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.Coloring.VertexColoring | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 35
} | [
{
"pp": "case map_rel'\nV : Type u\nG : SimpleGraph V\nn : ℕ\nβ : Type u_3\nf : V ↪ β\ninst✝ : NeZero n\nC : G.Coloring (Fin n)\n⊢ ∀ {a b : β},\n (SimpleGraph.map (⇑f) G).Adj a b →\n (completeGraph (Fin n)).Adj (extend (⇑f) (⇑C) (const β default) a) (extend (⇑f) (⇑C) (const β default) b)",
"usedCons... | intro a b ⟨_, _, _, hadj, ha, hb⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\nι : Type u_1\nf : ι → V\nhf : Pairwise fun i j ↦ G.Adj (f i) (f j)\nC : G.Coloring (Fin n)\n⊢ Nat.card ι ≤ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 339,
"column": 6
} | {
"line": 339,
"column": 32
} | [
{
"pp": "n r k i : ℕ\n𝒜 : Finset (Finset (Fin n))\nhir : i ≤ r\nhrk : r ≤ k\nhkn : k ≤ n\nh₁ : Set.Sized r ↑𝒜\nh₂ : k.choose r ≤ #𝒜\nrange'k : Finset (Fin n) := (range k).attachFin ⋯\n𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k\nthis✝ : Set.Sized r ↑𝒞\nA B : Finset (Fin n)\nhA : A ⊆ range'k ∧ #A ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 371,
"column": 64
} | {
"line": 371,
"column": 75
} | [
{
"pp": "n : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\nh1r : r > 0\nsize : (n - 1).choose (r - 1) < #𝒜\nthis✝¹ : Disjoint 𝒜 (∂^[n - 2 * r] 𝒜ᶜˢ)\nthis✝ : r ≤ n\nthis : 1 ≤ n\nz : (n - 1).choose (n - r) < #𝒜ᶜˢ\n⊢ Set.Sized (n - r) ↑𝒜ᶜˢ",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 355,
"column": 2
} | {
"line": 358,
"column": 48
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable G.chromaticNumber.toNat",
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"SimpleGraph.Colorable.chromaticNumber_eq_sInf",
"setOf",
"Classical.propDecidable",
"Membership.m... | classical
rw [hc.chromaticNumber_eq_sInf, Nat.sInf_def]
· apply Nat.find_spec
· exact colorable_set_nonempty_of_colorable hc | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 355,
"column": 2
} | {
"line": 358,
"column": 48
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable G.chromaticNumber.toNat",
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"SimpleGraph.Colorable.chromaticNumber_eq_sInf",
"setOf",
"Classical.propDecidable",
"Membership.m... | classical
rw [hc.chromaticNumber_eq_sInf, Nat.sInf_def]
· apply Nat.find_spec
· exact colorable_set_nonempty_of_colorable hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 355,
"column": 2
} | {
"line": 358,
"column": 48
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable G.chromaticNumber.toNat",
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"SimpleGraph.Colorable.chromaticNumber_eq_sInf",
"setOf",
"Classical.propDecidable",
"Membership.m... | classical
rw [hc.chromaticNumber_eq_sInf, Nat.sInf_def]
· apply Nat.find_spec
· exact colorable_set_nonempty_of_colorable hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 774,
"column": 4
} | {
"line": 774,
"column": 15
} | [
{
"pp": "case h\nV : Type u\nG : SimpleGraph V\nv w v' w' : V\np : (G.deleteEdges {s(v, w)}).Walk v' w'\nh : s(v, w) ∈ p.edges\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 424,
"column": 34
} | {
"line": 424,
"column": 45
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nα : Type u_2\ninst✝ : Fintype α\ni : ℕ\nC : G.Coloring (Fin i)\nh : i < card α\n⊢ card (Fin i) ≤ card α",
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"congrArg",
"Fintype.card",
"id",
"LE.le",
"instLENat",
"Fin.fint... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 26
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\nα : Type u_2\ninst✝ : Fintype α\ni : ℕ\nC : G.Coloring (Fin i)\nh : i < card α\nhC : Surjective (⇑⋯.some ∘ ⇑C)\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 473,
