module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 340,
"column": 41
} | {
"line": 348,
"column": 90
} | {
"line": 350,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝³ : DecidableEq α\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq β\nA : Finset α\nB : Finset β\nf : α → β\nhf : IsMulFreimanHom 2 (↑A) (↑B) f\nhf' : Set.BijOn f ↑A ↑B\n⊢ mulRothNumber B ≤ mulRothNumber A",
"ppTerm": "?m.21",
"assigned": true,... | [] | by
obtain ⟨s, hsB, hcard, hs⟩ := mulRothNumber_spec B
have hsA : invFunOn f A '' s ⊆ A :=
(hf'.surjOn.mapsTo_invFunOn.mono (coe_subset.2 hsB) Subset.rfl).image_subset
have hfsA : Set.SurjOn f A s := hf'.surjOn.mono Subset.rfl (coe_subset.2 hsB)
rw [← hcard, ← s.card_image_of_injOn ((invFunOn_injOn_image f _... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 354,
"column": 2
} | {
"line": 354,
"column": 72
} | {
"line": 355,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝³ : DecidableEq α\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq β\nA : Finset α\nB : Finset β\nf : α → β\nhf : IsMulFreimanIso 2 (↑A) (↑B) f\n⊢ mulRothNumber A = mulRothNumber B",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_2\nβ : Type u_3\ninst✝³ : DecidableEq α\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq β\nA : Finset α\nB : Finset β\nf : α → β\nhf : IsMulFreimanIso 2 (↑A) (↑B) f\n⊢ mulRothNumber A ≤ mulRothNumber B"
] | refine le_antisymm ?_ (hf.isMulFreimanHom.mulRothNumber_mono hf.bijOn) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Additive.AP.Three.Defs | {
"line": 375,
"column": 2
} | {
"line": 380,
"column": 58
} | {
"line": 381,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CancelCommMonoid α\ns : Finset α\na : α\n⊢ mulRothNumber (map (mulLeftEmbedding a) s) ≤ mulRothNumber s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CancelCommMonoid.toCommMonoid",
"in... | [
"case refine_2\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CancelCommMonoid α\ns : Finset α\na : α\n⊢ mulRothNumber s ≤ mulRothNumber (map (mulLeftEmbedding a) s)"
] | · obtain ⟨u, hus, hcard, hu⟩ := mulRothNumber_spec (s.map <| mulLeftEmbedding a)
rw [subset_map_iff] at hus
obtain ⟨u, hus, rfl⟩ := hus
rw [coe_map] at hu
rw [← hcard, card_map]
exact (threeGPFree_smul_set.1 hu).le_mulRothNumber hus | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 9
} | {
"line": 71,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : TopologicalSpace E\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs₀ : IsClosed s\nhs₁ : StrictConvex 𝕜 s\na : E\nha : a ∈ frontier s\nc : E\nhc : c ∈ frontier s\nhb : (1 / 2... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : TopologicalSpace E\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs₀ : IsClosed s\nhs₁ : StrictConvex 𝕜 s\na : E\nha : a ∈ frontier s\nc : E\nhc : c ∈ frontier s\nhb : (1 / 2) • a + (1 /... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 33
} | {
"line": 76,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\nr : ℝ\n⊢ ThreeAPFree (sphere x r)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real",
"ThreeAPFree",
"Real.instZero",
"AddCommGroup.toAddGroup",... | [
"case inl\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\n⊢ ThreeAPFree (sphere x 0)",
"case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ ThreeAPFree (sphere x r)... | obtain rfl | hr := eq_or_ne r 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 269,
"column": 16
} | {
"line": 269,
"column": 25
} | {
"line": 269,
"column": 26
} | [
{
"pp": "n d : ℕ\nhd : d ≠ 0\nhn : 2 ≤ n\n⊢ ↑d ^ (n - 2) / ↑n = ↑(d ^ n) / (↑n * ↑(d ^ 2))",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHDiv",
"HMul.hMul",
"congrArg",
"Nat.instM... | [
"n d : ℕ\nhd : d ≠ 0\nhn : 2 ≤ n\n⊢ ↑d ^ (n - 2) / ↑n = ↑d ^ n / (↑n * ↑(d ^ 2))"
] | cast_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 370,
"column": 6
} | {
"line": 372,
"column": 47
} | {
"line": 373,
"column": 4
} | [
{
"pp": "N : ℕ\nhN₃ : 8 ≤ N\nhN₀ : 0 < ↑N\nthis : ↑(nValue N) ≤ 2 * √(log ↑N)\n⊢ log 2 * 2 ≤ √(log ↑N)",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.... | [] | apply log_two_mul_two_le_sqrt_log_eight.trans
apply Real.sqrt_le_sqrt
exact log_le_log (by simp) (mod_cast hN₃) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 370,
"column": 6
} | {
"line": 372,
"column": 47
} | {
"line": 373,
"column": 4
} | [
{
"pp": "N : ℕ\nhN₃ : 8 ≤ N\nhN₀ : 0 < ↑N\nthis : ↑(nValue N) ≤ 2 * √(log ↑N)\n⊢ log 2 * 2 ≤ √(log ↑N)",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.... | [] | apply log_two_mul_two_le_sqrt_log_eight.trans
apply Real.sqrt_le_sqrt
exact log_le_log (by simp) (mod_cast hN₃) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.FreimanHom | {
"line": 443,
"column": 4
} | {
"line": 443,
"column": 23
} | {
"line": 444,
"column": 4
} | [
{
"pp": "k m n : ℕ\nhm : m ≠ 0\nhkmn : m * k ≤ n\ns t : Multiset (Fin (n + 1))\nhsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k\nhtA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k\nhs : s.card = m\nht : t.card = m\nthis : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum\n⊢ (map v... | [
"k m n : ℕ\nhm : m ≠ 0\nhkmn : m * k ≤ n\ns t : Multiset (Fin (n + 1))\nhsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k\nhtA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k\nhs : s.card = m\nht : t.card = m\nthis : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum\n⊢ (map val s).sum = ... | rw [← this, ← this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Additive.AP.Three.Behrend | {
"line": 486,
"column": 4
} | {
"line": 486,
"column": 45
} | {
"line": 487,
"column": 4
} | [
{
"pp": "case inr.inr\nN : ℕ\nhN : N > 0\nh₁ : N < 4096\n⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ ↑(rothNumberNat N)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"Nat.instAtLeastTwoHAddOfNat",
"Behrend.lower_bound_le_one",
"LT.lt.le",
"r... | [
"case inr.inr\nN : ℕ\nhN : N > 0\nh₁ : N < 4096\n⊢ 1 ≤ ↑(rothNumberNat N)"
] | apply (lower_bound_le_one hN h₁.le).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 19
} | {
"line": 148,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\n⊢ CovBySMul G (K ^ (m - 1) * L ^ (n - 1... | [
"G : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\nthis : 1 ≤ K\n⊢ CovBySMul G (K ^ (m - 1) * L ^ (n -... | have := hA.one_le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Additive.ApproximateSubgroup | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 58
