module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SetFamily.Kleitman | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 56
} | [
{
"pp": "case inl\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ i ∈ s, (↑(f i)).Intersecting\nthis : DecidableEq ι\nhs : Fintype.card α ≤ #s\n⊢ #{⊥}ᶜ = 2 ^ Fintype.card α - 1",
"usedConstants": [
"Finset.fin... | rw [card_compl, Fintype.card_finset, card_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SetFamily.Kleitman | {
"line": 70,
"column": 16
} | {
"line": 70,
"column": 24
} | [
{
"pp": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 115,
"column": 16
} | {
"line": 115,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CommSemiring β\na : α\ns : Finset α\nha : a ∉ s\n𝒜 : Finset (Finset α)\nf : Finset α → β\n⊢ ∑ t ∈ 𝒜 with t.erase a = s, f t = (if s ∈ 𝒜 then f s else 0) + if insert a s ∈ 𝒜 then f (insert a s) else 0",
"usedConstants": [
"Eq.mpr"... | filter_collapse_eq ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Intersecting | {
"line": 107,
"column": 2
} | {
"line": 112,
"column": 100
} | [
{
"pp": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Set α\nhs : s.Intersecting\nh : ∀ (t : Set α), t.Intersecting → s ⊆ t → s = t\n⊢ IsUpperSet s",
"usedConstants": [
"Eq.mpr",
"Set.Intersecting.ne_bot",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreord... | classical
rintro a b hab ha
rw [h (Insert.insert b s) _ (subset_insert _ _)]
· exact mem_insert _ _
exact
hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SetFamily.Intersecting | {
"line": 107,
"column": 2
} | {
"line": 112,
"column": 100
} | [
{
"pp": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Set α\nhs : s.Intersecting\nh : ∀ (t : Set α), t.Intersecting → s ⊆ t → s = t\n⊢ IsUpperSet s",
"usedConstants": [
"Eq.mpr",
"Set.Intersecting.ne_bot",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreord... | classical
rintro a b hab ha
rw [h (Insert.insert b s) _ (subset_insert _ _)]
· exact mem_insert _ _
exact
hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Intersecting | {
"line": 107,
"column": 2
} | {
"line": 112,
"column": 100
} | [
{
"pp": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns : Set α\nhs : s.Intersecting\nh : ∀ (t : Set α), t.Intersecting → s ⊆ t → s = t\n⊢ IsUpperSet s",
"usedConstants": [
"Eq.mpr",
"Set.Intersecting.ne_bot",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreord... | classical
rintro a b hab ha
rw [h (Insert.insert b s) _ (subset_insert _ _)]
· exact mem_insert _ _
exact
hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 200,
"column": 10
} | {
"line": 200,
"column": 19
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 204,
"column": 10
} | {
"line": 204,
"column": 19
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 209,
"column": 15
} | {
"line": 209,
"column": 24
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 95,
"column": 4
} | {
"line": 98,
"column": 44
} | [
{
"pp": "case h.mpr.inr.inr.inl\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\ncards' : #(s.erase (s.min' hs)) = #t\nk : α\nhks : k ∈ s.erase (s.min' hs)\nhkt : k ∉ t\nz : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))\nj : α := tᶜ.min' ⋯\nhjk : j ≤ ... | apply mem_insert_of_mem
rw [z lt]
refine mem_erase_of_ne_of_mem (lt_of_le_of_lt ?_ lt).ne' ha
apply min'_le _ _ (mem_of_mem_erase ‹_›) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 95,
"column": 4
} | {
"line": 98,
"column": 44
} | [
{
"pp": "case h.mpr.inr.inr.inl\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\ncards' : #(s.erase (s.min' hs)) = #t\nk : α\nhks : k ∈ s.erase (s.min' hs)\nhkt : k ∉ t\nz : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))\nj : α := tᶜ.min' ⋯\nhjk : j ≤ ... | apply mem_insert_of_mem
rw [z lt]
refine mem_erase_of_ne_of_mem (lt_of_le_of_lt ?_ lt).ne' ha
apply min'_le _ _ (mem_of_mem_erase ‹_›) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 110,
"column": 36
} | {
"line": 110,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\n𝒜 : Finset (Finset α)\ninst✝ : Finite α\nval✝ : Fintype α\nh₁ : IsInitSeg 𝒜 0\ns : Finset α\nhs : s ∈ 𝒜\n⊢ s ∈ {∅}",
"usedConstants": [
"Eq.mpr",
"Finset.mem_singleton",
"Preorder.toLT",
"Equiv.instEquivLike",
"Set.Sized",
... | rw [mem_singleton, ← Finset.card_eq_zero]; exact h₁.1 hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 110,
"column": 36
} | {
"line": 110,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\n𝒜 : Finset (Finset α)\ninst✝ : Finite α\nval✝ : Fintype α\nh₁ : IsInitSeg 𝒜 0\ns : Finset α\nhs : s ∈ 𝒜\n⊢ s ∈ {∅}",
"usedConstants": [
"Eq.mpr",
"Finset.mem_singleton",
"Preorder.toLT",
"Equiv.instEquivLike",
"Set.Sized",
... | rw [mem_singleton, ← Finset.card_eq_zero]; exact h₁.1 hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 220,
"column": 10
} | {
"line": 220,
"column": 19
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 224,