"column": 49
} | {
"line": 473,
"column": 60
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Fintype V\nh : G.chromaticNumber = ↑(card V)\nhh : G ≠ ⊤\na b : V\nhne : a ≠ b\nright✝ : ¬G.Adj a b\nthis : G.Coloring ↥(Finset.univ.erase b)\n⊢ G.Colorable (card V - 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 514,
"column": 50
} | {
"line": 514,
"column": 61
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nh : G = ⊥\nh' : ¬IsEmpty V\n⊢ Nonempty V",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 529,
"column": 8
} | {
"line": 529,
"column": 19
} | [
{
"pp": "V : Type u_4\nW : Type u_5\ninst✝¹ : Nonempty V\ninst✝ : Nonempty W\n⊢ (completeBipartiteGraph V W).Colorable 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 832,
"column": 6
} | {
"line": 832,
"column": 32
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w u : V\nc : G.Walk u u\nhc : c.IsCycle\nhe : s(v, w) ∈ c.edges\nhb : ∀ (p : G.Walk v w), s(v, w) ∈ p.edges\np : G.Walk w v\n⊢ s(w, v) ∈ p.edges",
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"Membership.mem",
"id",
"Sym2.e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 15
} | [
{
"pp": "case neg\nV : Type u_4\nW : Type u_5\ninst✝¹ : Nonempty V\ninst✝ : Nonempty W\nC : (completeBipartiteGraph V W).Coloring (Fin 2)\nb : Fin 2\nv : V\nw : W\nh : (completeBipartiteGraph V W).Adj (Sum.inl v) (Sum.inr w)\nhe : ¬C (Sum.inl v) = b\nhe' : ¬C (Sum.inr w) = b\n⊢ ∃ a, C a = b",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\ns : Finset V\nh : G.IsClique ↑s\nhc : G.Colorable n\n⊢ s.card ≤ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 544,
"column": 69
} | {
"line": 544,
"column": 91
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\ns : Finset V\nh : G.IsClique ↑s\nhc : G.Colorable n\n⊢ Pairwise fun i j ↦ G.Adj ↑i ↑j",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.Adj",
"Subtype.forall._simp_1",
"Membership.mem",
"id",
"Subt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 551,
"column": 75
} | {
"line": 551,
"column": 97
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ns : Finset V\nh : G.IsClique ↑s\n⊢ Pairwise fun i j ↦ G.Adj ↑i ↑j",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.Adj",
"Subtype.forall._simp_1",
"Membership.mem",
"id",
"Subtype",
"Subtype.mk",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 572,
"column": 2
} | {
"line": 572,
"column": 13
} | [
{
"pp": "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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 280,
"column": 47
} | {
"line": 280,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : ExistsAddOfLE β\na : α\nu : Finset α\nhu : a ∉ u\nf₁ f₂ f₃ f₄ : Finset α → β\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆ insert... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 281,
"column": 47
} | {
"line": 281,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : ExistsAddOfLE β\na : α\nu : Finset α\nhu : a ∉ u\nf₁ f₂ f₃ f₄ : Finset α → β\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆ insert... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 282,
"column": 4
} | {
"line": 283,
"column": 33
} | [
{
"pp": "case insert\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : ExistsAddOfLE β\na : α\nu : Finset α\nhu : a ∉ u\nf₁ f₂ f₃ f₄ : Finset α → β\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β\ninst✝³ : LinearOrder β\ninst✝² : IsStrictOrderedRing β\ninst✝¹ : ExistsAddOfLE β\nf₁ f₂ f₃ f₄ : α → β\ninst✝ : DecidableEq α\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β\ninst✝³ : LinearOrder β\ninst✝² : IsStrictOrderedRing β\ninst✝¹ : ExistsAddOfLE β\nf₁ f₂ f₃ f₄ : α → β\ninst✝ : DecidableEq α\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 916,