} | {
"line": 150,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\nthis : 1 ≤ K\nf : G → G → G\nhf : ∀ (a ... | [
"case refine_1\nG : Type u_1\ninst✝ : Group G\nA B : Set G\nK L : ℝ\nm n : ℕ\nhA : IsApproximateSubgroup K A\nhB : IsApproximateSubgroup L B\nhm : 2 ≤ m\nhn : 2 ≤ n\nF₁ : Finset G\nhF₁ : ↑(#F₁) ≤ K\nhAF₁ : A ^ 2 ⊆ ↑F₁ • A\nF₂ : Finset G\nhF₂ : ↑(#F₂) ≤ L\nhBF₂ : B ^ 2 ⊆ ↑F₂ • B\nthis : 1 ≤ K\nf : G → G → G\nhf : ∀ ... | refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Dart | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 19
} | {
"line": 106,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ ∀ {d : G.Dart} {u v : V}, d.edge = s(u, v) ↔ d.toProd = (u, v) ∨ d.toProd = (v, u)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.Dart.casesOn",
"Sym2.mk",
"SimpleGraph.Adj",
"Prod.mk",
"SimpleGraph.Dar... | [
"V : Type u_1\nG : SimpleGraph V\np : V × V\nh : G.Adj p.1 p.2\nu✝ v✝ : V\n⊢ { toProd := p, adj := h }.edge = s(u✝, v✝) ↔\n { toProd := p, adj := h }.toProd = (u✝, v✝) ∨ { toProd := p, adj := h }.toProd = (v✝, u✝)"
] | rintro ⟨p, h⟩ _ _ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.SimpleGraph.Finite | {
"line": 579,
"column": 38
} | {
"line": 581,
"column": 26
} | {
"line": 583,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nW : Type u_2\nG' : SimpleGraph W\nf : G ≃g G'\ninst✝¹ : Fintype ↑G.edgeSet\ninst✝ : Fintype ↑G'.edgeSet\n⊢ #G.edgeFinset = #G'.edgeFinset",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Simpl... | [] | by
apply Finset.card_eq_of_equiv
simpa using f.mapEdgeSet | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Partition.Equipartition | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 41
} | {
"line": 92,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\nhP : P.IsEquipartition\nz : #({x ∈ P.parts | #x = #s / #P.parts + 1}) + #P.parts * (#s / #P.parts) = #s\n⊢ #({p ∈ P.parts | #p = #s / #P.parts + 1}) = #s % #P.parts",
"ppTerm": "?m.214",
"assigned": true,
"usedConstants"... | [] | rw [← add_left_inj, Nat.mod_add_div, z] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Partition.Equipartition | {
"line": 153,
"column": 29
} | {
"line": 153,
"column": 53
} | {
"line": 153,
"column": 54
} | [
{
"pp": "case h\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\nhP : P.IsEquipartition\nf : ↥s ≃ (t : ↥P.parts) × Fin #↑t\nhf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst\ng : ↥P.parts ≃ Fin #P.parts\nhg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.part... | [
"case h\nα : Type u_1\ninst✝ : DecidableEq α\ns : Finset α\nP : Finpartition s\nhP : P.IsEquipartition\nf : ↥s ≃ (t : ↥P.parts) × Fin #↑t\nhf : ∀ (a b : ↥s), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst\ng : ↥P.parts ≃ Fin #P.parts\nhg : ∀ (t : ↥P.parts), #↑t = #s / #P.parts + 1 ↔ ↑(g t) < #s % #P.parts\nz : ↥s → ... | Nat.mod_eq_of_lt (gl b), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 38
} | {
"line": 326,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na : EInt\nX : C\n⊢ IsIso ((t.eTruncLTι a).app ((t.eTruncLT.obj ... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nt : TStructure C\ninst✝ : IsTriangulated C\na : EInt\nX : C\n⊢ IsIso ((t.eTruncLT.obj a).map ((t.eTruncLTι a).app X))"
... | ← eTruncLT_obj_map_eTruncLTι_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 49
} | {
"line": 92,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}.biUnion id) ≤ m) ∧... | [] | rw [card_sdiff_of_subset ‹t ⊆ s›, htn, hn₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Bound | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 9
} | {
"line": 232,
"column": 2
} | [
{
"pp": "case inr\nι : Type u_2\n𝕜 : Type u_3\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns t : Finset ι\nx : 𝕜\nhst : s ⊆ t\nf : ι → 𝕜\nd : 𝕜\nhx : 0 ≤ x\nhs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|\nht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2\nhscard : 0 < ↑(#s)\n... | [
"case inr\nι : Type u_2\n𝕜 : Type u_3\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns t : Finset ι\nx : 𝕜\nhst : s ⊆ t\nf : ι → 𝕜\nd : 𝕜\nhx : 0 ≤ x\nhs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|\nht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2\nhscard : 0 < ↑(#s)\nhtcard : 0 <... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 77
} | {
"line": 181,
"column": 2
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : s₂.Nonempty\nht₂ : t₂.Nonempty\n⊢ edgeDensity r s₂ t₂ - ↑(#s₂) / ↑(#s₁) * (↑(#t₂) / ↑(#t₁)) * edgeDensity r s₂ t₂ ≤\n 1 - ↑(#s₂) / ↑(#s₁) * (↑(#t... | [
"case refine_1\nα : Type u_4\nβ : Type u_5\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : s₂.Nonempty\nht₂ : t₂.Nonempty\n⊢ edgeDensity r s₂ t₂ - ↑(#s₂) / ↑(#s₁) * (↑(#t₂) / ↑(#t₁)) * edgeDensity r s₂ t₂ ≤\n (1 - ↑(#s₂) / ↑(#s₁) * (... | refine le_trans ?_ (mul_le_of_le_one_right ?_ (edgeDensity_le_one r s₂ t₂)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 31
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}.biUnion ... | [] | exact hab.resolve_left ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 31
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}.biUnion ... | [] | exact hab.resolve_left ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 31
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nm : ℕ\nm_pos : m > 0\ns : Finset α\nih :\n ∀ t ⊂ s,\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = #t →\n ∃ Q,\n (∀ x ∈ Q.parts, #x = m ∨ #x = m + 1) ∧\n (∀ x ∈ P.parts, #(x \\ {y ∈ Q.parts | y ⊆ x}.biUnion ... | [] | exact hab.resolve_left ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 235,
"column": 2
} | {
"line": 238,
"column": 47
} | {
"line": 240,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhδ : 0 ≤ δ\nhscard : (1 - δ) * ↑(#s₁) ≤ ↑(#s... | [] | refine (abs_sub _ _).trans (add_le_add (le_trans ?_ h) (le_trans ?_ h)) <;>
· rw [abs_of_nonneg]
· exact mod_cast edgeDensity_le_one r _ _
· exact mod_cast edgeDensity_nonneg r _ _ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Density | {
"line": 339,
"column": 8
} | {
"line": 339,
"column": 18
} | {
"line": 339,
"column": 19
} | [
{
"pp": "α : Type u_4\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\ns t : Finset α\ninst✝ : DecidableEq α\nh : Disjoint s t\nx : α × α\nhx : x.1 ∈ s ∧ x.2 ∈ t\n⊢ Gᶜ.Adj x.1 x.2 ↔ ¬G.Adj x.1 x.2",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Com... | [