"column": 10
} | {
"line": 224,
"column": 19
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 229,
"column": 15
} | {
"line": 229,
"column": 24
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 110,
"column": 22
} | {
"line": 110,
"column": 70
} | [
{
"pp": "case mpr.tail\nV : Type u\nG : SimpleGraph V\nu v b✝ c✝ : V\na✝ : Relation.ReflTransGen G.Adj u b✝\nha : G.Adj b✝ c✝\nhr : G.Reachable u b✝\n⊢ G.Reachable u c✝",
"usedConstants": [
"SimpleGraph.Walk",
"SimpleGraph.Reachable.trans",
"Nonempty.intro",
"SimpleGraph.Walk.cons",
... | exact Reachable.trans hr ⟨Walk.cons ha Walk.nil⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 110,
"column": 22
} | {
"line": 110,
"column": 70
} | [
{
"pp": "case mpr.tail\nV : Type u\nG : SimpleGraph V\nu v b✝ c✝ : V\na✝ : Relation.ReflTransGen G.Adj u b✝\nha : G.Adj b✝ c✝\nhr : G.Reachable u b✝\n⊢ G.Reachable u c✝",
"usedConstants": [
"SimpleGraph.Walk",
"SimpleGraph.Reachable.trans",
"Nonempty.intro",
"SimpleGraph.Walk.cons",
... | exact Reachable.trans hr ⟨Walk.cons ha Walk.nil⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 110,
"column": 22
} | {
"line": 110,
"column": 70
} | [
{
"pp": "case mpr.tail\nV : Type u\nG : SimpleGraph V\nu v b✝ c✝ : V\na✝ : Relation.ReflTransGen G.Adj u b✝\nha : G.Adj b✝ c✝\nhr : G.Reachable u b✝\n⊢ G.Reachable u c✝",
"usedConstants": [
"SimpleGraph.Walk",
"SimpleGraph.Reachable.trans",
"Nonempty.intro",
"SimpleGraph.Walk.cons",
... | exact Reachable.trans hr ⟨Walk.cons ha Walk.nil⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 177,
"column": 2
} | {
"line": 179,
"column": 26
} | [
{
"pp": "V : Type u\nu v : V\n⊢ (completeGraph V).Reachable u v",
"usedConstants": [
"SimpleGraph.Walk",
"SimpleGraph.completeGraph",
"Ne",
"Nonempty.intro",
"Or.casesOn",
"SimpleGraph",
"SimpleGraph.Walk.cons",
"BooleanAlgebra.toTop",
"SimpleGraph.Walk.... | obtain rfl | huv := eq_or_ne u v
· simp
· exact ⟨.cons huv .nil⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 177,
"column": 2
} | {
"line": 179,
"column": 26
} | [
{
"pp": "V : Type u\nu v : V\n⊢ (completeGraph V).Reachable u v",
"usedConstants": [
"SimpleGraph.Walk",
"SimpleGraph.completeGraph",
"Ne",
"Nonempty.intro",
"Or.casesOn",
"SimpleGraph",
"SimpleGraph.Walk.cons",
"BooleanAlgebra.toTop",
"SimpleGraph.Walk.... | obtain rfl | huv := eq_or_ne u v
· simp
· exact ⟨.cons huv .nil⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 50
} | [
{
"pp": "case mpr\nV : Type u\nG : SimpleGraph V\nv : V\nh : ∀ (w : V), G.Reachable v w\n⊢ G.Preconnected ∧ Nonempty V",
"usedConstants": [
"SimpleGraph.Reachable.symm",
"SimpleGraph.Preconnected",
"SimpleGraph.Reachable.trans",
"Nonempty.intro",
"And.intro",
"Nonempty"
... | exact ⟨fun u w => (h u).symm.trans (h w), ⟨v⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 72
} | [
{
"pp": "case inr\nn r : ℕ\n𝒜 : Finset (Finset (Fin n))\nh : Set.Sized r ↑𝒜\nusable : Finset (Finset (Fin n) × Finset (Fin n)) := {t | UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜}\nhusable : usable.Nonempty\nU V : Finset (Fin n)\nhUV : (U, V) ∈ univ ∧ UsefulCompression (U, V).1 (U, V).2 ∧ ¬IsCompress... | have p1 : #(∂ (𝓒 U V 𝒜)) ≤ #(∂ 𝒜) := compression_improved _ hUV.2.1 h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 793,
"column": 26
} | {
"line": 793,
"column": 42
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : G.Walk v w), s(v, w) ∈ p.edges\nc : G.Walk u u\nhc : c.IsTrail\nhe : s(v, w) ∈ c.edges\nhw : w ∈ (c.takeUntil v ⋯).support\nhv : v ∈ c.support\npuw : G.Walk u w := (c.takeUntil v hv).takeUntil w hw\npwv : G.Walk w v := (c.take... | List.mem_reverse | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 28
} | [
{
"pp": "case h\nV : Type u_1\nw : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhw : w ∈ t\nv : V\n⊢ G.Adj v w → v ∈ s",
"usedConstants": [
"SimpleGraph.IsBipartiteWith.mem_of_mem_adj'"
]
}
] | exact h.mem_of_mem_adj' hw | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 28
} | [
{
"pp": "case h\nV : Type u_1\nw : V\nG : SimpleGraph V\ns t : Finset V\ninst✝¹ : Fintype ↑(G.neighborSet w)\ninst✝ : DecidableRel G.Adj\nh : G.IsBipartiteWith ↑s ↑t\nhw : w ∈ t\nv : V\n⊢ G.Adj v w → v ∈ s",
"usedConstants": [
"Finset",
"SimpleGraph.IsBipartiteWith.mem_of_mem_adj'",
"SetLi... | exact h.mem_of_mem_adj' hw | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 509,
"column": 6
} | {
"line": 509,
"column": 49
} | [
{
"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... | rcases l with l | l <;> rcases r with r | r | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 513,
"column": 6
} | {
"line": 513,
"column": 49
} | [
{
"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... | rcases l with l | l <;> rcases r with r | r | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 51
} | [
{
"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\n⊢ False",
"usedConstants": [
