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

⊦ reachable(r, r0, h) ∧ ce.nxt = (h((h(r1)).nxt)).nxt ∧ r0 ≠ (h(r1)).nxt ∧ (h(r1)).nxt ∈ h ∧ (h(r1)).nxt ≠ null ∧ r1 ≠ null ∧ r1 ∈ h → (reachable(r, r0, h(r1 := ce)) ↔ true)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

ce.nxt = (h((h(r1)).nxt)).nxt, (h(r1)).nxt ≠ null, ¬(reachable(r, r0, h(r1 := ce)) ↔ true), r0 ≠ (h(r1)).nxt, r1 ≠ null, reachable(r, r0, h), (h(r1)).nxt ∈ h, r1 ∈ h ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

ce.nxt = (h((h(r1)).nxt)).nxt, (h(r1)).nxt ≠ null, r0 ≠ (h(r1)).nxt, r1 ≠ null, reachable(r, r0, h), r1 ∈ h, (h(r1)).nxt ∈ h, ¬ reachable(r, r0, h(r1 := ce)) ⊦

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, r0, r], <all paths>

Additional information

Used terms: h, r0, r

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

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

Lemma reachable-def

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

ce.nxt = (h((h(r1)).nxt)).nxt, (r ' + x).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r1 ≠ null, path(r ' + x, h), r1 ∈ h, (h(r1)).nxt ∈ h, ¬ reachable(r, (r ' + x).last, h(r1 := ce)) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

indvar: x

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; List[A] freely generated by [], +; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws)

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

ce.nxt = (h((h(r1)).nxt)).nxt, (r ' + (r0 + x0)).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r1 ≠ null, path(r ' + (r0 + x0), h), (h(r1)).nxt ∈ h, r1 ∈ h, ¬ reachable(r, (r ' + (r0 + x0)).last, h(r1 := ce)), ∀ ce, h, r1, r. ¬ ( ce.nxt = (h((h(r1)).nxt)).nxt ∧ (r ' + x0).last ≠ (h(r1)).nxt ∧ (h(r1)).nxt ≠ null ∧ r1 ≠ null ∧ path(r ' + x0, h) ∧ r1 ∈ h ∧ (h(r1)).nxt ∈ h ∧ ¬ reachable(r, (r ' + x0).last, h(r1 := ce))) ⊦

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 ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r)).nxt ' + x0, h), r ∈ h, (h(r1)).nxt ∈ h, r1 ∈ h, ¬ reachable(r, ((h(r)).nxt ' + x0).last, h(r1 := ce)), ∀ ce0, h0, r2, r3. ce0.nxt ≠ (h0((h0(r2)).nxt)).nxt ∨ (r3 ' + x0).last = (h0(r2)).nxt ∨ (h0(r2)).nxt = null ∨ r2 = null ∨ ¬ path(r3 ' + x0, h0) ∨ ¬ r2 ∈ h0 ∨ ¬ (h0(r2)).nxt ∈ h0 ∨ reachable(r3, (r3 ' + x0).last, h0(r2 := ce0)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: ce, h, r1, (h(r)).nxt abort: F discard: T computed: T

Additional information

Used terms: ce, h, r1, (h(r)).nxt

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r)).nxt ' + x0, h), r1 ∈ h, (h(r1)).nxt ∈ h, r ∈ h, ¬ reachable(r, ((h(r)).nxt ' + x0).last, h(r1 := ce)), ce.nxt ≠ (h((h(r1)).nxt)).nxt ∨ ((h(r)).nxt ' + x0).last = (h(r1)).nxt ∨ (h(r1)).nxt = null ∨ r1 = null ∨ ¬ path((h(r)).nxt ' + x0, h) ∨ ¬ r1 ∈ h ∨ ¬ (h(r1)).nxt ∈ h ∨ reachable((h(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r)).nxt ' + x0, h), reachable((h(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)), r ∈ h, (h(r1)).nxt ∈ h, r1 ∈ h, ¬ reachable(r, ((h(r)).nxt ' + x0).last, h(r1 := ce)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: fromnext-reachable-rev, seq:⊦ reachable((h(r)).nxt, r0, h) ∧ r ≠ r0 ∧ r ≠ null ∧ r ∈ h → (reachable(r, r0, h) ↔ true), subst:[h, r0, r] → [h(r1 := ce), ((h(r)).nxt ' + x0).last, r], <all paths>