"α : Type u_4\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\ns t : Finset α\ninst✝ : DecidableEq α\nh : Disjoint s t\nx : α × α\nhx : x.1 ∈ s ∧ x.2 ∈ t\n⊢ x.1 ≠ x.2 ∧ ¬G.Adj x.1 x.2 ↔ ¬G.Adj x.1 x.2"
] | compl_adj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 89,
"column": 15
} | {
"line": 91,
"column": 13
} | {
"line": 92,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nv✝ w✝ : V\nh : s(v, w) = s(v✝, w✝)\n⊢ G.Adj v✝ w✝",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"SimpleGraph.Adj",
"Membership.mem",
... | [] | by
rw [← G.mem_edgeSet, ← h]
exact hvw | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 452,
"column": 2
} | {
"line": 454,
"column": 48
} | {
"line": 456,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ Function.Injective fun G' ↦ (G'.verts, G'.spanningCoe)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"congrArg",
"SimpleGraph.Subgraph",
"SimpleGraph.Subgraph.spanningCoe_inj",
"Eq.mp",
"Prod.ext_iff",
"Prod.mk... | [] | intro G₁ G₂ h
rw [Prod.ext_iff] at h
exact Subgraph.ext h.1 (spanningCoe_inj.1 h.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 452,
"column": 2
} | {
"line": 454,
"column": 48
} | {
"line": 456,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ Function.Injective fun G' ↦ (G'.verts, G'.spanningCoe)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"congrArg",
"SimpleGraph.Subgraph",
"SimpleGraph.Subgraph.spanningCoe_inj",
"Eq.mp",
"Prod.ext_iff",
"Prod.mk... | [] | intro G₁ G₂ h
rw [Prod.ext_iff] at h
exact Subgraph.ext h.1 (spanningCoe_inj.1 h.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 61
} | {
"line": 523,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : Fintype W\n⊢ 0 < G.labelledCopyCount H ↔ H ⊑ G",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Copy.0.SimpleGraph.labelledCopyCount_pos._... | [] | simp [labelledCopyCount, IsContained, Fintype.card_pos_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 61
} | {
"line": 523,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : Fintype W\n⊢ 0 < G.labelledCopyCount H ↔ H ⊑ G",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Copy.0.SimpleGraph.labelledCopyCount_pos._... | [] | simp [labelledCopyCount, IsContained, Fintype.card_pos_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 61
} | {
"line": 523,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : Fintype W\n⊢ 0 < G.labelledCopyCount H ↔ H ⊑ G",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Copy.0.SimpleGraph.labelledCopyCount_pos._... | [] | simp [labelledCopyCount, IsContained, Fintype.card_pos_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 76
} | {
"line": 553,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : Fintype W\n⊢ G.copyCount H ≤ G.labelledCopyCount H",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.decidableInjectiveFintype",
"Finset.univ",
"R... | [] | classical rw [copyCount_eq_card_image_copyToSubgraph]; exact card_image_le | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 76
} | {
"line": 553,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : Fintype W\n⊢ G.copyCount H ≤ G.labelledCopyCount H",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.decidableInjectiveFintype",
"Finset.univ",
"R... | [] | classical rw [copyCount_eq_card_image_copyToSubgraph]; exact card_image_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 76
} | {
"line": 553,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝¹ : Fintype V\ninst✝ : Fintype W\n⊢ G.copyCount H ≤ G.labelledCopyCount H",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.decidableInjectiveFintype",
"Finset.univ",
"R... | [] | classical rw [copyCount_eq_card_image_copyToSubgraph]; exact card_image_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 63
} | {
"line": 204,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑(#s) ≤ ↑m + 1... | [] | · exact mod_cast card_le_m_add_one_of_mem_chunk_parts (h𝒜 hs) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 627,
"column": 73
} | {
"line": 632,
"column": 87
} | {
"line": 634,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nhH : H ≠ ⊥\n⊢ G.killCopies H = G ↔ H.Free G",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"SimpleGraph.Free",
"Eq.mpr",
"Exists.choose_spec",
"SimpleGraph.Is... | [] | by
simp only [killCopies_of_ne_bot hH, Set.disjoint_left, isContained_iff_exists_iso_subgraph,
@forall_comm _ G.Subgraph, deleteEdges_eq_self, Set.mem_iUnion,
not_exists, not_nonempty_iff, Nonempty.forall, Free]
exact forall_congr' fun G' ↦ ⟨fun h ↦ ⟨fun f ↦ h _
(Subgraph.edgeSet_subset _ <| (aux hH ⟨f⟩... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 31
} | {
"line": 156,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nx✝¹ x✝ : V\n⊢ Decidable ((edge s t).Adj x✝¹ x✝)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"SimpleGraph.edge",
"congrArg",
"SimpleGraph.Adj",
"Dec... | [] | rw [edge_adj]; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 31
} | {
"line": 156,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : V\ninst✝ : DecidableEq V\nx✝¹ x✝ : V\n⊢ Decidable ((edge s t).Adj x✝¹ x✝)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"SimpleGraph.edge",
"congrArg",
"SimpleGraph.Adj",
"Dec... | [] | rw [edge_adj]; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 199,
"column": 11
} | {
"line": 199,
"column": 25
} | {
"line": 199,
"column": 26
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ ⨆ e ∈ G.edgeSet, fromEdgeSet {e} = G",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iSup",
"SimpleGraph.fromEdgeSet",
"Membership.mem",
"Set.instSingletonSet",
"id",
"SimpleGraph.edgeSet",
... | [
"V : Type u_1\nG : SimpleGraph V\n⊢ (⨆ e ∈ G.edgeSet, fromEdgeSet {e}).edgeSet = G.edgeSet"
] | ← edgeSet_inj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Operations | {
"line": 203,
"column": 82
} | {
"line": 205,
"column": 93
} | {
"line": 207,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ sSup {x | ∃ u v, ∃ (_ : G.Adj u v), edge u v = x} = G",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.edge",
"_private.Mathlib.Combinatorics.SimpleGraph.Operations.0.SimpleGraph.sSup_edge_eq._simp_1_2",
... | [] | by
refine .trans ?_ G.biSup_fromEdgeSet_singleton_eq
simp_rw [edge, ← iSup_subtype'', iSup, Set.range, Subtype.exists, Sym2.exists, mem_edgeSet] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 944,
"column": 41
} | {
"line": 947,
"column": 37
} | {
