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Proof step 1 from proof acyclic-rem-not-reachable (conclusion) (back to tree)

∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), reachable(r, r0, h), (h(r0)).nxt ≠ null, (h(r0)).nxt ∈ h, ce.nxt = (h((h(r0)).nxt)).nxt ⊦ ¬ reachable(r, (h(r0)).nxt, h(r0 := ce))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, reachable(r, (h(r0)).nxt, h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, reachable(r, r0, h), reachable(r, (h(r0)).nxt, h(r0 := ce)), (h(r0)).nxt ∈ h, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-nodups-def, seq:⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x)), subst:[h, r0, r] → [h(r0 := ce), (h(r0)).nxt, r], <all paths>

Additional information

Used terms: h(r0 := ce), (h(r0)).nxt, r

Proof step 4 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))

Lemma reachable-nodups-def

Proof step 5 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, reachable(r, r0, h), (h(r0)).nxt ∈ h, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), ∃ x. path(r ' + x, h(r0 := ce)) ∧ (r ' + x).last = (h(r0)).nxt ∧ ¬ dups(r ' + x) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s)

Proof step 6 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(r ' + x).last = (h(r0)).nxt, ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, path(r ' + x, h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: polylist-last, name: lapp, seq: ⊦ y ≠ [] → (x + y).last = y.last, subst:[y, x] → [x, r ']; [A] → [Ref], <all paths>

Additional information

Used terms: x, r '

Proof step 7 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof step 8 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(r ' + x).last = (h(r0)).nxt, ce.nxt = (h((h(r0)).nxt)).nxt, x = [], (h(r0)).nxt ≠ null, path(r ' + x, h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h; ⊦ ¬ a ∈ []; (s) ⊦ ¬ dups([]); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ x + [] = x; (s) ⊦ a '.last = a; (s)

Proof step 9 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, reachable((h(r0)).nxt, r0, h), r0 ∈ h, (h(r0)).nxt ∈ h(r0 := ce), (h(r0)).nxt ∈ h, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: r0

Additional information

Used terms: r0

Proof step 10 from proof acyclic-rem-not-reachable (previous step) (back to tree)

x.last = (h(r0)).nxt, ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, x ≠ [], path(r ' + x, h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof acyclic-rem-not-reachable (previous step) (back to tree)

x.last = (h(r0)).nxt, ce.nxt = (h(x.last)).nxt, x.last ≠ null, x ≠ [], path(r ' + x, h(r0 := ce)), reachable(r, r0, h), x.last ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: polylist-last, name:last-elim, seq:x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a ', subst:[x, a, y] → [x, r1, y]; [A] → [Ref]

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x, a, y

Proof step 12 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h(r1)).nxt, r1 = (h(r0)).nxt, x = y + r1 ', r1 ≠ null, x ≠ [], path(r ' + x, h(r0 := ce)), reachable(r, r0, h), r1 ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 13 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 ∈ y

Additional information

Used formulas: r0 ∈ y

Proof step 14 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, r0 ∈ y, ¬ dups(y), ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: polylist, name: lastoc-rew, seq: ⊦ a ∈ x ↔ (∃ y, z. x = y + a ' + z ∧ ¬ a ∈ z), subst:[x, a] → [y, r0]; [A] → [Ref], <all paths>

Additional information

Used terms: y, r0

Proof step 15 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ y, z. x = y + a ' + z ∧ ¬ a ∈ z)

A spec-lemma

Proof step 16 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), ∃ y0, z. y = y0 + r0 ' + z ∧ ¬ r0 ∈ z ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws)

Proof step 17 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ r0, r ≠ (h(r0)).nxt, r ≠ r0, path(r ' + (y0 + r0 ' + y1) + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y1), ¬ r0 ∈ y1, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ (h(r0)).nxt ∈ y1, ¬ r ∈ y0, ¬ r ∈ y1, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: path(r0 ' + y1 + (h(r0)).nxt ', h(r0 := ce))

Additional information

Used simplifier rules: path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); a: ⊦ (x + y) + z = x + y + z; from specification polylist Used formulas: path(r0 ' + y1 + (h(r0)).nxt ', h(r0 := ce))

Proof step 18 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ r0, r ≠ (h(r0)).nxt, r ≠ r0, path(r ' + (y0 + r0 ' + y1) + (h(r0)).nxt ', h(r0 := ce)), path(r0 ' + y1 + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y1), ¬ r0 ∈ y1, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ (h(r0)).nxt ∈ y1, ¬ r ∈ y0, ¬ r ∈ y1, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: y1

