module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 13
} | {
"line": 322,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\n⊢ w ∈ p.support ↔ w = v ∨ ∃ e ∈ p.edges, w ∈ e",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"Membership.mem",
"Exists",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\n⊢ w ∈ nil.support ↔ w = u✝ ∨ ∃ e ∈ nil.edges, w ∈ e",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : w ∈ p✝.support ↔ w = w✝ ∨ ∃ e ∈ p✝.edges, w ∈ e\n⊢ w ∈ (cons h✝ p✝).support ↔ w = w✝ ∨ ∃ e ∈ (co... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 87,
"column": 55
} | {
"line": 87,
"column": 66
} | {
"line": 87,
"column": 67
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\n⊢ (Walk.map f p).length = p.length",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.instFunLike",
"SimpleGraph.Walk.length",
... | [
"case nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ : V\n⊢ (Walk.map f nil).length = nil.length",
"case cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ :... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 91,
"column": 58
} | {
"line": 91,
"column": 69
} | {
"line": 91,
"column": 70
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v w : V\np : G.Walk u v\nq : G.Walk v w\n⊢ Walk.map f (p.append q) = (Walk.map f p).append (Walk.map f q)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.ins... | [
"case nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v w u✝ : V\nq : G.Walk u✝ w\n⊢ Walk.map f (nil.append q) = (Walk.map f nil).append (Walk.map f q)",
"case cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 94,
"column": 64
} | {
"line": 94,
"column": 75
} | {
"line": 94,
"column": 76
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\n⊢ (Walk.map f p).reverse = Walk.map f p.reverse",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.instFunLike",
"SimpleGraph.Adj",... | [
"case nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ : V\n⊢ (Walk.map f nil).reverse = Walk.map f nil.reverse",
"case cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ v✝ w✝ : V\nh✝ : G.Ad... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 97,
"column": 64
} | {
"line": 97,
"column": 75
} | {
"line": 97,
"column": 76
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\n⊢ (Walk.map f p).support = List.map (⇑f) p.support",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.instFunLike",
"List.map",
... | [
"case nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ : V\n⊢ (Walk.map f nil).support = List.map (⇑f) nil.support",
"case cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ v✝ w✝ : V\nh✝ : G... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 100,
"column": 66
} | {
"line": 100,
"column": 77
} | {
"line": 100,
"column": 78
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\n⊢ (Walk.map f p).darts = List.map f.mapDart p.darts",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.instFunLike",
"List.map",
... | [
"case nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ : V\n⊢ (Walk.map f nil).darts = List.map f.mapDart nil.darts",
"case cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ v✝ w✝ : V\nh✝ : ... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 13
} | {
"line": 104,
"column": 14
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\n⊢ (Walk.map f p).edges = List.map (Sym2.map ⇑f) p.edges",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"Sym2.map",
"RelHom.instFunLike"... | [
"case nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ : V\n⊢ (Walk.map f nil).edges = List.map (Sym2.map ⇑f) nil.edges",
"case cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nf : G →g G'\nu v : V\np : G.Walk u v\nu✝ v✝ w✝ : V\nh... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 13
} | {
"line": 166,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.transfer G ⋯ = p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Walk.rec",
"SimpleGraph.Walk.transfer",
"Eq",
"SimpleGraph.Walk.edge... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nu✝ : V\n⊢ nil.transfer G ⋯ = nil",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.transfer G ⋯ = p✝\n⊢ (cons h✝ p✝).transfer G ⋯ = cons h✝ p✝"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 13
} | {
"line": 171,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nhp : ∀ e ∈ p.edges, e ∈ H.edgeSet\nGH : G ≤ H\n⊢ p.transfer H hp = Walk.map (Hom.ofLE GH) p",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"SimpleGraph.Adj",
"Simp... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nGH : G ≤ H\nu✝ : V\nhp : ∀ e ∈ nil.edges, e ∈ H.edgeSet\n⊢ nil.transfer H hp = Walk.map (Hom.ofLE GH) nil",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nGH : G ≤ H\nu✝ v✝ w✝ : V\nh✝ : G.... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 13
} | {
"line": 176,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nhp : ∀ e ∈ p.edges, e ∈ H.edgeSet\n⊢ (p.transfer H hp).edges = p.edges",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"Membership.mem",
"Simpl... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ : V\nhp : ∀ e ∈ nil.edges, e ∈ H.edgeSet\n⊢ (nil.transfer H hp).edges = nil.edges",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_i... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 13
} | {
"line": 184,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nhp : ∀ e ∈ p.edges, e ∈ H.edgeSet\n⊢ (p.transfer H hp).support = p.support",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk"... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ : V\nhp : ∀ e ∈ nil.edges, e ∈ H.edgeSet\n⊢ (nil.transfer H hp).support = nil.support",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 13
} | {
"line": 189,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nhp : ∀ e ∈ p.edges, e ∈ H.edgeSet\n⊢ (p.transfer H hp).length = p.length",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ : V\nhp : ∀ e ∈ nil.edges, e ∈ H.edgeSet\n⊢ (nil.transfer H hp).length = nil.length",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 13
} | {
"line": 194,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nhp : ∀ e ∈ p.edges, e ∈ H.edgeSet\nK : SimpleGraph V\nhp' : ∀ e ∈ (p.transfer H hp).edges, e ∈ K.edgeSet\n⊢ (p.transfer H hp).transfer K hp' = p.transfer K ⋯",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH K : SimpleGraph V\nu✝ : V\nhp : ∀ e ∈ nil.edges, e ∈ H.edgeSet\nhp' : ∀ e ∈ (nil.transfer H hp).edges, e ∈ K.edgeSet\n⊢ (nil.transfer H hp).transfer K hp' = nil.transfer K ⋯",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 13
} | {
"line": 113,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.append nil = p",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Walk.nil",
"SimpleGraph.Walk.rec",
"Eq",
"SimpleGraph.Walk.append"
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ nil.append nil = nil",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.append nil = p✝\n⊢ (cons h✝ p✝).append nil = cons h✝ p✝"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 13
} | {