"line": 949,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\nH : G.Subgraph\nh : H.Adj v w\n⊢ G.subgraphOfAdj ⋯ ≤ H",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"SimpleGraph.Subgraph.adj_sub",
"_private.Mathlib.Combinatorics.SimpleGraph.Subgraph.0.SimpleGraph.subgraphOfAdj_le_of_adj._pr... | [] | by
constructor
· grind [subgraphOfAdj_verts, h.fst_mem, h.snd_mem]
· grind [subgraphOfAdj_adj, h.symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 255,
"column": 2
} | {
"line": 256,
"column": 75
} | {
"line": 257,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | [
"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V ∈ P.parts\n... | conv_rhs => -- Porting note: LHS and RHS need separate treatment to get the desired form
simp only [SimpleGraph.edgeDensity_def, sum_div, Rat.cast_div, div_div] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 40
} | {
"line": 333,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.... | [] | apply this.trans
gcongr <;> [sz_positivity; norm_num] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 331,
"column": 4
} | {
"line": 332,
"column": 40
} | {
"line": 333,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.... | [] | apply this.trans
gcongr <;> [sz_positivity; norm_num] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 39
} | {
"line": 492,
"column": 2
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu✝ v✝ : V\np : G.Walk u✝ v✝\nf : G →g G'\nh : (Walk.map f p).IsPath\nv : V\nhv : v ∈ {w | w ∈ p.support}\nu : Fin p.support.length\nhu : p.support.get u ∈ {w | w ∈ p.support}\nhf : f (p.support.get u) = f v\n⊢ p.support.get u = v",
"p... | [
"V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu✝ v✝ : V\np : G.Walk u✝ v✝\nf : G →g G'\nh : (Walk.map f p).IsPath\nu : Fin p.support.length\nhu : p.support.get u ∈ {w | w ∈ p.support}\nv : Fin p.support.length\nhv : p.support.get v ∈ {w | w ∈ p.support}\nhf : f (p.support.get u) = f (p.support.g... | obtain ⟨v, rfl⟩ := List.get_of_mem hv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 36
} | {
"line": 455,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nn : ℕ\nf : α ↪ β\ninst✝ : Nonempty α\n⊢ (SimpleGraph.map (⇑f) G).CliqueFree n ↔ G.CliqueFree n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"instOf... | [
"case inl\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nn : ℕ\nf : α ↪ β\ninst✝ : Nonempty α\nhle : n ≤ 1\n⊢ (SimpleGraph.map (⇑f) G).CliqueFree n ↔ G.CliqueFree n",
"case inr\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nn : ℕ\nf : α ↪ β\ninst✝ : Nonempty α\nhlt : 1 < n\n⊢ (SimpleGraph.map (⇑f) G).CliqueFree... | obtain (hle | hlt) := le_or_gt n 1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 173,
"column": 90
} | {
"line": 174,
"column": 25
} | {
"line": 174,
"column": 25
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\na b c : α\nhab : G.Adj a b\nhac : G.Adj a c\nhbc : G.Adj b c\n⊢ #{s(b, c)} + 1 = 2",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
rw [card_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 959,
"column": 34
} | {
"line": 965,
"column": 49
} | {
"line": 967,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nv' : V\nhvv' : G.Adj v v'\nw : G.Walk v' v\nhw : (cons hvv' w).IsCircuit\n⊢ (cons hvv' w).cycleBypass.IsCycle",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.IsPath.isTrail",
"False",
... | [] | by
dsimp [cycleBypass]
refine ⟨⟨(bypass_isPath _).isTrail.cons _ fun hvv' ↦ ?_, by simp⟩, ?_⟩
· simp only [isCircuit_def, isTrail_cons, ne_eq, reduceCtorEq, not_false_eq_true,
and_true] at hw
exact hw.2 <| edges_bypass_subset_edges _ hvv'
· simpa using (bypass_isPath _).support_nodup | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 1014,
"column": 2
} | {
"line": 1014,
"column": 27
} | {
"line": 1016,
"column": 0
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu : V\np : G.Walk u u\nhp : (Walk.map f p).IsTrail ∧ ¬p.Nil\n⊢ p.IsTrail ∧ ¬p.Nil",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.instFunLike",
"SimpleGr... | [] | exact hp.imp_left .of_map | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 492,
"column": 74
} | {
"line": 492,
"column": 83
} | {
"line": 492,
"column": 84
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤... | [
"case refine_3\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈... | cast_pow, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Counting | {
"line": 141,
"column": 35
} | {
"line": 144,
"column": 30
} | {
"line": 146,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : Disjoint s t\nhsu : Disjoint s u\nhtu : Disjoint t u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nh : {x₁, y₁, z₁} = {x₂, y₂, z₂}\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)",
"ppTerm": "?... | [] | by
simp only [Finset.Subset.antisymm_iff, subset_iff, mem_insert, mem_singleton, forall_eq_or_imp,
forall_eq] at h
grind [Finset.disjoint_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 9
} | {
"line": 62,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : AddCommGroup G\nA : Finset (G × G)\nε : ℝ\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)\n⊢ ε * ((↑(Fintype.card G) + ↑(Fintype.card G) + ↑(Fintype.card G)) ^ 2 / 9) ≤ ↑(#A)",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
... | [
"G : Type u_1\ninst✝² : AddCommGroup G\nA : Finset (G × G)\nε : ℝ\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)\n⊢ ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 43
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nhA : A.Nonempty\n⊢ σₘ[A, B] * ↑(#A) = ↑(#(A * B))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"div_mul_cancel₀",
"Preor... | [] | exact div_mul_cancel₀ _ (by positivity) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 43
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nhA : A.Nonempty\n⊢ σₘ[A, B] * ↑(#A) = ↑(#(A * B))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"div_mul_cancel₀",
"Preor... | [] | exact div_mul_cancel₀ _ (by positivity) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 43
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nhA : A.Nonempty\n⊢ σₘ[A, B] * ↑(#A) = ↑(#(A * B))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"div_mul_cancel₀",
"Preor... | [] | exact div_mul_cancel₀ _ (by positivity) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 43
} | {
"line": 82,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nhA : A.Nonempty\n⊢ δₘ[A, B] * ↑(#A) = ↑(#(A / B))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"div_mul_cancel₀",