"SimpleGraph.Walk.reachable",
"SimpleGraph.... | obtain ⟨s, _⟩ := p₂.reachable.exists_path_of_dist | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 47
} | [
{
"pp": "case refine_1\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.IsCompleteBetwe... | rcases l with l | l <;> rcases r with r | r | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 182,
"column": 25
} | {
"line": 182,
"column": 35
} | [
{
"pp": "case a.inl\nα : Type u_1\nV : Type u_2\ninst✝¹ : DecidableEq V\ninst✝ : Ring α\ni j : V\nh✝ : i = j\n⊢ adjMatrix α ⊤ i j = of (fun i j ↦ if i = j then 0 else 1) i j",
"usedConstants": [
"Equiv.instEquivLike",
"SimpleGraph.adjMatrix.congr_simp",
"Ring.toNonAssocRing",
"Simple... | simp [‹_›] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 182,
"column": 25
} | {
"line": 182,
"column": 35
} | [
{
"pp": "case a.inr\nα : Type u_1\nV : Type u_2\ninst✝¹ : DecidableEq V\ninst✝ : Ring α\ni j : V\nh✝ : i ≠ j\n⊢ adjMatrix α ⊤ i j = of (fun i j ↦ if i = j then 0 else 1) i j",
"usedConstants": [
"False",
"Equiv.instEquivLike",
"eq_false",
"Ring.toNonAssocRing",
"SimpleGraph.Top... | simp [‹_›] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Prod | {
"line": 281,
"column": 4
} | {
"line": 282,
"column": 59
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nH : SimpleGraph β\nx y : α × β\ntop_case : (G □ H).edist x y = ⊤ ↔ G.edist x.1 y.1 = ⊤ ∨ H.edist x.2 y.2 = ⊤\nh : ¬(G □ H).edist x y = ⊤\nrGH : G.edist x.1 y.1 ≠ ⊤ ∧ H.edist x.2 y.2 ≠ ⊤\nwG : G.Walk x.1 y.1\nhwG : ↑wG.length = G.edist x.1 y.1\nwH... | have w_len : w_app.length = wG.length + wH.length := by
unfold w_app Walk.boxProdLeft Walk.boxProdRight; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 14
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nh : ENat.card V ≤ 2\n⊢ G.IsAcyclic",
"usedConstants": []
}
] | intro v p hp | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 76
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\nu v : V\nhuv : ¬u = v\nhadj : ¬G.Adj u v\nhacyc : (G ⊔ fromEdgeSet {s(u, v)}).IsAcyclic\nhreach : G.Reachable u v\nthis : (G ⊔ fromEdgeSet {s(u, v)}).IsBridge s(u, v)\n⊢ False",
"usedConstants": [
"Sym2.mk",
"SimpleGraph.fromEdgeSet",
"Si... | refine isBridge_iff.mp this |>.right <| hreach.mono <| Eq.le <| Eq.symm ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 57
} | [
{
"pp": "case refine_1\nn r : ℕ\nh : r ≠ 0 ∧ r < n\n⊢ ∃ x x_1, ↑x % r = ↑x_1 % r ∧ x ≠ x_1 ∨ ↑x % r ≠ ↑x_1 % r ∧ x = x_1",
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Exists",
"Fin.mk",
"Nat.instMod",
"instHMod",
"Ne",
"instOfNatNat",
... | use ⟨0, (Nat.pos_of_ne_zero h.1).trans h.2⟩, ⟨r, h.2⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Combinatorics.SimpleGraph.ConcreteColorings | {
"line": 66,
"column": 4
} | {
"line": 67,
"column": 44
} | [
{
"pp": "case a\nn : ℕ\nh : 2 ≤ n\nhc : (pathGraph n).Colorable (Fintype.card Bool)\n⊢ 2 ≤ (pathGraph n).chromaticNumber",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.0.SimpleGraph.chromaticNumber_pathGraph._simp_1_1",
"False",
"SimpleGraph.two_le_chrom... | have hadj : (pathGraph n).Adj ⟨0, Nat.zero_lt_of_lt h⟩ ⟨1, h⟩ := by simp [pathGraph_adj]
exact two_le_chromaticNumber_of_adj hadj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.ConcreteColorings | {
"line": 66,
"column": 4
} | {
"line": 67,
"column": 44
} | [
{
"pp": "case a\nn : ℕ\nh : 2 ≤ n\nhc : (pathGraph n).Colorable (Fintype.card Bool)\n⊢ 2 ≤ (pathGraph n).chromaticNumber",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.0.SimpleGraph.chromaticNumber_pathGraph._simp_1_1",
"False",
"SimpleGraph.two_le_chrom... | have hadj : (pathGraph n).Adj ⟨0, Nat.zero_lt_of_lt h⟩ ⟨1, h⟩ := by simp [pathGraph_adj]
exact two_le_chromaticNumber_of_adj hadj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 338,
"column": 39
} | {
"line": 338,
"column": 41
} | [
{
"pp": "case inr.a\nn r c : ℕ\nhr : r > 0\nw x : ℕ\nmw : w ∈ {i ∈ range r | c ≡ n + i [MOD r]}\n⊢ x ∈ {i ∈ range r | c ≡ n + i [MOD r]} → w = x",
"usedConstants": [
"Nat.instDecidableModEq",
"Finset",
"Membership.mem",
"Finset.range",
"instHAdd",
"Nat.ModEq",
"Fins... | mx | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 353,
"column": 11
} | {
"line": 353,
"column": 19
} | [
{
"pp": "n r : ℕ\n⊢ 2 * #(turanGraph (n + r) r).edgeFinset = 2 * (#(turanGraph n r).edgeFinset + n * (r - 1) + r.choose 2)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Nat.choose",
"HMul.hMul",
"SimpleGraph.turanGraph",
"SimpleGraph.instDecidableRelFinAdjTu... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 366,
"column": 4
} | {
"line": 369,
"column": 43
} | [
{
"pp": "case e_a.e_a\nn r : ℕ\n⊢ (∑ x ∈ range r, ∑ w ∈ range r, if (n + x) % r ≠ (n + w) % r then 1 else 0) = 2 * r.choose 2",
"usedConstants": [
"Nat.even_mul_pred_self",
"Eq.mpr",
"instDecidableNot",
"instHSMul",
"Nat.choose",
"instHDiv",
"instSMulOfMul",
