module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 168,
"column": 31
} | {
"line": 168,
"column": 53
} | {
"line": 168,
"column": 53
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv w : V\nh✝ : v ≠ w\nh : edge v w ≤ G\n⊢ (edge v w).Adj v w",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"False",
"SimpleGraph.edge",
"eq_false",
"congrArg",
"and_self",
"SimpleGraph.Adj",
"false_and",
... | [] | by simp_all [edge_adj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 128,
"column": 96
} | {
"line": 130,
"column": 90
} | {
"line": 132,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\nh : n < p.darts.length\n⊢ p.darts[n] = { fst := p.getVert n, snd := p.getVert (n + 1), adj := ⋯ }",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.snd_darts_getElem",
"congrArg",
"Sim... | [] | by
rw [p.length_darts] at h
ext <;> simp [p.getVert_eq_support_getElem (le_of_lt h), p.getVert_eq_support_getElem h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 209,
"column": 19
} | {
"line": 209,
"column": 28
} | {
"line": 211,
"column": 0
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\nih : p✝.reverse.reverse = p✝\n⊢ (cons h✝ p✝).reverse.reverse = cons h✝ p✝",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj.symm",
"congrArg",
"SimpleG... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 209,
"column": 19
} | {
"line": 209,
"column": 28
} | {
"line": 211,
"column": 0
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\nih : p✝.reverse.reverse = p✝\n⊢ (cons h✝ p✝).reverse.reverse = cons h✝ p✝",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj.symm",
"congrArg",
"SimpleG... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 209,
"column": 19
} | {
"line": 209,
"column": 28
} | {
"line": 211,
"column": 0
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\nih : p✝.reverse.reverse = p✝\n⊢ (cons h✝ p✝).reverse.reverse = cons h✝ p✝",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj.symm",
"congrArg",
"SimpleG... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 13
} | {
"line": 226,
"column": 6
} | [
{
"pp": "case succ.cons\nV : Type u_1\nG : SimpleGraph V\nu v : V\nn k✝ k : ℕ\nh : n ≤ k\nv✝ : V\nh✝ : G.Adj u v✝\np✝ : G.Walk v✝ v\nih : ((cons h✝ p✝).take n).IsSubwalk ((cons h✝ p✝).take k)\n⊢ ((cons h✝ p✝).take k).IsSubwalk ((cons h✝ p✝).take (k + 1))",
"ppTerm": "?succ.cons",
"assigned": true,
"... | [
"case succ.cons.zero\nV : Type u_1\nG : SimpleGraph V\nu v : V\nn k : ℕ\nv✝ : V\nh✝ : G.Adj u v✝\np✝ : G.Walk v✝ v\nh : n ≤ 0\nih : ((cons h✝ p✝).take n).IsSubwalk ((cons h✝ p✝).take 0)\n⊢ ((cons h✝ p✝).take 0).IsSubwalk ((cons h✝ p✝).take (0 + 1))",
"case succ.cons.succ\nV : Type u_1\nG : SimpleGraph V\nu v : V\... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 200,
"column": 47
} | {
"line": 201,
"column": 91
} | {
"line": 203,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w u : V\ninst✝ : DecidableEq V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.dropUntil u h).support <:+ p.support",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"List.suffix_refl",
"Eq.mpr",
"List.drop_suffix",
"congrArg",
... | [] | by
grw [dropUntil_eq_drop, support_copy, drop_support_eq_support_drop_min, List.drop_suffix] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 77
} | {
"line": 223,
"column": 4
} | [
{
"pp": "α : Type u_1\nG H : SimpleGraph α\ninst✝³ : Fintype α\ninst✝² : DecidableRel G.Adj\ninst✝¹ : DecidableRel H.Adj\ninst✝ : DecidableEq α\ntris : Finset (Finset α)\nhtris : tris ⊆ G.cliqueFinset 3\npd : (↑tris).Pairwise fun x y ↦ (↑x ∩ ↑y).Subsingleton\nhHG : H ≤ G\nhH : H.CliqueFree 3\nhG : #(G.edgeFinse... | [
"α : Type u_1\nG H : SimpleGraph α\ninst✝³ : Fintype α\ninst✝² : DecidableRel G.Adj\ninst✝¹ : DecidableRel H.Adj\ninst✝ : DecidableEq α\ntris : Finset (Finset α)\nhtris : tris ⊆ G.cliqueFinset 3\npd : (↑tris).Pairwise fun x y ↦ (↑x ∩ ↑y).Subsingleton\nhHG : H ≤ G\nhH : H.CliqueFree 3\nhG : #(G.edgeFinset \\ H.edgeF... | simp only [not_and, mem_sdiff, not_not, mem_edgeFinset, mem_edgeSet] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 335,
"column": 96
} | {
"line": 340,
"column": 100
} | {
"line": 342,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\n⊢ (∃ s, G.IsNClique 3 s) ↔ ∃ u w, w.IsCycle ∧ w.length = 3",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Adj.symm",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.is3Clique_iff_exists_cyc... | [] | by
classical
simp_rw [is3Clique_iff, isCycle_def]
exact
⟨(fun ⟨_, a, _, _, hab, hac, hbc, _⟩ => ⟨a, cons hab (cons hbc (cons hac.symm nil)), by aesop⟩),
(fun ⟨_, .cons hab (.cons hbc (.cons hca nil)), _, _⟩ => ⟨_, _, _, _, hab, hca.symm, hbc, rfl⟩)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 82
} | {
"line": 528,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\ns t : α\nh : ¬(G.replaceVertex s t).CliqueFree n\nφ : Fin n ↪ α\nhφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b\nmt : ∀ (x : Fin n), φ x ≠ t\n⊢ ∀ {a b : Fin n}, G.Adj (φ a) (φ b) ↔ (c... | [
"case neg\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\ns t : α\nh : ¬(G.replaceVertex s t).CliqueFree n\nφ : Fin n ↪ α\nhφ : ∀ {a b : Fin n}, G.Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b\nmt : ∀ (x : Fin n), φ x ≠ t\n⊢ ∀ {a b : Fin n}, G.Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b"
] | conv at hφ => enter [a, b]; rw [G.adj_replaceVertex_iff_of_ne _ (mt a) (mt b)] | Lean.Elab.Tactic.Conv.evalConv | Lean.Parser.Tactic.Conv.conv |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 792,
"column": 90
} | {
"line": 799,
"column": 57
} | {
"line": 801,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np q : G.Walk u v\nhp : p.IsPath\nhq : q.IsPath\nh : p ≠ q\n⊢ ∃ w ∈ p.support, w ∈ q.support ∧ ∃ c, c.IsCycle ∧ c.support <+ p.support ++ q.support.reverse.tail",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",... | [] | by
have ⟨u', v', p', q', hp', hq', hcyc⟩ := hp.exists_isCycle_of_ne hq h
use u', hp'.support_subset p'.start_mem_support, hq'.support_subset q'.start_mem_support
refine ⟨_, hcyc, ?_⟩
rw [support_append, support_reverse]
refine .append ?_ <| .tail <| .reverse ?_
· exact isSubwalk_iff_support_isInfix.mp hp' |... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 846,
"column": 91
} | {
"line": 853,
"column": 11
} | {
"line": 855,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\n⊢ p.bypass.support <+ p.support",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.cons_sublist_cons._simp_1",
"instLawfulBEq",
"congrArg",
"SimpleGraph.Adj",
... | [] | by
induction p with
| nil => simp!