"line": 117,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w x : V\np : G.Walk u v\nq : G.Walk v w\nr : G.Walk w x\n⊢ p.append (q.append r) = (p.append q).append r",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Walk.rec",
"Eq",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w x : V\nr : G.Walk w x\nu✝ : V\nq : G.Walk u✝ w\n⊢ nil.append (q.append r) = (nil.append q).append r",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w x : V\nr : G.Walk w x\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (q : G.Walk w✝ w), p✝.append (q... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 13
} | {
"line": 202,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nw : V\nq : G.Walk v w\nhpq : ∀ e ∈ (p.append q).edges, e ∈ H.edgeSet\n⊢ (p.append q).transfer H hpq = (p.transfer H ⋯).append (q.transfer H ⋯)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"SimpleGr... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nw u✝ : V\nq : G.Walk u✝ w\nhpq : ∀ e ∈ (nil.append q).edges, e ∈ H.edgeSet\n⊢ (nil.append q).transfer H hpq = (nil.transfer H ⋯).append (q.transfer H ⋯)",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : S... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Maps | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 13
} | {
"line": 209,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nhp : ∀ e ∈ p.edges, e ∈ H.edgeSet\n⊢ (p.transfer H hp).reverse = p.reverse.transfer H ⋯",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.reverse_transfer._proof_2",
"SimpleGraph... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ : V\nhp : ∀ e ∈ nil.edges, e ∈ H.edgeSet\n⊢ (nil.transfer H hp).reverse = nil.reverse.transfer H ⋯",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nH : SimpleGraph V\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : ... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 13
} | {
"line": 229,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nq : G.Walk v w\n⊢ (p.append q).length = p.length + q.length",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.length",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"instHAdd",
"HAdd.h... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\nq : G.Walk u✝ w\n⊢ (nil.append q).length = nil.length + q.length",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (q : G.Walk w✝ w), (p✝.append q).length = p✝.length + q.length\nq : G.Walk w✝ w\n... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | {
"line": 96,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.takeUntil u h).append (p.dropUntil u h) = p",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",... | [
"case nil\nV : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w u✝ : V\nh : u ∈ nil.support\n⊢ (nil.takeUntil u h).append (nil.dropUntil u h) = nil",
"case cons\nV : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (h : u ∈ p✝.support), (p... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 13
} | {
"line": 331,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nh : G.Adj v w\n⊢ (p.concat h).support = p.support ++ [w]",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"SimpleGraph.Walk.concat",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\nh : G.Adj u✝ w\n⊢ (nil.concat h).support = nil.support ++ [w]",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (h : G.Adj w✝ w), (p✝.concat h).support = p✝.support ++ [w]\nh : G.Adj w✝ w\n⊢ ((con... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 13
} | {
"line": 146,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ List.count u (p.takeUntil u h).support = 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w u✝ : V\nh : u ∈ nil.support\n⊢ List.count u (nil.takeUntil u h).support = 1",
"case cons\nV : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (h : u ∈ p✝.support), List.count... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 13
} | {
"line": 341,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\np' : G.Walk v w\n⊢ (p.append p').support = p.support ++ p'.support.tail",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"List.tail",... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\np' : G.Walk u✝ w\n⊢ (nil.append p').support = nil.support ++ p'.support.tail",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (p' : G.Walk w✝ w), (p✝.append p').support = p✝.support ++ p'.support... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 13
} | {
"line": 345,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.reverse.support = p.support.reverse",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"List",
"SimpleGraph.Walk.rec",
"L... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ nil.reverse.support = nil.support.reverse",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.reverse.support = p✝.support.reverse\n⊢ (cons h✝ p✝).reverse.support = (cons h✝ p✝).support.reverse"
] | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 13
} | {
"line": 349,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\np' : G.Walk v w\n⊢ (p.append p').support = p.support.dropLast ++ p'.support",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk.support",
"SimpleGraph.Walk",
"List.dr... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\np' : G.Walk u✝ w\n⊢ (nil.append p').support = nil.support.dropLast ++ p'.support",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (p' : G.Walk w✝ w), (p✝.append p').support = p✝.support.dropLast ... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 54
} | {
"line": 143,
"column": 4
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nu v u' v' : V\np₁ : G.Walk u v\np₂ : G.Walk u' v'\nhnil : ¬p₁.Nil\nx✝ :\n ∃ k, p₁.darts.length + k ≤ p₂.darts.length ∧ ∀ (i : ℕ) (h : i < p₁.darts.length), p₂.darts[i + k]? = some p₁.darts[i]\nk : ℕ\nhk : p₁.darts.length + k ≤ p₂.darts.length\nh : ∀ (i : ℕ) (... | [] | grind [not_nil_iff_lt_length, snd_darts_getElem] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 13
} | {
"line": 437,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nh : G.Adj v w\n⊢ (p.concat h).darts = p.darts.concat { fst := v, snd := w, adj := h }",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"List.concat",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"Prod.mk",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\nh : G.Adj u✝ w\n⊢ (nil.concat h).darts = nil.darts.concat { fst := u✝, snd := w, adj := h }",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (h : G.Adj w✝ w), (p✝.concat h).darts = p✝.darts.conca... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 13
} | {
"line": 448,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\np' : G.Walk v w\n⊢ (p.append p').darts = p.darts ++ p'.darts",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Dart",
"SimpleGraph.Walk.darts",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\np' : G.Walk u✝ w\n⊢ (nil.append p').darts = nil.darts ++ p'.darts",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (p' : G.Walk w✝ w), (p✝.append p').darts = p✝.darts ++ p'.darts\np' : G.Walk w✝ ... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 453,
