[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 1 from proof add-preserves-reachable-rev (conclusion) (back to tree)

⊦ (h(r1)).nxt = (h(r2)).nxt ∧ r0 ≠ r2 ∧ ce.nxt = r2 ∧ ¬ reachable(r, r0, h) ∧ ¬ reachable(r, r2, h) ∧ r2 ≠ null ∧ r2 ∈ h ∧ reachable(r, r1, h) → (reachable(r, r0, h(r1 := ce)) ↔ false)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(r2)).nxt, ce.nxt = r2, ¬(reachable(r, r0, h(r1 := ce)) ↔ false), r0 ≠ r2, r2 ≠ null, reachable(r, r1, h), r2 ∈ h, ¬ reachable(r, r2, h), ¬ reachable(r, 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 3 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, ce.nxt ≠ null, r0 ≠ ce.nxt, reachable(r, r1, h), reachable(r, r0, h(r1 := ce)), ce.nxt ∈ h, r1 ∈ h, r0 ∈ h(r1 := ce), ¬ reachable(r, ce.nxt, h), ¬ reachable(r, r0, 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(r1 := ce), r0, r], <all paths>

Additional information

Used terms: h(r1 := ce), r0, r

Proof step 4 from proof add-preserves-reachable-rev (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 add-preserves-reachable-rev (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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; Reflist freely generated by [], +; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s) ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws)

Proof step 7 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, (r ' + (r0 + x0)).last ≠ ce.nxt, ce.nxt ≠ null, path(r ' + (r0 + x0), h(r1 := ce)), reachable(r, r1, h), (r ' + (r0 + x0)).last ∈ h(r1 := ce), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r, ce.nxt, h), ¬ reachable(r, (r ' + (r0 + x0)).last, h), ∀ h, r1, ce, r. ¬ ( (h(r1)).nxt = (h(ce.nxt)).nxt ∧ (r ' + x0).last ≠ ce.nxt ∧ ce.nxt ≠ null ∧ path(r ' + x0, h(r1 := ce)) ∧ reachable(r, r1, h) ∧ ce.nxt ∈ h ∧ r1 ∈ h ∧ (r ' + x0).last ∈ h(r1 := ce) ∧ ¬ reachable(r, ce.nxt, h) ∧ ¬ reachable(r, (r ' + x0).last, 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 list[reflist] ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

Proof step 8 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 13 terms: h, r1, ce, (h(r1 := ce)(r)).nxt abort: F discard: T computed: T

Additional information

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

Proof step 9 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: store[linheap], name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [r, ce, r1, h], <all paths>

Additional information

Used terms: r, ce, r1, h

Proof step 11 from proof add-preserves-reachable-rev (previous step) (back to tree)

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

Proof step 12 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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; ⊦ st(a := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s)

Proof step 13 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∃ r. reachable(r1, (h(r)).nxt, h) ∧ path(r ' + x0, h(r1 := ce)) ∧ ¬ reachable(r1, (r ' + x0).last, h)

Additional information

Used formulas: ∃ r. reachable(r1, (h(r)).nxt, h) ∧ path(r ' + x0, h(r1 := ce)) ∧ ¬ reachable(r1, (r ' + x0).last, h)

Proof step 14 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, (ce.nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r1 ≠ null, path(ce.nxt ' + x0, h(r1 := ce)), ce.nxt ∈ h, r1 ∈ h, (ce.nxt ' + x0).last ∈ h(r1 := ce), ¬ reachable(r1, ce.nxt, h), ¬ reachable(r1, (ce.nxt ' + x0).last, h), ∃ r. reachable(r1, (h(r)).nxt, h) ∧ path(r ' + x0, h(r1 := ce)) ∧ ¬ reachable(r1, (r ' + x0).last, 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 15 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 6, left 8, left 11

Additional information

Proof step 16 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule structural induction

And obtained the following new goals

Rule argument

indvar: x0

Additional information

Used simplifier rules: Reflist freely generated by [], +; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s)