Additional information

Used simplifier rules: List[A] freely generated by [], +;

Proof step 19 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, y1 = [], (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ r0, r ≠ (h(r0)).nxt, r ≠ r0, path(r0 ' + y1 + (h(r0)).nxt ', h(r0 := ce)), path(r ' + (y0 + r0 ' + y1) + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y1), ¬ r0 ∈ y1, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ (h(r0)).nxt ∈ y1, ¬ r ∈ y0, ¬ r ∈ y1, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ (h(r)).nxt = r0; ⊦ ¬ a ∈ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ x + [] = x; (s) ⊦ st(a := d)(a) = d; (s) ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s) ⊦ a ∈ st(a := d); (s) ⊦ [] + x = x; (s)

Proof step 20 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h(ce.nxt)).nxt, ce.nxt = (h(r0)).nxt, ce.nxt ≠ null, ce.nxt ≠ r0, r ≠ ce.nxt, r ≠ r0, r0 ≠ null, path(r ' + y0 + r0 ' + ce.nxt ', h(r0 := ce)), reachable(r, r0, h), ce.nxt ∈ h, ¬ dups(y0), ¬ r0 ∈ y0, ¬ ce.nxt ∈ y0, ¬ r ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 15 terms: ce.nxt abort: F discard: T computed: F

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; Used terms: ce.nxt

Proof step 21 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, y1 = r1 + y, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ r0, r ≠ (h(r0)).nxt, r ≠ r0, path(r0 ' + y1 + (h(r0)).nxt ', h(r0 := ce)), path(r ' + (y0 + r0 ' + y1) + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y1), ¬ r0 ∈ y1, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ (h(r0)).nxt ∈ y1, ¬ r ∈ y0, ¬ r ∈ y1, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws) ⊦ st(a := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s)

Proof step 22 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ ce.nxt, (h(r0)).nxt ≠ r0, (h(r0)).nxt ≠ null, r ≠ ce.nxt, r ≠ r0, r ≠ (h(r0)).nxt, r0 ≠ ce.nxt, r0 ≠ null, path(ce.nxt ' + y + (h(r0)).nxt ', h(r0 := ce)), path(r ' + y0 + r0 ' + (ce.nxt ' + y) + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y), ¬ r ∈ y, ¬ r ∈ y0, ¬ (h(r0)).nxt ∈ y, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: path-modify, seq: ⊦ ¬ r0 ∈ x → (path(x, h(r0 := ce)) ↔ path(x, h)), subst:[ce, r0, h, x] → [ce, r0, h, ce.nxt ' + y + (h(r0)).nxt '], <all paths>

Additional information

Used simplifier rules: ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce, r0, h, ce.nxt ' + y + (h(r0)).nxt '

Proof step 23 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ ¬ r0 ∈ x → (path(x, h(r0 := ce)) ↔ path(x, h))

Lemma path-modify

Proof step 24 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ ce.nxt, (h(r0)).nxt ≠ r0, (h(r0)).nxt ≠ null, r ≠ ce.nxt, r ≠ r0, r ≠ (h(r0)).nxt, r0 ≠ ce.nxt, r0 ≠ null, path(r ' + y0 + r0 ' + (ce.nxt ' + y) + (h(r0)).nxt ', h(r0 := ce)), path(ce.nxt ' + y + (h(r0)).nxt ', h), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y), ¬ r0 ∈ ce.nxt ' + y + (h(r0)).nxt ', ¬ r ∈ y, ¬ r ∈ y0, ¬ (h(r0)).nxt ∈ y, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s)

Proof step 25 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ ce.nxt, (h(r0)).nxt ≠ r0, (h(r0)).nxt ≠ null, r ≠ ce.nxt, r ≠ r0, r ≠ (h(r0)).nxt, r0 ≠ ce.nxt, r0 ≠ null, path(ce.nxt ' + y + (h(r0)).nxt ', h), path(r ' + y0 + r0 ' + (ce.nxt ' + y) + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y), ¬ r ∈ y, ¬ r ∈ y0, ¬ (h(r0)).nxt ∈ y, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 23 terms: (h(r0)).nxt abort: F discard: T computed: F