"column": 2
} | {
"line": 453,
"column": 13
} | {
"line": 453,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.reverse.darts = (List.map Dart.symm p.darts).reverse",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"List.map",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Dart",
"SimpleGraph.Walk.d... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v u✝ : V\n⊢ nil.reverse.darts = (List.map Dart.symm nil.darts).reverse",
"case cons\nV : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.reverse.darts = (List.map Dart.symm p✝.darts).reverse\n⊢ (cons h✝ p✝).reverse.darts = (L... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 382,
"column": 9
} | {
"line": 382,
"column": 19
} | {
"line": 382,
"column": 20
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu : V\nn : ℕ\np : G.Walk u u\nhn : n < p.length\nh : (p.take (p.length - 1)).IsPath\nthis : ((p.take (p.length - 1)).take n).IsPath\n⊢ (p.take n).IsPath",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"congrArg",
"SimpleGraph.Walk.length"... | [
"V : Type u\nG : SimpleGraph V\nu : V\nn : ℕ\np : G.Walk u u\nhn : n < p.length\nh : (p.take (p.length - 1)).IsPath\nthis : ((p.take (min (p.length - 1) n)).copy ⋯ ⋯).IsPath\n⊢ (p.take n).IsPath"
] | take_take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 754,
"column": 2
} | {
"line": 754,
"column": 13
} | {
"line": 755,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nt u v : V\np : G.Walk u v\nh : G.Adj v t\n⊢ (p.concat h).dropLast = p.copy ⋯ ⋯",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Walk.concat",
"SimpleGraph.Walk.copy",
"S... | [
"case nil\nV : Type u\nG : SimpleGraph V\nt u v u✝ : V\nh : G.Adj u✝ t\n⊢ (nil.concat h).dropLast = nil.copy ⋯ ⋯",
"case cons\nV : Type u\nG : SimpleGraph V\nt u v u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (h : G.Adj w✝ t), (p✝.concat h).dropLast = p✝.copy ⋯ ⋯\nh : G.Adj w✝ t\n⊢ ((cons h✝ p✝).c... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 496,
"column": 34
} | {
"line": 496,
"column": 45
} | {
"line": 496,
"column": 46
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nhp : p.IsPath\nhnil : ¬p.Nil\nhmem : ((u, w) = (u, p.snd) ∨ (u, w) = (u, p.snd).swap) ∨ s(u, w) ∈ p.tail.edges\n⊢ u ∉ p.tail.support",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Sym2.mk",
"SimpleGraph.Adj",
... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\nhp : Walk.nil.IsPath\nhnil : ¬Walk.nil.Nil\nhmem : ((u✝, w) = (u✝, Walk.nil.snd) ∨ (u✝, w) = (u✝, Walk.nil.snd).swap) ∨ s(u✝, w) ∈ Walk.nil.tail.edges\n⊢ u✝ ∉ Walk.nil.tail.support",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 216,
"column": 2
} | {
"line": 221,
"column": 51
} | {
"line": 223,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nv w : α\ns : Set α\nh : v ≠ w\nhv : G.IsClique (s \\ {v})\nhw : G.IsClique (s \\ {w})\n⊢ (G ⊔ edge v w).IsClique s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Sym2.Rel",
"Lattice.toSemilatticeSup",
... | [] | intro x hx y hy hxy
by_cases h' : x ∈ s \ {v} ∧ y ∈ s \ {v} ∨ x ∈ s \ {w} ∧ y ∈ s \ {w}
· obtain (⟨hx, hy⟩ | ⟨hx, hy⟩) := h'
· exact hv.mono le_sup_left hx hy hxy
· exact hw.mono le_sup_left hx hy hxy
· exact Or.inr ⟨by by_cases x = v <;> aesop, hxy⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 216,
"column": 2
} | {
"line": 221,
"column": 51
} | {
"line": 223,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nv w : α\ns : Set α\nh : v ≠ w\nhv : G.IsClique (s \\ {v})\nhw : G.IsClique (s \\ {w})\n⊢ (G ⊔ edge v w).IsClique s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Sym2.Rel",
"Lattice.toSemilatticeSup",
... | [] | intro x hx y hy hxy
by_cases h' : x ∈ s \ {v} ∧ y ∈ s \ {v} ∨ x ∈ s \ {w} ∧ y ∈ s \ {w}
· obtain (⟨hx, hy⟩ | ⟨hx, hy⟩) := h'
· exact hv.mono le_sup_left hx hy hxy
· exact hw.mono le_sup_left hx hy hxy
· exact Or.inr ⟨by by_cases x = v <;> aesop, hxy⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 572,
"column": 4
} | {
"line": 572,
"column": 79
} | {
"line": 573,
"column": 4
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v✝ : V\nh' : G.Adj u v✝\np : G.Walk v✝ u\nx✝ : (cons h' p).tail.IsPath ∧ 3 ≤ (cons h' p).length\nh₁ : (cons h' p).tail.IsPath\nh₂ : 3 ≤ (cons h' p).length\n⊢ (cons h' p).IsCycle",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"cong... | [
"case cons\nV : Type u\nG : SimpleGraph V\nu v✝ : V\nh' : G.Adj u v✝\np : G.Walk v✝ u\nx✝ : (cons h' p).tail.IsPath ∧ 3 ≤ (cons h' p).length\nh₂ : 3 ≤ p.length + 1\nh₁ : p.IsPath\n⊢ (cons h' p).IsCycle"
] | simp only [getVert_cons_succ, tail_cons, isPath_copy, length_cons] at h₁ h₂ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 661,
"column": 4
} | {
"line": 661,
"column": 58
} | {
"line": 662,
"column": 4
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : ∀ (v_1 : V) (w : G.Walk v_1 v_1), w.IsSubwalk p → w.Nil\ni j : ℕ\nx✝³ : i < p.support.length\nx✝² : j < p.support.length\nx✝¹ : i < j\nx✝ : p.support[i] = p.support[j]\np' : G.Walk ((p.take j).getVert i) (p.getVert j) := (p.take... | [
"case refine_2\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : ∀ (v_1 : V) (w : G.Walk v_1 v_1), w.IsSubwalk p → w.Nil\ni j : ℕ\nx✝³ : i < p.support.length\nx✝² : j < p.support.length\nx✝¹ : i < j\nx✝ : p.support[i] = p.support[j]\np' : G.Walk ((p.take j).getVert i) (p.getVert j) := (p.take j).drop i\n... | have : ¬p'.Nil := by grind [nil_drop_iff, take_length] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 708,
"column": 50
} | {
"line": 708,
"column": 69
} | {
"line": 710,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nh : G.Adj u v\n⊢ s(u, v) ∈ (↑(singleton h)).edges",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"False",
"Sym2.mk",
"congrArg",
"Membership.mem",
"List.not_mem_nil._simp_1",
"List.cons",
"List",
... | [] | by simp [singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 47
} | {
"line": 434,
"column": 2
} | [
{
"pp": "α : Type u_1\nn : ℕ\nh : 2 ≤ n\nt : Finset α\nht : ⊥.IsNClique n t\n⊢ False",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsNClique",
"Preorder.toLE",
"instOfNatNat",
"Bot.bot",
"LE.le",
"instLENat",
"SimpleGraph",
"An... | [
"α : Type u_1\nn : ℕ\nh : 2 ≤ n\nt : Finset α\nht : ⊥.IsNClique n t\nthis : 2 ≤ 1\n⊢ False"
] | have := le_trans h (isNClique_bot_iff.1 ht).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 530,
"column": 36
} | {
"line": 530,
"column": 81
} | {
"line": 530,
"column": 81
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\ns t : α\nh : ¬(G.replaceVertex s t).CliqueFree n\nφ : Fin n ↪ α\nhφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b\nmt : ∀ (x : Fin n), φ x ≠ t\na b : Fin n\n| (G.replaceVertex s t).Adj (φ a) (φ b... | [
"α : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\ns t : α\nh : ¬(G.replaceVertex s t).CliqueFree n\nφ : Fin n ↪ α\nhφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b\nmt : ∀ (x : Fin n), φ x ≠ t\na b : Fin n\n| G.Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj ... | G.adj_replaceVertex_iff_of_ne _ (mt a) (mt b) | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 77
