{
"pp": "case a\nV : Type u\ninst✝³ : Fintype V\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝¹ : DecidableEq V\nα : Type u_1\ninst✝ : Semiring α\nh : G.IsSRGWith n k ℓ μ\nv w : V\n⊢ (adjMatrix α G ^ 2) v w = (k • 1 + ℓ • adjMatrix α G + μ • adjMatrix α Gᶜ) v w",
"usedConstants": [
... | simp only [adjMatrix_pow_apply_eq_card_walk, Matrix.add_apply, Matrix.smul_apply,
adjMatrix_apply, compl_adj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Trails | {
"line": 92,
"column": 16
} | {
"line": 92,
"column": 45
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : G.Walk u v\nh : p.IsEulerian\ne : Sym2 V\nhe : e ∈ G.edgeSet\n⊢ e ∈ p.edges",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.UniversalVerts | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 77
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns : Set ↑G.deleteUniversalVerts.verts\n⊢ Disjoint (Subtype.val '' s) G.universalVerts",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Set.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 609,
"column": 47
} | {
"line": 609,
"column": 58
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : G'.Adj v w\nw' : V\nhw' : w ≠ w' ∧ G'.Adj v w'\n⊢ M.Adj v w ↔ ¬M.Adj v w'",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 617,
"column": 53
} | {
"line": 617,
"column": 64
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : G'.Adj v w\nw' : V\nhw' : w ≠ w' ∧ G'.Adj v w'\nhmadj : M.Adj v w ↔ ¬M.Adj v w'\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 624,
"column": 53
} | {
"line": 624,
"column": 64
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : ¬G'.Adj v w\ny : V\nhr : G'.Adj v y ∧ ¬M.Adj v y\nw' : V\nhw' : y ≠ w' ∧ G'.Adj v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 29
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nM' : G'.Subgraph\nhM : M.IsPerfectMatching\nhM' : M'.IsPerfectMatching\n⊢ (M.spanningCoe ∆ M'.spanningCoe).IsAlternating M'.spanningCoe",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 364,
"column": 10
} | {
"line": 364,
"column": 25
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 21
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 79,
"column": 4
} | {
"line": 80,
"column": 11
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nf : G ≃g H\nc : Set W\nh : G.IsVertexCover (⇑f ⁻¹' c)\n⊢ H.IsVertexCover c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 17
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nh : G.vertexCoverNum = 0\n⊢ G = ⊥",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 64
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\n⊢ G.IsVertexCover (Set.univ \\ {x})",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.vertexCoverNum_le_card_sub_one._proof_1_4"
... | grind [IsVertexCover, Adj.ne'] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 64
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\n⊢ G.IsVertexCover (Set.univ \\ {x})",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.vertexCoverNum_le_card_sub_one._proof_1_4"
... | grind [IsVertexCover, Adj.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 64
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\n⊢ G.IsVertexCover (Set.univ \\ {x})",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.vertexCoverNum_le_card_sub_one._proof_1_4"
... | grind [IsVertexCover, Adj.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 65
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\na✝ : Nontrivial V\nx : V\nn : ℕ\nhn : ∀ (i : Set V), G.IsVertexCover i → ↑n ≤ i.encard\nthis : ↑n ≤ (Set.univ \\ {x}).encard\n⊢ ↑n ≤ ENat.card V - 1",
"usedConstants": [
"Eq.mpr",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"instSubENat",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 400,
"column": 12
} | {
"line": 400,
"column": 52
} | [
{
"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... | rw [← hw.card_inter, card_eq_zero] at hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Tiling.Tile | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 49
} | [
{
"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\ng : G\npt : PlacedTile ps\na b : G\nr : a⁻¹ * b ∈ Subgroup.map (MulAction.stabilizer G ↑(↑ps pt.index)).subtype (↑ps pt.index).symmetries\n⊢ ((fun h ↦ { index := pt.index, groupElts := ↑(g * h) }) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Young.SemistandardTableau | {
"line": 71,
"column": 4
} | {
"line": 73,
"column": 9
} | [
{
"pp": "μ : YoungDiagram\nT T' : SemistandardYoungTableau μ\nh : T.entry = T'.entry\n⊢ T = T'",
"usedConstants": [
"YoungDiagram",
"SemistandardYoungTableau.mk",
"Membership.mem",
"Eq.rec",
"Prod.mk",
"instOfNatNat",
"LE.le",
"instLENat",
"Nat",
"... | cases T
cases T'
congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Young.SemistandardTableau | {
"line": 71,
"column": 4
} | {
"line": 73,
"column": 9
} | [
{
"pp": "μ : YoungDiagram\nT T' : SemistandardYoungTableau μ\nh : T.entry = T'.entry\n⊢ T = T'",
"usedConstants": [
"YoungDiagram",
"SemistandardYoungTableau.mk",
"Membership.mem",
"Eq.rec",
"Prod.mk",
"instOfNatNat",
"LE.le",
"instLENat",
"Nat",
"... | cases T
cases T'
congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Young.YoungDiagram | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 54
} | [
{
"pp": "case cells.h\nμ : YoungDiagram\ni j : ℕ\n⊢ (∃ (_ : i < μ.colLen 0), j < μ.rowLen i) ↔ (i, j) ∈ μ",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Young.YoungDiagram.0.YoungDiagram.ofRowLens_to_rowLens_eq_self._simp_1_4",
"YoungDiagram",
"Iff.of_eq",
"cong... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 78
} | [
{
"pp": "case refine_4\nV : Type u_1\nG G' : SimpleGraph V\nx b a c : V\nM : (G ⊔ edge a c).Subgraph\np : G'.Walk a x\nhp : p.IsPath\nhcalt : G'.IsAlternating M.spanningCoe\nhM2nadj : ¬M.Adj x a\nhpac : p.toSubgraph.Adj a c\nhnpxb : ¬p.toSubgraph.Adj x b\nhM2ac : M.Adj a c\nhgadj : G.Adj x a\nhnxc : x ≠ c\nhnab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 133,
"column": 29
} | {
"line": 134,
"column": 42
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nhveven : Even (Nat.card V)\nh : ¬G.IsTutteViolator G.universalVerts\nh' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp\nval✝ : Fintype V\nM : G.Subgraph\nhM : M.IsMatching\nhsub : M.vertsᶜ ⊆ G.univer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 142,
"column": 47
} | {
"line": 142,
"column": 58
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nhodd : Odd (Nat.card V)\n⊢ Odd (Nat.card ↑(⊤.deleteVerts ∅).verts)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
"Set.univ",
"Odd",
"Set.Elem",
"id",
"Nat.card",
"SimpleGrap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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