"Preor... | [] | exact div_mul_cancel₀ _ (by positivity) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 43
} | {
"line": 82,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nhA : A.Nonempty\n⊢ δₘ[A, B] * ↑(#A) = ↑(#(A / B))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"div_mul_cancel₀",
"Preor... | [] | exact div_mul_cancel₀ _ (by positivity) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 43
} | {
"line": 82,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\nhA : A.Nonempty\n⊢ δₘ[A, B] * ↑(#A) = ↑(#(A / B))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"div_mul_cancel₀",
"Preor... | [] | exact div_mul_cancel₀ _ (by positivity) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 70
} | {
"line": 95,
"column": 71
} | [
{
"pp": "G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : 1 < triangleRemovalBound (ε / 9) * 27 * ↑(Fintype.card G)\nA : Finset (G × G)\nhAε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)\nhA : IsCornerFree ↑A\nhε₁ : ε ≤ 1\nthis : NoAccidental (triangleIndices A)\nh₁ : triangleRemovalBound... | [
"case e'_3\nG : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : 1 < triangleRemovalBound (ε / 9) * 27 * ↑(Fintype.card G)\nA : Finset (G × G)\nhAε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)\nhA : IsCornerFree ↑A\nhε₁ : ε ≤ 1\nthis : NoAccidental (triangleIndices A)\nh₁ : triangleRemovalBound ... | convert! h₁.trans (Nat.cast_le.2 <| card_le_univ _) using 1 <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 44
} | {
"line": 112,
"column": 4
} | [
{
"pp": "n : ℕ\nε : ℝ\nhε : 0 < ε\nhn : cornersTheoremBound (ε / 9) ≤ n\nA : Finset (ℕ × ℕ)\nhAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)\nhAε : ε * ↑n ^ 2 ≤ ↑(#A)\nhA : IsCornerFree ↑A\n⊢ ∀ a ∈ ↑A, Prod.map Fin.val Fin.val (Prod.map Nat.cast Nat.cast a) = id a",
"ppTerm": "?m.109",
"assigned": true,
"usedCo... | [
"n : ℕ\nε : ℝ\nhε : 0 < ε\nhn : cornersTheoremBound (ε / 9) ≤ n\nA : Finset (ℕ × ℕ)\nhAn : ↑A ⊆ ↑(range n) ×ˢ ↑(range n)\nhAε : ε * ↑n ^ 2 ≤ ↑(#A)\nhA : IsCornerFree ↑A\n⊢ ∀ (a b : ℕ), (a, b) ∈ A → a < 2 * n + 1 ∧ b < 2 * n + 1"
] | simp only [mem_coe, Nat.succ_eq_add_one, Prod.map_apply, Fin.val_natCast, id_eq, Prod.forall,
Prod.mk.injEq, Nat.mod_succ_eq_iff_lt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 90,
"column": 49
} | {
"line": 90,
"column": 61
} | {
"line": 90,
"column": 62
} | [
{
"pp": "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\nd : σ →₀ ℕ\nhd : d ∈ f.support\ni : σ\nhi : d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Su... | [
"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\nd : σ →₀ ℕ\nhd : d ∈ f.support\ni : σ\nhi : d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Subtype.val = ... | univ_unique, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 191,
"column": 71
} | {
"line": 194,
"column": 87
} | {
"line": 196,
"column": 0
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf₁ f₂ : MvPolynomial σ K\nh : f₁.totalDegree + f₂.totalDegree < Fintype.card σ\n⊢ p ∣ Fintype.card { x // (eval x) f₁ = 0 ∧ (eval x) f₂ = 0 }",... | [] | by
let F : Bool → MvPolynomial σ K := fun b => cond b f₂ f₁
have : (∑ b : Bool, (F b).totalDegree) < Fintype.card σ := (add_comm _ _).trans_lt h
simpa only [Bool.forall_bool] using! char_dvd_card_solutions_of_fintype_sum_lt p this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 172,
"column": 28
} | {
"line": 172,
"column": 53
} | {
"line": 172,
"column": 53
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n)... | [] | by rw [card_cons, h𝒜card] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Colex | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 63
} | {
"line": 382,
"column": 4
} | [
{
"pp": "case inr.inr.inl\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\na : α\nhcard : #s ≤ #t\nha : a ∈ s\nht : t.Nonempty\nm : α := t.min' ht\nh' : s ≠ t\nhwt : m ∈ t\nhws : m ∉ s\nhw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)\nhaw : a < m\nthis : t.erase m ⊆ s.erase a\n⊢ #(s.erase a) ≤ #(t.erase m)",
"... | [
"case inr.inr.inl\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\na : α\nhcard : #s ≤ #t\nha : a ∈ s\nht : t.Nonempty\nm : α := t.min' ht\nh' : s ≠ t\nhwt : m ∈ t\nhws : m ∉ s\nhw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)\nhaw : a < m\nthis : t.erase m ⊆ s.erase a\n⊢ #s - 1 ≤ #t - 1"
] | rw [card_erase_of_mem ha, card_erase_of_mem (min'_mem _ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 237,
"column": 42
} | {
"line": 237,
"column": 54
} | {
"line": 237,
"column": 54
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : D ≠ D'\na : K\nhD : a • D.rep = D'.rep\n⊢ mk K D'.rep ⋯ = mk K D.rep ⋯",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Projectivization.mk",
"Eq.mpr",
... | [
"K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : D ≠ D'\na : K\nhD : a • D.rep = D'.rep\n⊢ ∃ a, a • D.rep = D'.rep"
] | mk_eq_mk_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Colex | {
"line": 478,
"column": 75
} | {
"line": 481,
"column": 45
} | {
"line": 483,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\n𝒜 : Finset (Finset α)\nr : ℕ\ninst✝ : Fintype α\n⊢ IsInitSeg 𝒜 r ∧ 𝒜.Nonempty ↔ ∃ s, #s = r ∧ 𝒜 = initSeg s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finset",
"Finset.Colex.IsInitSeg.exists_initSeg",
"Exists",
... | [] | by
refine ⟨fun h𝒜 ↦ h𝒜.1.exists_initSeg h𝒜.2, ?_⟩
rintro ⟨s, rfl, rfl⟩
exact ⟨isInitSeg_initSeg, initSeg_nonempty⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Configuration | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 63
} | {
"line": 126,
"column": 4
} | [
{
"pp": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : Nondegenerate P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nh : Fintype.card L ≤ Fintype.card P\n⊢ ∃ f, Function.Injective f ∧ ∀ (l : L), f l ∉ l",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"instDecidableNot",
... | [
"P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : Nondegenerate P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nh : Fintype.card L ≤ Fintype.card P\nt : L → Finset P := fun l ↦ {p | p ∉ l}.toFinset\n⊢ ∃ f, Function.Injective f ∧ ∀ (l : L), f l ∉ l"
] | let t : L → Finset P := fun l => Set.toFinset { p | p ∉ l } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Projectivization.Constructions | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 22
} | {