"... | rw [mul_comm 2, Nat.choose_two_right, Nat.div_two_mul_two_of_even (Nat.even_mul_pred_self r)]
conv_rhs => enter [1]; rw [← card_range r]
rw [← smul_eq_mul, ← sum_const]
congr!; exact sum_ne_add_mod_eq_sub_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 366,
"column": 4
} | {
"line": 369,
"column": 43
} | [
{
"pp": "case e_a.e_a\nn r : ℕ\n⊢ (∑ x ∈ range r, ∑ w ∈ range r, if (n + x) % r ≠ (n + w) % r then 1 else 0) = 2 * r.choose 2",
"usedConstants": [
"Nat.even_mul_pred_self",
"Eq.mpr",
"instDecidableNot",
"instHSMul",
"Nat.choose",
"instHDiv",
"instSMulOfMul",
"... | rw [mul_comm 2, Nat.choose_two_right, Nat.div_two_mul_two_of_even (Nat.even_mul_pred_self r)]
conv_rhs => enter [1]; rw [← card_range r]
rw [← smul_eq_mul, ← sum_const]
congr!; exact sum_ne_add_mod_eq_sub_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 410,
"column": 35
} | {
"line": 410,
"column": 43
} | [
{
"pp": "n r : ℕ\n⊢ 2 * r * ((n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) + (n % r).choose 2) ≤ (r - 1) * n ^ 2",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Nat.choose",
"instHDiv",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"HDi... | mul_add, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 427,
"column": 24
} | {
"line": 427,
"column": 33
} | [
{
"pp": "case inl\nV : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\ncf : Fintype.card V = 0\n⊢ #G.edgeFinset ≤ (Fintype.card V ^ 2 - (Fintype.card V % 0) ^ 2) * 0 / (2 * 0) + (Fintype.card V % 0).choose 2",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 157,
"column": 46
} | {
"line": 158,
"column": 45
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : Finite ι\ns : ι → Set α\n⊢ (⋃ i, s i).encard ≤ ∑ᶠ (i : ι), (s i).encard",
"usedConstants": [
"Set.encard",
"Iff.of_eq",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"finsum",
"Membership.mem",
"Eq.mp",
"LE.... | by
simpa using finite_univ.encard_biUnion_le s | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.LapMatrix | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 16
} | [
{
"pp": "case e_a.e_f.h\nV : Type u_1\nR : Type u_2\ninst✝⁴ : Fintype V\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : DecidableEq V\ninst✝¹ : Field R\ninst✝ : CharZero R\nx : V → R\ni : V\n⊢ (∑ x_1, if G.Adj i x_1 then x i * x i - x i * x x_1 + (x x_1 * x x_1 - x x_1 * x i) else 0 + 0) =\n ∑ j, i... | congr 2 with j | Batteries.Tactic._aux_Batteries_Tactic_Congr___macroRules_Batteries_Tactic_congrConfigWith_1 | Batteries.Tactic.congrConfigWith |
Mathlib.Combinatorics.SimpleGraph.LapMatrix | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 19
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nx : V → ℝ\nh : ∀ (i j : V), G.Adj i j → x i = x j\ni j u✝ v✝ w✝ : V\nhA : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\nh' : x v✝ = x w✝\n⊢ x u✝ = x w✝",
"usedConstants": [
"Real",
"Eq.... | | cons hA _ h' => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Combinatorics.SimpleGraph.LapMatrix | {
"line": 170,
"column": 48
} | {
"line": 172,
"column": 31
} | [
{
"pp": "V : Type u_1\ninst✝³ : Fintype V\ninst✝² : DecidableEq V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq G.ConnectedComponent\nc : G.ConnectedComponent\n⊢ (fun i ↦ if G.connectedComponentMk i = c then 1 else 0) ∈ (toLin' (lapMatrix ℝ G)).ker",
"usedConstants": [
"Eq.mpr",... | by
rw [LinearMap.mem_ker, toLin'_apply, lapMatrix_mulVec_eq_zero_iff_forall_reachable]
grind [ConnectedComponent.eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 32
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nn : ℕ\nh₁ : G.vertexCoverNum ≤ ↑n\nh₂ : ↑n ≤ ENat.card V\ns : Set V\nhs₁ : s.encard = G.vertexCoverNum\nhs₂ : G.IsVertexCover s\nr : Set V\nhr₁ : s ⊆ r\nleft✝ : r ⊆ Set.univ\nhr₃ : r.encard = ↑n\n⊢ ∃ s, s.encard = ↑n ∧ G.IsVertexCover s",
"usedConstants": [
"S... | exact ⟨r, hr₃, hs₂.subset hr₁⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 8
} | [
{
"pp": "case e_s\nV : Type u_1\nG : SimpleGraph V\nc : G.ConnectedComponent\nh : G.IsCycles\nv w : V\nhw : v ∈ c.supp ∧ G.Adj v w\n⊢ c.toSimpleGraph.spanningCoe.neighborSet v = G.neighborSet v",
"usedConstants": [
"Set.ext",
"SimpleGraph.neighborSet",
"SimpleGraph.ConnectedComponent.instS... | ext w' | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 82
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nv w : V\nhcyc : G.IsCycles\np : G.Walk v w\nhp : p.IsPath\nn : ℕ\nhnl : n ≤ p.length\nhw : w ≠ p.getVert n\nhadj : G.Adj v (p.getVert n)\nhn : ¬(n = 0 ∨ n = p.length)\ne : ↑(G.neighborSet (p.getVert n)) ≃ ↑(p.toSubgraph.neighborSet (p.getVert... | rw [Subgraph.adj_comm, Subgraph.adj_iff_of_neighborSet_equiv e (Set.toFinite _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 526,
"column": 10
} | {
"line": 526,
"column": 16