| cons _ _ ih =>
dsimp! only
split_ifs
· exact support_dropUntil_suffix_support .. |>.sublist.trans ih |>.cons _
· simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 873,
"column": 6
} | {
"line": 873,
"column": 73
} | {
"line": 874,
"column": 4
} | [
{
"pp": "case pos\nV : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nu✝ v✝ w✝ : V\nh✝¹ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\nih : p✝.bypass.darts <+ p✝.darts\nh✝ : u✝ ∈ p✝.bypass.support\n⊢ (p✝.bypass.dropUntil u✝ h✝).darts <+ { fst := u✝, snd := v✝, adj := h✝¹ } :: p✝.darts",
"ppTerm": "?pos✝",
... | [] | exact darts_dropUntil_suffix_darts .. |>.sublist.trans ih |>.cons _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 873,
"column": 6
} | {
"line": 873,
"column": 73
} | {
"line": 874,
"column": 4
} | [
{
"pp": "case pos\nV : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nu✝ v✝ w✝ : V\nh✝¹ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\nih : p✝.bypass.darts <+ p✝.darts\nh✝ : u✝ ∈ p✝.bypass.support\n⊢ (p✝.bypass.dropUntil u✝ h✝).darts <+ { fst := u✝, snd := v✝, adj := h✝¹ } :: p✝.darts",
"ppTerm": "?pos✝",
... | [] | exact darts_dropUntil_suffix_darts .. |>.sublist.trans ih |>.cons _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 873,
"column": 6
} | {
"line": 873,
"column": 73
} | {
"line": 874,
"column": 4
} | [
{
"pp": "case pos\nV : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nu✝ v✝ w✝ : V\nh✝¹ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\nih : p✝.bypass.darts <+ p✝.darts\nh✝ : u✝ ∈ p✝.bypass.support\n⊢ (p✝.bypass.dropUntil u✝ h✝).darts <+ { fst := u✝, snd := v✝, adj := h✝¹ } :: p✝.darts",
"ppTerm": "?pos✝",
... | [] | exact darts_dropUntil_suffix_darts .. |>.sublist.trans ih |>.cons _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 59,
"column": 4
} | {
"line": 62,
"column": 27
} | {
"line": 63,
"column": 2
} | [] | [
"K : Type u_1\nσ : Type u_2\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : f.totalDegree < (q - 1) * Fintype.card σ\nthis : DecidableEq K\n⊢ ∀ x ∈ f.support, ∑ x_1, coeff x f * ∏ i, x_1 i ^ x i = 0"
] | ∑ x, eval x f = ∑ x : σ → K, ∑ d ∈ f.support, f.coeff d * ∏ i, x i ^ d i := by
simp only [eval_eq']
_ = ∑ d ∈ f.support, ∑ x : σ → K, f.coeff d * ∏ i, x i ^ d i := sum_comm
_ = 0 := sum_eq_zero ?_ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Data.Finset.Slice | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 58
} | {
"line": 106,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nh𝒜 : Set.Sized r ↑𝒜\n⊢ #𝒜 ≤ #(powersetCard r univ)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.Sized",
"Finset.univ",
"Finset",
"Finset.card_le_card",
"Finset... | [] | exact card_le_card (subset_powersetCard_univ_iff.mpr h𝒜) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Derangements.Basic | {
"line": 136,
"column": 2
} | {
"line": 144,
"column": 35
} | {
"line": 145,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ fiber (some a) → f ∈ {f | fixedPoints ⇑f ⊆ {a}}",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"HEq.refl",
"False.elim",
"setOf",
... | [
"case mpr\nα : Type u_1\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ {f | fixedPoints ⇑f ⊆ {a}} → f ∈ fiber (some a)"
] | · rw [RemoveNone.mem_fiber]
rintro ⟨F, F_derangement, F_none, rfl⟩ x x_fixed
rw [mem_fixedPoints_iff] at x_fixed
apply_fun some at x_fixed
rcases Fx : F (some x) with - | y
· rwa [removeNone_none F Fx, F_none, Option.some_inj, eq_comm] at x_fixed
· exfalso
rw [removeNone_some F ⟨y, Fx⟩] at... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 187,
"column": 10
} | {
"line": 187,
"column": 34
} | {
"line": 187,
"column": 35
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na✝ b✝ c✝ d✝ x y : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₁ : ∀ x ∈ A, ∀ y ∈ A, 1 / 2 * ↑(#A) < ↑(#(x •> A ∩ y •> A))\na : G\nha : a⁻¹ ∈ A\nb : G\nhb : b ∈ A\nc : G\nhc : c⁻¹ ∈ A\nd : G\nhd : d ∈ A\nh₂ : 1 / 2 *... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na✝ b✝ c✝ d✝ x y : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₁ : ∀ x ∈ A, ∀ y ∈ A, 1 / 2 * ↑(#A) < ↑(#(x •> A ∩ y •> A))\na : G\nha : a⁻¹ ∈ A\nb : G\nhb : b ∈ A\nc : G\nhc : c⁻¹ ∈ A\nd : G\nhd : d ∈ A\nh₂ : 1 / 2 * ↑(#A) < ↑(#... | ← Nat.cast_inj (R := ℚ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 236,
"column": 37
} | {
"line": 236,
"column": 69
} | {
"line": 236,
"column": 70
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\na : G\nha : a ∈ A\n⊢ ↑A⁻¹ * ↑A * (↑A⁻¹ * ↑A) = ↑A⁻¹ * ↑A",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\na : G\nha : a ∈ A\n⊢ ↑(A.invMulSubgroup h) * ↑(A.invMulSubgroup h) = ↑(A.invMulSubgroup h)"
] | ← invMulSubgroup_eq_inv_mul _ h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 255,
"column": 37
} | {
"line": 255,
"column": 69
} | {
"line": 255,
"column": 70
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\na : G\nha : a ∈ A\nz b : G\nhb : b ∈ A\nc : G\nhc : c ∈ A\nhz : a * (b⁻¹ * c * a) = z\nl : Finset G := A ∩ (z * a⁻¹) •> (A⁻¹ * A)\nr : Finset G := a •> (A⁻¹ * A) ∩ z •> A⁻¹\n⊢ ↑A⁻¹ * ↑A * (↑A⁻¹ * ↑A) =... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\na : G\nha : a ∈ A\nz b : G\nhb : b ∈ A\nc : G\nhc : c ∈ A\nhz : a * (b⁻¹ * c * a) = z\nl : Finset G := A ∩ (z * a⁻¹) •> (A⁻¹ * A)\nr : Finset G := a •> (A⁻¹ * A) ∩ z •> A⁻¹\n⊢ ↑(A.invMulSubgroup h) * ↑(A.invMulSub... | ← invMulSubgroup_eq_inv_mul _ h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 22
} | {
"line": 191,
"column": 22
} | [
{
"pp": "p : DyckWord\ni : ℕ\nlb : 0 < i\nub : i < 1 + (↑p).length + 1\n⊢ count D (List.take (i - 1) ↑p) < count U (List.take (i - 1) ↑p) + 1",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"DyckStep.U",
"congrArg",
"HSub.h... | [
"p : DyckWord\ni : ℕ\nlb : 0 < i\nub : i < 1 + (↑p).length + 1\n⊢ count D (List.take (i - 1) ↑p) ≤ count U (List.take (i - 1) ↑p)"
] | Nat.lt_add_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 186,
"column": 80
} | {
"line": 192,
"column": 30
} | {
"line": 192,
"column": 30
} | [
{
"pp": "p : DyckWord\ni : ℕ\nlb : 0 < i\nub : i < (↑p.nest).length\n⊢ count D (List.take i ↑p.nest) < count U (List.take i ↑p.nest)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"False",