Additional information

Used terms: (h(r0)).nxt

Proof step 26 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ ce.nxt, (h(r0)).nxt ≠ r0, (h(r0)).nxt ≠ null, r ≠ ce.nxt, r ≠ r0, r ≠ (h(r0)).nxt, r0 ≠ ce.nxt, r0 ≠ null, path(r ' + y0 + r0 ' + (ce.nxt ' + y) + (h(r0)).nxt ', h(r0 := ce)), path(ce.nxt ' + y + (h(r0)).nxt ', h), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y), ¬ r ∈ y, ¬ r ∈ y0, ¬ (h(r0)).nxt ∈ y, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, (h(r0)).nxt ∈ h → ¬ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ ce.nxt, (h(r0)).nxt ≠ r0, (h(r0)).nxt ≠ null, r ≠ ce.nxt, r ≠ r0, r ≠ (h(r0)).nxt, r0 ≠ ce.nxt, r0 ≠ null, path(ce.nxt ' + y + (h(r0)).nxt ', h), path(r ' + y0 + r0 ' + (ce.nxt ' + y) + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y), ¬ reachable(ce.nxt, (h(r0)).nxt, h), ¬ r ∈ y, ¬ r ∈ y0, ¬ (h(r0)).nxt ∈ y, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-def, seq: ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0), subst:[h, r0, r] → [h, (h(r0)).nxt, ce.nxt], <all paths>

Additional information

Used terms: h, (h(r0)).nxt, ce.nxt

Proof step 28 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)

Lemma reachable-def

Proof step 29 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ ce.nxt, (h(r0)).nxt ≠ r0, (h(r0)).nxt ≠ null, r ≠ ce.nxt, r ≠ r0, r ≠ (h(r0)).nxt, r0 ≠ ce.nxt, r0 ≠ null, path(r ' + y0 + r0 ' + (ce.nxt ' + y) + (h(r0)).nxt ', h(r0 := ce)), path(ce.nxt ' + y + (h(r0)).nxt ', h), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y), ¬ r ∈ y, ¬ r ∈ y0, ¬ (h(r0)).nxt ∈ y, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ∀ x. ¬(path(ce.nxt ' + x, h) ∧ (ce.nxt ' + x).last = (h(r0)).nxt) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 30 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ ce.nxt, (h(r0)).nxt ≠ r0, (h(r0)).nxt ≠ null, r ≠ ce.nxt, r ≠ r0, r ≠ (h(r0)).nxt, r0 ≠ ce.nxt, r0 ≠ null, path(ce.nxt ' + y + (h(r0)).nxt ', h), path(r ' + y0 + r0 ' + (ce.nxt ' + y) + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y0 + y), ¬ r ∈ y, ¬ r ∈ y0, ¬ (h(r0)).nxt ∈ y, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ∀ x. ¬ path(ce.nxt ' + x, h) ∨ (ce.nxt ' + x).last ≠ (h(r0)).nxt ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 23 terms: y + (h(r0)).nxt '

Additional information

Used simplifier rules: ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: y + (h(r0)).nxt '

Proof step 31 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: path-modify, seq: ⊦ ¬ r0 ∈ x → (path(x, h(r0 := ce)) ↔ path(x, h)), subst:[x, h, r0, ce] → [r ' + y + (h(r0)).nxt ', h, r0, ce], <all paths>

Additional information

Used terms: r ' + y + (h(r0)).nxt ', h, r0, ce

Proof step 32 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ ¬ r0 ∈ x → (path(x, h(r0 := ce)) ↔ path(x, h))

Lemma path-modify

Proof step 33 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), r0 ∈ r ' + y + (h(r0)).nxt ', (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s)

Proof step 34 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), r0 = r ∨ r0 = (h(r0)).nxt ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: r0 ≠ r

Additional information

Used formulas: r0 ≠ r

Proof step 35 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), r0 = r ∨ r0 = (h(r0)).nxt ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 36 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h(ce.nxt)).nxt, ce.nxt ≠ null, r ≠ ce.nxt, path(r ' + y + ce.nxt ', h(ce.nxt := ce)), reachable(r, ce.nxt, h), ce.nxt ∈ h, ¬ dups(y), ¬ r ∈ y, ¬ ce.nxt ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 10 terms: ce.nxt abort: F discard: T computed: F

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; Used terms: ce.nxt

Proof step 37 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, r0 = r, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h(r0 := ce)), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), r0 = r ∨ r0 = (h(r0)).nxt ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h;

Proof step 38 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h(r)).nxt, r ≠ null, path(r ' + y + (h(r)).nxt ', h(r := ce)), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(y), ¬ (h(r)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: y

Additional information

Used simplifier rules: List[A] freely generated by [], +;

Proof step 39 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, y = [], (h(r)).nxt ≠ null, r ≠ (h(r)).nxt, r ≠ null, path(r ' + y + (h(r)).nxt ', h(r := ce)), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(y), ¬ (h(r)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ (h(r)).nxt = r0; ⊦ ¬ a ∈ []; (s) ⊦ ¬ dups([]); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ st(a := d)(a) = d; (s) ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s) ⊦ a ∈ st(a := d); (s) ⊦ [] + x = x; (s)