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv : V\ninst✝ : Finite V\nc : G.ConnectedComponent\nh : G.IsCycles\nhv : v ∈ c.supp\nw : V\nhw : w ∈ G.neighborSet v\nu : V\np : G.Walk u u\nhp : p.IsCycle ∧ s(v, w) ∈ p.edges\nhvp : v ∈ p.support\nc' : G.ConnectedComponent\nhc' : p.toSubgraph.verts = c'.supp\n⊢ v ∈ c'.s... | ← hc', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 390,
"column": 79
} | {
"line": 390,
"column": 87
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 225,
"column": 4
} | {
"line": 226,
"column": 71
} | [
{
"pp": "case refine_2\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nx a b c : V\nM1 : (G ⊔ edge x b).Subgraph\nM2 : (G ⊔ edge a c).Subgraph\nhxa : G.Adj x a\nhab : G.Adj a b\nhnGxb : ¬G.Adj x b\nhnGac : ¬G.Adj a c\nhnxb : x ≠ b\nhnxc : x ≠ c\nhnac : a ≠ c\nhnbc : b ≠ c\nhM1 : M1.IsPerfectMatching\nhM2 : ... | · simp [htw _ (by simp) (Walk.mem_support_of_adj_toSubgraph h.symm),
htw _ (by simp) (Walk.mem_support_of_adj_toSubgraph h)] at hnxb | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Computability.Primrec.List | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝ : Primcodable α\nl : List (List α)\n⊢ List.foldr (fun b s ↦ (fun x1 x2 ↦ x1 ++ x2) (l, b, s).2.1 (l, b, s).2.2) [] (id l) = l.flatten",
"usedConstants": [
"congrArg",
"id",
"Prod.mk",
"List.rec",
"Prod.fst",
"List.cons",
"instHAppendOfAp... | dsimp; induction l <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Primrec.List | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝ : Primcodable α\nl : List (List α)\n⊢ List.foldr (fun b s ↦ (fun x1 x2 ↦ x1 ++ x2) (l, b, s).2.1 (l, b, s).2.2) [] (id l) = l.flatten",
"usedConstants": [
"congrArg",
"id",
"Prod.mk",
"List.rec",
"Prod.fst",
"List.cons",
"instHAppendOfAp... | dsimp; induction l <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.Primrec.List | {
"line": 257,
"column": 16
} | {
"line": 257,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : Primcodable α\nl : List α\n⊢ List.foldr (fun b s ↦ (fun a b ↦ (a, b).2.2.succ) (l, b, s).1 (l, b, s).2) 0 (id l) = l.length",
"usedConstants": [
"congrArg",
"id",
"Prod.mk",
"instOfNatNat",
"List.rec",
"Prod.fst",
"List.cons",
"L... | dsimp; induction l <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Primrec.List | {
"line": 257,
"column": 16
} | {
"line": 257,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : Primcodable α\nl : List α\n⊢ List.foldr (fun b s ↦ (fun a b ↦ (a, b).2.2.succ) (l, b, s).1 (l, b, s).2) 0 (id l) = l.length",
"usedConstants": [
"congrArg",
"id",
"Prod.mk",
"instOfNatNat",
"List.rec",
"Prod.fst",
"List.cons",
"L... | dsimp; induction l <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.PSub | {
"line": 62,
"column": 12
} | {
"line": 62,
"column": 53
} | [
{
"pp": "m : ℕ\n⊢ ppred 0 = some m ↔ m.succ = 0",
"usedConstants": [
"Option.ctorIdx",
"False.elim",
"noConfusion_of_Nat",
"Option.some",
"instOfNatNat",
"Nat",
"Iff.intro",
"Nat.ctorIdx",
"Nat.ppred",
"OfNat.ofNat",
"Nat.succ",
"Eq",
... | constructor <;> intro h <;> contradiction | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Nat.PSub | {
"line": 62,
"column": 12
} | {
"line": 62,
"column": 53
} | [
{
"pp": "m : ℕ\n⊢ ppred 0 = some m ↔ m.succ = 0",
"usedConstants": [
"Option.ctorIdx",
"False.elim",
"noConfusion_of_Nat",
"Option.some",
"instOfNatNat",
"Nat",
"Iff.intro",
"Nat.ctorIdx",
"Nat.ppred",
"OfNat.ofNat",
"Nat.succ",
"Eq",
... | constructor <;> intro h <;> contradiction | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.PSub | {
"line": 62,
"column": 12
} | {
"line": 62,
"column": 53
} | [
{
"pp": "m : ℕ\n⊢ ppred 0 = some m ↔ m.succ = 0",
"usedConstants": [
"Option.ctorIdx",
"False.elim",
"noConfusion_of_Nat",
"Option.some",
"instOfNatNat",
"Nat",
"Iff.intro",
"Nat.ctorIdx",
"Nat.ppred",
"OfNat.ofNat",
"Nat.succ",
"Eq",
... | constructor <;> intro h <;> contradiction | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.Primrec.List | {
"line": 354,
"column": 22
} | {
"line": 354,
"column": 57
} | [
{
"pp": "case zero\nβ : Type u_2\nσ : Type u_4\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : β → σ\nm : β → ℕ\nl : β → List β\ng : β → List σ → Option σ\nhm : Primrec m\nhl : Primrec l\nhg : Primrec₂ g\nOrd : ∀ (b b' : β), b' ∈ l b → m b' < m b\nH : ∀ (b : β), g b (List.map f (l b)) = some (f b)\nthis✝ : ... | simpa [graph] using bindList_eq_nil | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Computability.Partrec | {
"line": 581,
"column": 2
} | {
"line": 582,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nσ : Type u_4\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : α → β → σ\n⊢ (Computable₂ fun a n ↦ Option.map (f a) (decode n)) ↔ Computable₂ f",
"usedConstants": [
"Eq.mpr",
"Primcodable.ofDenumerable",
"HEq.refl",
"Opti... | convert (bind_decode_iff (f := fun a => Option.some ∘ f a)).trans option_some_iff
apply Option.map_eq_bind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Partrec | {
"line": 581,
"column": 2
} | {