"line": 129,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nv w : ℙ F (Fin 3 → F)\nh : v ≠ w\n⊢ v.orthogonal (v.cross w)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Projectivization.orthogonal_comm",
"Projectivization.orthogonal",
... | [
"F : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nv w : ℙ F (Fin 3 → F)\nh : v ≠ w\n⊢ (v.cross w).orthogonal v"
] | rw [orthogonal_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Projectivization.Constructions | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 22
} | {
"line": 134,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nv w : ℙ F (Fin 3 → F)\nh : v ≠ w\n⊢ w.orthogonal (v.cross w)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Projectivization.orthogonal_comm",
"Projectivization.orthogonal",
... | [
"F : Type u_1\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nv w : ℙ F (Fin 3 → F)\nh : v ≠ w\n⊢ (v.cross w).orthogonal w"
] | rw [orthogonal_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Configuration | {
"line": 199,
"column": 55
} | {
"line": 199,
"column": 79
} | {
"line": 199,
"column": 79
} | [
{
"pp": "case neg.intro\nP : Type u_1\nL : Type u_2\ninst✝² : Membership P L\ninst✝¹ : HasLines P L\np : P\nl : L\nh : p ∉ l\ninst✝ : Finite { l // p ∈ l }\nhf : ¬Infinite { p // p ∈ l }\nthis : Fintype { p // p ∈ l }\nval✝ : Fintype { l // p ∈ l }\n⊢ Fintype.card { p // p ∈ l } ≤ Nat.card { l // p ∈ l }",
... | [
"case neg.intro\nP : Type u_1\nL : Type u_2\ninst✝² : Membership P L\ninst✝¹ : HasLines P L\np : P\nl : L\nh : p ∉ l\ninst✝ : Finite { l // p ∈ l }\nhf : ¬Infinite { p // p ∈ l }\nthis : Fintype { p // p ∈ l }\nval✝ : Fintype { l // p ∈ l }\n⊢ Fintype.card { p // p ∈ l } ≤ Fintype.card { l // p ∈ l }"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 157,
"column": 14
} | {
"line": 159,
"column": 48
} | {
"line": 160,
"column": 2
} | [
{
"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)\n⊢ 1 ∈ ↑A⁻¹ * ↑A",
"ppTerm": "?m.519",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"MulOne.toOne",
"inv_mul_c... | [] | by
have ⟨x, hx⟩ : A.Nonempty := nonempty_of_doubling h
exact ⟨x⁻¹, inv_mem_inv hx, x, by simp [hx]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Configuration | {
"line": 498,
"column": 56
} | {
"line": 498,
"column": 73
} | {
"line": 498,
"column": 74
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\na b c d : Fin 3 → K\nhac : a ⬝ᵥ c = 0\nhbc : b ⬝ᵥ c = 0\nhad : a ⬝ᵥ d = 0\nhbd : b ⬝ᵥ d = 0\nh : LinearIndependent K (of ![a, b]).row ∧ LinearIndependent K (of ![c, d]).row\nA : Matrix (Fin 2) (Fin 3) K := of ![a, b]\nB : Matrix (Fin 2) (Fin 3) K := of ![c, d]\nhAB : Fint... | [
"K : Type u_3\ninst✝ : Field K\na b c d : Fin 3 → K\nhac : a ⬝ᵥ c = 0\nhbc : b ⬝ᵥ c = 0\nhad : a ⬝ᵥ d = 0\nhbd : b ⬝ᵥ d = 0\nh : LinearIndependent K (of ![a, b]).row ∧ LinearIndependent K (of ![c, d]).row\nA : Matrix (Fin 2) (Fin 3) K := of ![a, b]\nB : Matrix (Fin 2) (Fin 3) K := of ![c, d]\nhAB : (Nat.succ 0).suc... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 15
} | {
"line": 78,
"column": 0
} | [
{
"pp": "x : ℕ\nhx : x ≠ 0\nc : ℕ\n⊢ (x * c + x)! = (x * (c + 1))!",
"ppTerm": "?m.356",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.RingNF.add_assoc_rev",
"Mathlib.T... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 88,
"column": 8
} | {
"line": 88,
"column": 43
} | {
"line": 89,
"column": 8
} | [
{
"pp": "case pos\nm : Multiset ℕ\nthis : ?m.115\nhm : 0 ∈ m.toFinset\n⊢ (∏ m_1 ∈ m.toFinset, m_1 ! ^ count m_1 m) *\n ∏ x ∈ m.toFinset.erase 0, (count x m)! * ∏ j ∈ Finset.range (count x m), (j * x + x - 1).choose (x - 1) =\n ∏ i ∈ m.toFinset, (i * count i m)!",
"ppTerm": "?pos✝",
"assigned": t... | [
"case pos\nm : Multiset ℕ\nthis : ?m.115\nhm : 0 ∈ m.toFinset\n⊢ (∏ x ∈ m.toFinset.erase 0, x ! ^ count x m) * 0! ^ count 0 m *\n ∏ x ∈ m.toFinset.erase 0, (count x m)! * ∏ j ∈ Finset.range (count x m), (j * x + x - 1).choose (x - 1) =\n ∏ i ∈ m.toFinset, (i * count i m)!"
] | rw [← Finset.prod_erase_mul _ _ hm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 89,
"column": 8
} | {
"line": 89,
"column": 43
} | {
"line": 90,
"column": 8
} | [
{
"pp": "case pos\nm : Multiset ℕ\nthis : ?m.115\nhm : 0 ∈ m.toFinset\n⊢ (∏ x ∈ m.toFinset.erase 0, x ! ^ count x m) * 0! ^ count 0 m *\n ∏ x ∈ m.toFinset.erase 0, (count x m)! * ∏ j ∈ Finset.range (count x m), (j * x + x - 1).choose (x - 1) =\n ∏ i ∈ m.toFinset, (i * count i m)!",
"ppTerm": "?pos✝"... | [
"case pos\nm : Multiset ℕ\nthis : ?m.115\nhm : 0 ∈ m.toFinset\n⊢ (∏ x ∈ m.toFinset.erase 0, x ! ^ count x m) * 0! ^ count 0 m *\n ∏ x ∈ m.toFinset.erase 0, (count x m)! * ∏ j ∈ Finset.range (count x m), (j * x + x - 1).choose (x - 1) =\n (∏ x ∈ m.toFinset.erase 0, (x * count x m)!) * (0 * count 0 m)!"
] | rw [← Finset.prod_erase_mul _ _ hm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 87,
"column": 6
} | {
"line": 94,
"column": 18
} | {
"line": 95,
"column": 4
} | [
{
"pp": "m : Multiset ℕ\nthis : ?m.115\n⊢ (∏ m_1 ∈ m.toFinset, m_1 ! ^ count m_1 m) *\n ∏ x ∈ m.toFinset.erase 0, (count x m)! * ∏ j ∈ Finset.range (count x m), (j * x + x - 1).choose (x - 1) =\n ∏ i ∈ m.toFinset, (i * count i m)!",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
... | [] | by_cases hm : 0 ∈ m.toFinset
· rw [← Finset.prod_erase_mul _ _ hm]
rw [← Finset.prod_erase_mul _ _ hm]
simp only [factorial_zero, one_pow, mul_one, zero_mul]
exact this
· nth_rewrite 1 [← Finset.erase_eq_of_notMem hm]
nth_rewrite 3 [← Finset.erase_eq_of_notMem hm]
exa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 87,
"column": 6
} | {
"line": 94,
"column": 18
} | {
"line": 95,
"column": 4
} | [
{
"pp": "m : Multiset ℕ\nthis : ?m.115\n⊢ (∏ m_1 ∈ m.toFinset, m_1 ! ^ count m_1 m) *\n ∏ x ∈ m.toFinset.erase 0, (count x m)! * ∏ j ∈ Finset.range (count x m), (j * x + x - 1).choose (x - 1) =\n ∏ i ∈ m.toFinset, (i * count i m)!",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
... | [] | by_cases hm : 0 ∈ m.toFinset
· rw [← Finset.prod_erase_mul _ _ hm]
rw [← Finset.prod_erase_mul _ _ hm]
simp only [factorial_zero, one_pow, mul_one, zero_mul]
exact this
· nth_rewrite 1 [← Finset.erase_eq_of_notMem hm]
nth_rewrite 3 [← Finset.erase_eq_of_notMem hm]
exa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 300,
"column": 33
} | {
"line": 300,
"column": 75
} | {
"line": 301,
"column": 6
} | [
{
"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 •> (A⁻¹ * A) * A⁻¹ ⊆ A * (A⁻¹ * A) * A⁻¹",
"ppTerm": "?m.129",
"assigned": true,
"usedConstants": [
"le_refl",
"instHSMul",