Proof step 17 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, (ce.nxt ' + (r0 + x)).last ≠ ce.nxt, ce.nxt ≠ null, r1 ≠ null, path(r ' + (r0 + x), h(r1 := ce)), reachable(r1, (h(r)).nxt, h), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, ce.nxt, h), ¬ reachable(r1, (ce.nxt ' + (r0 + x)).last, h), ¬ reachable(r1, (r ' + (r0 + x)).last, h), ∀ h, r1, ce, r. ¬ ( (h(r1)).nxt = (h(ce.nxt)).nxt ∧ (ce.nxt ' + x).last ≠ ce.nxt ∧ ce.nxt ≠ null ∧ r1 ≠ null ∧ path(r ' + x, h(r1 := ce)) ∧ reachable(r1, (h(r)).nxt, h) ∧ ce.nxt ∈ h ∧ r1 ∈ h ∧ ¬ reachable(r1, ce.nxt, h) ∧ ¬ reachable(r1, (ce.nxt ' + x).last, h) ∧ ¬ reachable(r1, (r ' + x).last, 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 list[reflist] ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

Proof step 18 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: store[linheap], name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [r, ce, r1, h], <all paths>

Additional information

Used terms: r, ce, r1, h

Proof step 19 from proof add-preserves-reachable-rev (previous step) (back to tree)

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

Proof step 20 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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; ⊦ st(a := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s)

Proof step 21 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: h, r1, ce, ce.nxt abort: F discard: T computed: T

Additional information

Used simplifier rules: reachnext: ⊦ reachable(r0, (h(r0)).nxt, h) ↔ r0 ≠ null ∧ r0 ∈ h ∧ (h(r0)).nxt ∈ h ∧ (h(r0)).nxt ≠ null; Used terms: h, r1, ce, ce.nxt

Proof step 22 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s)

Proof step 23 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: x = []

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s) Used formulas: x = []

Proof step 24 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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 list[reflist] ⊦ x ≠ [] → (a ' + x).last = x.last; (s)

Proof step 25 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 15 terms: h, r1, ce, (h(r)).nxt abort: F discard: T computed: T

Additional information

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

Proof step 26 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

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 27 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x

Additional information

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

Proof step 28 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, x = r0 + x0, x.last ≠ ce.nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, r1 ≠ null, x ≠ [], path((h(r)).nxt ' + x, h(r1 := ce)), reachable(r1, (h(r)).nxt, h), r ∈ h, ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, ce.nxt, h), ¬ reachable(r1, x.last, h) ⊦ (h((h(r)).nxt)).nxt ≠ null ∧ (h((h(r)).nxt)).nxt ∈ 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 = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 29 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: store[linheap], name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [(h(r)).nxt, ce, r1, h], <all paths>

Additional information

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

Proof step 30 from proof add-preserves-reachable-rev (previous step) (back to tree)

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

Proof step 31 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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; ⊦ st(a := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s)

Proof step 32 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x0

Additional information

Used simplifier rules: Reflist freely generated by [], +; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s)

Proof step 33 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h((h(r)).nxt)).nxt = (h(ce.nxt)).nxt, x0 = r0 + x, ce.nxt ' + x0 ≠ [], (ce.nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, (h(r)).nxt ≠ null, r ≠ null, r ≠ (h(r)).nxt, path(ce.nxt ' + x0, h((h(r)).nxt := ce)), (h(r)).nxt ∈ h, ce.nxt ∈ h, r ∈ h, ¬ reachable((h(r)).nxt, ce.nxt, h), ¬ reachable((h(r)).nxt, (ce.nxt ' + x0).last, h) ⊦ (h(ce.nxt)).nxt ≠ null ∧ (h(ce.nxt)).nxt ∈ 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 list[reflist] ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

Proof step 34 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x

Additional information

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

Proof step 35 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h((h(r)).nxt)).nxt = (h(ce.nxt)).nxt, x = [], ce.nxt ' + (h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x ≠ [], ((h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x).last ≠ ce.nxt, (h(r)).nxt ≠ null, ce.nxt ≠ null, r ≠ (h(r)).nxt, r ≠ null, path((h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x, h((h(r)).nxt := ce)), r ∈ h, ce.nxt ∈ h, (h(r)).nxt ∈ h, ce.nxt ∈ h((h(r)).nxt := ce), ¬ reachable((h(r)).nxt, ((h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x).last, h), ¬ reachable((h(r)).nxt, ce.nxt, h) ⊦ (h(ce.nxt)).nxt ≠ null ∧ (h(ce.nxt)).nxt ∈ 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; ⊦ a '.last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + [] = x; (s)

