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

⊦ h0 ⊥ h1 ∧ ls(r, x)(h0) → (((ls(r, y) * hP))(h0 ∪ h1) ↔ x = y ∧ hP(h1))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

¬(((ls(r, y) * hP))(h0 ∪ h1) ↔ x = y ∧ hP(h1)), ls(r, x)(h0), h0 ⊥ h1 ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Used simplifier rules: ⊦ hP0(h0) ∧ hP1(h1) ∧ h0 ⊥ h1 → ((hP0 * hP1))(h0 ∪ h1); (ss) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep

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

((ls(r, y) * hP))(h0 ∪ h1), ls(r, x)(h0), h0 ⊥ h1 ⊦ x = y ∧ hP(h1)

User applied rule structural induction

And obtained the following new goals

Rule argument

indvar: y

Additional information

Used simplifier rules: ls-base: ⊦ ls(r, []) = (emp ∧ ⌜r = null⌝ ); ls-null: ⊦ ls(null, x) = (emp ∧ ⌜x = []⌝ ); list freely generated by [], +;; ⊦ (⌜bv⌝ ∧ hP0) * hP1 = (⌜bv⌝ ∧ hP0 * hP1); (s) ⊦ emp * hP = hP; (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep ⊦ ((hP0 ∧ hP1))(h) ↔ hP0(h) ∧ hP1(h); (s) ⊦ emp(h) ↔ h = ∅; (s) ⊦ ⌜bv⌝ (h) ↔ bv; (s) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification heap-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification heap-sep ⊦ ∅ ∪ h = h; (s) ⊦ ∅ ⊥ h; (s)

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

((ls(r, a + y0) * hP))(h0 ∪ h1), ls(r, x)(h0), h0 ⊥ h1, ∀ r, hP, h0, h1, x. ((ls(r, y0) * hP))(h0 ∪ h1) ∧ ls(r, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ x = a + y0 ∧ hP(h1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ls-rec: ⊦ ls(r, a ' + x) = _∃(λ r0. r => a × r0 * ls(r0, x)); ⊦ a + x = a ' + x; (s) sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep ⊦ ((_∃ rP * hP))(h) ↔ (∃ r. ((rP(r) * hP))(h)); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

(((r => a × r0 * ls(r0, y0)) * hP))(h0 ∪ h1), ls(r, x)(h0), h0 ⊥ h1, ∀ r, hP, h0, h1, x. ((ls(r, y0) * hP))(h0 ∪ h1) ∧ ls(r, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ x = a ' + y0 ∧ hP(h1)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x

Additional information

Used simplifier rules: ls-base: ⊦ ls(r, []) = (emp ∧ ⌜r = null⌝ ); list freely generated by [], +;; ⊦ ((hP0 ∧ hP1))(h) ↔ hP0(h) ∧ hP1(h); (s) ⊦ emp(h) ↔ h = ∅; (s) ⊦ ⌜bv⌝ (h) ↔ bv; (s) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification heap-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification heap-sep ⊦ a ' + x ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ∅ ⊥ h; (s) ⊦ ∅ ∪ h = h; (s) ⊦ null => o = ⌜false⌝ ; (s) ⊦ ⌜false⌝ = false; (s) ⊦ false * hP = false; (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep ⊦ ¬ false(h); (s)

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

x = a0 + x0, (((r => a × r0 * ls(r0, y0)) * hP))(h0 ∪ h1), ls(r, x)(h0), h0 ⊥ h1, ∀ r, hP, h0, h1, x. ((ls(r, y0) * hP))(h0 ∪ h1) ∧ ls(r, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ x = a ' + y0 ∧ hP(h1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ls-rec: ⊦ ls(r, a ' + x) = _∃(λ r0. r => a × r0 * ls(r0, x)); ⊦ a + x = a ' + x; (s) ⊦ (_∃ rP)(h) ↔ (∃ r. rP(r)(h)); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