Additional information

Used terms: h(r1 := ce), ((h(r)).nxt ' + x0).last, r

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

⊦ reachable((h(r)).nxt, r0, h) ∧ r ≠ r0 ∧ r ≠ null ∧ r ∈ h → (reachable(r, r0, h) ↔ true)

Lemma fromnext-reachable-rev

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r)).nxt ' + x0, h), reachable((h(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)), r1 ∈ h, (h(r1)).nxt ∈ h, r ∈ h, ¬ reachable(r, ((h(r)).nxt ' + x0).last, h(r1 := ce)) ⊦ reachable((h(r1 := ce)(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)) ∧ r ≠ ((h(r)).nxt ' + x0).last ∧ r ≠ null ∧ r ∈ h(r1 := ce)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws)

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r)).nxt ' + x0, h), reachable((h(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)), r1 ∈ h, (h(r1)).nxt ∈ h, r ∈ h, r ∈ h(r1 := ce), ¬ reachable(r, ((h(r)).nxt ' + x0).last, h(r1 := ce)) ⊦ r ≠ ((h(r)).nxt ' + x0).last ∧ reachable((h(r1 := ce)(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

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

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r)).nxt ' + x0, h), reachable((h(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)), r ∈ h(r1 := ce), r ∈ h, (h(r1)).nxt ∈ h, r1 ∈ h, ¬ reachable((h(r1 := ce)(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)), ¬ reachable(r, ((h(r)).nxt ' + x0).last, h(r1 := ce)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: polystore, name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [r, ce, r1, h]; [A, B] → [Ref, Cell], <all paths>

Additional information

Used terms: r, ce, r1, h

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

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

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

ce.nxt = (h((h(r1)).nxt)).nxt, r = r1, ((h(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r)).nxt ' + x0, h), reachable((h(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)), r1 ∈ h, (h(r1)).nxt ∈ h, r ∈ h, r ∈ h(r1 := ce), ¬ reachable((h(r1 := ce)(r)).nxt, ((h(r)).nxt ' + x0).last, h(r1 := ce)), ¬ reachable(r, ((h(r)).nxt ' + x0).last, h(r1 := ce)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ st(a := d)(a) = d; (s)

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r1)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r1 ≠ null, path((h(r1)).nxt ' + x0, h), reachable((h(r1)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), r1 ∈ h(r1 := ce), (h(r1)).nxt ∈ h, r1 ∈ h, ¬ reachable(r1, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), ¬ reachable(ce.nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fromnext-reachable, seq: ⊦ reachable(r, r0, h) ∧ r ≠ r0 → (reachable((h(r)).nxt, r0, h) ↔ true), subst:[r, r0, h] → [(h(r1)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)]

Additional information

Used terms: (h(r1)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)

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

⊦ reachable(r, r0, h) ∧ r ≠ r0 → (reachable((h(r)).nxt, r0, h) ↔ true)

Lemma fromnext-reachable

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r1)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r1 ≠ null, path((h(r1)).nxt ' + x0, h), reachable((h(r1)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), r1 ∈ h, (h(r1)).nxt ∈ h, r1 ∈ h(r1 := ce), ¬ reachable(r1, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), ¬ reachable(ce.nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), reachable((h(r1)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)) ∧ (h(r1)).nxt ≠ ((h(r1)).nxt ' + x0).last → (reachable((h(r1 := ce)((h(r1)).nxt)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)) ↔ true) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r1)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r1 ≠ null, path((h(r1)).nxt ' + x0, h), reachable((h(r1)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), reachable((h(r1 := ce)((h(r1)).nxt)).nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), r1 ∈ h(r1 := ce), (h(r1)).nxt ∈ h, r1 ∈ h, ¬ reachable(r1, ((h(r1)).nxt ' + x0).last, h(r1 := ce)), ¬ reachable(ce.nxt, ((h(r1)).nxt ' + x0).last, h(r1 := ce)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: polystore, name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [(h(r1)).nxt, ce, r1, h]; [A, B] → [Ref, Cell], <all paths>

Additional information

Used terms: (h(r1)).nxt, ce, r1, h

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

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

Proof Tree for rem-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