"column": 2
} | {
"line": 916,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nw : G.Walk u v\ne : Sym2 V\nhuv : ¬(G.deleteEdges {e}).Reachable u v\n⊢ e ∈ w.edges",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 926,
"column": 8
} | {
"line": 926,
"column": 19
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v x y : V\nw : G.Walk u v\nhw : w.IsTrail\nhuy : ¬(G.deleteEdges {s(x, y)}).Reachable u y\nhvy : ¬(G.deleteEdges {s(x, y)}).Reachable v y\nhxy : s(x, y) ∈ w.edges\n⊢ s(x, y) ∈ (w.dropUntil y ⋯).edges",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 321,
"column": 4
} | {
"line": 322,
"column": 11
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ✝ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β✝\ninst✝³ : LinearOrder β✝\ninst✝² : IsStrictOrderedRing β✝\ninst✝¹ : ExistsAddOfLE β✝\nf₁ f₂ f₃ f₄ : α → β✝\ninst✝ : DecidableEq α\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ (a b : α), f₁ a * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 944,
"column": 6
} | {
"line": 944,
"column": 17
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhxu : x ≠ u\nhxv : x ≠ v\nhx : (G.neighborSet x).Subsingleton\nhxw : x ∈ w.support\ny : V\nhxy : G.Adj x y\np : (G.deleteEdges {s(y, x)}).Walk u x\n⊢ G.Adj x p.penultimate ∧ ?m.117 ∧ ¬p.penultimate = y",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 944,
"column": 6
} | {
"line": 944,
"column": 17
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhxu : x ≠ u\nhxv : x ≠ v\nhx : (G.neighborSet x).Subsingleton\nhxw : x ∈ w.support\ny : V\nhxy : G.Adj x y\np : (G.deleteEdges {s(y, x)}).Walk v x\n⊢ G.Adj x p.penultimate ∧ ?m.184 ∧ ¬p.penultimate = y",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 42
} | [
{
"pp": "case refine_1.inl.inr\nα : Type u_1\nβ✝ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β✝\ninst✝³ : LinearOrder β✝\ninst✝² : IsStrictOrderedRing β✝\ninst✝¹ : ExistsAddOfLE β✝\nf₁ f₂ f₃ f₄ : α → β✝\ninst✝ : DecidableEq α\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ (a b : α),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 40
} | [
{
"pp": "case refine_1.inr\nα : Type u_1\nβ✝ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β✝\ninst✝³ : LinearOrder β✝\ninst✝² : IsStrictOrderedRing β✝\ninst✝¹ : ExistsAddOfLE β✝\nf₁ f₂ f₃ f₄ : α → β✝\ninst✝ : DecidableEq α\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ (a b : α), f₁ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 93
} | [
{
"pp": "case refine_1.inr\nα : Type u_1\nβ✝ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β✝\ninst✝³ : LinearOrder β✝\ninst✝² : IsStrictOrderedRing β✝\ninst✝¹ : ExistsAddOfLE β✝\nf₁ f₂ f₃ f₄ : α → β✝\ninst✝ : DecidableEq α\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ (a b : α), f₁ ... | · simpa [extend_apply' _ _ _ hs] using mul_nonneg
(extend_nonneg (fun a : L ↦ h₃ a) le_rfl _) (extend_nonneg (fun a : L ↦ h₄ a) le_rfl _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : DecidableEq α\ns t : Finset α\n⊢ #s * #t ≤ #(s ⊼ t) * #(s ⊻ t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 347,
"column": 12
} | {
"line": 347,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β\ninst✝³ : LinearOrder β\ninst✝² : IsStrictOrderedRing β\ninst✝¹ : ExistsAddOfLE β\nf₁ f₂ f₃ f₄ : α → β\ninst✝ : Fintype α\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 33
} | [
{
"pp": "case inr.inr.refine_2\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : DistribLattice α\ninst✝⁴ : CommSemiring β\ninst✝³ : LinearOrder β\ninst✝² : IsStrictOrderedRing β\ninst✝¹ : ExistsAddOfLE β\nf g μ : α → β\ninst✝ : Fintype α\nhμ₀ : 0 ≤ μ\nhf✝ : 0 ≤ f\nhg✝ : 0 ≤ g\nhμ : Monotone μ\nhfg : ∑ a, f a = ∑ a, g a\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 23