Proof step 40 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h(ce.nxt)).nxt, ce.nxt = (h(r)).nxt, ce.nxt ≠ null, r ≠ ce.nxt, r ≠ null, ce.nxt ∈ h, r ∈ h, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: ce.nxt abort: F discard: T computed: F

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; Used terms: ce.nxt

Proof step 41 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, y = r0 + y0, (h(r)).nxt ≠ null, r ≠ (h(r)).nxt, r ≠ null, path(r ' + y + (h(r)).nxt ', h(r := ce)), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(y), ¬ (h(r)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s) ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s) ⊦ st(a := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s)

Proof step 42 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ ce.nxt, (h(r)).nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ (h(r)).nxt, path(ce.nxt ' + y0 + (h(r)).nxt ', h(r := ce)), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(y0), ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ ce.nxt ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: path-modify, seq: ⊦ ¬ r0 ∈ x → (path(x, h(r0 := ce)) ↔ path(x, h)), subst:[ce, r0, h, x] → [ce, r, h, ce.nxt ' + y0 + (h(r)).nxt '], <all paths>

Additional information

Used simplifier rules: ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce, r, h, ce.nxt ' + y0 + (h(r)).nxt '

Proof step 43 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ ¬ r0 ∈ x → (path(x, h(r0 := ce)) ↔ path(x, h))

Lemma path-modify

Proof step 44 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ ce.nxt, (h(r)).nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ (h(r)).nxt, path(ce.nxt ' + y0 + (h(r)).nxt ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(y0), ¬ r ∈ ce.nxt ' + y0 + (h(r)).nxt ', ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ ce.nxt ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s)

Proof step 45 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ ce.nxt, (h(r)).nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ (h(r)).nxt, path(ce.nxt ' + y0 + (h(r)).nxt ', h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(y0), ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ ce.nxt ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags:

Additional information

Proof step 46 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h((h(r)).nxt)).nxt, r ≠ null, r ≠ (h(r)).nxt, path((h((h(r)).nxt)).nxt ' + y0 + (h(r)).nxt ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(y0), ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ (h((h(r)).nxt)).nxt ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: (h(r)).nxt abort: F discard: T computed: F

Additional information

Used terms: (h(r)).nxt

Proof step 47 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ null, r ≠ (h((h(r)).nxt)).nxt, r ≠ null, r ≠ (h(r)).nxt, path((h((h(r)).nxt)).nxt ' + y0 + (h(r)).nxt ', h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(y0), ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ (h((h(r)).nxt)).nxt ∈ y0, (h(r)).nxt ∈ h → ¬ reachable((h((h(r)).nxt)).nxt, (h(r)).nxt, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 48 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ ce.nxt, (h(r)).nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ (h(r)).nxt, path(ce.nxt ' + y0 + (h(r)).nxt ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(y0), ¬ reachable(ce.nxt, (h(r)).nxt, h), ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ ce.nxt ∈ y0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-def, seq: ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0), subst:[h, r0, r] → [h, (h(r)).nxt, ce.nxt], <all paths>

Additional information

Used terms: h, (h(r)).nxt, ce.nxt

Proof step 49 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)

Lemma reachable-def

Proof step 50 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ ce.nxt, (h(r)).nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ (h(r)).nxt, path(ce.nxt ' + y0 + (h(r)).nxt ', h), r ∈ h, (h(r)).nxt ∈ h, ¬ dups(y0), ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ ce.nxt ∈ y0, ∀ x. ¬(path(ce.nxt ' + x, h) ∧ (ce.nxt ' + x).last = (h(r)).nxt) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 51 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r)).nxt)).nxt, (h(r)).nxt ≠ ce.nxt, (h(r)).nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ (h(r)).nxt, path(ce.nxt ' + y0 + (h(r)).nxt ', h), (h(r)).nxt ∈ h, r ∈ h, ¬ dups(y0), ¬ r ∈ y0, ¬ (h(r)).nxt ∈ y0, ¬ ce.nxt ∈ y0, ∀ x. ¬ path(ce.nxt ' + x, h) ∨ (ce.nxt ' + x).last ≠ (h(r)).nxt ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 14 terms: y0 + (h(r)).nxt '

Additional information

Used simplifier rules: ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: y0 + (h(r)).nxt '

Proof step 52 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ r ' + y + (h(r0)).nxt ', ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss)