Proof step 36 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h((h(r)).nxt)).nxt = (h(ce.nxt)).nxt, ce.nxt ' + (h((h(r)).nxt := ce)(ce.nxt)).nxt ' ≠ [], (h(r)).nxt ≠ null, ce.nxt ≠ null, (h((h(r)).nxt := ce)(ce.nxt)).nxt ≠ ce.nxt, (h((h(r)).nxt := ce)(ce.nxt)).nxt ≠ null, r ≠ (h(r)).nxt, r ≠ null, ce.nxt ∈ h((h(r)).nxt := ce), (h(r)).nxt ∈ h, ce.nxt ∈ h, r ∈ h, (h((h(r)).nxt := ce)(ce.nxt)).nxt ∈ h((h(r)).nxt := ce), ¬ reachable((h(r)).nxt, ce.nxt, h), ¬ reachable((h(r)).nxt, (h((h(r)).nxt := ce)(ce.nxt)).nxt, h) ⊦ (h(ce.nxt)).nxt ≠ null ∧ (h(ce.nxt)).nxt ∈ h

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: store[linheap], name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [ce.nxt, ce, (h(r)).nxt, h], <all paths>

Additional information

Used simplifier rules: ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws) ⊦ a = b → st(b := d)(a) = d; (s) Used terms: ce.nxt, ce, (h(r)).nxt, h

Proof step 37 from proof add-preserves-reachable-rev (previous step) (back to tree)

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

Proof step 38 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h((h(r)).nxt)).nxt = (h(ce.nxt)).nxt, x = r0 + x1, ce.nxt ' + (h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x ≠ [], ((h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x).last ≠ ce.nxt, (h(r)).nxt ≠ null, ce.nxt ≠ null, r ≠ (h(r)).nxt, r ≠ null, path((h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x, h((h(r)).nxt := ce)), r ∈ h, ce.nxt ∈ h, (h(r)).nxt ∈ h, ce.nxt ∈ h((h(r)).nxt := ce), ¬ reachable((h(r)).nxt, ((h((h(r)).nxt := ce)(ce.nxt)).nxt ' + x).last, h), ¬ reachable((h(r)).nxt, ce.nxt, h) ⊦ (h(ce.nxt)).nxt ≠ null ∧ (h(ce.nxt)).nxt ∈ 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 list[reflist] ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

Proof step 39 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h((h(r)).nxt)).nxt = (h(ce.nxt)).nxt, ce.nxt ' + (h((h(r)).nxt := ce)(ce.nxt)).nxt ' + (h((h(r)).nxt := ce)((h((h(r)).nxt := ce)(ce.nxt)).nxt)).nxt ' + x1 ≠ [], ((h((h(r)).nxt := ce)((h((h(r)).nxt := ce)(ce.nxt)).nxt)).nxt ' + x1).last ≠ ce.nxt, (h((h(r)).nxt := ce)(ce.nxt)).nxt ≠ null, ce.nxt ≠ null, (h(r)).nxt ≠ null, r ≠ null, r ≠ (h(r)).nxt, path((h((h(r)).nxt := ce)((h((h(r)).nxt := ce)(ce.nxt)).nxt)).nxt ' + x1, h((h(r)).nxt := ce)), (h((h(r)).nxt := ce)(ce.nxt)).nxt ∈ h((h(r)).nxt := ce), r ∈ h, ce.nxt ∈ h, (h(r)).nxt ∈ h, ce.nxt ∈ h((h(r)).nxt := ce), ¬ reachable((h(r)).nxt, ((h((h(r)).nxt := ce)((h((h(r)).nxt := ce)(ce.nxt)).nxt)).nxt ' + x1).last, h), ¬ reachable((h(r)).nxt, ce.nxt, h) ⊦ (h(ce.nxt)).nxt ≠ null ∧ (h(ce.nxt)).nxt ∈ h

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: store[linheap], name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [ce.nxt, ce, (h(r)).nxt, h], <all paths>

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws) ⊦ a = b → st(b := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s) Used terms: ce.nxt, ce, (h(r)).nxt, h

Proof step 40 from proof add-preserves-reachable-rev (previous step) (back to tree)

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

Proof step 41 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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 42 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule constructor cut

And closed the goal!