"instLawfulBEq",
"DyckStep.U",
... | [] | by
simp_rw [nest, length_append, length_singleton] at ub ⊢
rw [take_append_of_le_length (by rw [singleton_append, length_cons]; lia),
take_append, take_of_length_le (by rw [length_singleton]; lia),
length_singleton, singleton_append, count_cons_of_ne (by simp), count_cons_self,
Nat.lt_add_one_iff]
exa... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 312,
"column": 23
} | {
"line": 312,
"column": 55
} | {
"line": 312,
"column": 56
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nha : a ∈ A\n⊢ ↑A⁻¹ * (↑A * (↑A⁻¹ * ↑A)) = ↑A * ↑A⁻¹",
"ppTerm": "?m.378",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"DivI... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nha : a ∈ A\n⊢ ↑A⁻¹ * (↑A * ↑(A.invMulSubgroup h)) = ↑A * ↑A⁻¹"
] | ← invMulSubgroup_eq_inv_mul _ h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 313,
"column": 10
} | {
"line": 313,
"column": 42
} | {
"line": 313,
"column": 43
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nha : a ∈ A\n⊢ ↑A⁻¹ * ↑A * ↑(A.invMulSubgroup h) = ↑A * ↑A⁻¹",
"ppTerm": "?m.389",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nha : a ∈ A\n⊢ ↑(A.invMulSubgroup h) * ↑(A.invMulSubgroup h) = ↑A * ↑A⁻¹"
] | ← invMulSubgroup_eq_inv_mul _ h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 73
} | {
"line": 392,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA : Finset G\nhK₁ : 1 < K\nhKφ : K < φ\nhA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)\nhA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)\nK_pos hK₀ : 0 < K\nhKφ' : 0 < φ - K\nhKψ' : 0 < K - ψ\nhK₂' : 0 < 2 - K\n⊢ ∃ H x Z, ↑(#Z) ≤ (2 - K) * K / ((φ - K) * (K - ψ)) ∧ ↑H * ↑Z... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA : Finset G\nhK₁ : 1 < K\nhKφ : K < φ\nhA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)\nhA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)\nK_pos hK₀ : 0 < K\nhKφ' : 0 < φ - K\nhKψ' : 0 < K - ψ\nhK₂' : 0 < 2 - K\nconst_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))\n⊢ ∃ H x Z, ↑(#Z) ≤ (2 -... | have const_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ)) := by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 276,
"column": 20
} | {
"line": 276,
"column": 66
} | {
"line": 277,
"column": 2
} | [
{
"pp": "F : Type u_1\n𝕜 : Type u_2\n𝕝 : Type u_3\n𝕞 : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : Preorder α\ninst✝⁴ : LocallyFiniteOrder α\ninst✝³ : DecidableEq α\ninst✝² : Semiring 𝕜\ninst✝¹ : Semiring 𝕝\ninst✝ : Module 𝕜 𝕝\nf : IncidenceAlgebra 𝕝 α\n⊢ 0 • f = 0",
"ppTerm": "?m.153",
"assi... | [] | ext; exact sum_eq_zero fun x _ ↦ zero_smul _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 276,
"column": 20
} | {
"line": 276,
"column": 66
} | {
"line": 277,
"column": 2
} | [
{
"pp": "F : Type u_1\n𝕜 : Type u_2\n𝕝 : Type u_3\n𝕞 : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : Preorder α\ninst✝⁴ : LocallyFiniteOrder α\ninst✝³ : DecidableEq α\ninst✝² : Semiring 𝕜\ninst✝¹ : Semiring 𝕝\ninst✝ : Module 𝕜 𝕝\nf : IncidenceAlgebra 𝕝 α\n⊢ 0 • f = 0",
"ppTerm": "?m.153",
"assi... | [] | ext; exact sum_eq_zero fun x _ ↦ zero_smul _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 455,
"column": 64
} | {
"line": 465,
"column": 61
} | {
"line": 467,
"column": 0
} | [
{
"pp": "p q : DyckWord\npq : p ≤ q\n⊢ p.semilength ≤ q.semilength",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"DyckWord.insidePart_zero",
"DyckWord.semilength_outsidePart_lt",
"le_refl",
"congrArg",
"Preorder.toLE",
"Eq.rec",
"Eq.mp",
"D... | [] | by
induction pq with
| refl => rfl
| tail _ mq ih =>
rename_i m r _
rcases eq_or_ne r 0 with rfl | hr
· rw [insidePart_zero, outsidePart_zero, or_self] at mq
rwa [mq] at ih
· rcases mq with hm | hm
· exact ih.trans (hm ▸ semilength_insidePart_lt hr).le
· exact ih.trans (hm ▸ semi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 491,
"column": 17
} | {
"line": 491,
"column": 41
} | {
"line": 491,
"column": 41
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\nhab : a ≠ b\n⊢ (mu' 𝕜) a b = -∑ x ∈ Ioc a b, (mu' 𝕜) x b",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toN... | [
"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\nhab : a ≠ b\n⊢ -∑ x ∈ Ioc a b, (mu' 𝕜) x b = -∑ x ∈ Ioc a b, (mu' 𝕜) x b"
] | mu'_eq_sum_Ioc_of_ne hab | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 14
} | {
"line": 98,
"column": 2
} | [
{
"pp": "case zero\nhn : 1 < 0\n⊢ (0 + 1).smallSchroder = 3 * smallSchroder 0 + 2 * ∑ i ∈ Ioo 0 (0 - 1), (i + 1).smallSchroder * (0 - i).smallSchroder",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"False",
"Nat.instMulZeroClass",
"LinearOrde... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 14
} | {
"line": 98,
"column": 2
} | [
{
"pp": "case zero\nhn : 1 < 0\n⊢ (0 + 1).smallSchroder = 3 * smallSchroder 0 + 2 * ∑ i ∈ Ioo 0 (0 - 1), (i + 1).smallSchroder * (0 - i).smallSchroder",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"False",
"Nat.instMulZeroClass",
"LinearOrde... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 14
} | {
"line": 98,
"column": 2
} | [
{
"pp": "case zero\nhn : 1 < 0\n⊢ (0 + 1).smallSchroder = 3 * smallSchroder 0 + 2 * ∑ i ∈ Ioo 0 (0 - 1), (i + 1).smallSchroder * (0 - i).smallSchroder",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"False",
"Nat.instMulZeroClass",
"LinearOrde... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 14
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case succ.zero\nhn : 1 < 0 + 1\n⊢ (0 + 1 + 1).smallSchroder =\n 3 * (0 + 1).smallSchroder + 2 * ∑ i ∈ Ioo 0 (0 + 1 - 1), (i + 1).smallSchroder * (0 + 1 - i).smallSchroder",
"ppTerm": "?succ.zero",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"congrArg",
... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 14
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case succ.zero\nhn : 1 < 0 + 1\n⊢ (0 + 1 + 1).smallSchroder =\n 3 * (0 + 1).smallSchroder + 2 * ∑ i ∈ Ioo 0 (0 + 1 - 1), (i + 1).smallSchroder * (0 + 1 - i).smallSchroder",