} | [
{
"pp": "V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhv : v ∈ s\nhadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s\nnhv : v ∉ t\n⊢ w ∈ t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 23
} | [
{
"pp": "V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhw : w ∈ t\nhadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s\nnhw : w ∉ s\n⊢ v ∈ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 67
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : GeneralizedBooleanAlgebra α\ns t : Finset α\nthis :\n ∀ (s t : Finset α),\n map { toFun := ⇑liftLatticeHom, inj' := ⋯ } (s \\\\ t) =\n map { toFun := ⇑liftLatticeHom, inj' := ⋯ } s \\\\ map { toFun := ⇑liftLatticeHom, inj' := ⋯ } t\n⊢ #s * #t ≤ #(s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 399,
"column": 4
} | {
"line": 399,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : GeneralizedBooleanAlgebra α\ns : Finset α\n⊢ #s ^ 2 ≤ #(s \\\\ s) ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 13
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\ninst✝ : Fintype ↑(G.neighborSet u)\nh : G.IsEdgeReachable k u v\nhuv : u ≠ v\nhh : (G.incidenceSet u).encard < ↑k\nw : (G.deleteEdges (G.incidenceSet u)).Walk u v\nh✝ : w.IsPath\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 17
} | [
{
"pp": "V : 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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 17
} | [
{
"pp": "V : 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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 15
} | [
{
"pp": "case refine_2.inl\nV : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\nhk : k ≠ 0\nh : ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeReachable k u v\nhs : ∅.encard < ↑(k + 1)\n⊢ (G.deleteEdges ∅).Reachable u v",
"usedConstants": [
"SimpleGraph.deleteEdges",
"Eq.mpr",
"congrArg",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 122,
"column": 78
} | {
"line": 123,
"column": 79
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nk : ℕ\nhk : k ≠ 0\n⊢ G.IsEdgeConnected (k + 1) ↔ ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeConnected k",
"usedConstants": [
"SimpleGraph.IsEdgeReachable",
"SimpleGraph.deleteEdges",
"SimpleGraph.isEdgeReachable_add_one",
"congrArg",
"fo... | by
simp [IsEdgeConnected, isEdgeReachable_add_one hk, forall_comm (α := Sym2 _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 351,
"column": 8
} | {
"line": 351,
"column": 33
} | [
{
"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... | ← Sum.inl_getLeft s₁ hs₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 15
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nu v : V\nhne : u ≠ v\nh : G.IsEdgeReachable 2 u v\nw : G.Walk u v\nhw : w.IsPath\nthis✝ : G.Adj u w.snd\nhs : {s(u, w.snd)}.encard < ↑2\nhh : s(u, w.snd) ∈ w.tail.edges\nthis : u = w.getVert 2\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 177,
"column": 23
} | {
"line": 177,
"column": 56
} | [
{
"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\n⊢ ?m.25",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 182,
"column": 8
} | {
"line": 182,
"column": 19
} | [
{
"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\nhe' : ¬(G.deleteEdges {e}).Reachable v x\nhy : x ∈ w.support\n⊢ e ∈ (w.dropUntil x hy).edges",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 440,
"column": 2
} | {
"line": 440,
"column": 34
} | [
{
"pp": "V : Type u_1\nv : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhv : v ∈ s\n⊢ G.neighborFinset v ⊆ (between (↑s) (↑s)ᶜ G).neighborFinset v ∪ s",