} | {
"line": 805,
"column": 4
} | [
{
"pp": "case neg\nV : Type u\nG : SimpleGraph V\ns : ℕ\nih :\n ∀ m < s,\n ∀ {u v : V} {p q : G.Walk u v},\n p.IsPath →\n q.IsPath →\n p ≠ q → p.length = m → ∃ u' v' p' q', p'.IsSubwalk p ∧ q'.IsSubwalk q ∧ (p'.append q'.reverse).IsCycle\nu v : V\np q : G.Walk u v\nhp : p.IsPath\nhq : q... | [
"case neg.refine_1\nV : Type u\nG : SimpleGraph V\ns : ℕ\nih :\n ∀ m < s,\n ∀ {u v : V} {p q : G.Walk u v},\n p.IsPath →\n q.IsPath →\n p ≠ q → p.length = m → ∃ u' v' p' q', p'.IsSubwalk p ∧ q'.IsSubwalk q ∧ (p'.append q'.reverse).IsCycle\nu v : V\np q : G.Walk u v\nhp : p.IsPath\nhq : q.Is... | refine hp.isCycle_append (isPath_reverse_iff q |>.mpr hq) (fun _ ↦ ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 162,
"column": 47
} | {
"line": 178,
"column": 70
} | {
"line": 180,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\n⊢ #G.edgeFinset = 3 * #(G.cliqueFinset 3)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Triangle.Basic... | [] | by
refine hG.edgeDisjointTriangles.card_edgeFinset_le.antisymm' ?_
rw [← mul_comm, ← mul_one #_]
refine card_mul_le_card_mul (fun e s ↦ e ∈ s.sym2) ?_ ?_
· simpa [Sym2.forall, Nat.one_le_iff_ne_zero, -Finset.card_eq_zero, Finset.card_ne_zero,
Finset.Nonempty]
using hG.2
simp only [mem_cliqueFins... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 928,
"column": 2
} | {
"line": 928,
"column": 13
} | {
"line": 928,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\nhp : p.IsPath\n⊢ p.bypass = p",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Walk.bypass",
"SimpleGraph.Walk.cons",
"Si... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nu✝ : V\nhp : Walk.nil.IsPath\n⊢ Walk.nil.bypass = Walk.nil",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : p✝.IsPath → p✝.bypass = p✝\nhp : (cons h✝ p✝)... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 1139,
"column": 4
} | {
"line": 1141,
"column": 36
} | {
"line": 1143,
"column": 0
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu : V\nH : SimpleGraph V\nv✝ : V\nh✝ : G.Adj u v✝\nq : G.Walk v✝ u\nqc : (cons h✝ q).IsCycle\nhq : ∀ e ∈ (cons h✝ q).edges, e ∈ H.edgeSet\n⊢ ((cons h✝ q).transfer H hq).IsCycle",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [] | simp only [edges_cons, List.mem_cons, forall_eq_or_imp] at hq
simp only [Walk.transfer, cons_isCycle_iff, edges_transfer q hq.2] at qc ⊢
exact ⟨qc.1.transfer hq.2, qc.2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 1139,
"column": 4
} | {
"line": 1141,
"column": 36
} | {
"line": 1143,
"column": 0
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu : V\nH : SimpleGraph V\nv✝ : V\nh✝ : G.Adj u v✝\nq : G.Walk v✝ u\nqc : (cons h✝ q).IsCycle\nhq : ∀ e ∈ (cons h✝ q).edges, e ∈ H.edgeSet\n⊢ ((cons h✝ q).transfer H hq).IsCycle",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [] | simp only [edges_cons, List.mem_cons, forall_eq_or_imp] at hq
simp only [Walk.transfer, cons_isCycle_iff, edges_transfer q hq.2] at qc ⊢
exact ⟨qc.1.transfer hq.2, qc.2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 71,
"column": 2
} | {
"line": 78,
"column": 56
} | {
"line": 80,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset α\nP : Finpartition univ\nε : ℝ\nhP₁ : P.IsEquipartition\nhP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊\nhX : s ∈ P.parts\n⊢ ↑(Fintype.card α) / (2 * ↑(bound (ε / 8) ⌈4 / ε⌉₊)) ≤ ↑(#s)",
"ppTerm": "?m.72",
"assigned": true,
"used... | [] | cases isEmpty_or_nonempty α
· simp [Fintype.card_eq_zero]
have := Finset.Nonempty.card_pos ⟨_, hX⟩
calc
_ ≤ card α / (2 * #P.parts : ℝ) := by gcongr
_ ≤ ↑(card α / #P.parts) :=
(div_le_iff₀' (by positivity)).2 <| mod_cast (aux ‹_› P.card_parts_le_card).le
_ ≤ (#s : ℝ) := mod_cast hP₁.average_le_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 71,
"column": 2
} | {
"line": 78,
"column": 56
} | {
"line": 80,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset α\nP : Finpartition univ\nε : ℝ\nhP₁ : P.IsEquipartition\nhP₃ : #P.parts ≤ bound (ε / 8) ⌈4 / ε⌉₊\nhX : s ∈ P.parts\n⊢ ↑(Fintype.card α) / (2 * ↑(bound (ε / 8) ⌈4 / ε⌉₊)) ≤ ↑(#s)",
"ppTerm": "?m.72",
"assigned": true,
"used... | [] | cases isEmpty_or_nonempty α
· simp [Fintype.card_eq_zero]
have := Finset.Nonempty.card_pos ⟨_, hX⟩
calc
_ ≤ card α / (2 * #P.parts : ℝ) := by gcongr
_ ≤ ↑(card α / #P.parts) :=
(div_le_iff₀' (by positivity)).2 <| mod_cast (aux ‹_› P.card_parts_le_card).le
_ ≤ (#s : ℝ) := mod_cast hP₁.average_le_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 98
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : G.FarFromTriangleFree ε\nh✝ : Nonempty α\nhε : ε ≤ 0\n⊢ triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(#(G.cliqueFinset 3))",
"ppTerm": "?inr.inl",
"assigned": ... | [
"case inr.inr\nα : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : G.FarFromTriangleFree ε\nh✝ : Nonempty α\nhε : 0 < ε\n⊢ triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(#(G.cliqueFinset 3))"
] | · apply (mul_nonpos_of_nonpos_of_nonneg (triangleRemovalBound_nonpos hε) _).trans <;> positivity | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 444,
"column": 2
} | {
"line": 447,
"column": 67
} | {
"line": 448,
"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.... | [
"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.parts\nhV : ... | have hqt : |q - t| ≤ ε ^ 5 / 49 := by
have := average_density_near_total_density hPα hPε hε₁
(Subset.refl (chunk hP G ε hU).parts) (Subset.refl (chunk hP G ε hV).parts)
simpa [← sup_eq_biUnion, sup_parts, card_chunk (m_pos hPα).ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Additive.Dissociation | {
"line": 146,
"column": 2
} | {
"line": 147,
"column": 75
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset α\nd : ℕ\nhs : ∀ s' ⊆ s, MulDissociated ↑s' → #s' ≤ d\ns' : Finset α\nhs' : Maximal (fun x ↦ x ⊆ s ∧ MulDissociated ↑x) s'\na : α\nha : a ∈ s\nha' : a ∉ s'\n⊢ a ∈ s'.mulSpan",
"ppTerm": "?neg✝",
... | [
"case neg\nα : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset α\nd : ℕ\nhs : ∀ s' ⊆ s, MulDissociated ↑s' → #s' ≤ d\ns' : Finset α\nhs' : Maximal (fun x ↦ x ⊆ s ∧ MulDissociated ↑x) s'\na : α\nha : a ∈ s\nha' : a ∉ s'\nt u : Finset α\nht : ↑t ⊆ ↑(insert a s')\nhu : ↑u ⊆ ↑(inse... | obtain ⟨t, u, ht, hu, htu⟩ := not_mulDissociated_iff_exists_disjoint.1 fun h ↦
hs'.not_gt ⟨insert_subset_iff.2 ⟨ha, hs'.1.1⟩, h⟩ <| ssubset_insert ha' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 474,
"column": 6
} | {
"line": 476,
"column": 39
} | {
"line": 477,
"column": 6
} | [
{
"pp": "case h₂.refine_1\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 : ... | [
"case h₂.refine_2\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 : ... | · -- This seems faster than `exact div_nonneg (by positivity) (by positivity)` and *much*
-- (tens of seconds) faster than `positivity` on its own.
