Proof step 1 from proof union-left-singleton (conclusion) (back to tree)

r ≠ null ⊦ r ∈ h → (∅ ++ r)(r := h(r)) ∪ (h -- r) = h

System applied rule normalize

And obtained the following new goals

Proof step 2:
r ≠ null, (∅ ++ r)(r := h(r)) ∪ (h -- r) ≠ h, r ∈ h ⊦

Rule argument

Additional information

Proof step 2 from proof union-left-singleton (previous step) (back to tree)

r ≠ null, (∅ ++ r)(r := h(r)) ∪ (h -- r) ≠ h, r ∈ h ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ h0 = h1 ↔ (∀ r. (r ∈ h0 ↔ r ∈ h1) ∧ (r ∈ h0 → h0(r) = h1(r)))
Proof step 4:
r ≠ null, r ∈ h, ∃ r0. ¬((r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) ↔ r0 ∈ h) ∧ (r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) → ((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) = h(r0))) ⊦

Rule argument

spec: heap, name: ext, seq: ⊦ h0 = h1 ↔ (∀ r. (r ∈ h0 ↔ r ∈ h1) ∧ (r ∈ h0 → h0(r) = h1(r))), subst:[h1, h0] → [h, (∅ ++ r)(r := h(r)) ∪ (h -- r)], <all paths>

Additional information

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

Proof step 3 from proof union-left-singleton (previous step) (back to tree)

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

A spec-lemma

Proof step 4 from proof union-left-singleton (previous step) (back to tree)

r ≠ null, r ∈ h, ∃ r0. ¬((r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) ↔ r0 ∈ h) ∧ (r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) → ((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) = h(r0))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
r ∈ h ⊦ (r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) ↔ r0 ∈ h) ∧ (r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) → ((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) = h(r0))

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ h → r ≠ null; (s)

Proof step 5 from proof union-left-singleton (previous step) (back to tree)

r ∈ h ⊦ (r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) ↔ r0 ∈ h) ∧ (r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r) → ((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) = h(r0))

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1)
Proof step 7:
r ∈ h, (∅ ++ r)(r := h(r)) ⊥ h -- r ⊦ (r0 ∈ (∅ ++ r)(r := h(r)) ∨ r0 ∈ h -- r ↔ r0 ∈ h) ∧ (r0 ∈ (∅ ++ r)(r := h(r)) ∨ r0 ∈ h -- r → ((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) = h(r0))

Rule argument

spec: <toplevel>, name: in-uni, seq: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1), subst:[h1, h0, r] → [h -- r, (∅ ++ r)(r := h(r)), r0], <all paths>

Additional information

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

Proof step 6 from proof union-left-singleton (previous step) (back to tree)

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

Lemma in-uni

Proof step 7 from proof union-left-singleton (previous step) (back to tree)

r ∈ h, (∅ ++ r)(r := h(r)) ⊥ h -- r ⊦ (r0 ∈ (∅ ++ r)(r := h(r)) ∨ r0 ∈ h -- r ↔ r0 ∈ h) ∧ (r0 ∈ (∅ ++ r)(r := h(r)) ∨ r0 ∈ h -- r → ((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) = h(r0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) ≠ h(r0), r ∈ h, r0 ∈ h ⊦

Rule argument

Additional information

Used simplifier rules: disj-singleton-write: r ≠ null ⊦ (∅ ++ r)(r := o) ⊥ h ↔ ¬ r ∈ h; ⊦ 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)

Proof step 8 from proof union-left-singleton (previous step) (back to tree)

((∅ ++ r)(r := h(r)) ∪ (h -- r))(r0) ≠ h(r0), r ∈ h, r0 ∈ h ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 9:
r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r))
Proof step 10:
(r0 ∈ (∅ ++ r)(r := h(r)) ⊃ (∅ ++ r)(r := h(r))(r0);(h -- r)(r0)) ≠ h(r0), r ∈ h, r0 ∈ h, r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r), (∅ ++ r)(r := h(r)) ⊥ h -- r ⊦

Rule argument

spec: <toplevel>, name: at-uni, seq:r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r)), subst:[r, h1, h0] → [r0, h -- r, (∅ ++ r)(r := h(r))], <all paths>

Additional information

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

Proof step 9 from proof union-left-singleton (previous step) (back to tree)

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

Lemma at-uni

Proof step 10 from proof union-left-singleton (previous step) (back to tree)

(r0 ∈ (∅ ++ r)(r := h(r)) ⊃ (∅ ++ r)(r := h(r))(r0);(h -- r)(r0)) ≠ h(r0), r ∈ h, r0 ∈ h, r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r), (∅ ++ r)(r := h(r)) ⊥ h -- r ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
(r0 = r ⊃ h(r);h(r0)) ≠ h(r0), r ∈ h, r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r), r0 ∈ h ⊦

Rule argument

Additional information

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

Proof step 11 from proof union-left-singleton (previous step) (back to tree)

(r0 = r ⊃ h(r);h(r0)) ≠ h(r0), r ∈ h, r0 ∈ (∅ ++ r)(r := h(r)) ∪ (h -- r), r0 ∈ h ⊦

Heuristic nonsplitting cut applied rule cut formula

And closed the goal!

Rule argument

fma: r0 = r

Additional information

Used formulas: r0 = r

Proof Tree for union-left-singleton (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.