"ppTerm": "?succ.zero",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"congrArg",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 14
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case succ.zero\nhn : 1 < 0 + 1\n⊢ (0 + 1 + 1).smallSchroder =\n 3 * (0 + 1).smallSchroder + 2 * ∑ i ∈ Ioo 0 (0 + 1 - 1), (i + 1).smallSchroder * (0 + 1 - i).smallSchroder",
"ppTerm": "?succ.zero",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"congrArg",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 627,
"column": 15
} | {
"line": 627,
"column": 39
} | {
"line": 629,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\nβ : Type u_6\ninst✝⁴ : Ring 𝕜\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : DecidableLE α\ninst✝ : DecidableLE β\nx y : α × β\nhxy : x ≤ y\n⊢ ((zeta 𝕜).prod (zeta 𝕜)) x y = (zeta 𝕜) x y",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"H... | [] | simp [hxy, hxy.1, hxy.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 132,
"column": 4
} | {
"line": 134,
"column": 24
} | {
"line": 136,
"column": 0
} | [
{
"pp": "case refine_3\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\n⊢ ∀ (x : ↑(Function.mulSupport fun i ↦ 1 - (X ^ (i + 1)) ^ m)),\n (if m ∣ (↑x + 1) * m - 1 + 1 then 1 * (1 ... | [] | intro i
have : (i + 1) * m - 1 + 1 = (i + 1) * m := by grind
simp [this, pow_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 132,
"column": 4
} | {
"line": 134,
"column": 24
} | {
"line": 136,
"column": 0
} | [
{
"pp": "case refine_3\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\n⊢ ∀ (x : ↑(Function.mulSupport fun i ↦ 1 - (X ^ (i + 1)) ^ m)),\n (if m ∣ (↑x + 1) * m - 1 + 1 then 1 * (1 ... | [] | intro i
have : (i + 1) * m - 1 + 1 = (i + 1) * m := by grind
simp [this, pow_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | {
"line": 167,
"column": 66
} | {
"line": 169,
"column": 90
} | {
"line": 171,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : CommRing α\ns : Finset α\ninst✝¹ : Fact (IsUnit 2)\nhs : ThreeAPFree ↑s\ninst✝ : DecidableEq α\n⊢ #(graph (triangleIndices s)).edgeFinset = 3 * Fintype.card α * #s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sem... | [] | by
haveI := noAccidental hs
rw [(locallyLinear hs).card_edgeFinset, card_triangles, card_triangleIndices, mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Graph.Maps | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 20
} | {
"line": 83,
"column": 0
} | [
{
"pp": "α : Type u_1\nα' : Type u_2\nβ : Type u_4\nG : Graph α β\nu v : α\nf : α → α'\ne : β\nh : G.IsLink e u v\n⊢ (map f G).Adj (f u) (f v)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Graph.IsLink.map",
"Graph.IsLink",
"Exists.intro",
"Graph.map"
],
... | [] | exact ⟨e, h.map f⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Hindman | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 12
} | {
"line": 133,
"column": 13
} | [
{
"pp": "case h\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm✝ : M\na : Stream' M\nm : M\nh✝ : FP a.tail m\nn : ℕ\nhn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail\n⊢ ∀ m' ∈ FP (Stream'.drop (n + 1) a), m * m' ∈ FP a",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [],
"use... | [
"case h\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm✝ : M\na : Stream' M\nm : M\nh✝ : FP a.tail m\nn : ℕ\nhn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail\nm' : M\n⊢ m' ∈ FP (Stream'.drop (n + 1) a) → m * m' ∈ FP a"
] | intro m' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.Hindman | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 12
} | {
"line": 138,
"column": 13
} | [
{
"pp": "case h\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm✝ : M\na : Stream' M\nm : M\nh✝ : FP a.tail m\nn : ℕ\nhn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail\n⊢ ∀ m' ∈ FP (Stream'.drop (n + 1) a), a.head * m * m' ∈ FP a",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [],... | [
"case h\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm✝ : M\na : Stream' M\nm : M\nh✝ : FP a.tail m\nn : ℕ\nhn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail\nm' : M\n⊢ m' ∈ FP (Stream'.drop (n + 1) a) → a.head * m * m' ∈ FP a"
] | intro m' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.Hindman | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 34
} | {
"line": 192,
"column": 2
} | [
{
"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 * ... | [] | intro m hm
exact this _ m hm ⟨s₀, sU⟩ rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Hindman | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 34
} | {
"line": 192,
"column": 2
} | [
{
"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 * ... | [] | intro m hm
exact this _ m hm ⟨s₀, sU⟩ rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.HalesJewett | {
"line": 507,
"column": 2
} | {
"line": 507,
"column": 37
} | {
"line": 508,
"column": 2
} | [
{
"pp": "α : Type u_5\nκ : Type u_6\nη : Type u_7\ninst✝² : Finite α\ninst✝¹ : Finite κ\ninst✝ : Finite η\nι : Type\nιfin : Fintype ι\nhι : ∀ (C : (ι → α) → κ), ∃ l, IsMono C l\n⊢ ∃ n, ∀ (C : (Fin n → α) → κ), ∃ l, IsMono C l",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Combinato... | [
"α : Type u_5\nκ : Type u_6\nη : Type u_7\ninst✝² : Finite α\ninst✝¹ : Finite κ\ninst✝ : Finite η\nι : Type\nιfin : Fintype ι\nhι : ∀ (C : (ι → α) → κ), ∃ l, IsMono C l\nC : (Fin (Fintype.card ι) → α) → κ\n⊢ ∃ l, IsMono C l"
] | refine ⟨Fintype.card ι, fun C ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 95,
"column": 49
} | {
"line": 95,
"column": 68
} | {
"line": 95,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC : Set α\n⊢ M✶ ＼ C = M✶ ↔ Disjoint C M.E",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Matroid... | [
"α : Type u_1\nM : Matroid α\nC : Set α\n⊢ Disjoint C M✶.E ↔ Disjoint C M.E"
] | delete_eq_self_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 48
} | {
"line": 182,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.IsBasis I X\nJ : Set α\nhJ : J ⊆ (M✶ ＼ I ＼ (X \\ I)).E\nhss : X \\ I ⊆ (M✶ ＼ I).coloops\n⊢ (M✶ ＼ I).Indep J ∧ Disjoint J (X \\ I) ↔ Disjoint J (X \\ I) ∧ (M✶ ＼ I).Indep (J ∪ X \\ I)",