Proof step 53 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h), reachable(r, r0, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-nodups-def, seq:⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x)), subst:[h, r0, r] → [h, r0, r], <all paths>

Additional information

Used terms: h, r0, r

Proof step 54 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))

Lemma reachable-nodups-def

Proof step 55 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), ∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s)

Proof step 56 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h((r ' + x).last)).nxt)).nxt, (r ' + x).last ≠ r, (r ' + x).last ≠ (h((r ' + x).last)).nxt, (h((r ' + x).last)).nxt ≠ null, r ≠ (h((r ' + x).last)).nxt, path(r ' + x, h), path(r ' + y + (h((r ' + x).last)).nxt ', h), (h((r ' + x).last)).nxt ∈ h, ¬ dups(y), ¬ dups(x), ¬ (r ' + x).last ∈ y, ¬ (h((r ' + x).last)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: polylist-last, name: lapp, seq: ⊦ y ≠ [] → (x + y).last = y.last, subst:[y, x] → [x, r ']; [A] → [Ref], <all paths>

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) Used terms: x, r '

Proof step 57 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof step 58 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(x.last)).nxt)).nxt, x.last ≠ r, x.last ≠ (h(x.last)).nxt, (h(x.last)).nxt ≠ null, r ≠ (h(x.last)).nxt, x ≠ [], path(r ' + y + (h(x.last)).nxt ', h), path(r ' + x, h), (h(x.last)).nxt ∈ h, ¬ dups(y), ¬ dups(x), ¬ x.last ∈ y, ¬ (h(x.last)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 59 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(x.last)).nxt)).nxt, x.last ≠ r, x.last ≠ (h(x.last)).nxt, (h(x.last)).nxt ≠ null, r ≠ (h(x.last)).nxt, x ≠ [], path(r ' + x, h), path(r ' + y + (h(x.last)).nxt ', h), (h(x.last)).nxt ∈ h, ¬ dups(y), ¬ dups(x), ¬ x.last ∈ y, ¬ (h(x.last)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: polylist-last, name:last-elim, seq:x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a ', subst:[x, a, y] → [x, r0, y0]; [A] → [Ref]

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x, a, y0

Proof step 60 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, x = y0 + r0 ', (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, x ≠ [], path(r ' + y + (h(r0)).nxt ', h), path(r ' + x, h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ dups(x), ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ x, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 61 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, path(r ' + y0 + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:linearity-path, seq:path(r ' + x, h), path(r ' + y, h) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0), subst:[r, x, h, y] → [r, y + (h(r0)).nxt ', h, y0 + r0 ']

Additional information

Used terms: r, y + (h(r0)).nxt ', h, y0 + r0 '

Proof step 62 from proof acyclic-rem-not-reachable (previous step) (back to tree)

path(r ' + x, h), path(r ' + y, h) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0)

Lemma linearity-path

Proof step 63 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h), path(r ' + y0 + r0 ', h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), y + (h(r0)).nxt ' = y0 + r0 ' ∨ (∃ r, y1. y + (h(r0)).nxt ' = (y0 + r0 ') + r ' + y1) ∨ (∃ r, x0. y0 + r0 ' = (y + (h(r0)).nxt ') + r ' + x0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist a: ⊦ ¬ dups(x + y) → ¬ dups(x) ∧ ¬ dups(y); (forward) from specification polylist-dup ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws) ⊦ x + a ' ≠ []; (s) ⊦ x = x + y ↔ y = []; (s) ⊦ x + y = x + z ↔ y = z; (s) ⊦ x + y = z + y ↔ x = z; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ x + a ' = y + a0 ' ↔ x = y ∧ a = a0; (ws) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification polylist inex: ⊦ x + a ' = y + a0 ' + z ∧ a ≠ a0 → (∃ z0. z = z0 + a '); (forward) from specification polylist in-16: ⊦ x + a ' = y + a0 ' + z ∧ a ≠ a0 → a ∈ z; (forward) from specification polylist in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification polylist ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ ¬ a0 ∈ y ∧ a0 ≠ a → y + a ' ≠ y0 + a0 ' + z; (s)

Proof step 64 from proof acyclic-rem-not-reachable (previous step) (back to tree)

r1 ' + x0 = y1 + r0 ', ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, path(r ' + (y + (h(r0)).nxt ' + y1) + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y + y1), ¬ dups(y), ¬ dups(y1), ¬ (h(r0)).nxt ∈ y1, ¬ r0 ∈ y1, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y1, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), r1 ∈ y1 ∨ r1 = r0, r0 = r1 ∨ r0 ∈ x0 ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: y1