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"congrArg",
"Compl.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 447,
"column": 4
} | {
"line": 447,
"column": 15
} | [
{
"pp": "V : Type u_1\nv : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhv : v ∈ s\n⊢ (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Finset",
"BooleanAlgebra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 34
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ G.neighborFinset w ⊆ (between (↑s) (↑s)ᶜ G).neighborFinset w ∪ sᶜ",
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"congrArg",
"Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 456,
"column": 80
} | {
"line": 456,
"column": 91
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ w ∈ (↑s)ᶜ",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"congrArg",
"Compl.compl",
"Finset",
"Membership.mem",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 463,
"column": 4
} | {
"line": 463,
"column": 15
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Finset",
"BooleanAlgebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 36
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nhuv : G.Adj u v\n⊢ (⊤.induce {u, v}).Connected",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
"Set.instSingletonSet",
"id",
"Insert.insert",
"SimpleGraph.Subgraph.instTop",
"Set.instInsert",
... | rw [← subgraphOfAdj_eq_induce huv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 18
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nH : G.Subgraph\nh : H.Preconnected\nv : ↑H.verts\ninst✝ : Fintype ↑(H.neighborSet ↑v)\nhv : H.degree ↑v = 0\nhn : H.verts.Nontrivial\nthis : Nontrivial ↑H.verts\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 131,
"column": 21
} | {
"line": 131,
"column": 32
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nH : G.Subgraph\nhc : H.Connected\nh : ∀ v ∈ H.verts, ∀ (w : V), G.Adj v w → H.Adj v w\nv : V\nhv : v ∈ H.verts\nw : V\nhw : w ∈ H.verts\n⊢ G.Reachable w v",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 549,
"column": 18
} | {
"line": 549,
"column": 29
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nx✝ : V × V\nh :\n x✝ ∈\n {x |\n match x with\n | (x, y) => G.Adj x y}\n⊢ (fun x x_1 ↦\n match x, x_1 with\n | (v, w), x => s(Sum.inl v, Sum.inr w))\n x✝ h ∈\n G.bipartiteDoubleCover.edge... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 553,
"column": 22
} | {
"line": 553,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv w : V ⊕ V\nval✝¹ val✝ : V\nhe : G.bipartiteDoubleCover.Adj (Sum.inl val✝¹) (Sum.inr val✝)\n⊢ ∃ a,\n ∃ (ha :\n a ∈\n {x |\n match x with\n | (x, y) => G.Adj x y}),\n (fun x x_1 ↦\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 554,
"column": 22
} | {
"line": 554,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv w : V ⊕ V\nval✝¹ val✝ : V\nhe : G.bipartiteDoubleCover.Adj (Sum.inr val✝¹) (Sum.inl val✝)\n⊢ ∃ a,\n ∃ (ha :\n a ∈\n {x |\n match x with\n | (x, y) => G.Adj x y}),\n (fun x x_1 ↦\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nw : G.Walk u v\nh : ¬w.Nil\n⊢ w.toSubgraph.Adj u w.snd",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 281,
"column": 2
} | {
"line": 282,
"column": 9
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nw : G.Walk u v\nh : 0 < w.length\n⊢ w.toSubgraph.Adj w.penultimate v",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 15
} | [
{
"pp": "case left\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\ninst✝ : DecidableEq V\n⊢ p.bypass.toSubgraph.verts ⊆ p.toSubgraph.verts",
"usedConstants": [
"Eq.mpr",
"congrArg",
"setOf",
"SimpleGraph.Walk.support",
"Membership.mem",