Rule argument

var: x0

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; reach-reachnext: ⊦ reachable(r, r1, h) → (reachable(r, (h(r1)).nxt, h) ↔ ¬((h(r1)).nxt = null ∨ ¬ (h(r1)).nxt ∈ h)); path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); Reflist freely generated by [], +; ⊦ a + x = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws) ⊦ a = b → st(b := d)(a) = d; (s) ⊦ x + [] = x; (s) ⊦ a '.last = a; (s)

Proof step 43 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: ce.nxt abort: F discard: T computed: T

Additional information

Used terms: ce.nxt

Proof step 44 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 45 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, (ce.nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r1 ≠ null, path(ce.nxt ' + x0, h(r1 := ce)), ce.nxt ∈ h, r1 ∈ h, (ce.nxt ' + x0).last ∈ h(r1 := ce), ¬ reachable(r1, ce.nxt, h), ¬ reachable(r1, (ce.nxt ' + x0).last, h), ¬ reachable(r1, (h(r1)).nxt, 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(r1)).nxt, r1], <all paths>

Additional information

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

Proof step 46 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Lemma reachable-def

Proof step 47 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: (h(r1)).nxt ' abort: F discard: T computed: F

Additional information

Used terms: (h(r1)).nxt '

Proof step 48 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ (x + a ').last = a; (s)

Proof step 49 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x0

Additional information

Used simplifier rules: Reflist freely generated by [], +; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s)

Proof step 50 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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 list[reflist] ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

Proof step 51 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: store[linheap], name: At-other, seq: ⊦ a ≠ b → st(b := d)(a) = st(a), subst:[a, d, b, st] → [ce.nxt, ce, r1, h], <all paths>

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ a ∈ st(a := d); (s) ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws) ⊦ st(a := d)(a) = d; (s) Used terms: ce.nxt, ce, r1, h

Proof step 52 from proof add-preserves-reachable-rev (previous step) (back to tree)

⊦ a ≠ b → st(b := d)(a) = st(a)

A spec-lemma

Proof step 53 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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 54 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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(r1)).nxt ' + x).last, r1], <all paths>

Additional information

Used terms: h, ((h(r1)).nxt ' + x).last, r1

Proof step 55 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Lemma reachable-def

Proof step 56 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: (h(r1)).nxt ' abort: F discard: T computed: F

Additional information

Used terms: (h(r1)).nxt '

Proof step 57 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 58 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule constructor cut

And closed the goal!

Rule argument

var: x

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ (h(r)).nxt = r0 ∧ path(r0 ' + x, h); Reflist freely generated by [], +; ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws) ⊦ a = b → st(b := d)(a) = d; (s) ⊦ a '.last = a; (s) ⊦ x + [] = x; (s)

Proof step 59 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 14

Additional information

Used simplifier rules: fromnext-reachable: ⊦ reachable(r, r0, h) ∧ r ≠ r0 → (reachable((h(r)).nxt, r0, h) ↔ true); ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s)

Proof step 60 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path((h(r)).nxt ' + x0, h(r1 := ce)), reachable(r, r1, h), reachable((h(r)).nxt, ce.nxt, h), r ∈ h(r1 := ce), ((h(r)).nxt ' + x0).last ∈ h(r1 := ce), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r, ((h(r)).nxt ' + x0).last, h), ¬ reachable(r, ce.nxt, 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; ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s)

Proof step 61 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: x0 = []

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; reachnext: ⊦ reachable(r0, (h(r0)).nxt, h) ↔ r0 ≠ null ∧ r0 ∈ h ∧ (h(r0)).nxt ∈ h ∧ (h(r0)).nxt ≠ null; startnotinH-notreachable: ⊦ ¬ r ∈ h → (reachable(r, r0, h) ↔ false); ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s) Used formulas: x0 = []

Proof step 62 from proof add-preserves-reachable-rev (previous step) (back to tree)

(h(r1)).nxt = (h(ce.nxt)).nxt, ((h(r)).nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, x0 ≠ [], path((h(r)).nxt ' + x0, h(r1 := ce)), reachable(r, r1, h), reachable((h(r)).nxt, ce.nxt, h), ce.nxt ∈ h, r ∈ h, ((h(r)).nxt ' + x0).last ∈ h(r1 := ce), r1 ∈ h, ¬ reachable(r, ((h(r)).nxt ' + x0).last, h), ¬ reachable(r, ce.nxt, 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 list[reflist] ⊦ x ≠ [] → (a ' + x).last = x.last; (s)

Proof step 63 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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, ce.nxt, r], <all paths>

Additional information

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

Proof step 64 from proof add-preserves-reachable-rev (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 65 from proof add-preserves-reachable-rev (previous step) (back to tree)

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

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, ((h(r)).nxt ' + x0).last, r], <all paths>

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h; self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws) Used terms: h, ((h(r)).nxt ' + x0).last, r

Proof step 66 from proof add-preserves-reachable-rev (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 Tree for add-preserves-reachable-rev (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