apply div_nonneg <;> positivity | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.Randomisation | {
"line": 51,
"column": 2
} | {
"line": 52,
"column": 56
} | {
"line": 53,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : AddCommGroup G\nc : AddChar G ℂ → ℝ\nd : AddChar G ℂ → ℂ\nhcd : AddDissociated {ψ | d ψ ≠ 0}\nt : Finset (AddChar G ℂ)\nht : t ≠ ∅\nu : Finset (AddChar G ℂ)\nx✝ : u ∈ t.powerset\n⊢ ((∏ ψ ∈ u, d ψ) * ∏ ψ ∈ t \\ u, (starRingEnd ℂ) (d ψ)) * 𝔼 a, (∑ ψ ∈ u, ψ - ∑ ψ... | [
"G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : AddCommGroup G\nc : AddChar G ℂ → ℝ\nd : AddChar G ℂ → ℂ\nhcd : AddDissociated {ψ | d ψ ≠ 0}\nt : Finset (AddChar G ℂ)\nht : t ≠ ∅\nu : Finset (AddChar G ℂ)\nx✝ : u ∈ t.powerset\n⊢ ((∀ a ∈ u, d a ≠ 0) ∧ ∀ a ∈ t \\ u, (starRingEnd ℂ) (d a) ≠ 0) → ∑ ψ ∈ u, ψ ≠ ∑ ψ ∈ t \\ u, ... | rw [mul_eq_zero, AddChar.expect_eq_zero_iff_ne_zero, sub_ne_zero, or_iff_not_imp_left, ← Ne,
mul_ne_zero_iff, prod_ne_zero_iff, prod_ne_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 51
} | {
"line": 167,
"column": 4
} | [
{
"pp": "case composite.succ\nm : 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 ... | [
"case composite.succ\nm : 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 ... | obtain ⟨t₀, ht₀, ht₀card, ht₀sum⟩ := ihn a this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Colex | {
"line": 177,
"column": 2
} | {
"line": 181,
"column": 13
} | {
"line": 183,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\ns : Finset α\na : α\n⊢ toColex s < toColex {a} ↔ ∀ b ∈ s, b < a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finset.Colex.toColex_le_singleton",
"Iff.mpr",
"lt_iff_le_and_ne",
"Eq.mpr",
"False",
"Preorde... | [] | rw [lt_iff_le_and_ne, toColex_le_singleton, ne_eq, toColex_inj]
refine ⟨fun h b hb ↦ (h.1 _ hb).1.lt_of_ne ?_,
fun h ↦ ⟨fun b hb ↦ ⟨(h _ hb).le, fun ha ↦ (lt_irrefl _ <| h _ ha).elim⟩, ?_⟩⟩ <;> rintro rfl
· refine h.2 <| eq_singleton_iff_unique_mem.2 ⟨hb, fun c hc ↦ (h.1 _ hc).2 hb⟩
· simp at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Colex | {
"line": 177,
"column": 2
} | {
"line": 181,
"column": 13
} | {
"line": 183,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\ns : Finset α\na : α\n⊢ toColex s < toColex {a} ↔ ∀ b ∈ s, b < a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finset.Colex.toColex_le_singleton",
"Iff.mpr",
"lt_iff_le_and_ne",
"Eq.mpr",
"False",
"Preorde... | [] | rw [lt_iff_le_and_ne, toColex_le_singleton, ne_eq, toColex_inj]
refine ⟨fun h b hb ↦ (h.1 _ hb).1.lt_of_ne ?_,
fun h ↦ ⟨fun b hb ↦ ⟨(h _ hb).le, fun ha ↦ (lt_irrefl _ <| h _ ha).elim⟩, ?_⟩⟩ <;> rintro rfl
· refine h.2 <| eq_singleton_iff_unique_mem.2 ⟨hb, fun c hc ↦ (h.1 _ hc).2 hb⟩
· simp at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CrossProduct | {
"line": 157,
"column": 8
} | {
"line": 157,
"column": 11
} | {
"line": 157,
"column": 12
} | [
{
"pp": "case pos\nF : Type u_2\ninst✝ : Field F\nv w : Fin 3 → F\nhv : v = 0\n⊢ ¬LinearIndependent F ![v, w] ↔ (crossProduct v) w = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Algebra.to_smulCommClass",
"Semiring.toModule",
... | [
"case pos\nF : Type u_2\ninst✝ : Field F\nv w : Fin 3 → F\nhv : v = 0\n⊢ ¬LinearIndependent F ![0, w] ↔ (crossProduct 0) w = 0"
] | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 172,
"column": 14
} | {
"line": 172,
"column": 16
} | {
"line": 172,
"column": 17
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na✝ b c✝ d x y : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₁ : ∀ x ∈ A, ∀ y ∈ A, 1 / 2 * ↑(#A) < ↑(#(x •> A ∩ y •> A))\na c : G\n⊢ a ∈ A⁻¹ * A → c ∈ A⁻¹ * A → a * c ∈ A⁻¹ * A",
"ppTerm": "?m.104",
"assigned... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na✝ b c✝ d x y : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₁ : ∀ x ∈ A, ∀ y ∈ A, 1 / 2 * ↑(#A) < ↑(#(x •> A ∩ y •> A))\na c : G\nha : a ∈ A⁻¹ * A\n⊢ c ∈ A⁻¹ * A → a * c ∈ A⁻¹ * A"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.Digraph.Basic | {
"line": 60,
"column": 19
} | {
"line": 64,
"column": 65
} | {
"line": 66,
"column": 0
} | [
{
"pp": "V : Type u_1\nadj adj' : V → V → Bool\n⊢ (fun x ↦ { Adj := fun v w ↦ x v w = true }) adj = (fun x ↦ { Adj := fun v w ↦ x v w = true }) adj' → adj = adj'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Digraph",
"congrArg",
"Digraph.mk",
"Di... | [] | by
simp_rw [mk.injEq]
intro h
funext v w
simpa only [eq_iff_iff, Bool.coe_iff_coe] using congr($h v w) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Configuration | {
"line": 438,
"column": 2
} | {
"line": 462,
"column": 93
} | {
"line": 464,
"column": 0
} | [
{
"pp": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Fintype P\ninst✝ : Finite L\n⊢ Fintype.card P = order P L ^ 2 + order P L + 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Fintype.card_congr",
"Iff.mpr",
"Eq.mpr",
... | [] | cases nonempty_fintype L
obtain ⟨p, -⟩ := @exists_config P L _ _
let ϕ : { q // q ≠ p } ≃ Σ l : { l : L // p ∈ l }, { q // q ∈ l.1 ∧ q ≠ p } :=
{ toFun := fun q => ⟨⟨mkLine q.2, (mkLine_ax q.2).2⟩, q, (mkLine_ax q.2).1, q.2⟩
invFun := fun lq => ⟨lq.2, lq.2.2.2⟩
right_inv := fun lq =>
Sigma.s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Configuration | {
"line": 438,
"column": 2
} | {
"line": 462,
"column": 93
} | {
"line": 464,
"column": 0
} | [
{
"pp": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : ProjectivePlane P L\ninst✝¹ : Fintype P\ninst✝ : Finite L\n⊢ Fintype.card P = order P L ^ 2 + order P L + 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Fintype.card_congr",
"Iff.mpr",
"Eq.mpr",
... | [] | cases nonempty_fintype L
obtain ⟨p, -⟩ := @exists_config P L _ _
let ϕ : { q // q ≠ p } ≃ Σ l : { l : L // p ∈ l }, { q // q ∈ l.1 ∧ q ≠ p } :=
{ toFun := fun q => ⟨⟨mkLine q.2, (mkLine_ax q.2).2⟩, q, (mkLine_ax q.2).1, q.2⟩
invFun := fun lq => ⟨lq.2, lq.2.2.2⟩
right_inv := fun lq =>
Sigma.s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 219,
"column": 14
} | {
"line": 219,
"column": 16
} | {
"line": 219,
"column": 17
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₀ : A.Nonempty\nh₁ : ∀ a ∈ A⁻¹ * A, 1 / 2 * ↑(#A) < ↑(#({xy ∈ A ×ˢ A | xy.1 * xy.2⁻¹ = a}))\na b : G\n⊢ a ∈ A → b ∈ A → ∃ y ∈ A, ∃ z, (∃ y ∈ A, y⁻¹ = z) ∧ y * z = a * b⁻¹",
"ppTerm": "?m.183",
... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₀ : A.Nonempty\nh₁ : ∀ a ∈ A⁻¹ * A, 1 / 2 * ↑(#A) < ↑(#({xy ∈ A ×ˢ A | xy.1 * xy.2⁻¹ = a}))\na b : G\nha : a ∈ A\n⊢ b ∈ A → ∃ y ∈ A, ∃ z, (∃ y ∈ A, y⁻¹ = z) ∧ y * z = a * b⁻¹"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 25
} | {
"line": 345,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nc : Set G\nx : G\nhc : c ∈ orbit Gᵐᵒᵖ ↑H\nhx : x ∈ c\n⊢ ↑H <• x = c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"instHSMul",
"PreOpposite.casesOn",
"MulOpposite",
"Membership.mem",
"DivInvMonoid.toMon... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nx a : G\nhx : x ∈ (fun m ↦ m •> ↑H) { unop' := a }\n⊢ ↑H <• x = (fun m ↦ m •> ↑H) { unop' := a }"
