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Proof step 1 from proof removed-stays-reachable (conclusion) (back to tree)

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof removed-stays-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 ⊦ reachable(r, h[r0].nxt, h[r0, ce]) ↔ h[h[r0].nxt].nxt = h[r0].nxt ∨ r ≠ r0 ∧ r = h[r0].nxt ∨ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h) ∧ ¬ reachable(h[h[r0].nxt].nxt, r0, h) ∨ (∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h;

Proof step 3 from proof removed-stays-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, r0 ∈ h ⊦ reachable(r, h[r0].nxt, h[r0, ce]) ↔ ce.nxt = h[r0].nxt ∨ h[r0].nxt ≠ r0 ∧ r = h[r0].nxt ∨ reachable(ce.nxt, h[r0].nxt, h) ∧ ¬ reachable(ce.nxt, r0, h) ∨ (∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:rem-stays-reachable-aux, seq:reachable(r, r0, h), h[r0].nxt ≠ null, h[r0].nxt ∈ h, h[r0].nxt ≠ h[h[r0].nxt].nxt, ce.nxt = h[h[r0].nxt].nxt ⊦ reachable(r, h[r0].nxt, h[r0, ce]) ↔ h[h[r0].nxt].nxt = h[r0].nxt ∨ r ≠ r0 ∧ r = h[r0].nxt ∨ (∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) ∨ (∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x), subst:[ce, r0, h, r] → [ce, r0, h, r], <all paths>

Additional information

Used terms: ce, r0, h, r

Proof step 4 from proof removed-stays-reachable (previous step) (back to tree)

reachable(r, r0, h), h[r0].nxt ≠ null, h[r0].nxt ∈ h, h[r0].nxt ≠ h[h[r0].nxt].nxt, ce.nxt = h[h[r0].nxt].nxt ⊦ reachable(r, h[r0].nxt, h[r0, ce]) ↔ h[h[r0].nxt].nxt = h[r0].nxt ∨ r ≠ r0 ∧ r = h[r0].nxt ∨ (∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) ∨ (∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x)

Lemma rem-stays-reachable-aux

Proof step 5 from proof removed-stays-reachable (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof removed-stays-reachable (previous step) (back to tree)

ce.nxt = h[ce.nxt].nxt, ce.nxt = h[r0].nxt, ce.nxt ≠ null, reachable(r, r0, h), r0 ∈ h, ce.nxt ∈ h, ¬ reachable(r, ce.nxt, h[r0, ce]) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: reachable(r, r0, h[r0, ce])

Additional information

Used simplifier rules: changenonnxt-reachable: ⊦ reachable(r, r0, h) ∧ ce.nxt = h[r1].nxt → (reachable(r, r0, h[r1, ce]) ↔ true); Used formulas: reachable(r, r0, h[r0, ce])

Proof step 7 from proof removed-stays-reachable (previous step) (back to tree)

ce.nxt = h[r0].nxt, ce.nxt = h[ce.nxt].nxt, ce.nxt ≠ null, reachable(r, r0, h), reachable(r, r0, h[r0, ce]), ce.nxt ∈ h, r0 ∈ h, ¬ reachable(r, ce.nxt, h[r0, ce]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h; ⊦ a ∈ st[a, d]; (s)

Proof step 8 from proof removed-stays-reachable (previous step) (back to tree)

ce.nxt = h[r0].nxt, ce.nxt = h[ce.nxt].nxt, ce.nxt ≠ null, reachable(r, r0, h), reachable(r, r0, h[r0, ce]), ce.nxt ∈ h, r0 ∈ h, ¬ reachable(r, ce.nxt, h[r0, ce]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:reach-reachnext, seq:⊦ reachable(r, r1, h) → (reachable(r, h[r1].nxt, h) ↔ ¬(h[r1].nxt = null ∨ ¬ h[r1].nxt ∈ h)), subst:[r, r1, h] → [r, r0, h[r0, ce]]

Additional information

Used simplifier rules: changenonnxt-reachable: ⊦ reachable(r, r0, h) ∧ ce.nxt = h[r1].nxt → (reachable(r, r0, h[r1, ce]) ↔ true); endnotinH-notreachable: ⊦ ¬ r0 ∈ h → (reachable(r, r0, h) ↔ false); ⊦ st[a, d][a] = d; (s) ⊦ a ∈ st[b, d] ↔ a = b ∨ a ∈ st; (ws) Used terms: r, r0, h[r0, ce]

Proof step 9 from proof removed-stays-reachable (previous step) (back to tree)

⊦ reachable(r, r1, h) → (reachable(r, h[r1].nxt, h) ↔ ¬(h[r1].nxt = null ∨ ¬ h[r1].nxt ∈ h))

Lemma reach-reachnext

Proof step 10 from proof removed-stays-reachable (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof removed-stays-reachable (previous step) (back to tree)

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:rem-stays-reachable-aux1, seq:r0 ∈ h, h[r0].nxt ≠ null, h[r0].nxt ∈ h, h[r0].nxt ≠ h[h[r0].nxt].nxt ⊦ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h) ∧ ¬ reachable(h[h[r0].nxt].nxt, r0, h) ↔ (∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x), subst:[r0, h] → [r0, h]

Additional information

Used terms: r0, h

Proof step 12 from proof removed-stays-reachable (previous step) (back to tree)

r0 ∈ h, h[r0].nxt ≠ null, h[r0].nxt ∈ h, h[r0].nxt ≠ h[h[r0].nxt].nxt ⊦ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h) ∧ ¬ reachable(h[h[r0].nxt].nxt, r0, h) ↔ (∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x)

