Proof step 1 from proof reachable-split-last (conclusion) (back to tree)

⊦ r ≠ r0 → (reachable(r, r0, h) ↔ (∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null))

System applied rule normalize

And obtained the following new goals

Proof step 2:
r ≠ r0 ⊦ reachable(r, r0, h) ↔ (∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null)

Rule argument

Additional information

Proof step 2 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0 ⊦ reachable(r, r0, h) ↔ (∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
r ≠ r0 ⊦ reachable(r, r0, h) ↔ (∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null)

Rule argument

Additional information

Proof step 3 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0 ⊦ reachable(r, r0, h) ↔ (∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null)

User applied rule case distinction

And obtained the following new goals

Proof step 4:
r ≠ r0, reachable(r, r0, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Rule argument

pos: right 1

Additional information

Used simplifier rules: reach-reachnext: ⊦ reachable(r, r1, h) → (reachable(r, h[r1].nxt, h) ↔ ¬(h[r1].nxt = null ∨ ¬ h[r1].nxt ∈ h));

Proof step 4 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0, reachable(r, r0, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
r ≠ r0, reachable(r, r0, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Rule argument

Additional information

Proof step 5 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0, reachable(r, r0, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)
Proof step 7:
r ≠ r0, ∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

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, r0, r], <all paths>

Additional information

Used terms: h, r0, r

Proof step 6 from proof reachable-split-last (previous step) (back to tree)

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

Lemma reachable-def

Proof step 7 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0, ∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
r ≠ (r ' + x).last, path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = (r ' + x).last ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Rule argument

Additional information

Proof step 8 from proof reachable-split-last (previous step) (back to tree)

r ≠ (r ' + x).last, path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = (r ' + x).last ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ y ≠ [] → (x + y).last = y.last
Proof step 10:
r ≠ x.last, x ≠ [], path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = x.last ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

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 9 from proof reachable-split-last (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof step 10 from proof reachable-split-last (previous step) (back to tree)

r ≠ x.last, x ≠ [], path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = x.last ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
r ≠ x.last, x ≠ [], path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = x.last ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Rule argument

Additional information

Proof step 11 from proof reachable-split-last (previous step) (back to tree)

r ≠ x.last, x ≠ [], path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = x.last ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

User applied rule apply elim lemma

And obtained the following new goals

Proof step 12:
x = y + r0 ', r ≠ r0, x ≠ [], path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

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, 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 reachable-split-last (previous step) (back to tree)

x = y + r0 ', r ≠ r0, x ≠ [], path(r ' + x, h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
r ≠ r0, path(r ' + y + r0 ', h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ x + a ' ≠ []; (s)

Proof step 13 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0, path(r ' + y + r0 ', h) ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = r0 ∧ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

User applied rule exists right

And obtained the following new goals

Proof step 14:
r ≠ r0, path(r ' + y + r0 ', h) ⊦ reachable(r, (r ' + y).last, h) ∧ h[(r ' + y).last].nxt = r0 ∧ h[(r ' + y).last].nxt ∈ h ∧ h[(r ' + y).last].nxt ≠ null

Rule argument

pos: right 1 terms: (r ' + y).last abort: F discard: T computed: F

Additional information

Used terms: (r ' + y).last

Proof step 14 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0, path(r ' + y + r0 ', h) ⊦ reachable(r, (r ' + y).last, h) ∧ h[(r ' + y).last].nxt = r0 ∧ h[(r ' + y).last].nxt ∈ h ∧ h[(r ' + y).last].nxt ≠ null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
r ≠ r0, path(r ' + y + r0 ', h) ⊦ reachable(r, (r ' + y).last, h) ∧ h[(r ' + y).last].nxt ∈ h ∧ h[(r ' + y).last].nxt ≠ null

Rule argument

Additional information

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

Proof step 15 from proof reachable-split-last (previous step) (back to tree)

r ≠ r0, path(r ' + y + r0 ', h) ⊦ reachable(r, (r ' + y).last, h) ∧ h[(r ' + y).last].nxt ∈ h ∧ h[(r ' + y).last].nxt ≠ null

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 16:
⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ h[(r ' + x).last].nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null
Proof step 17:
h[(r ' + y).last].nxt = r0, r ≠ r0, r0 ≠ null, path(r ' + y, h), r0 ∈ h ⊦ reachable(r, (r ' + y).last, h) ∧ h[(r ' + y).last].nxt ∈ h ∧ h[(r ' + y).last].nxt ≠ null

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, r], <all paths>

Additional information

Used terms: h, r0, y, r

Proof step 16 from proof reachable-split-last (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 17 from proof reachable-split-last (previous step) (back to tree)

h[(r ' + y).last].nxt = r0, r ≠ r0, r0 ≠ null, path(r ' + y, h), r0 ∈ h ⊦ reachable(r, (r ' + y).last, h) ∧ h[(r ' + y).last].nxt ∈ h ∧ h[(r ' + y).last].nxt ≠ null

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
h[(r ' + y).last].nxt ≠ null, r ≠ h[(r ' + y).last].nxt, path(r ' + y, h), h[(r ' + y).last].nxt ∈ h, ¬ reachable(r, (r ' + y).last, h) ⊦

Rule argument

Additional information

Proof step 18 from proof reachable-split-last (previous step) (back to tree)

h[(r ' + y).last].nxt ≠ null, r ≠ h[(r ' + y).last].nxt, path(r ' + y, h), h[(r ' + y).last].nxt ∈ h, ¬ reachable(r, (r ' + y).last, h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 19:
⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)
Proof step 20:
h[(r ' + y).last].nxt ≠ null, r ≠ h[(r ' + y).last].nxt, path(r ' + y, h), h[(r ' + y).last].nxt ∈ h, ∀ x. ¬(path(r ' + x, h) ∧ (r ' + x).last = (r ' + y).last) ⊦

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, (r ' + y).last, r], <all paths>

Additional information

Used terms: h, (r ' + y).last, r

Proof step 19 from proof reachable-split-last (previous step) (back to tree)

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

Lemma reachable-def

Proof step 20 from proof reachable-split-last (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
h[(r ' + y).last].nxt ≠ null, r ≠ h[(r ' + y).last].nxt, path(r ' + y, h), h[(r ' + y).last].nxt ∈ h, ∀ x. ¬ path(r ' + x, h) ∨ (r ' + x).last ≠ (r ' + y).last ⊦

Rule argument

Additional information

Proof step 21 from proof reachable-split-last (previous step) (back to tree)

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: y

Additional information

Used terms: y

Proof Tree for reachable-split-last (View conclusion)

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