"ppTerm": "?m.140",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.IsBasis I X\nJ : Set α\nhJ : J ⊆ (M✶ ＼ I ＼ (X \\ I)).E\nhss : X \\ I ⊆ (M✶ ＼ I).coloops\n⊢ (M✶ ＼ I).Indep J ∧ Disjoint J (X \\ I) ↔ Disjoint J (X \\ I) ∧ (M✶ ＼ I).Indep J"
] | union_indep_iff_indep_of_subset_coloops hss, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 197,
"column": 77
} | {
"line": 201,
"column": 58
} | {
"line": 202,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\nthis : ∀ ⦃K : Set α⦄, Disjoint K J → M.Indep (K ∪ J) → K ⊆ X → I ⊆ K ∪ J → K ⊆ I\n⊢ (M ／ J).IsBasis' (I \\ J) (X \\ J)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplet... | [] | by
simpa +contextual [IsBasis', (hIX.indep.subset hJI).contract_indep_iff,
subset_sdiff, maximal_subset_iff, disjoint_sdiff_left,
union_eq_self_of_subset_right hJI, hIX.indep, sdiff_subset.trans hIX.subset,
sdiff_subset_iff, subset_antisymm_iff, union_comm J] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 42
} | {
"line": 553,
"column": 0
} | [
{
"pp": "α : Type u_1\nR C : Set α\nM : Matroid α\nhCR : C ⊆ R\n⊢ (M ↾ R) ／ C = (M ↾ (R \\ C ∪ C)) ／ C",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.union_eq_self_of_subset_right",
"Set.instUnion",
"Set.sdiff_union_self",
"SDiff.sdiff",
... | [] | simp [union_eq_self_of_subset_right hCR] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 19
} | {
"line": 172,
"column": 4
} | [
{
"pp": "case h.left\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nb : ι → MvPolynomial σ R\nhb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))\nf : MvPolynomial σ R\nhb' : ∀ (i : ι), m.degree (b i) ≠ 0\nhf0 : ¬f = 0\ni : ι\nhf : m.degree (b i) ≤ m.degree f\ndeg_reduce : m.toSyn... | [] | simp [reduce] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 40
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nσ✝ : Type u_2\ninst✝¹ : Finite σ✝\ninst✝ : IsDomain R\nσ τ : Type u_2\ne : σ ≃ τ\nh :\n ∀ (P : MvPolynomial σ R) (S : σ → Finset R),\n (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0\nP : MvPolynomial τ R\nS : ... | [] | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Optimization.ValuedCSP | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 51
} | {
"line": 182,
"column": 2
} | [
{
"pp": "D : Type u_1\nC : Type u_3\ninst✝² : AddCommMonoid C\ninst✝¹ : PartialOrder C\ninst✝ : IsOrderedCancelAddMonoid C\nf : (Fin 2 → D) → C\nω : FractionalOperation D 2\nvalid : ω.IsValid\nsymmega : ω.IsSymmetric\na b : D\ncontr : 2 • (Multiset.map f (ω.tt ![![a, b], ![b, a]])).sum ≤ ω.size • 2 • f ![a, b]\... | [] | simp [FractionalOperation.tt, Multiset.map_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Optimization.ValuedCSP | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 15
} | {
"line": 184,
"column": 16
} | [
{
"pp": "D : Type u_1\nC : Type u_3\ninst✝² : AddCommMonoid C\ninst✝¹ : PartialOrder C\ninst✝ : IsOrderedCancelAddMonoid C\nf : (Fin 2 → D) → C\nω : FractionalOperation D 2\nvalid : ω.IsValid\nsymmega : ω.IsSymmetric\na b : D\ncontr : 2 • (Multiset.map f (ω.tt ![![a, b], ![b, a]])).sum ≤ ω.size • 2 • f ![a, b]\... | [
"D : Type u_1\nC : Type u_3\ninst✝² : AddCommMonoid C\ninst✝¹ : PartialOrder C\ninst✝ : IsOrderedCancelAddMonoid C\nf : (Fin 2 → D) → C\nω : FractionalOperation D 2\nvalid : ω.IsValid\nsymmega : ω.IsSymmetric\na b : D\ncontr : 2 • (Multiset.map f (ω.tt ![![a, b], ![b, a]])).sum ≤ ω.size • 2 • f ![a, b]\nhab : a ≠ b... | rhs_swap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 75,
"column": 20
} | {
"line": 75,
"column": 29
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : a ∈ p'.vertices\n⊢ a ∈ (p'.cons e).vertices",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"true_or",
"Membership.mem",
"List.not_mem_nil... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 75,
"column": 20
} | {
"line": 75,
"column": 29
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : a ∈ p'.vertices\n⊢ a ∈ (p'.cons e).vertices",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"true_or",
"Membership.mem",
"List.not_mem_nil... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 75,
"column": 20
} | {
"line": 75,
"column": 29
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : a ∈ p'.vertices\n⊢ a ∈ (p'.cons e).vertices",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"true_or",
"Membership.mem",
"List.not_mem_nil... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 82,
"column": 20
} | {
"line": 82,
"column": 29
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : p'.vertices.head? = some a\n⊢ (p'.cons e).vertices.head? = some a",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"List.head?",
"congrArg",
"List.head?_append",
"Opt... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 82,
"column": 20
} | {
"line": 82,
"column": 29
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : p'.vertices.head? = some a\n⊢ (p'.cons e).vertices.head? = some a",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"List.head?",
"congrArg",
"List.head?_append",
"Opt... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 82,
"column": 20
} | {
"line": 82,
"column": 29
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : p'.vertices.head? = some a\n⊢ (p'.cons e).vertices.head? = some a",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"List.head?",
"congrArg",
"List.head?_append",
"Opt... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 96,
"column": 20
} | {
"line": 96,
"column": 29
} | {
"line": 98,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : p'.vertices[0] = a\n⊢ (p'.cons e).vertices[0] = a",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Nat.instCanonicallyOrde... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 96,
"column": 20
} | {
"line": 96,
"column": 29
} | {
"line": 98,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : p'.vertices[0] = a\n⊢ (p'.cons e).vertices[0] = a",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Nat.instCanonicallyOrde... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 96,
"column": 20
} | {
"line": 96,
"column": 29
} | {