"line": 582,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nσ : Type u_4\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : α → β → σ\n⊢ (Computable₂ fun a n ↦ Option.map (f a) (decode n)) ↔ Computable₂ f",
"usedConstants": [
"Eq.mpr",
"Primcodable.ofDenumerable",
"HEq.refl",
"Opti... | convert (bind_decode_iff (f := fun a => Option.some ∘ f a)).trans option_some_iff
apply Option.map_eq_bind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.Primrec.List | {
"line": 492,
"column": 96
} | {
"line": 496,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : α → β → Prop\ninst✝¹ : Primcodable α\ninst✝ : Primcodable β\nhf : PrimrecRel R\n⊢ PrimrecRel fun L b ↦ ∀ a ∈ L, R a b",
"usedConstants": [
"PrimrecRel.of_eq",
"Primrec.list_length",
"congrArg",
"Primcodable.ofDenumerable",
"PrimrecRel.li... | by
classical
have h (L) (b) : (List.filter (R · b) L).length = L.length ↔ ∀ a ∈ L, R a b := by simp
apply PrimrecRel.of_eq ?_ h
exact (Primrec.eq.comp (list_length.comp <| PrimrecRel.listFilter hf) (.comp list_length fst)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Primrec.List | {
"line": 731,
"column": 21
} | {
"line": 731,
"column": 38
} | [
{
"pp": "n : ℕ\nf : List.Vector ℕ n → ℕ\nhf : Primrec' f\ns : Primrec' fun v ↦ (f v).sqrt\nfss : Primrec' fun v ↦ f v - (f v).sqrt * (f v).sqrt\nv : List.Vector ℕ n\n⊢ (if f v - (f v).sqrt * (f v).sqrt < (f v).sqrt then f v - (f v).sqrt * (f v).sqrt else (f v).sqrt) =\n (if f v - (f v).sqrt * (f v).sqrt < (f... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Computability.Partrec | {
"line": 755,
"column": 26
} | {
"line": 755,
"column": 58
} | [
{
"pp": "α : Type u_1\nσ : Type u_4\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α →. σ ⊕ α\nhf : Partrec f\nF : α → ℕ →. σ ⊕ α :=\n fun a n ↦ Nat.rec (Part.some (Sum.inr a)) (fun x IH ↦ IH.bind fun s ↦ Sum.casesOn s (fun x ↦ Part.some s) f) n\nhF : Partrec₂ F\np : α → ℕ → Part Bool := fun a n ↦ Part.ma... | by simpa [p] using fix_aux f _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Primrec.List | {
"line": 737,
"column": 21
} | {
"line": 737,
"column": 38
} | [
{
"pp": "n : ℕ\nf : List.Vector ℕ n → ℕ\nhf : Primrec' f\ns : Primrec' fun v ↦ (f v).sqrt\nfss : Primrec' fun v ↦ f v - (f v).sqrt * (f v).sqrt\nv : List.Vector ℕ n\n⊢ (if f v - (f v).sqrt * (f v).sqrt < (f v).sqrt then (f v).sqrt else f v - (f v).sqrt * (f v).sqrt - (f v).sqrt) =\n (if f v - (f v).sqrt * (f... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 209,
"column": 86
} | {
"line": 222,
"column": 19
} | [
{
"pp": "f : ℝ → ℝ\nhf : GrowsPolynomially f\n⊢ (∀ᶠ (x : ℝ) in atTop, f x = 0) ∨ (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0",
"usedConstants": [
"Filter.eventually_and",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"_private.Mathlib.Computabili... | by
by_cases! h : ∃ᶠ x in atTop, f x = 0
· exact Or.inl <| eventually_zero_of_frequently_zero hf h
· cases eventually_atTop_nonneg_or_nonpos hf with
| inl h' =>
refine Or.inr (Or.inl ?_)
simp only [lt_iff_le_and_ne]
rw [eventually_and]
exact ⟨h', by filter_upwards [h] with x hx; exact h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 346,
"column": 24
} | {
"line": 346,
"column": 51
} | [
{
"pp": "f g : ℝ → ℝ\nhf✝¹ : GrowsPolynomially f\nhg✝¹ : GrowsPolynomially g\nhf'✝ : 0 ≤ᶠ[atTop] f\nhg'✝ : 0 ≤ᶠ[atTop] g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nc₃ : ℝ\nhc₃_mem : c₃... | gcongr; exact le_of_lt hb.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 346,
"column": 24
} | {
"line": 346,
"column": 51
} | [
{
"pp": "f g : ℝ → ℝ\nhf✝¹ : GrowsPolynomially f\nhg✝¹ : GrowsPolynomially g\nhf'✝ : 0 ≤ᶠ[atTop] f\nhg'✝ : 0 ≤ᶠ[atTop] g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nc₃ : ℝ\nhc₃_mem : c₃... | gcongr; exact le_of_lt hb.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 129,
"column": 10
} | {
"line": 129,
"column": 34
} | [
{
"pp": "p : ℝ\nhp : p ≠ 0\n⊢ (fun z ↦ z ^ (p - 1) / log z ^ 2) =o[atTop] fun z ↦ z ^ (p - 1) / 1",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instDivInvMo... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nq : ℝ → ℝ\nhq_diff : DifferentiableOn ℝ q (Set.Ioi 1)\nhq_poly : GrowsPolynomially fun x ↦ ‖deriv q x‖\ni : α\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nhb_lt_on... | obtain ⟨c₁, _, c₂, _, hq_poly⟩ := hq_poly b' hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Computability.PartrecCode | {
"line": 644,
"column": 6
} | {
"line": 649,
"column": 26
} | [
{
"pp": "k k₂ : ℕ\nhl : k + 1 ≤ k₂ + 1\nhl' : k ≤ k₂\nthis :\n ∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},\n k ≤ k₂ →\n (x ∈ o₁ → x ∈ o₂) →\n (x ∈ do\n guard (n ≤ k)\n o₁) →\n x ∈ do\n guard (n ≤ k₂)\n o₂\ncf : Code\nhf : ∀ (n x : ℕ), evaln (k + 1) c... | simp? [Bind.bind, Option.bind_eq_some_iff] at h ⊢ says
simp only [unpaired, bind, pair_unpair, Option.pure_def, Option.mem_def,
Option.bind_eq_some_iff] at h ⊢