Additional information

Used simplifier rules: List[A] freely generated by [], +;

Proof step 65 from proof acyclic-rem-not-reachable (previous step) (back to tree)

r1 ' + x0 = y1 + r0 ', ce.nxt = (h((h(r0)).nxt)).nxt, y1 = [], (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h), path(r ' + (y + (h(r0)).nxt ' + y1) + r0 ', h), (h(r0)).nxt ∈ h, ¬ dups(y + y1), ¬ dups(y), ¬ dups(y1), ¬ (h(r0)).nxt ∈ y1, ¬ r0 ∈ y1, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y1, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), r0 = r1 ∨ r0 ∈ x0, r1 ∈ y1 ∨ r1 = r0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ []; (s) ⊦ ¬ dups([]); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ x + [] = x; (s) ⊦ [] + x = x; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws)

Proof step 66 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ (h(r0)).nxt, r0 ≠ r, path(r ' + y + (h(r0)).nxt ' + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y), ¬ r ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r0 ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: path-split-last, seq:⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ (h((r ' + x).last)).nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null, subst:[h, r0, x, r] → [h, r0, y + (h(r0)).nxt ', r], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: h, r0, y + (h(r0)).nxt ', r

Proof step 67 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ (h((r ' + x).last)).nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null

Lemma path-split-last

Proof step 68 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(h((r ' + y + (h(r0)).nxt ').last)).nxt = r0, ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ null, path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ r ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r0 ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (a ' + x + b ').last = b; (s)

Proof step 69 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(h((h(ce.nxt)).nxt)).nxt = ce.nxt, ce.nxt ≠ null, ce.nxt ≠ r, ce.nxt ≠ (h(ce.nxt)).nxt, (h(ce.nxt)).nxt ≠ null, r ≠ (h(ce.nxt)).nxt, path(r ' + y + (h(ce.nxt)).nxt ', h), (h(ce.nxt)).nxt ∈ h, ce.nxt ∈ h, ¬ dups(y), ¬ ce.nxt ∈ y, ¬ (h(ce.nxt)).nxt ∈ y, ¬ r ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: ce.nxt abort: F discard: T computed: F

Additional information

Used terms: ce.nxt

Proof step 70 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(h((h(ce.nxt)).nxt)).nxt = ce.nxt, ce.nxt ≠ null, ce.nxt ≠ r, ce.nxt ≠ (h(ce.nxt)).nxt, (h(ce.nxt)).nxt ≠ null, r ≠ (h(ce.nxt)).nxt, path(r ' + y + (h(ce.nxt)).nxt ', h), ce.nxt ∈ h, (h(ce.nxt)).nxt ∈ h, ¬ dups(y), ¬ ce.nxt ∈ y, ¬ (h(ce.nxt)).nxt ∈ y, ¬ r ∈ y, ce.nxt ∈ h → ¬ reachable((h(ce.nxt)).nxt, ce.nxt, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 71 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(h((h(ce.nxt)).nxt)).nxt = ce.nxt, ce.nxt ≠ null, ce.nxt ≠ r, ce.nxt ≠ (h(ce.nxt)).nxt, (h(ce.nxt)).nxt ≠ null, r ≠ (h(ce.nxt)).nxt, path(r ' + y + (h(ce.nxt)).nxt ', h), (h(ce.nxt)).nxt ∈ h, ce.nxt ∈ h, ¬ dups(y), ¬ reachable((h(ce.nxt)).nxt, ce.nxt, h), ¬ ce.nxt ∈ y, ¬ (h(ce.nxt)).nxt ∈ y, ¬ r ∈ y ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-def, seq: ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0), subst:[h, r0, r] → [h, ce.nxt, (h(ce.nxt)).nxt], <all paths>

Additional information

Used terms: h, ce.nxt, (h(ce.nxt)).nxt

Proof step 72 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)

Lemma reachable-def

Proof step 73 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(h((h(ce.nxt)).nxt)).nxt = ce.nxt, ce.nxt ≠ null, ce.nxt ≠ r, ce.nxt ≠ (h(ce.nxt)).nxt, (h(ce.nxt)).nxt ≠ null, r ≠ (h(ce.nxt)).nxt, path(r ' + y + (h(ce.nxt)).nxt ', h), ce.nxt ∈ h, (h(ce.nxt)).nxt ∈ h, ¬ dups(y), ¬ ce.nxt ∈ y, ¬ (h(ce.nxt)).nxt ∈ y, ¬ r ∈ y, ∀ x. ¬(path((h(ce.nxt)).nxt ' + x, h) ∧ ((h(ce.nxt)).nxt ' + x).last = ce.nxt) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 74 from proof acyclic-rem-not-reachable (previous step) (back to tree)