"line": 98,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b b✝ c✝ : V\np' : Path a b✝\ne : b✝ ⟶ c✝\nih : p'.vertices[0] = a\n⊢ (p'.cons e).vertices[0] = a",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Nat.instCanonicallyOrde... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 115,
"column": 20
} | {
"line": 115,
"column": 29
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b c : V\np : Path a b\nb✝ c✝ : V\nq' : Path b b✝\ne : b✝ ⟶ c✝\nih : (p.comp q').vertices = p.vertices.dropLast ++ q'.vertices\n⊢ (p.comp (q'.cons e)).vertices = p.vertices.dropLast ++ (q'.cons e).vertices",
"ppTerm": "?cons",
"assigned": true,
"u... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 115,
"column": 20
} | {
"line": 115,
"column": 29
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b c : V\np : Path a b\nb✝ c✝ : V\nq' : Path b b✝\ne : b✝ ⟶ c✝\nih : (p.comp q').vertices = p.vertices.dropLast ++ q'.vertices\n⊢ (p.comp (q'.cons e)).vertices = p.vertices.dropLast ++ (q'.cons e).vertices",
"ppTerm": "?cons",
"assigned": true,
"u... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 115,
"column": 20
} | {
"line": 115,
"column": 29
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b c : V\np : Path a b\nb✝ c✝ : V\nq' : Path b b✝\ne : b✝ ⟶ c✝\nih : (p.comp q').vertices = p.vertices.dropLast ++ q'.vertices\n⊢ (p.comp (q'.cons e)).vertices = p.vertices.dropLast ++ (q'.cons e).vertices",
"ppTerm": "?cons",
"assigned": true,
"u... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Schnirelmann | {
"line": 290,
"column": 17
} | {
"line": 290,
"column": 27
} | {
"line": 290,
"column": 27
} | [
{
"pp": "A B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∈ A\n⊢ n ∈ A",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [] | simp [hnA] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Schnirelmann | {
"line": 290,
"column": 17
} | {
"line": 290,
"column": 27
} | {
"line": 290,
"column": 27
} | [
{
"pp": "A B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∈ A\n⊢ n ∈ A",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [] | simp [hnA] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Schnirelmann | {
"line": 290,
"column": 17
} | {
"line": 290,
"column": 27
} | {
"line": 290,
"column": 27
} | [
{
"pp": "A B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∈ A\n⊢ n ∈ A",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [] | simp [hnA] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 19
} | {
"line": 169,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\np : Finset (Finset α) → Prop\n𝒜 : Finset (Finset α)\nempty : p ∅\nsingleton_empty : p {∅}\nsubfamily : ∀ (a : α) ⦃𝒜 : Finset (Finset α)⦄, p (nonMemberSubfamily a 𝒜) → p (memberSubfamily a 𝒜) → p 𝒜\n⊢ p 𝒜",
"ppTerm": "?m.12",
"assigned": true,
"used... | [
"α : Type u_1\ninst✝ : DecidableEq α\np : Finset (Finset α) → Prop\n𝒜 : Finset (Finset α)\nempty : p ∅\nsingleton_empty : p {∅}\nsubfamily : ∀ (a : α) ⦃𝒜 : Finset (Finset α)⦄, p (nonMemberSubfamily a 𝒜) → p (memberSubfamily a 𝒜) → p 𝒜\nu : Finset α := 𝒜.sup id\n⊢ p 𝒜"
] | set u := 𝒜.sup id | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Data.Finset.Sups | {
"line": 656,
"column": 69
} | {
"line": 656,
"column": 74
} | {
"line": 656,
"column": 74
} | [
{
"pp": "α : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nn : ℕ\nh𝒜 : Set.Sized n ↑𝒜\ns : Finset α\nhs : s ∈ 𝒜\n⊢ Fintype.card α - #s = Fintype.card α - n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",... | [
"α : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nn : ℕ\nh𝒜 : Set.Sized n ↑𝒜\ns : Finset α\nhs : s ∈ 𝒜\n⊢ Fintype.card α - n = Fintype.card α - n"
] | h𝒜 hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 212,
"column": 2
} | {
"line": 213,
"column": 35
} | {
"line": 214,
"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\nha : a ∉ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ Disjoint v a",
"ppTerm": "?pos✝",
"assigned":... | [
"case neg\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\nha : a ∉ s\nb : α\nhb : b ∈ s\nh✝ : ¬(Disjoint u b ∧ v ≤ b)\nhba : b = a\n⊢ Disjoint v a"
] | · rw [← hba]
exact disjoint_sdiff_self_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Birkhoff | {
"line": 195,
"column": 8
} | {
"line": 195,
"column": 72
} | {
"line": 196,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝³ : DistribLattice α\ninst✝² : Fintype α\ninst✝¹ : DecidablePred SupIrred\ninst✝ : OrderBot α\na : α\n⊢ (fun s ↦ (↑s).toFinset.sup Subtype.val) ((fun a ↦ { carrier := {b | ↑b ≤ a}, lower' := ⋯ }) a) = a",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Latt... | [
"α : Type u_1\ninst✝³ : DistribLattice α\ninst✝² : Fintype α\ninst✝¹ : DecidablePred SupIrred\ninst✝ : OrderBot α\na : α\n⊢ a ≤ (fun s ↦ (↑s).toFinset.sup Subtype.val) ((fun a ↦ { carrier := {b | ↑b ≤ a}, lower' := ⋯ }) a)"
] | refine le_antisymm (Finset.sup_le fun b ↦ Set.mem_toFinset.1) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SetFamily.Intersecting | {
"line": 118,
"column": 4
} | {
"line": 123,
"column": 100
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Finset α\nhs : (↑s).Intersecting\nh : ∀ (t : Finset α), (↑t).Intersecting → s ⊆ t → s = t\n⊢ IsUpperSet ↑s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Intersecting.ne_bot",
"Finse... | [] | rintro a b hab ha
rw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]
· exact mem_insert_self _ _
rw [coe_insert]
exact
hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Intersecting | {
"line": 118,
"column": 4
} | {
"line": 123,
"column": 100
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Finset α\nhs : (↑s).Intersecting\nh : ∀ (t : Finset α), (↑t).Intersecting → s ⊆ t → s = t\n⊢ IsUpperSet ↑s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Intersecting.ne_bot",
"Finse... | [] | rintro a b hab ha
rw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]
· exact mem_insert_self _ _
rw [coe_insert]
exact
hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Intersecting | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 30