refine h.imp fun x => And.imp (hf _ _) ?_
by_cases x0 : x = 0 <;> simp [x0]
exact evaln_mono hl' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.PartrecCode | {
"line": 644,
"column": 6
} | {
"line": 649,
"column": 26
} | [
{
"pp": "k k₂ : ℕ\nhl : k + 1 ≤ k₂ + 1\nhl' : k ≤ k₂\nthis :\n ∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},\n k ≤ k₂ →\n (x ∈ o₁ → x ∈ o₂) →\n (x ∈ do\n guard (n ≤ k)\n o₁) →\n x ∈ do\n guard (n ≤ k₂)\n o₂\ncf : Code\nhf : ∀ (n x : ℕ), evaln (k + 1) c... | simp? [Bind.bind, Option.bind_eq_some_iff] at h ⊢ says
simp only [unpaired, bind, pair_unpair, Option.pure_def, Option.mem_def,
Option.bind_eq_some_iff] at h ⊢
refine h.imp fun x => And.imp (hf _ _) ?_
by_cases x0 : x = 0 <;> simp [x0]
exact evaln_mono hl' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 394,
"column": 11
} | {
"line": 397,
"column": 37
} | [] | f x + g x ≥ f x - ‖g x‖ := by
rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le (g x)
_ ≥ f x - 1 / 2 * f x := by gcongr
_ = 1 / 2 * f x := by ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 112,
"column": 55
} | {
"line": 112,
"column": 79
} | [
{
"pp": "⊢ (fun n ↦ ↑n / log ↑n ^ 2) = fun n ↦ ↑n * (log ↑n ^ 2)⁻¹",
"usedConstants": [
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"MulOne.toMul",
"HDiv.hDiv",... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 112,
"column": 55
} | {
"line": 112,
"column": 79
} | [
{
"pp": "⊢ (fun n ↦ ↑n / log ↑n ^ 2) = fun n ↦ ↑n * (log ↑n ^ 2)⁻¹",
"usedConstants": [
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"MulOne.toMul",
"HDiv.hDiv",... | simp_rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 112,
"column": 55
} | {
"line": 112,
"column": 79
} | [
{
"pp": "⊢ (fun n ↦ ↑n / log ↑n ^ 2) = fun n ↦ ↑n * (log ↑n ^ 2)⁻¹",
"usedConstants": [
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"MulOne.toMul",
"HDiv.hDiv",... | simp_rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 53
} | [
{
"pp": "c : ℝ\nhc : c < 1\n⊢ Tendsto (fun x ↦ 1 - ε x) atTop (𝓝 1)",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instRCLike",
"congrArg",
"Real.instSub",
"NeZero.charZero_one",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub",
"PseudoMetricSpace.toUnif... | rw [← isEquivalent_const_iff_tendsto one_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 403,
"column": 39
} | {
"line": 403,
"column": 57
} | [
{
"pp": "⊢ (fun z ↦ -deriv ε z) =o[atTop] fun x ↦ x⁻¹",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.denselyNormedField",
"congrArg",
"Real.instInv",
"deriv",
"NormedSpace.toModule",
"Pseud... | isLittleO_neg_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.PartrecCode | {
"line": 936,
"column": 6
} | {
"line": 936,
"column": 42
} | [
{
"pp": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : ℕ\n⊢ n ∈ List.range k →\n Nat.rec Option.none\n (fun n_1 n_ih ↦\n rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)\n (fun cf cg x x_1 ↦ do... | generalize ofNat Code p.unpair.2 = c | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 568,
"column": 30
} | {
"line": 568,
"column": 37
} | [
{
"pp": "case h\nf : ℝ → ℝ\np : ℝ\nhf : GrowsPolynomially f\nhf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : 0 < c₁\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nhc₁p : 0 < c₁ ^ p\nhc₂p : 0 < c₂ ^ p\... | hf_pos₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 663,
"column": 62
} | {
"line": 664,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nc₁ : ℝ\nhc₁_mem : c₁ ∈ Set.Ioo 0 1\nhc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁... | by
gcongr; simp only [tsub_le_iff_right, le_add_iff_nonneg_right, Nat.cast_nonneg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 487,
"column": 7
} | {
"line": 546,
"column": 55
} | [] | T n
_ = (∑ i, a i * T (r i n)) + g n := R.h_rec n (by grind)
_ ≤ (∑ i, a i * (C * ((1 - ε (r i n)) * asympBound g a b (r i n)))) + g n := by
-- Apply the induction hypothesis
gcongr (∑ i, a i * ?_) + g n with i _
· exact le_of_lt <| R.a_pos _
· exact h_ind (r i n) (by grind)
_ = (∑ i... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Data.Nat.Bitwise | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 19
} | [
{
"pp": "f : Bool → Bool → Bool\n⊢ 0 = if f true false = true then 0 else 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instDecidableEqBool",
"instOfNatNat",
"if_pos",
"dite",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 596,
"column": 57
} | {
"line": 596,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 652,
"column": 29
} | {
"line": 652,
"column": 56
} | [
{
"pp": "f g : ℝ → ℝ\nhg✝ : GrowsPolynomially g\nhf : f =Θ[atTop] g\nhf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_pos : 0 < b\nc₁ : ℝ\nhc₁_pos : 0 < c₁\nhf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖\nc₂ : ℝ\nhc₂_pos : 0 < c₂\nhf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖\nc₃ : ℝ\nhc₃_... | gcongr; exact le_of_lt hb.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 652,