] | obtain ⟨⟨a⟩, rfl⟩ := hc | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 367,
"column": 2
} | {
"line": 373,
"column": 10
} | {
"line": 375,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : DecidableEq G\nH : Subgroup G\ninst✝ : Fintype ↥H\nZ : Finset G\nhZ : Set.InjOn (fun x ↦ ↑H <• x) ↑Z\n⊢ Fintype.card ↥H * #Z = #((↑H).toFinset * Z)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.instSProd",
... | [] | rw [card_mul_iff.2]
· simp
rintro ⟨h₁, z₁⟩ ⟨hh₁, hz₁⟩ ⟨h₂, z₂⟩ ⟨hh₂, hz₂⟩ h
simp only [Set.coe_toFinset, SetLike.mem_coe] at *
obtain rfl := hZ hz₁ hz₂ <| (rightCoset_eq_iff _).2 <| by
simpa [eq_inv_mul_iff_mul_eq.2 h, mul_assoc] using mul_mem (inv_mem hh₂) hh₁
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 367,
"column": 2
} | {
"line": 373,
"column": 10
} | {
"line": 375,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : DecidableEq G\nH : Subgroup G\ninst✝ : Fintype ↥H\nZ : Finset G\nhZ : Set.InjOn (fun x ↦ ↑H <• x) ↑Z\n⊢ Fintype.card ↥H * #Z = #((↑H).toFinset * Z)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.instSProd",
... | [] | rw [card_mul_iff.2]
· simp
rintro ⟨h₁, z₁⟩ ⟨hh₁, hz₁⟩ ⟨h₂, z₂⟩ ⟨hh₂, hz₂⟩ h
simp only [Set.coe_toFinset, SetLike.mem_coe] at *
obtain rfl := hZ hz₁ hz₂ <| (rightCoset_eq_iff _).2 <| by
simpa [eq_inv_mul_iff_mul_eq.2 h, mul_assoc] using mul_mem (inv_mem hh₂) hh₁
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 217,
"column": 76
} | {
"line": 217,
"column": 86
} | {
"line": 218,
"column": 6
} | [
{
"pp": "p q : DyckWord\nhn : p.IsNested\ni : ℕ\nh : ↑p ≠ []\nl1 : List.take 1 ↑p = [(↑p).head h]\nl3 : (↑p).length - 1 = (↑p).length - 1 - 1 + 1\nj : ℕ := min (1 + i) ((↑p).length - 1)\nub : j < (↑p).length\nlb : 0 < j\neq :\n count D (List.take 1 (List.take j ↑p)) + count D (List.drop 1 (List.take j ↑p)) <\n... | [
"p q : DyckWord\nhn : p.IsNested\ni : ℕ\nh : ↑p ≠ []\nl1 : List.take 1 ↑p = [(↑p).head h]\nl3 : (↑p).length - 1 = (↑p).length - 1 - 1 + 1\nj : ℕ := min (1 + i) ((↑p).length - 1)\nub : j < (↑p).length\nlb : 0 < j\neq :\n count D (List.take (min 1 j) ↑p) + count D (List.drop 1 (List.take j ↑p)) <\n count U (List.... | take_take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 191,
"column": 52
} | {
"line": 195,
"column": 47
} | {
"line": 197,
"column": 0
} | [
{
"pp": "m n : ℕ\nhn : n ≠ 0\n⊢ m.uniformBell n = (m * n)! / (n ! ^ m * m !)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"instHDiv",
"HMul.hMul",
"congrArg",
"Nat.uniformBell_mul_eq",
"Nat.instMonoid",
"mul_as... | [] | by
rw [eq_comm]
apply Nat.div_eq_of_eq_mul_left
· exact Nat.mul_pos (Nat.pow_pos n.factorial_pos) m.factorial_pos
· rw [← mul_assoc, ← uniformBell_mul_eq _ hn] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 18
} | {
"line": 374,
"column": 4
} | [
{
"pp": "case pos\nF : 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\nhab : a = b\nh : ¬1 = 0\n⊢ a ≤ b",
"ppTerm": "?pos✝",
"assigned": true,... | [
"case neg\nF : 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\nhab : ¬a = b\nh : ¬-∑ x ∈ (Ico a b).attach, muFun 𝕜 a ↑x = 0\n⊢ a ≤ b"
] | · exact hab.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 20
} | {
"line": 434,
"column": 6
} | [
{
"pp": "case pos\nF : 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\nhab : a = b\nh : ¬1 = 0\n⊢ a ≤ b",
"ppTerm": "?pos✝",
"assigned": true,... | [
"case neg\nF : 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\nhab : ¬a = b\nh : ¬-∑ x ∈ (Ioc a b).attach, muFun' 𝕜 b ↑x = 0\n⊢ a ≤ b"
] | · exact hab.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 39
} | {
"line": 309,
"column": 4
} | [
{
"pp": "case neg\np q : DyckWord\nh : ¬p = 0\nu : ↑(p + q) = ↑p ++ ↑q\n⊢ p.firstReturn < (range (↑(p + q)).length).length",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instAddDyckWord",
"congrArg",
"List.length_range",
"DyckWord",
"List.len... | [
"case neg\np q : DyckWord\nh : ¬p = 0\nu : ↑(p + q) = ↑p ++ ↑q\n⊢ p.firstReturn < (↑p).length + (↑q).length"
] | rw [length_range, u, length_append] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 54
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case neg\np q : DyckWord\nh : ¬p = 0\nu : ↑(p + q) = ↑p ++ ↑q\n⊢ p.firstReturn < (↑p).length + (↑q).length",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"DyckWord.firstReturn_lt_length",
"DyckWord.firstReturn",
"Nat.lt_add_right",
"DyckWord.toList",
... | [] | exact Nat.lt_add_right _ (firstReturn_lt_length h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 337,
"column": 10
} | {
"line": 337,
"column": 20
} | {
"line": 337,
"column": 21
} | [
{
"pp": "p q : DyckWord\nh✝ : p ≠ 0\nh : ¬p = 0\ni : ℕ\nlb : 0 < i\nub : i < p.firstReturn + 1\n⊢ count D (List.take i (List.take (p.firstReturn + 1) ↑p)) < count U (List.take i (List.take (p.firstReturn + 1) ↑p))",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instD... | [
"p q : DyckWord\nh✝ : p ≠ 0\nh : ¬p = 0\ni : ℕ\nlb : 0 < i\nub : i < p.firstReturn + 1\n⊢ count D (List.take (min i (p.firstReturn + 1)) ↑p) < count U (List.take (min i (p.firstReturn + 1)) ↑p)"
] | take_take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 146,
"column": 2
} | {
"line": 168,
"column": 9
} | {
"line": 170,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nn : ℕ\np : n.Partition\ns : Finset ℕ\nhs : Icc 1 n ⊆ s\nhs0 : 0 ∉ s\n⊢ (Multiset.toFinsupp p.parts).prod f =\n ∏ i ∈ s, (coeff (p.toFinsuppAntidiag i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1... | [] | simp_rw [Finsupp.prod, Multiset.toFinsupp_support, Multiset.toFinsupp_apply]
apply prod_subset_one_on_sdiff
· grind
· intro x hx
rw [mem_sdiff, Multiset.mem_toFinset] at hx
have hx0 : x ≠ 0 := fun h ↦ hs0 (h ▸ hx.1)
have hsum := (summable_genFun_term' f hx0).map_tsum _
(WithPiTopology.continuous... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 146,
"column": 2
} | {
"line": 168,
"column": 9
} | {
"line": 170,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nn : ℕ\np : n.Partition\ns : Finset ℕ\nhs : Icc 1 n ⊆ s\nhs0 : 0 ∉ s\n⊢ (Multiset.toFinsupp p.parts).prod f =\n ∏ i ∈ s, (coeff (p.toFinsuppAntidiag i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1... | [] | simp_rw [Finsupp.prod, Multiset.toFinsupp_support, Multiset.toFinsupp_apply]
apply prod_subset_one_on_sdiff
· grind
· intro x hx
rw [mem_sdiff, Multiset.mem_toFinset] at hx
have hx0 : x ≠ 0 := fun h ↦ hs0 (h ▸ hx.1)
have hsum := (summable_genFun_term' f hx0).map_tsum _
(WithPiTopology.continuous... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 640,
"column": 2
} | {
"line": 641,
"column": 60
} | {
"line": 642,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nS : Finset G\nhK : K < 1\nhS : S.Nonempty\nH : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A\nex : Finset G → ℝ := expansion K S\nκ : ℝ := connectivity K S\nκ_add_one_pos : 0 < κ + 1\none_sub_K_pos : 0 < 1 - K\nt : ℕ := ⌊(κ + 1) / (1 - K)... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nS : Finset G\nhK : K < 1\nhS : S.Nonempty\nH : ∀ (A : Finset G), A.Nonempty → ¬IsFragment K S A\nex : Finset G → ℝ := expansion K S\nκ : ℝ := connectivity K S\nκ_add_one_pos : 0 < κ + 1\none_sub_K_pos : 0 < 1 - K\nt : ℕ := ⌊(κ + 1) / (1 - K)⌋₊\nlargeA :... | let M := {x ∈ ((Icc #S (t * #S)).map Nat.castEmbedding -