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanAlgebra α\ns : Finset α\nhs : (↑s).Intersecting\nx : α\nhx' : x ∈ s\nhx : { toFun := compl, inj' := ⋯ } x ∈ s\nhxc : { toFun := compl, inj' := ⋯ } x ∈ map { toFun := compl, inj' := ⋯ } s\n⊢ False",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"F... | [] | exact hs.compl_notMem hx' hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.Kleitman | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 15
} | {
"line": 79,
"column": 16
} | [
{
"pp": "case inr.cons.refine_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fin... | [
"case inr.cons.refine_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card α ... | mul_tsub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Kleitman | {
"line": 79,
"column": 64
} | {
"line": 79,
"column": 73
} | {
"line": 80,
"column": 4
} | [
{
"pp": "case inr.cons.refine_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fin... | [
"case inr.cons.refine_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card α ... | mul_tsub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 67,
"column": 49
} | {
"line": 67,
"column": 74
} | {
"line": 67,
"column": 75
} | [
{
"pp": "case mp.left\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\na : α\nha : a ∉ t\nhst : #s = #(insert a t)\nhts : toColex (insert a t) ≤ toColex s\n⊢ #(insert a t) - 1 = #t",
"ppTerm": "?mp.left",
"assigned": true,
"usedConstants": [
... | [
"case mp.left\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\na : α\nha : a ∉ t\nhst : #s = #(insert a t)\nhts : toColex (insert a t) ≤ toColex s\n⊢ #t + 1 - 1 = #t"
] | card_insert_of_notMem ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Kleitman | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 15
} | {
"line": 86,
"column": 16
} | [
{
"pp": "case h₂\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card α - 2... | [
"case h₂\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.... | mul_tsub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 69,
"column": 11
} | {
"line": 69,
"column": 63
} | {
"line": 69,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\nh : 𝒜.Shatters s\n⊢ image (fun t ↦ s ∩ t) 𝒜 = s.powerset",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_image",
"congrArg",
"Finset",
"Iff.rfl",
... | [] | by ext t; rw [mem_image, mem_powerset, h.subset_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 11
} | {
"line": 123,
"column": 11
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CommSemiring β\n𝒜 : Finset (Finset α)\na : α\nf : Finset α → β\ns t u : Finset α\nha : a ∉ s\nht : t ∈ 𝒜\nhu : u ∈ 𝒜\nhts : t = s\nhus : u = insert a s\n⊢ collapse 𝒜 a f s = f t + f u",
"ppTerm": "?m.26",
"assigned": true,
"use... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CommSemiring β\n𝒜 : Finset (Finset α)\na : α\nf : Finset α → β\nt u : Finset α\nht : t ∈ 𝒜\nhu : u ∈ 𝒜\nha : a ∉ t\nhus : u = insert a t\n⊢ collapse 𝒜 a f t = f t + f u"
] | subst hts | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 11
} | {
"line": 130,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : CommSemiring β\ninst✝¹ : LinearOrder β\ninst✝ : IsStrictOrderedRing β\n𝒜 : Finset (Finset α)\na : α\nf : Finset α → β\ns t : Finset α\nha : a ∉ s\nhf : 0 ≤ f\nhts : t = s\nht : t ∈ 𝒜\n⊢ f t ≤ collapse 𝒜 a f s",
"ppTerm": "?m.20",
"... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : CommSemiring β\ninst✝¹ : LinearOrder β\ninst✝ : IsStrictOrderedRing β\n𝒜 : Finset (Finset α)\na : α\nf : Finset α → β\nt : Finset α\nhf : 0 ≤ f\nht : t ∈ 𝒜\nha : a ∉ t\n⊢ f t ≤ collapse 𝒜 a f t"
] | subst hts | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 158,
"column": 8
} | {
"line": 158,
"column": 33
} | {
"line": 158,
"column": 34
} | [
{
"pp": "case neg\nα : Type u_2\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\nhs : #s = k\nt : Finset α\nht : t ∈ 𝒜\nhst✝ : s ⊆ t\nh : s ∉ 𝒜\na : α\nha : a ∉ s\nhst : insert a s ⊆ t\n⊢ #(insert a s) = k + 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.... | [
"case neg\nα : Type u_2\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\nhs : #s = k\nt : Finset α\nht : t ∈ 𝒜\nhst✝ : s ⊆ t\nh : s ∉ 𝒜\na : α\nha : a ∉ s\nhst : insert a s ⊆ t\n⊢ #s + 1 = k + 1"
] | card_insert_of_notMem ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 199,
"column": 2
} | {
"line": 201,
"column": 83
} | {
"line": 203,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\n𝒜 : Finset (Finset α)\ninst✝ : Fintype α\n⊢ #𝒜.shatterer ≤ ∑ k ∈ Iic 𝒜.vcDim, (Fintype.card α).choose k",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finset.Shatters.card_le_vcDim",
"Eq.mpr",
"Nat.choos... | [] | simp_rw [← card_univ, ← card_powersetCard]
refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le
exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 199,
"column": 2
} | {
"line": 201,
"column": 83
} | {
"line": 203,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\n𝒜 : Finset (Finset α)\ninst✝ : Fintype α\n⊢ #𝒜.shatterer ≤ ∑ k ∈ Iic 𝒜.vcDim, (Fintype.card α).choose k",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finset.Shatters.card_le_vcDim",
"Eq.mpr",
"Nat.choos... | [] | simp_rw [← card_univ, ← card_powersetCard]
refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le
exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 763,
"column": 2
} | {
"line": 763,
"column": 22
} | {
"line": 763,
"column": 22
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ne : Sym2 V\nhe : G.IsBridge e\n⊢ Nontrivial V",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Sym2.mk",
"SimpleGraph.IsBridge",
"Sym2.ind",
"Eq.ndrec",
"Eq.refl",
"Eq.symm",
"Eq",
"S... | [
"case h\nV : Type u\nG : SimpleGraph V\nu v : V\nhe : G.IsBridge s(u, v)\n⊢ Nontrivial V"
] | cases e with | h u v => _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 573,
"column": 6
} | {
"line": 573,
"column": 11
} | {
"line": 573,
"column": 11
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\nhc : G.chromaticNumber < ↑n\nhne : ↑G.chromaticNumber.toNat = G.chromaticNumber\nm : ℕ\nhc' : G.Colorable m\nthis : G.Colorable G.chromaticNumber.toNat\n⊢ G.chromaticNumber.toNat < n",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"ENat.... | [