Lemma rem-stays-reachable-aux1

Proof step 13 from proof removed-stays-reachable (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 14 from proof removed-stays-reachable (previous step) (back to tree)

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:rem-stays-reachable-aux2, seq:reachable(r, r0, h), h[r0].nxt ≠ null, h[r0].nxt ∈ h ⊦ (∃ r1. reachable(r, r1, h) ∧ h[r0].nxt = h[r1].nxt ∧ ¬ reachable(r0, r1, h)) ↔ (∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x), subst:[r, r0, h] → [r, r0, h]

Additional information

Used terms: r, r0, h

Proof step 15 from proof removed-stays-reachable (previous step) (back to tree)

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

Lemma rem-stays-reachable-aux2

Proof step 16 from proof removed-stays-reachable (previous step) (back to tree)

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 17 from proof removed-stays-reachable (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 18 from proof removed-stays-reachable (previous step) (back to tree)

ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, reachable(r, r0, h), h[r0].nxt ∈ h, r0 ∈ h, (∃ r1. reachable(r, r1, h) ∧ h[r0].nxt = h[r1].nxt ∧ ¬ reachable(r0, r1, h)) ↔ (∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x), h[r0].nxt = r0 ∨ r ≠ h[r0].nxt, path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x ∨ path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0 ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h), ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, reachable(ce.nxt, h[r0].nxt, h) ∧ ¬ reachable(ce.nxt, r0, h)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 7

Additional information

Proof step 19 from proof removed-stays-reachable (previous step) (back to tree)

ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, reachable(r, r0, h), r0 ∈ h, h[r0].nxt ∈ h, path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x ∨ path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, h[r0].nxt = r0 ∨ r ≠ h[r0].nxt ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h), ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x, ∃ r1. reachable(r, r1, h) ∧ h[r0].nxt = h[r1].nxt ∧ ¬ reachable(r0, r1, h), reachable(ce.nxt, h[r0].nxt, h) ∧ ¬ reachable(ce.nxt, r0, h)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 7

Additional information

Proof step 20 from proof removed-stays-reachable (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 21 from proof removed-stays-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, r0 ≠ r, path(r ' + x + h[r0].nxt ', h), reachable(r, r0, h), r0 ∈ h, h[r0].nxt ∈ h, ¬ r0 ∈ x ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h), ∃ r1. reachable(r, r1, h) ∧ h[r0].nxt = h[r1].nxt ∧ ¬ reachable(r0, r1, h), ∃ x0. path(r ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, reachable(ce.nxt, h[r0].nxt, h) ∧ ¬ reachable(ce.nxt, r0, h)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 4 terms: x

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: x

Proof step 22 from proof removed-stays-reachable (previous step) (back to tree)

ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, path(h[r0].nxt ' + x0 + h[r0].nxt ', h), reachable(r, r0, h), h[r0].nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x0, h[r0].nxt = r0 ∨ r ≠ h[r0].nxt ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r0].nxt = h[r1].nxt ∧ ¬ reachable(r0, r1, h), ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x, ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h), ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, reachable(ce.nxt, h[r0].nxt, h) ∧ ¬ reachable(ce.nxt, r0, h)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 4 terms: x0

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: x0

Proof step 23 from proof removed-stays-reachable (previous step) (back to tree)

ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, reachable(r, r0, h), r0 ∈ h, h[r0].nxt ∈ h, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x, ∃ r1. reachable(r, r1, h) ∧ h[r0].nxt = h[r1].nxt ∧ ¬ reachable(r0, r1, h), path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x ∨ path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, h[r0].nxt = r0 ∨ r ≠ h[r0].nxt ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h), reachable(ce.nxt, h[r0].nxt, h) ∧ ¬ reachable(ce.nxt, r0, h)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h;

Proof step 24 from proof removed-stays-reachable (previous step) (back to tree)

h[r0].nxt = h[r1].nxt, ce.nxt = h[h[r1].nxt].nxt, h[r1].nxt ≠ null, h[r1].nxt ≠ ce.nxt, r ≠ h[r1].nxt, r0 ≠ r, path(r ' + x1 + h[r1].nxt ', h), reachable(r, r1, h), reachable(r, r0, h), h[r1].nxt ∈ h, r0 ∈ h, r1 ∈ h, ¬ reachable(r0, r1, h), ¬ r0 ∈ x1, path(r ' + x + h[r1].nxt ', h) ∧ ¬ r0 ∈ x ∨ path(h[r1].nxt ' + x0 + h[r1].nxt ', h) ∧ ¬ r0 ∈ x0 ⊦ ∃ r1. reachable(r, r1, h) ∧ h[r1].nxt = h[r0].nxt ∧ ¬ reachable(r0, r1, h), ∃ x. path(h[r1].nxt ' + x + h[r1].nxt ', h) ∧ ¬ r0 ∈ x, reachable(ce.nxt, h[r1].nxt, h) ∧ ¬ reachable(ce.nxt, r0, h)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: r1

Additional information

Used terms: r1

Proof step 25 from proof removed-stays-reachable (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h;

Proof step 26 from proof removed-stays-reachable (previous step) (back to tree)

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

User applied rule exists right

And closed the goal!

Rule argument

pos: right 2 terms: r1 abort: F discard: T computed: T

Additional information

Used terms: r1

Proof Tree for removed-stays-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