K • (Icc 1 t).map Nat.castEmbedding : Finset ℝ) | κ < x} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 752,
"column": 4
} | {
"line": 759,
"column": 55
} | {
"line": 761,
"column": 2
} | [
{
"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... | [] | calc
ε / 2 * (Fintype.card H)
_ = ε / 2 * #(H : Set G).toFinset := by
simp only [Set.toFinset_card, SetLike.coe_sort_coe]
_ = (1 - K) * #(H : Set G).toFinset := by ring
_ ≤ ex (Set.toFinset H) := mul_card_le_expansion hS
_ ≤ (1 - ε / 2) * #S ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 768,
"column": 17
} | {
"line": 768,
"column": 52
} | {
"line": 769,
"column": 6
} | [
{
"pp": "case e_a\nG : 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\nh... | [] | simpa using congr(($hHZS).toFinset) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 25
} | {
"line": 109,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤ G\ne : β\nhe : e ∈ E(H)\nx y : α\n⊢ H.IsLink e x y ↔ G.IsLink e x y",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Graph.IsSubgraph.isLink_iff"
],
"usedFVars": [
"α",
"β",
"x",
"y",
... | [] | exact isLink_iff hHG he | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Semigroup | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 90
} | {
"line": 52,
"column": 8
} | [
{
"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,... | [
"case ht.refine_1\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 (... | refine ⟨N_closed.inter ((T1Space.t1 m).preimage (continuous_const_mul m)), ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Hypergraph.Basic | {
"line": 143,
"column": 49
} | {
"line": 143,
"column": 64
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\ne f : Set α\nH : Hypergraph α\nh : H.EAdj e f\n⊢ H.EAdj f e",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.Hypergraph.Basic.0.Hypergraph.EAdj.symm._proof_1_2"
],
"usedFVars": [
"α",
"e",
"f",
"H",
... | [] | by grind [EAdj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Hindman | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 32
} | {
"line": 147,
"column": 0
} | [
{
"pp": "case h\nM : Type u_1\ninst✝ : Semigroup M\na✝ : Stream' M\nm✝ : M\na : Stream' M\nm : M\nh✝ : FP a.tail m\nn : ℕ\nhn : ∀ m' ∈ FP (Stream'.drop n a.tail), m * m' ∈ FP a.tail\nm' : M\nhm' : m' ∈ FP (Stream'.drop (n + 1) a)\n⊢ a.head * (m * m') ∈ FP a",
"ppTerm": "?h",
"assigned": true,
"usedC... | [] | exact FP.cons _ _ (hn _ hm') | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Hindman | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 54
} | {
"line": 177,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := ⋯\nU : Ultrafilter M\nhU : ∀ (i : ℕ), U ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : ∀ (i : ℕ), V ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}\nn : ℕ\nm : M\nhm : m ∈ FP (Stream'.drop n a... | [] | simpa only [Stream'.drop_drop, add_comm] using hm' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 177,
"column": 4
} | {
"line": 179,
"column": 28
} | {
"line": 180,
"column": 4
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.IsBasis I X\nJ : Set α\nhJ : J ⊆ (M✶ ＼ I ＼ (X \\ I)).E\ne : α\nhe : e ∈ X \\ I\n⊢ e ∈ (M✶ ＼ I).coloops",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"ChainCompletePartialOrder.instOfCompl... | [
"α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.IsBasis I X\nJ : Set α\nhJ : J ⊆ (M✶ ＼ I ＼ (X \\ I)).E\ne : α\nhe : e ∈ X \\ I\n⊢ e ∈ M.closure X \\ I"
] | rw [← dual_contract, dual_coloops, ← IsLoop, ← singleton_dep, hI.indep.contract_dep_iff,
singleton_union, and_iff_right (by simpa using he.2), hI.indep.insert_dep_iff,
hI.closure_eq_closure] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 65,
"column": 4
} | {
"line": 69,
"column": 50
} | {
"line": 71,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nM✝ M : (i : ι) → Matroid (α i)\nI : Set ((i : ι) × α i)\n⊢ (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)) ↔ ∃ B, (∀ (i : ι), (M i).IsBase (Sigma.mk i ⁻¹' B)) ∧ I ⊆ B",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePa... | [] | refine ⟨fun h ↦ ?_, fun ⟨B, hB, hIB⟩ i ↦ (hB i).indep.subset (preimage_mono hIB)⟩
choose Bs hBs using fun i ↦ (h i).exists_isBase_superset
refine ⟨univ.sigma Bs, fun i ↦ by simpa using (hBs i).1, ?_⟩
rw [← univ_sigma_preimage_mk I]
refine sigma_mono rfl.subset fun i ↦ (hBs i).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 65,
"column": 4
} | {
"line": 69,
"column": 50
} | {
"line": 71,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nM✝ M : (i : ι) → Matroid (α i)\nI : Set ((i : ι) × α i)\n⊢ (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)) ↔ ∃ B, (∀ (i : ι), (M i).IsBase (Sigma.mk i ⁻¹' B)) ∧ I ⊆ B",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePa... | [] | refine ⟨fun h ↦ ?_, fun ⟨B, hB, hIB⟩ i ↦ (hB i).indep.subset (preimage_mono hIB)⟩
choose Bs hBs using fun i ↦ (h i).exists_isBase_superset
refine ⟨univ.sigma Bs, fun i ↦ by simpa using (hBs i).1, ?_⟩
rw [← univ_sigma_preimage_mk I]
refine sigma_mono rfl.subset fun i ↦ (hBs i).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 63
} | {
"line": 280,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhI : M.IsBasis I X\n⊢ (M ／ X).Indep (J \\ X) ↔ M.Indep (J \\ X ∪ I)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.IsBasis.contract_indep_iff",
... | [] | rw [hI.contract_indep_iff, and_iff_left disjoint_sdiff_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 63
} | {
"line": 280,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhI : M.IsBasis I X\n⊢ (M ／ X).Indep (J \\ X) ↔ M.Indep (J \\ X ∪ I)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.IsBasis.contract_indep_iff",
... | [] | rw [hI.contract_indep_iff, and_iff_left disjoint_sdiff_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 63
} | {
"line": 280,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhI : M.IsBasis I X\n⊢ (M ／ X).Indep (J \\ X) ↔ M.Indep (J \\ X ∪ I)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.IsBasis.contract_indep_iff",
... | [] | rw [hI.contract_indep_iff, and_iff_left disjoint_sdiff_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 63
} | {
"line": 287,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhI : M.IsBasis' I X\n⊢ (M ／ X).Indep (J \\ X) ↔ M.Indep (J \\ X ∪ I)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistrib... | [] | rw [hI.contract_indep_iff, and_iff_left disjoint_sdiff_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 63
} | {
"line": 287,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhI : M.IsBasis' I X\n⊢ (M ／ X).Indep (J \\ X) ↔ M.Indep (J \\ X ∪ I)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistrib... | [] | rw [hI.contract_indep_iff, and_iff_left disjoint_sdiff_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 63
} | {