"V : Type u\nG : SimpleGraph V\nn : ℕ\nhc : ↑G.chromaticNumber.toNat < ↑n\nhne : ↑G.chromaticNumber.toNat = G.chromaticNumber\nm : ℕ\nhc' : G.Colorable m\nthis : G.Colorable G.chromaticNumber.toNat\n⊢ G.chromaticNumber.toNat < n"
] | ← hne | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 27
} | {
"line": 112,
"column": 0
} | [
{
"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 ∈ t",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsBipartiteWith.mem_of_mem_adj"
],
"usedFVars": [
"V",
"v",
"w",
... | [] | exact h.mem_of_mem_adj hv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 27
} | {
"line": 173,
"column": 0
} | [
{
"pp": "V : Type u_1\nv : V\nG : SimpleGraph V\ns t : Finset V\ninst✝¹ : Fintype ↑(G.neighborSet v)\ninst✝ : DecidableRel G.Adj\nh : G.IsBipartiteWith ↑s ↑t\nhv : v ∈ s\nw : V\n⊢ G.Adj v w → w ∈ t",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Finset",
"SimpleGraph.IsBiparti... | [] | exact h.mem_of_mem_adj hv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 42
} | {
"line": 73,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\n⊢ G.IsEdgeReachable 1 u v ↔ G.Reachable u v",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsEdgeReachable",
"SimpleGraph.deleteEdges",
"Set.encard",
"NeZero.one",
"instAddMonoidWithOneENat",
... | [] | simp [IsEdgeReachable, Order.lt_one_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 42
} | {
"line": 73,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\n⊢ G.IsEdgeReachable 1 u v ↔ G.Reachable u v",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsEdgeReachable",
"SimpleGraph.deleteEdges",
"Set.encard",
"NeZero.one",
"instAddMonoidWithOneENat",
... | [] | simp [IsEdgeReachable, Order.lt_one_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 42
} | {
"line": 73,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\n⊢ G.IsEdgeReachable 1 u v ↔ G.Reachable u v",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsEdgeReachable",
"SimpleGraph.deleteEdges",
"Set.encard",
"NeZero.one",
"instAddMonoidWithOneENat",
... | [] | simp [IsEdgeReachable, Order.lt_one_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 49
} | {
"line": 187,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhuv : G.IsEdgeReachable 2 u v\nhuy : ¬G.IsEdgeReachable 2 u x\ne : Sym2 V\nhe : ¬(G.deleteEdges {e}).Reachable u x\n⊢ x ∉ w.support",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsEdge... | [
"V : Type u_1\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhuv : G.IsEdgeReachable 2 u v\nhuy : ¬G.IsEdgeReachable 2 u x\ne : Sym2 V\nhe : ¬(G.deleteEdges {e}).Reachable u x\nhe' : ¬(G.deleteEdges {e}).Reachable v x\n⊢ x ∉ w.support"
] | have he' : ¬ (G.deleteEdges {e}).Reachable v x := fun hvy ↦
he <| (isEdgeReachable_two.1 huv _).trans hvy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 352,
"column": 20
} | {
"line": 352,
"column": 32
} | {
"line": 352,
"column": 32
} | [
{
"pp": "case inl\nV : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nleft right : Finset V\ncard_left : #left = Fintype.card α\ncard_right : #right = Fintype.card β\nh : G.IsCompleteBetween ↑left ↑right\nthis✝ : Nonempty (α ↪ ↥left)\nfα : α... | [
"case inl\nV : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nleft right : Finset V\ncard_left : #left = Fintype.card α\ncard_right : #right = Fintype.card β\nh : G.IsCompleteBetween ↑left ↑right\nthis✝ : Nonempty (α ↪ ↥left)\nfα : α ↪ ↥left := ... | Sum.elim_inr | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 49
} | {
"line": 374,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nG' : SimpleGraph V\nh : G ≤ G'\nhr : G.Reachable u v\n⊢ G'.dist u v ≤ G.dist u v",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"SimpleGraph.dist",
"SimpleGraph.Walk.length",
"SimpleGraph.Walk",
"Exists",
"LE.... | [
"V : Type u_1\nG : SimpleGraph V\nu v : V\nG' : SimpleGraph V\nh : G ≤ G'\nhr : G.Reachable u v\nw✝ : G.Walk u v\nhw : w✝.length = G.dist u v\n⊢ G'.dist u v ≤ G.dist u v"
] | obtain ⟨_, hw⟩ := hr.exists_walk_length_eq_dist | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 532,
"column": 2
} | {
"line": 535,
"column": 40
} | {
"line": 537,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : Nontrivial V\ninst✝ : DecidableRel G.Adj\nh : G.IsTree\n⊢ ∃ v, G.degree v = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"SimpleGraph.neighborSet",
... | [] | obtain ⟨v, hv⟩ := G.exists_minimal_degree_vertex
use v
rw [← hv]
exact h.minDegree_eq_one_of_nontrivial | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 532,
"column": 2
} | {
"line": 535,
"column": 40
} | {
"line": 537,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : Nontrivial V\ninst✝ : DecidableRel G.Adj\nh : G.IsTree\n⊢ ∃ v, G.degree v = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"SimpleGraph.neighborSet",
... | [] | obtain ⟨v, hv⟩ := G.exists_minimal_degree_vertex
use v
rw [← hv]
exact h.minDegree_eq_one_of_nontrivial | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Extremal.Basic | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 17
} | {
"line": 179,
"column": 18
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ G.IsExtremal H.Free ↔ H.Free G ∧ #G.edgeFinset = extremalNumber (Fintype.card V) H",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"SimpleGraph.Free",
"E... | [
"V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ (H.Free G ∧ ∀ ⦃G' : SimpleGraph V⦄ [inst : DecidableRel G'.Adj], H.Free G' → #G'.edgeFinset ≤ #G.edgeFinset) ↔\n H.Free G ∧ #G.edgeFinset = extremalNumber (Fintype.card V) H"
] | IsExtremal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 155,
"column": 57
} | {
"line": 155,
"column": 77
} | {
"line": 155,
"column": 77
} | [
{
"pp": "α : Type u\nG : SimpleGraph α\nx✝ : ∃ v w₁ w₂, G.IsPathGraph3Compl v w₁ w₂\nw✝² w✝¹ w✝ : α\nh1 : G.Adj w✝¹ w✝\nh2 : ¬G.Adj w✝² w✝¹\nh3 : ¬G.Adj w✝² w✝\nh : G.IsCompleteMultipartite\n⊢ ¬G.Adj w✝¹ w✝²",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"congrArg",
"SimpleGra... | [] | rwa [adj_comm] at h2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
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