"instSMulOfMul",
"... | [] | by grw [smul_finset_subset_mul (by simpa)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 125,
"column": 12
} | {
"line": 125,
"column": 44
} | {
"line": 125,
"column": 44
} | [
{
"pp": "case pos\nm : Multiset ℕ\na : ℕ\nha : a ≠ 0\nrest : ℕ := ∏ j ∈ (m.toFinset.erase 0).erase a, (count j m)!\nhrest : rest = ∏ j ∈ ((a ::ₘ m).toFinset.erase 0).erase a, (count j (a ::ₘ m))!\nc : ℕ := (map (fun x ↦ x !) m).prod * (count a m)! * rest\nhmem : a ∈ m.toFinset.erase 0\n⊢ (count a m)! * rest = ∏... | [
"case pos\nm : Multiset ℕ\na : ℕ\nha : a ≠ 0\nrest : ℕ := ∏ j ∈ (m.toFinset.erase 0).erase a, (count j m)!\nhrest : rest = ∏ j ∈ ((a ::ₘ m).toFinset.erase 0).erase a, (count j (a ::ₘ m))!\nc : ℕ := (map (fun x ↦ x !) m).prod * (count a m)! * rest\nhmem : a ∈ m.toFinset.erase 0\n⊢ (count a m)! * rest = (count a m)! ... | ← Finset.mul_prod_erase _ _ hmem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 85
} | {
"line": 213,
"column": 85
} | [
{
"pp": "n : ℕ\n⊢ ∑ i ∈ Finset.range (n + 1), n.choose i * (n - i).bell = ∑ ij ∈ Finset.antidiagonal n, n.choose ij.1 * ij.2.bell",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Finset.Nat.sum_antidiagonal_eq_sum_range_succ",
"Eq.mpr",
"Nat.choose",
"HMul.hMul",
... | [
"n : ℕ\n⊢ ∑ ij ∈ Finset.antidiagonal n, n.choose ij.1 * ij.2.bell = ∑ ij ∈ Finset.antidiagonal n, n.choose ij.1 * ij.2.bell"
] | ← Finset.Nat.sum_antidiagonal_eq_sum_range_succ (fun x y ↦ choose n x * y.bell) n | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 30
} | {
"line": 290,
"column": 31
} | [
{
"pp": "p : DyckWord\ni : ℕ\nhi : i < p.firstReturn\nne :\n decide (count U (List.take ((range (↑p).length)[i] + 1) ↑p) = count D (List.take ((range (↑p).length)[i] + 1) ↑p)) =\n false\n⊢ count D (List.take (i + 1) ↑p) < count U (List.take (i + 1) ↑p)",
"ppTerm": "?m.32",
"assigned": true,
"use... | [
"p : DyckWord\ni : ℕ\nhi : i < p.firstReturn\nne : ¬count U (List.take ((range (↑p).length)[i] + 1) ↑p) = count D (List.take ((range (↑p).length)[i] + 1) ↑p)\n⊢ count D (List.take (i + 1) ↑p) < count U (List.take (i + 1) ↑p)"
] | decide_eq_false_iff_not, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 308,
"column": 19
} | {
"line": 308,
"column": 97
} | {
"line": 310,
"column": 0
} | [
{
"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 : PartialOrder α\ninst✝⁴ : LocallyFiniteOrder α\ninst✝³ : DecidableEq α\ninst✝² : CommSemiring 𝕜\ninst✝¹ : CommSemiring 𝕝\ninst✝ : Algebra 𝕜 𝕝\nc : 𝕜\nf : IncidenceAlgebra 𝕝 α\n⊢ c • f = { toFun := fun c... | [] | by classical ext a b hab; simp [if_pos hab, constSMul_apply, Algebra.smul_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 371,
"column": 8
} | {
"line": 371,
"column": 19
} | {
"line": 371,
"column": 19
} | [
{
"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✝⁴ : AddCommGroup 𝕜\ninst✝³ : One 𝕜\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\nh : ¬muFun 𝕜 a b = 0\n⊢ a ≤ b",
"ppTerm": "?m.24",
"assigned": true,
"usedCo... | [
"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✝⁴ : AddCommGroup 𝕜\ninst✝³ : One 𝕜\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\nh : ¬(if a = b then 1 else -∑ x ∈ (Ico a b).attach, muFun 𝕜 a ↑x) = 0\n⊢ a ≤ b"
] | muFun_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 317,
"column": 4
} | {
"line": 323,
"column": 9
} | {
"line": 324,
"column": 2
} | [
{
"pp": "case right\np : DyckWord\nu : ↑p.nest = U :: ↑p ++ [D]\n⊢ ∀ j < (↑p).length + 1,\n decide (count U (List.take (j + 1) (U :: ↑p ++ [D])) = count D (List.take (j + 1) (U :: ↑p ++ [D]))) = false",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableE... | [] | · intro j hj
simp_rw [cons_append, take_succ_cons, count_cons, beq_self_eq_true, ite_true,
beq_iff_eq, reduceCtorEq, ite_false, take_append,
show j - p.toList.length = 0 by lia, take_zero, append_nil]
have := p.count_D_le_count_U j
simp only [add_zero, decide_eq_false_iff_not, ne_eq]
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 423,
"column": 81
} | {
"line": 424,
"column": 13
} | {
"line": 426,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝⁴ : AddCommGroup 𝕜\ninst✝³ : One 𝕜\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\n⊢ muFun' 𝕜 b a = if a = b then 1 else -∑ x ∈ (Ioc a b).attach, muFun' 𝕜 b ↑x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": ... | [] | by
rw [muFun'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Pentagonal | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 34
} | {
"line": 59,
"column": 0
} | [
{
"pp": "x y : ℤ\nh : (3 * (x + y) - 1) * (x - y) = 0\n⊢ x = y",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"MulZeroClass.toMul",
"HSub.hSub",
"Int",
"_private.Mathlib.Combinatorics.Enumerative.Pentagonal.0.p... | [] | cases mul_eq_zero.mp h <;> grind | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 752,
"column": 8
} | {
"line": 752,
"column": 60
} | {
"line": 752,
"column": 60
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nS : Finset G\nε : ℝ\nhε₀ : 0 < ε\nhε₁ : ε ≤ 1\nhS : S.Nonempty\nK : ℝ := 1 - ε / 2\nhK : K < 1\nex : Finset G → ℝ := expansion K S\nκ : ℝ := connectivity K S\nH : Subgroup G\nw✝ : Fintype ↥H\nhH : IsAtom K S (↑H).toFinset\nZ : Finset G\nhZS : Z ⊆ S... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nS : Finset G\nε : ℝ\nhε₀ : 0 < ε\nhε₁ : ε ≤ 1\nhS : S.Nonempty\nK : ℝ := 1 - ε / 2\nhK : K < 1\nex : Finset G → ℝ := expansion K S\nκ : ℝ := connectivity K S\nH : Subgroup G\nw✝ : Fintype ↥H\nhH : IsAtom K S (↑H).toFinset\nZ : Finset G\nhZS : Z ⊆ S\nhHZS : ↑H ... | ← mul_le_mul_iff_right₀ (a := ε / 2) (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 148,
"column": 82
} | {
"line": 151,
"column": 29
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤ G\n⊢ EqOn H.IsLoopAt G.IsLoopAt E(H)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Graph.IsLoopAt",
"Membership.mem",
"funext",
"Graph.edgeSet",
"propext",
"Graph.IsSubgraph.isLoopAt_cong... | [] | by
rintro e he
ext x
exact hHG.isLoopAt_congr he | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Extremal.RuzsaSzemeredi | {
"line": 199,
"column": 6
} | {
"line": 199,
"column": 13
} | {
"line": 201,
"column": 0
} | [
{
"pp": "n : ℕ\nα : Type := Fin (2 * n + 1)\nthis✝ : Coprime 2 (2 * n + 1)\nthis : Fact (IsUnit 2)\n⊢ ruzsaSzemerediNumberNat (2 * n + 1 + (2 * n + 1 + (2 * n + 1))) = ruzsaSzemerediNumberNat (6 * n + 3)",