"line": 287,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhI : M.IsBasis' I X\n⊢ (M ／ X).Indep (J \\ X) ↔ M.Indep (J \\ X ∪ I)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistrib... | [] | rw [hI.contract_indep_iff, and_iff_left disjoint_sdiff_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 150,
"column": 6
} | {
"line": 151,
"column": 20
} | {
"line": 152,
"column": 4
} | [
{
"pp": "case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nb : ι → MvPolynomial σ R\nhb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))\nf : MvPolynomial σ R\ni : ι\nhb0 : m.degree (b i) = 0\nj : ι\nhj : ¬j = i\n⊢ m.toSyn (m.degree (b j * (Finsupp.single i (⋯.unit⁻¹ • f)) j... | [] | · simp only [Finsupp.single_eq_of_ne hj, mul_zero, degree_zero, map_zero]
apply bot_le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 192,
"column": 6
} | {
"line": 192,
"column": 33
} | {
"line": 193,
"column": 6
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nb : ι → MvPolynomial σ R\nhb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))\nf : MvPolynomial σ R\nhb' : ∀ (i : ι), m.degree (b i) ≠ 0\nhf0 : ¬f = 0\nhf : ∀ (i : ι), ¬m.degree (b i) ≤ m.degree f\nthis :\n ∃ g' r',\n m.su... | [
"σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nb : ι → MvPolynomial σ R\nhb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))\nf : MvPolynomial σ R\nhb' : ∀ (i : ι), m.degree (b i) ≠ 0\nhf0 : ¬f = 0\nhf : ∀ (i : ι), ¬m.degree (b i) ≤ m.degree f\ng' : ι →₀ MvPolynomial σ R\nr' : MvPolyn... | obtain ⟨g', r', H'⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 218,
"column": 4
} | {
"line": 222,
"column": 77
} | {
"line": 223,
"column": 2
} | [
{
"pp": "case nil\nV : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ nil.vertices\n⊢ ∃ p₁ p₂, nil = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"False",
"Quiver.Path.nil",
"congrArg",
"Membership.mem",
... | [] | have hxa : v = a := by
simpa [vertices_nil, List.mem_singleton] using hv
subst hxa
exact ⟨Path.nil, Path.nil, by simp only [comp_nil],
by simp only [vertices_nil, tail_cons, not_mem_nil, not_false_eq_true]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 218,
"column": 4
} | {
"line": 222,
"column": 77
} | {
"line": 223,
"column": 2
} | [
{
"pp": "case nil\nV : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ nil.vertices\n⊢ ∃ p₁ p₂, nil = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"False",
"Quiver.Path.nil",
"congrArg",
"Membership.mem",
... | [] | have hxa : v = a := by
simpa [vertices_nil, List.mem_singleton] using hv
subst hxa
exact ⟨Path.nil, Path.nil, by simp only [comp_nil],
by simp only [vertices_nil, tail_cons, not_mem_nil, not_false_eq_true]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 48
} | {
"line": 241,
"column": 6
} | [
{
"pp": "case cons.inl\nV : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv b✝ c✝ : V\npPrev : Path a b✝\ne : b✝ ⟶ c✝\nih : v ∈ pPrev.vertices → ∃ p₁ p₂, pPrev = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail\nhv : v ∈ (pPrev.cons e).vertices\nh_case₁ : v = (pPrev.cons e).end → ∃ p₁ p₂, pPrev.cons e = p₁.comp p₂ ∧ ¬v ... | [
"case pos\nV : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv b✝ c✝ : V\npPrev : Path a b✝\ne : b✝ ⟶ c✝\nih : v ∈ pPrev.vertices → ∃ p₁ p₂, pPrev = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail\nhv : v ∈ (pPrev.cons e).vertices\nh_case₁ : v = (pPrev.cons e).end → ∃ p₁ p₂, pPrev.cons e = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tai... | by_cases h_eq_end : v = (pPrev.cons e).end | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 293,
"column": 53
} | {
"line": 294,
"column": 88
} | {
"line": 296,
"column": 0
} | [
{
"pp": "U : Type u_1\ninst✝⁴ : Quiver U\nV : Type u_2\ninst✝³ : Quiver V\nφ : U ⥤q V\ninst✝² : HasInvolutiveReverse U\ninst✝¹ : HasInvolutiveReverse V\ninst✝ : φ.MapReverse\nu : U\n⊢ Bijective (φ.costar u) ↔ Bijective (φ.star u)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | by
rw [Prefunctor.costar_conj_star φ, EquivLike.comp_bijective, EquivLike.bijective_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 61,
"column": 28
} | {
"line": 61,
"column": 38
} | {
"line": 61,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ image (fun s ↦ s.erase a) ({s ∈ 𝒜 | a ∈ s}) ↔ insert a s ∈ 𝒜 ∧ a ∉ s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
... | [
"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\n⊢ (∃ a_1 ∈ {s ∈ 𝒜 | a ∈ s}, a_1.erase a = s) ↔ insert a s ∈ 𝒜 ∧ a ∉ s"
] | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 97,
"column": 2
} | {
"line": 99,
"column": 44
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\ninst✝ : DecidableEq α\na : α\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ insert a s ∈ 𝒜 ∧ a ∉ s ∨ s ∈ 𝒜 ∧ a ∉ s → ∃ a_2 ∈ 𝒜, a_2.erase a = s",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Exists",
"Insert.i... | [
"case mpr\nα : Type u_1\ninst✝ : DecidableEq α\na : α\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ a_1 ∈ 𝒜, a_1.erase a = s) → insert a s ∈ 𝒜 ∧ a ∉ s ∨ s ∈ 𝒜 ∧ a ∉ s"
] | · rintro (h | h)
· exact ⟨_, h.1, erase_insert h.2⟩
· exact ⟨_, h.1, erase_eq_of_notMem h.2⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 228,
"column": 51
} | {
"line": 228,
"column": 61
} | {
"line": 228,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∈ image (fun s ↦ s.erase a) 𝒜 ∧ s ∉ 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∉ 𝒜 ∧ insert a s ∈ 𝒜",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congr... | [
"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ (∃ a_1 ∈ 𝒜, a_1.erase a = s) ∧ s ∉ 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∉ 𝒜 ∧ insert a s ∈ 𝒜"
] | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 75
} | {
"line": 285,
"column": 2
} | [
{
"pp": "⊢ schnirelmannDensity (Set.ofPred Odd) = 2⁻¹",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Set.ext",
"Set.ofPred",
"Odd",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"HMod.hMod",
"Nat",
"Nat.instSemiring",
"OfNat.ofNat"... | [
"h : Set.ofPred Odd = {n | n % 2 = 1}\n⊢ schnirelmannDensity (Set.ofPred Odd) = 2⁻¹"
] | have h : Set.ofPred Odd = {n | n % 2 = 1} := Set.ext fun _ => Nat.odd_iff | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Schnirelmann | {
"line": 315,
"column": 50
} | {
"line": 315,
"column": 77
} | {
"line": 316,
"column": 4
} | [
{
"pp": "A B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∉ A\nhnB : n ∉ B\nf : ℕ ⊕ ℕ → ℕ :=\n fun x ↦\n match x with\n | Sum.inl x => x\n | Sum.inr y =>... | [] | gcongr; grind [mem_disjSum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Schnirelmann | {
"line": 315,
"column": 50
} | {
"line": 315,
"column": 77
} | {
"line": 316,
"column": 4
} | [
{
"pp": "A B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∉ A\nhnB : n ∉ B\nf : ℕ ⊕ ℕ → ℕ :=\n fun x ↦\n match x with\n | Sum.inl x => x\n | Sum.inr y =>... | [] | gcongr; grind [mem_disjSum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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