(h((h(ce.nxt)).nxt)).nxt = ce.nxt, ce.nxt ≠ null, ce.nxt ≠ r, ce.nxt ≠ (h(ce.nxt)).nxt, (h(ce.nxt)).nxt ≠ null, r ≠ (h(ce.nxt)).nxt, path(r ' + y + (h(ce.nxt)).nxt ', h), (h(ce.nxt)).nxt ∈ h, ce.nxt ∈ h, ¬ dups(y), ¬ ce.nxt ∈ y, ¬ (h(ce.nxt)).nxt ∈ y, ¬ r ∈ y, ∀ x. ¬ path((h(ce.nxt)).nxt ' + x, h) ∨ ((h(ce.nxt)).nxt ' + x).last ≠ ce.nxt ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 14 terms: ce.nxt ' abort: F discard: T computed: F

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ (h(r)).nxt = r0; ⊦ (x + a ').last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce.nxt '

Proof step 75 from proof acyclic-rem-not-reachable (previous step) (back to tree)

r1 ' + x0 = y1 + r0 ', ce.nxt = (h((h(r0)).nxt)).nxt, y1 = r2 + y0, (h(r0)).nxt ≠ null, r ≠ (h(r0)).nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, path(r ' + y + (h(r0)).nxt ', h), path(r ' + (y + (h(r0)).nxt ' + y1) + r0 ', h), (h(r0)).nxt ∈ h, ¬ dups(y + y1), ¬ dups(y), ¬ dups(y1), ¬ (h(r0)).nxt ∈ y1, ¬ r0 ∈ y1, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y1, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), r0 = r1 ∨ r0 ∈ x0, r1 ∈ y1 ∨ r1 = r0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) a: ⊦ ¬ dups(x + y) → ¬ dups(x) ∧ ¬ dups(y); (forward) from specification polylist-dup in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification polylist ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a + x = a ' + x; (s) ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws)

Proof step 76 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r2, (h(r0)).nxt ≠ null, r ≠ r2, r ≠ (h(r0)).nxt, r0 ≠ r2, r0 ≠ (h(r0)).nxt, r0 ≠ r, path(r ' + y + (h(r0)).nxt ' + (r2 ' + y0) + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y0), ¬ dups(y), ¬ dups(y + y0), ¬ r ∈ y0, ¬ r ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r2 ∈ y0, ¬ r2 ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 23 terms: (h(r0)).nxt abort: F discard: F computed: F

Additional information

Used terms: (h(r0)).nxt

Proof step 77 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r2, (h(r0)).nxt ≠ null, r ≠ r2, r ≠ (h(r0)).nxt, r0 ≠ r2, r0 ≠ (h(r0)).nxt, r0 ≠ r, path(r ' + y + (h(r0)).nxt ', h), path(r ' + y + (h(r0)).nxt ' + (r2 ' + y0) + r0 ', h), (h(r0)).nxt ∈ h, ¬ dups(y0), ¬ dups(y), ¬ dups(y + y0), ¬ r ∈ y0, ¬ r ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r2 ∈ y0, ¬ r2 ∈ y, (h(r0)).nxt ∈ h → ¬ reachable((h((h(r0)).nxt)).nxt, (h(r0)).nxt, h), ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 78 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r2, (h(r0)).nxt ≠ null, r ≠ r2, r ≠ (h(r0)).nxt, r0 ≠ r2, r0 ≠ (h(r0)).nxt, r0 ≠ r, path(r ' + y + (h(r0)).nxt ' + (r2 ' + y0) + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y0), ¬ dups(y), ¬ dups(y + y0), ¬ reachable(ce.nxt, (h(r0)).nxt, h), ¬ r ∈ y0, ¬ r ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r2 ∈ y0, ¬ r2 ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: reachable-def, seq: ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0), subst:[h, r0, r] → [h, (h(r0)).nxt, ce.nxt], <all paths>

Additional information

Used terms: h, (h(r0)).nxt, ce.nxt

Proof step 79 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)

Lemma reachable-def

Proof step 80 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r2, (h(r0)).nxt ≠ null, r ≠ r2, r ≠ (h(r0)).nxt, r0 ≠ r2, r0 ≠ (h(r0)).nxt, r0 ≠ r, path(r ' + y + (h(r0)).nxt ', h), path(r ' + y + (h(r0)).nxt ' + (r2 ' + y0) + r0 ', h), (h(r0)).nxt ∈ h, ¬ dups(y0), ¬ dups(y), ¬ dups(y + y0), ¬ r ∈ y0, ¬ r ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r2 ∈ y0, ¬ r2 ∈ y, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), ∀ x. ¬(path(ce.nxt ' + x, h) ∧ (ce.nxt ' + x).last = (h(r0)).nxt) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: path((h(r0)).nxt ' + (r2 ' + y0) + r0 ', h)