"SimpleGraph.Walk.toSubgra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 46
} | [
{
"pp": "case right\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\ninst✝ : DecidableEq V\n⊢ ∀ ⦃v_1 w : V⦄, p.bypass.toSubgraph.Adj v_1 w → p.toSubgraph.Adj v_1 w",
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"Membership.mem",
"_private.Mathlib.Combinatorics.SimpleGraph.Connecti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nhp : p.IsPath\nhnp : ¬p.Nil\n⊢ p.toSubgraph.neighborSet v = {p.penultimate}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 101,
"column": 15
} | {
"line": 101,
"column": 25
} | [
{
"pp": "case inl\nV : Type u_1\nG : SimpleGraph V\nu v w : V\nhuv : G.edist u v = ⊤\n⊢ G.edist u w ≤ G.edist u v + G.edist v w",
"usedConstants": [
"instTopENat",
"congrArg",
"le_top._simp_2",
"PartialOrder.toPreorder",
"LinearOrderedAddCommMonoidWithTop.toOrderTop",
"Pr... | simp [huv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 101,
"column": 15
} | {
"line": 101,
"column": 25
} | [
{
"pp": "case inl\nV : Type u_1\nG : SimpleGraph V\nu v w : V\nhuv : G.edist u v = ⊤\n⊢ G.edist u w ≤ G.edist u v + G.edist v w",
"usedConstants": [
"instTopENat",
"congrArg",
"le_top._simp_2",
"PartialOrder.toPreorder",
"LinearOrderedAddCommMonoidWithTop.toOrderTop",
"Pr... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 101,
"column": 15
} | {
"line": 101,
"column": 25
} | [
{
"pp": "case inl\nV : Type u_1\nG : SimpleGraph V\nu v w : V\nhuv : G.edist u v = ⊤\n⊢ G.edist u w ≤ G.edist u v + G.edist v w",
"usedConstants": [
"instTopENat",
"congrArg",
"le_top._simp_2",
"PartialOrder.toPreorder",
"LinearOrderedAddCommMonoidWithTop.toOrderTop",
"Pr... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 31
} | {
"line": 573,
"column": 42
} | [
{
"pp": "case inl.inl.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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 14
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : u = v\n⊢ G.edist u v ≤ 1 ↔ G.Adj u v ∨ u = v",
"usedConstants": [
"False",
"instAddMonoidWithOneENat",
"congrArg",
"CommSemiring.toSemiring",
"instIsBotZeroClass",
"zero_le._simp_1",
"SimpleGraph.Adj... | simp [huv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 14
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : u = v\n⊢ G.edist u v ≤ 1 ↔ G.Adj u v ∨ u = v",
"usedConstants": [
"False",
"instAddMonoidWithOneENat",
"congrArg",
"CommSemiring.toSemiring",
"instIsBotZeroClass",
"zero_le._simp_1",
"SimpleGraph.Adj... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 14
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : u = v\n⊢ G.edist u v ≤ 1 ↔ G.Adj u v ∨ u = v",
"usedConstants": [
"False",
"instAddMonoidWithOneENat",
"congrArg",
"CommSemiring.toSemiring",
"instIsBotZeroClass",
"zero_le._simp_1",
"SimpleGraph.Adj... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 31
} | {
"line": 573,
"column": 42
} | [
{
"pp": "case inl.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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 53
} | {
"line": 573,
"column": 64
} | [
{
"pp": "case inl.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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 31
} | {
"line": 573,
"column": 42
} | [
{
"pp": "case inr.inl.inl\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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 53
} | {
"line": 573,
"column": 64
} | [
{
"pp": "case inr.inl.inl\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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 31
} | {
"line": 573,
"column": 42
} | [
{
"pp": "case inr.inr.inl\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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 79
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nh : 0 < sInf (Set.range Walk.length)\n⊢ Set.univ.Nonempty",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 577,
"column": 31
} | {
"line": 577,
"column": 42
} | [
{
"pp": "case inl.inl.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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 365,
"column": 59
} | {
"line": 365,
"column": 67
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v u' v' : V\np₁ : G.Walk u v\np₂ : G.Walk u' v'\nh₁ : p₁.length = G.dist u v\nhh : G.dist u' v' < p₂.length\nru : G.Walk u u'\nrv : G.Walk v' v\nh : p₁ = (ru.append p₂).append rv\ns : G.Walk u' v'\nh✝ : s.IsPath ∧ s.length = G.dist u' v'\nr : G.Walk u v := (ru.append ... | simp [r] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 577,
"column": 31
} | {
"line": 577,
"column": 42
} | [
{
"pp": "case inl.inr.inl\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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 577,
"column": 53
} | {
"line": 577,
"column": 64
} | [
{
"pp": "case inl.inr.inl\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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 384,
"column": 38
} | {
"line": 386,
"column": 9
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : p.length = G.dist v w\nhl : 1 < G.dist v w\nhnp : ¬p.Nil\n⊢ ¬p.tail.Nil",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Metric.0.SimpleGraph.Walk.exists_adj_adj_not_adj_ne._simp_1_2",
"Eq.mpr",
"Simple... | by
simp only [not_nil_iff_lt_length, ← p.length_tail_add_one hnp] at hp ⊢
lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 391,
"column": 6
} | {
"line": 391,
"column": 38
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : p.length = G.dist v w\nhl : 1 < G.dist v w\nhnp : ¬p.Nil\nthis✝ : p.tail.tail.length < p.tail.length\nthis : p.tail.length < p.length\nhv : v = p.getVert 2\n⊢ G.dist v w ≤ p.tail.tail.length",
"usedConstants": [
"Eq.mpr",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 390,
"column": 4
} | {
"line": 392,
"column": 7
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : p.length = G.dist v w\nhl : 1 < G.dist v w\nhnp : ¬p.Nil\nthis✝ : p.tail.tail.length < p.tail.length\nthis : p.tail.length < p.length\nhv : v = p.getVert 2\n⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p... | have : G.dist v w ≤ p.tail.tail.length := by
simpa [hv, p.getVert_tail] using dist_le p.tail.tail
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 390,
"column": 4
} | {
"line": 392,
"column": 7
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : p.length = G.dist v w\nhl : 1 < G.dist v w\nhnp : ¬p.Nil\nthis✝ : p.tail.tail.length < p.tail.length\nthis : p.tail.length < p.length\nhv : v = p.getVert 2\n⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p... | have : G.dist v w ≤ p.tail.tail.length := by
simpa [hv, p.getVert_tail] using dist_le p.tail.tail
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 47
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v v\nh : p.IsCycle\nhadj : p.toSubgraph.Adj v w\n⊢ ∃ p', p'.IsCycle ∧ p'.snd = w ∧ p'.toSubgraph.verts = p.toSubgraph.verts",
"usedConstants": [
"Membership.mem",
"SimpleGraph.Walk.toSubgraph",
"SimpleGraph.Subgraph.neighborSet",
... | have : w ∈ p.toSubgraph.neighborSet v := hadj | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 577,
"column": 31
} | {
"line": 577,
"column": 42
} | [
{
"pp": "case inr.inl.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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 577,
"column": 53
} | {
"line": 577,
"column": 64
} | [
{
"pp": "case inr.inl.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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 520,
"column": 61
} | {
"line": 520,
"column": 92
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : G.Walk u v\ns : Finset V\nh✝ : {x ∈ s | x ∈ p.support}.Nonempty\nx : V\nhxs : x ∈ s\nhx : x ∈ p.support\nh : {t ∈ s.erase x | t ∈ (p.takeUntil x hx).support} = ∅\nthis : {t ∈ s | t ∈ (p.takeUntil x hx).support} ⊆ {x}\n⊢ ∀ t ∈ s, t ∈ (p.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 577,
"column": 31
} | {
"line": 577,
"column": 42
} | [
{
"pp": "case inr.inr.inl\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.IsCompleteBe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 75
} | [
{
"pp": "case refine_2.patches\nV : Type u\nG : SimpleGraph V\nGpc : G.Preconnected\nt : Finset V\nu : V\nut : u ∈ t\nv : V\nhv : v ∈ ↑(t.biUnion fun v ↦ (Nonempty.some ⋯).support.toFinset)\n⊢ ∃ s' ⊆ ↑(t.biUnion fun v ↦ (Nonempty.some ⋯).support.toFinset),\n ∃ (hu' : u ∈ s') (hv' : v ∈ s'), (induce s' G).Rea... | simp only [Finset.mem_coe, Finset.mem_biUnion, List.mem_toFinset] at hv | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 17
} | [
{
"pp": "case refine_1.inl.inl\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.Connectivity.Subgraph | {
"line": 703,
"column": 18
} | {
"line": 703,
"column": 23
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nf : G'.coe →g G\nhpreconn : G''.Preconnected\nu' : V\nu : ↑G'.verts\nhu : u ∈ G''.verts\nhfu : f u = u'\nv' : V\nv : ↑G'.verts\nhv : v ∈ G''.verts\nhfv : f v = v'\n⊢ (Subgraph.map f G'').coe.Reachable ⟨f u, ⋯⟩ ⟨v', ⋯⟩",
"usedCon... | ← hfv | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 27
} | [
{
"pp": "M : Type u_1\ns : Set M\ninst✝ : Group M\nu v : M\n⊢ (mulCayley s).Adj u v ↔ u ≠ v ∧ (u⁻¹ * v ∈ s ∨ v⁻¹ * u ∈ s)",
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrArg",
"_private.Mathlib.Combinatorics.Sim... | simp [mulCayley_adj', ← eq_inv_mul_iff_mul_eq (b := u), ← inv_mul_eq_iff_eq_mul (a := v),
and_or_left, exists_or] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 27
} | [
{
"pp": "M : Type u_1\ns : Set M\ninst✝ : Group M\nu v : M\n⊢ (mulCayley s).Adj u v ↔ u ≠ v ∧ (u⁻¹ * v ∈ s ∨ v⁻¹ * u ∈ s)",
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrArg",
"_private.Mathlib.Combinatorics.Sim... | simp [mulCayley_adj', ← eq_inv_mul_iff_mul_eq (b := u), ← inv_mul_eq_iff_eq_mul (a := v),
and_or_left, exists_or] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 27
} | [
{
"pp": "M : Type u_1\ns : Set M\ninst✝ : Group M\nu v : M\n⊢ (mulCayley s).Adj u v ↔ u ≠ v ∧ (u⁻¹ * v ∈ s ∨ v⁻¹ * u ∈ s)",
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrArg",
"_private.Mathlib.Combinatorics.Sim... | simp [mulCayley_adj', ← eq_inv_mul_iff_mul_eq (b := u), ← inv_mul_eq_iff_eq_mul (a := v),
and_or_left, exists_or] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 581,
"column": 57
} | {
"line": 581,
"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.Bipartite | {
"line": 583,
"column": 58
} | {
"line": 583,
"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": 587,
"column": 8
} | {
"line": 587,
"column": 19
} | [
{
"pp": "case refine_1.inl.inr.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.Acyclic | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 15
} | [
{
"pp": "case cons\nV : Type u_1\nHs : Set (SimpleGraph V)\nhHs : Hs.Nonempty\nh_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) Hs\nu✝ v✝ u v w : V\np : (sSup Hs).Walk v w\nH₁ : SimpleGraph V\nhH₁ : H₁ ∈ Hs\nih : ∀ e ∈ p.edges, e ∈ H₁.edgeSet\nH₂ : SimpleGraph V\nhH₂ : H₂ ∈ Hs\nh_adj : H₂.Adj u v\nH : SimpleGraph V\nhH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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