"ppTerm": "?m.225",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiri... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Combinatorics.Graph.Basic | {
"line": 222,
"column": 2
} | {
"line": 228,
"column": 12
} | {
"line": 230,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx y : α\ne : β\nG : Graph α β\n⊢ G.IsLink e x y ↔ G.Inc e x ∧ G.Inc e y ∧ ∀ (z : α), G.Inc e z → z = x ∨ z = y",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Graph.Inc",
"Graph.IsLink.inc_left",
"Or.casesOn",
"And.casesOn",
... | [] | refine ⟨fun h ↦ ⟨h.inc_left, h.inc_right, fun z h' ↦ h'.eq_or_eq_of_isLink h⟩, ?_⟩
rintro ⟨⟨x', hx'⟩, ⟨y', hy'⟩, h⟩
obtain rfl | rfl := h _ hx'.inc_right
· obtain rfl | rfl := hx'.left_eq_or_eq hy'
· assumption
exact hy'.symm
assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Graph.Basic | {
"line": 222,
"column": 2
} | {
"line": 228,
"column": 12
} | {
"line": 230,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx y : α\ne : β\nG : Graph α β\n⊢ G.IsLink e x y ↔ G.Inc e x ∧ G.Inc e y ∧ ∀ (z : α), G.Inc e z → z = x ∨ z = y",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Graph.Inc",
"Graph.IsLink.inc_left",
"Or.casesOn",
"And.casesOn",
... | [] | refine ⟨fun h ↦ ⟨h.inc_left, h.inc_right, fun z h' ↦ h'.eq_or_eq_of_isLink h⟩, ?_⟩
rintro ⟨⟨x', hx'⟩, ⟨y', hy'⟩, h⟩
obtain rfl | rfl := h _ hx'.inc_right
· obtain rfl | rfl := hx'.left_eq_or_eq hy'
· assumption
exact hy'.symm
assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Semigroup | {
"line": 42,
"column": 8
} | {
"line": 44,
"column": 80
} | {
"line": 45,
"column": 6
} | [
{
"pp": "case ht\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r\nS : Set (Set M) := ⋯\nN : Set M\nhN : Minimal (fun x ↦ x ∈ S) N\nN_closed : IsClosed N\nN_mul : ∀ m ∈ N,... | [] | refine ⟨(continuous_const_mul m).isClosedMap _ N_closed, ⟨_, ⟨m, hm, rfl⟩⟩, ?_⟩
rintro _ ⟨m'', hm'', rfl⟩ _ ⟨m', hm', rfl⟩
exact ⟨m'' * m * m', N_mul _ (N_mul _ hm'' _ hm) _ hm', mul_assoc _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Semigroup | {
"line": 42,
"column": 8
} | {
"line": 44,
"column": 80
} | {
"line": 45,
"column": 6
} | [
{
"pp": "case ht\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r\nS : Set (Set M) := ⋯\nN : Set M\nhN : Minimal (fun x ↦ x ∈ S) N\nN_closed : IsClosed N\nN_mul : ∀ m ∈ N,... | [] | refine ⟨(continuous_const_mul m).isClosedMap _ N_closed, ⟨_, ⟨m, hm, rfl⟩⟩, ?_⟩
rintro _ ⟨m'', hm'', rfl⟩ _ ⟨m', hm', rfl⟩
exact ⟨m'' * m * m', N_mul _ (N_mul _ hm'' _ hm) _ hm', mul_assoc _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Semigroup | {
"line": 53,
"column": 10
} | {
"line": 53,
"column": 65
} | {
"line": 54,
"column": 10
} | [
{
"pp": "case ht.refine_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r\nS : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}\nN : Set M\nhN... | [
"case ht.refine_2\nM : Type u_1\ninst✝⁴ : Nonempty M\ninst✝³ : Semigroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : CompactSpace M\ninst✝ : T2Space M\ncontinuous_const_mul : ∀ (r : M), Continuous fun x ↦ x * r\nS : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}\nN : Set M\nhN : Minimal (... | rintro m'' ⟨mem'', eq'' : _ = m⟩ m' ⟨mem', eq' : _ = m⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.Hindman | {
"line": 185,
"column": 66
} | {
"line": 185,
"column": 86
} | {
"line": 185,
"column": 86
} | [
{
"pp": "M : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns₀ : Set M\nsU : s₀ ∈ U\ns : Set M\nhs : s ∈ U\n⊢ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s} ∈ ↑U",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
... | [] | rwa [← U_idem] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Combinatorics.Hindman | {
"line": 185,
"column": 66
} | {
"line": 185,
"column": 86
} | {
"line": 185,
"column": 86
} | [
{
"pp": "M : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns₀ : Set M\nsU : s₀ ∈ U\ns : Set M\nhs : s ∈ U\n⊢ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s} ∈ ↑U",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
... | [] | rwa [← U_idem] at hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Hindman | {
"line": 185,
"column": 66
} | {
"line": 185,
"column": 86
} | {
"line": 185,
"column": 86
} | [
{
"pp": "M : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns₀ : Set M\nsU : s₀ ∈ U\ns : Set M\nhs : s ∈ U\n⊢ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s} ∈ ↑U",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
... | [] | rwa [← U_idem] at hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 86
} | {
"line": 306,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis' I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\n⊢ (M ／ C).IsBasis' (I \\ C) (X \\ C)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.IsBasis'",... | [
"α : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis' I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\n⊢ (M ／ C).IsBasis (I \\ C) ((X ∩ M.E) \\ C)"
] | rw [isBasis'_iff_isBasis_inter_ground, contract_ground, ← sdiff_inter_distrib_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 119,
"column": 2
} | {
"line": 124,
"column": 19
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case right\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf b : MvPolynomial σ R\nhb : IsUnit (m.leadingCoeff b)\nhbf : m.degree b ≤ m.degree f\nhf : m.degree f ≠ 0\nH : m.degree f = m.degree ((monomial (m.degree f - m.degree b)) (↑hb.unit⁻¹ * m.leadingCoeff f)) + m.degree b\nH' ... | [] | · intro K
simp only [EmbeddingLike.apply_eq_iff_eq] at K
nth_rewrite 1 [← K] at H'
rw [← leadingCoeff, leadingCoeff_eq_zero_iff] at H'
rw [H', degree_zero] at K
exact hf K.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 117,
"column": 8
} | {
"line": 117,
"column": 76
} | {
"line": 118,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nσ✝ : Type u_2\ninst✝² : Finite σ✝\ninst✝¹ : IsDomain R\nσ : Type u_2\ninst✝ : Fintype σ\nh :\n ∀ (P : MvPolynomial σ R) (S : σ → Finset R),\n (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0\nP : MvPolynomial (O... | [
"R : Type u_1\ninst✝³ : CommRing R\nσ✝ : Type u_2\ninst✝² : Finite σ✝\ninst✝¹ : IsDomain R\nσ : Type u_2\ninst✝ : Fintype σ\nh :\n ∀ (P : MvPolynomial σ R) (S : σ → Finset R),\n (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0\nP : MvPolynomial (Option σ) R\n... | set n := (embDomain Function.Embedding.some m).update none d with hn | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
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