Additional information

Used simplifier rules: path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); a: ⊦ (x + y) + z = x + y + z; from specification polylist Used formulas: path((h(r0)).nxt ' + (r2 ' + y0) + r0 ', h)

Proof step 81 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ r2, (h(r0)).nxt ≠ null, r ≠ r2, r ≠ (h(r0)).nxt, r0 ≠ r2, r0 ≠ (h(r0)).nxt, r0 ≠ r, path(r ' + y + (h(r0)).nxt ' + (r2 ' + y0) + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), path((h(r0)).nxt ' + (r2 ' + y0) + r0 ', h), (h(r0)).nxt ∈ h, ¬ dups(y0), ¬ dups(y), ¬ dups(y + y0), ¬ r ∈ y0, ¬ r ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r2 ∈ y0, ¬ r2 ∈ y, ∀ x. ¬(path(ce.nxt ' + x, h) ∧ (ce.nxt ' + x).last = (h(r0)).nxt), ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification polylist

Proof step 82 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ ce.nxt, r ≠ (h(r0)).nxt, r ≠ ce.nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, r0 ≠ ce.nxt, path(ce.nxt ' + y0 + r0 ', h), path(r ' + y + (h(r0)).nxt ' + (ce.nxt ' + y0) + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y + y0), ¬ dups(y), ¬ dups(y0), ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y0, ∀ x. ¬ path(ce.nxt ' + x, h) ∨ (ce.nxt ' + x).last ≠ (h(r0)).nxt, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 24 terms: y0 + (r0 ' + (h(r0)).nxt) abort: F discard: T computed: F

Additional information

Used terms: y0 + (r0 ' + (h(r0)).nxt)

Proof step 83 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ ce.nxt, r ≠ (h(r0)).nxt, r ≠ ce.nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, r0 ≠ ce.nxt, path(r ' + y + (h(r0)).nxt ', h), path(r ' + y + (h(r0)).nxt ' + (ce.nxt ' + y0) + r0 ', h), path(ce.nxt ' + y0 + r0 ', h), (h(r0)).nxt ∈ h, ¬ dups(y + y0), ¬ dups(y), ¬ dups(y0), ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h), ¬ path(ce.nxt ' + y0 + (r0 ' + (h(r0)).nxt), h) ∨ (ce.nxt ' + y0 + (r0 ' + (h(r0)).nxt)).last ≠ (h(r0)).nxt ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (a ' + x + b ').last = b; (s) ⊦ x + a = x + a '; (s)

Proof step 84 from proof acyclic-rem-not-reachable (previous step) (back to tree)

ce.nxt = (h((h(r0)).nxt)).nxt, (h(r0)).nxt ≠ null, (h(r0)).nxt ≠ ce.nxt, r ≠ (h(r0)).nxt, r ≠ ce.nxt, r0 ≠ r, r0 ≠ (h(r0)).nxt, r0 ≠ ce.nxt, path(ce.nxt ' + y0 + r0 ', h), path(r ' + y + (h(r0)).nxt ' + (ce.nxt ' + y0) + r0 ', h), path(r ' + y + (h(r0)).nxt ', h), (h(r0)).nxt ∈ h, ¬ dups(y + y0), ¬ dups(y), ¬ dups(y0), ¬ path(ce.nxt ' + y0 + r0 ' + (h(r0)).nxt ', h), ¬ ce.nxt ∈ y, ¬ ce.nxt ∈ y0, ¬ (h(r0)).nxt ∈ y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ (h(r0)).nxt ∈ y, ¬ r ∈ y, ¬ r ∈ y0, ∀ r. r ∈ h → ¬ reachable((h(r)).nxt, r, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: path-split-last, seq:⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ (h((r ' + x).last)).nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null, subst:[r, x, r0, h] → [ce.nxt, y0 + r0 ', (h(r0)).nxt, h], <all paths>

Additional information

Used simplifier rules: path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce.nxt, y0 + r0 ', (h(r0)).nxt, h

Proof step 85 from proof acyclic-rem-not-reachable (previous step) (back to tree)

⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ (h((r ' + x).last)).nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null

Lemma path-split-last

Proof Tree for acyclic-rem-not-reachable (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree