Proof step 1 from proof exec-assign-wp (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬(hP(h1) ↔ (r.=>_ * (r => o1 -* hP))(h0)), h1 ≡ h0(r := o1), r ∈ h0 ⊦

Rule argument

Additional information

Proof step 2 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ¬(hP(h0(r := o1)) ↔ (r.=>_ * (r => o1 -* hP))(h0)), r ∈ h0 ⊦

Rule argument

Additional information

Used simplifier rules: eq: ⊦ h0 ≡ h1 ↔ h0 = h1;

Proof step 3 from proof exec-assign-wp (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 4:

  hP(h0(r := o1)), r ∈ h0, ¬ (r.=>_ * (r => o1 -* hP))(h0) ⊦

• Proof step 22:

  r ∈ h0, (r.=>_ * (r => o1 -* hP))(h0), ¬ hP(h0(r := o1)) ⊦

Rule argument

Additional information

Proof step 4 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 5:

  ⊦ h1 ≡ h0 -- r → ((r.=>_ * hP)(h0) ↔ hP(h1) ∧ r ∈ h0 ∧ ¬ r ∈ h1)

• Proof step 6:

  h0 -- r ≡ h0 -- r, hP(h0(r := o1)), r ∈ h0, ¬ (r => o1 -* hP)(h0 -- r) ∨ ¬ r ∈ h0 ∨ r ∈ h0 -- r ⊦

Rule argument

Additional information

Used simplifier rules: eq: ⊦ h0 ≡ h1 ↔ h0 = h1; Used terms: h0 -- r, h0, r, r => o1 -* hP

Proof step 5 from proof exec-assign-wp (previous step) (back to tree)

Lemma free-unknown

Proof step 6 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  hP(h0(r := o1)), r ∈ h0, ¬ (r => o1 -* hP)(h0 -- r) ⊦

Rule argument

Additional information

Used simplifier rules: eq: ⊦ h0 ≡ h1 ↔ h0 = h1; ⊦ r ∈ h -- r0 ↔ r ≠ r0 ∧ r ∈ h; (s)

Proof step 7 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 8:

  ⊦ (hP0 -* hP1)(h) ↔ (∀ h1. h1 ⊥ h ∧ hP0(h1) → hP1(h1 ∪ h))

• Proof step 9:

  hP(h0(r := o1)), r ∈ h0, ∃ h1. ¬(h1 ⊥ h0 -- r ∧ (r => o1)(h1) → hP(h1 ∪ (h0 -- r))) ⊦

Rule argument

Additional information

Used terms: h0 -- r, hP, r => o1

Proof step 8 from proof exec-assign-wp (previous step) (back to tree)

A spec-lemma

Proof step 9 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  h1 ⊥ h0 -- r, hP(h0(r := o1)), (r => o1)(h1), r ∈ h0, ¬ hP(h1 ∪ (h0 -- r)) ⊦

Rule argument

Additional information

Proof step 10 from proof exec-assign-wp (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 11:

  h1 ∪ (h0 -- r) ≠ h0(r := o1), h1 ⊥ h0 -- r, hP(h0(r := o1)), r ∈ h0, (r => o1)(h1), ¬ hP(h1 ∪ (h0 -- r)) ⊦

Rule argument

Additional information

Used formulas: h1 ∪ (h0 -- r) = h0(r := o1)

Proof step 11 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 12:

  ⊦ (r => o)(h) ↔ h = (∅ ++ r)(r := o) ∧ r ≠ null

• Proof step 13:

  h1 = (∅ ++ r)(r := o1), h1 ∪ (h0 -- r) ≠ h0(r := o1), r ≠ null, h1 ⊥ h0 -- r, hP(h0(r := o1)), r ∈ h0, ¬ hP(h1 ∪ (h0 -- r)) ⊦

Rule argument

Additional information

Used terms: h1, o1, r

Proof step 12 from proof exec-assign-wp (previous step) (back to tree)

A spec-lemma

Proof step 13 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  (∅ ++ r)(r := o1) ∪ (h0 -- r) ≠ h0(r := o1), hP(h0(r := o1)), r ∈ h0, ¬ hP((∅ ++ r)(r := o1) ∪ (h0 -- r)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ h -- r0 ↔ r ≠ r0 ∧ r ∈ h; (s) r ≠ null ⊦ (∅ ++ r)(r := o) ⊥ h ↔ ¬ r ∈ h; (s) ⊦ r ∈ h → r ≠ null; (s)

Proof step 14 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 15:

  ⊦ h0 = h1 ↔ (∀ r. (r ∈ h0 ↔ r ∈ h1) ∧ (r ∈ h0 → h0(r) = h1(r)))

• Proof step 16:

  hP(h0(r := o1)), r ∈ h0, ¬ hP((∅ ++ r)(r := o1) ∪ (h0 -- r)), ∃ r0. ¬ ( (r0 ∈ (∅ ++ r)(r := o1) ∪ (h0 -- r) ↔ r0 ∈ h0(r := o1)) ∧ (r0 ∈ (∅ ++ r)(r := o1) ∪ (h0 -- r) → ((∅ ++ r)(r := o1) ∪ (h0 -- r))(r0) = h0(r := o1)(r0))) ⊦

Rule argument

Additional information

Used terms: h0(r := o1), (∅ ++ r)(r := o1) ∪ (h0 -- r)

Proof step 15 from proof exec-assign-wp (previous step) (back to tree)

A spec-lemma

Proof step 16 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  hP(h0(r := o1)), r ∈ h0, ¬ hP((∅ ++ r)(r := o1) ∪ (h0 -- r)) ⊦ (r0 ∈ (∅ ++ r)(r := o1) ∪ (h0 -- r) ↔ r0 ∈ h0) ∧ (r0 ∈ (∅ ++ r)(r := o1) ∪ (h0 -- r) → ((∅ ++ r)(r := o1) ∪ (h0 -- r))(r0) = (r0 = r ⊃ o1;h0(r0)))

Rule argument

Additional information

Used simplifier rules: ⊦ h(r0 := o)(r) = (r = r0 ∧ r0 ∈ h ⊃ o;h(r)); (s) ⊦ r ∈ h(r0 := o) ↔ r ∈ h; (s)

Proof step 17 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 18:

  ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1)

• Proof step 19:

  hP(h0(r := o1)), r ∈ h0, (∅ ++ r)(r := o1) ⊥ h0 -- r, ¬ hP((∅ ++ r)(r := o1) ∪ (h0 -- r)) ⊦ (r0 ∈ (∅ ++ r)(r := o1) ∨ r0 ∈ h0 -- r ↔ r0 ∈ h0) ∧ (r0 ∈ (∅ ++ r)(r := o1) ∨ r0 ∈ h0 -- r → ((∅ ++ r)(r := o1) ∪ (h0 -- r))(r0) = (r0 = r ⊃ o1;h0(r0)))

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ h → r ≠ null; (s) r ≠ null ⊦ (∅ ++ r)(r := o) ⊥ h ↔ ¬ r ∈ h; (s) ⊦ r ∈ h -- r0 ↔ r ≠ r0 ∧ r ∈ h; (s) Used terms: h0 -- r, (∅ ++ r)(r := o1), r0

Proof step 18 from proof exec-assign-wp (previous step) (back to tree)

A spec-lemma

Proof step 19 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

  ((∅ ++ r)(r := o1) ∪ (h0 -- r))(r0) ≠ (r0 = r ⊃ o1;h0(r0)), hP(h0(r := o1)), r ∈ h0, r0 ∈ h0, ¬ hP((∅ ++ r)(r := o1) ∪ (h0 -- r)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ h -- r0 ↔ r ≠ r0 ∧ r ∈ h; (s) ⊦ ¬ r ∈ ∅; (s) r0 ≠ null, ¬ r0 ∈ h ⊦ r ∈ h ++ r0 ↔ r = r0 ∨ r ∈ h; (s) ⊦ r ∈ h → r ≠ null; (s) ⊦ r ∈ h(r0 := o) ↔ r ∈ h; (s) r ≠ null ⊦ (∅ ++ r)(r := o) ⊥ h ↔ ¬ r ∈ h; (s)

Proof step 20 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 21:

  r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r))

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ h(r0 := o) ↔ r ∈ h; (s) ⊦ h(r0 := o)(r) = (r = r0 ∧ r0 ∈ h ⊃ o;h(r)); (s) ⊦ r ∈ h → r ≠ null; (s) ⊦ ¬ r ∈ ∅; (s) r0 ≠ null, ¬ r0 ∈ h ⊦ r ∈ h ++ r0 ↔ r = r0 ∨ r ∈ h; (s) r ∈ h ⊦ r ≠ r0 → (h -- r0)(r) = h(r); (s) ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); (s) r ≠ null ⊦ (∅ ++ r)(r := o) ⊥ h ↔ ¬ r ∈ h; (s) ⊦ r ∈ h -- r0 ↔ r ≠ r0 ∧ r ∈ h; (s) Used terms: r0, h0 -- r, (∅ ++ r)(r := o1)

Proof step 21 from proof exec-assign-wp (previous step) (back to tree)

A spec-lemma

Proof step 22 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 23:

  (r.=>_ * (r => o1 -* hP))(h0), ¬ hP(h0(r := o1)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ (r.=>_ * hP)(h) → r ∈ h; (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep

Proof step 23 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 24:

  ⊦ (r.=>_ * hP)(h) ↔ (r => o1 * hP)(h(r := o1)) ∧ r ∈ h

• Proof step 25:

  (r => o1 * (r => o1 -* hP))(h0(r := o1)), r ∈ h0, ¬ hP(h0(r := o1)) ⊦

Rule argument

Additional information

Used terms: o1, h0, r => o1 -* hP, r

Proof step 24 from proof exec-assign-wp (previous step) (back to tree)

A spec-lemma

Proof step 25 from proof exec-assign-wp (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 26:

  r ∈ h0, (r => o1 * (r => o1 -* hP))(h0(r := o1)), ¬ hP(h0(r := o1)) ⊦

Rule argument

Additional information

Used simplifier rules: sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep map-write: r ∈ h ⊦ (r => o0 * hP)(h(r := o1)) → o0 = o1; (forward) from specification maplet

Proof step 26 from proof exec-assign-wp (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 27:

  ⊦ (hP0 * (hP0 -* hP1))(h) → hP1(h)

Rule argument

Additional information

Used terms: r => o1, hP, h0(r := o1)

Proof step 27 from proof exec-assign-wp (previous step) (back to tree)

A spec-lemma