"column": 29
} | {
"line": 652,
"column": 56
} | [
{
"pp": "f g : ℝ → ℝ\nhg✝ : GrowsPolynomially g\nhf : f =Θ[atTop] g\nhf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_pos : 0 < b\nc₁ : ℝ\nhc₁_pos : 0 < c₁\nhf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖\nc₂ : ℝ\nhc₂_pos : 0 < c₂\nhf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖\nc₃ : ℝ\nhc₃_... | gcongr; exact le_of_lt hb.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Bitwise | {
"line": 164,
"column": 42
} | {
"line": 164,
"column": 54
} | [
{
"pp": "b : Bool\nn : ℕ\nhn : (∀ (i : ℕ), n.testBit i = false) → n = 0\nh : ∀ (i : ℕ), (bit b n).testBit i = false\nthis : b = false\ni : ℕ\n⊢ n.testBit i = false",
"usedConstants": [
"Nat.bit",
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
... | ← h (i + 1), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Size | {
"line": 84,
"column": 6
} | {
"line": 88,
"column": 72
} | [
{
"pp": "case bit\nb : Bool\nm : ℕ\ne : m = 0 → b = true\nIH : ∀ {n : ℕ}, m < 2 ^ n → m.size ≤ n\nn : ℕ\nh : bit b m < 2 ^ n\n⊢ (bit b m).size ≤ n",
"usedConstants": [
"Nat.bit",
"instPowNat",
"Eq.mpr",
"congrArg",
"False.elim",
"Eq.mp",
"id",
"Ne",
"ins... | rw [← Nat.bit_ne_zero_iff] at e
rw [size_bit e]
cases n with
| zero => exact (e (Nat.lt_one_iff.mp h)).elim
| succ n => exact succ_le_succ (IH (bit_lt_two_pow_succ_iff.mp h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Size | {
"line": 84,
"column": 6
} | {
"line": 88,
"column": 72
} | [
{
"pp": "case bit\nb : Bool\nm : ℕ\ne : m = 0 → b = true\nIH : ∀ {n : ℕ}, m < 2 ^ n → m.size ≤ n\nn : ℕ\nh : bit b m < 2 ^ n\n⊢ (bit b m).size ≤ n",
"usedConstants": [
"Nat.bit",
"instPowNat",
"Eq.mpr",
"congrArg",
"False.elim",
"Eq.mp",
"id",
"Ne",
"ins... | rw [← Nat.bit_ne_zero_iff] at e
rw [size_bit e]
cases n with
| zero => exact (e (Nat.lt_one_iff.mp h)).elim
| succ n => exact succ_le_succ (IH (bit_lt_two_pow_succ_iff.mp h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 644,
"column": 6
} | {
"line": 645,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n... | gcongr
exact mul_nonneg (by grind +splitIndPred) g_pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 644,
"column": 6
} | {
"line": 645,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n... | gcongr
exact mul_nonneg (by grind +splitIndPred) g_pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 652,
"column": 6
} | {
"line": 652,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\n⊢ T =O[atTop] fun n ↦ (1 - ε ↑n) * asympBound g a b n",
"usedConstants": [
"AkraBazziRecurrence.T_isBigO_smoothingFn_mul_asympBound"
]
}
] | exact R.T_isBigO_smoothingFn_mul_asympBound | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 652,
"column": 6
} | {
"line": 652,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\n⊢ T =O[atTop] fun n ↦ (1 - ε ↑n) * asympBound g a b n",
"usedConstants": [
"AkraBazziRecurrence.T_isBigO_smoothingFn_mul_asympBound"
]
}
] | exact R.T_isBigO_smoothingFn_mul_asympBound | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 652,
"column": 6
} | {
"line": 652,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\n⊢ T =O[atTop] fun n ↦ (1 - ε ↑n) * asympBound g a b n",
"usedConstants": [
"AkraBazziRecurrence.T_isBigO_smoothingFn_mul_asympBound"
]
}
] | exact R.T_isBigO_smoothingFn_mul_asympBound | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.ReduceOption | {
"line": 57,
"column": 2
} | {
"line": 58,
"column": 43
} | [
{
"pp": "case mp\nα : Type u_1\nl : List (Option α)\n⊢ (∀ (a : Option α), a ∈ l → id a = none) → ∃ n, l = replicate n none",
"usedConstants": [
"List.replicate",
"List.eq_replicate_of_mem",
"Membership.mem",
"id",
"Option.none",
"List",
"List.instMembership",
... | · intro h
exact ⟨l.length, eq_replicate_of_mem h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.List.ReduceOption | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 63
} | [
{
"pp": "α : Type u_1\nl : List (Option α)\n⊢ l.reduceOption.length = l.length ↔ ∀ (x : Option α), x ∈ l → x.isSome = true",
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.reduceOption_length_eq",
"Iff.rfl",
"Membership.mem",
"id",
"Bool.true",
"List",
"... | rw [reduceOption_length_eq, List.length_filter_eq_length_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.ReduceOption | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 63
} | [
{
"pp": "α : Type u_1\nl : List (Option α)\n⊢ l.reduceOption.length = l.length ↔ ∀ (x : Option α), x ∈ l → x.isSome = true",
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.reduceOption_length_eq",
"Iff.rfl",
"Membership.mem",
"id",
"Bool.true",
"List",
"... | rw [reduceOption_length_eq, List.length_filter_eq_length_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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