(((r => a × r0 * ls(r0, y0)) * hP))(h0 ∪ h1), ((r => a0 × r1 * ls(r1, x0)))(h0), h0 ⊥ h1, ∀ r, hP, h0, h1, x. ((ls(r, y0) * hP))(h0 ∪ h1) ∧ ls(r, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ a0 ' + x0 = a ' + y0 ∧ hP(h1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: maplet[cell-heap], name:map-del, seq: ⊦ ((r => o * hP))(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h[r] = o, subst:[h, hP, o, r] → [h0 ∪ h1, hP * ls(r0, y0), a × r0, r], <all paths>

Additional information

Used simplifier rules: sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep Used terms: h0 ∪ h1, hP * ls(r0, y0), a × r0, r

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

⊦ ((r => o * hP))(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h[r] = o

A spec-lemma

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

(h0 ∪ h1)[r] = a × r0, ((r => a0 × r1 * ls(r1, x0)))(h0), ((hP * ls(r0, y0)))((h0 ∪ h1) -- r), r ∈ h0 ∪ h1, h0 ⊥ h1, ∀ r, hP, h0, h1, x. ((ls(r, y0) * hP))(h0 ∪ h1) ∧ ls(r, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ a0 ' + x0 = a ' + y0 ∧ hP(h1)

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 ⊦ a ' + x = b ' + y ↔ a = b ∧ x = y; (ws) ⊦ h0 ⊥ h1 → (h0 ∪ h1) -- r = (h0 -- r) ∪ (h1 -- r); (s) ⊦ ((r => o * hP))(h) → r ∈ h; (s) ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); (s) ⊦ a × r = a0 × r0 ↔ a = a0 ∧ r = r0; (ws) ⊦ ((r => o0 * hP))(h) → (h[r] = o1 ↔ o0 = o1); (s) r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)[r] = (r ∈ h0 ⊃ h0[r];h1[r]); (s) sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep

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

((hP * ls(r0, y0)))((h0 -- r) ∪ (h1 -- r)), ((r => a0 × r0 * ls(r0, x0)))(h0), h0 ⊥ h1, ∀ r, hP, h0, h1, x. ((ls(r, y0) * hP))(h0 ∪ h1) ∧ ls(r, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ x0 = y0 ∧ hP(h1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: maplet[cell-heap], name:map-del, seq: ⊦ ((r => o * hP))(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h[r] = o, subst:[h, hP, o, r] → [h0, ls(r0, x0), a0 × r0, r], <all paths>

Additional information

Used terms: h0, ls(r0, x0), a0 × r0, r

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

⊦ ((r => o * hP))(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h[r] = o

A spec-lemma

Proof step 12 from proof ls-union (previous step) (back to tree)

h0[r] = a0 × r0, ((hP * ls(r0, y0)))((h0 -- r) ∪ (h1 -- r)), ls(r0, x0)(h0 -- r), r ∈ h0, h0 ⊥ h1, ∀ r, hP, h0, h1, x. ((ls(r, y0) * hP))(h0 ∪ h1) ∧ ls(r, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ x0 = y0 ∧ hP(h1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: r0, hP, h0 -- r, h1 -- r, x0 abort: F discard: T computed: T

Additional information

Used terms: r0, hP, h0 -- r, h1 -- r, x0

Proof step 13 from proof ls-union (previous step) (back to tree)

h0[r] = a0 × r0, ((hP * ls(r0, y0)))((h0 -- r) ∪ (h1 -- r)), ls(r0, x0)(h0 -- r), r ∈ h0, h0 ⊥ h1, ((ls(r0, y0) * hP))((h0 -- r) ∪ (h1 -- r)) ∧ ls(r0, x0)(h0 -- r) ∧ h0 -- r ⊥ h1 -- r → x0 = y0 ∧ hP(h1 -- r) ⊦ x0 = y0 ∧ hP(h1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep

Proof step 14 from proof ls-union (previous step) (back to tree)

h0[r] = a0 × r0, ((hP * ls(r0, y0)))((h0 -- r) ∪ (h1 -- r)), ls(r0, x0)(h0 -- r), r ∈ h0, h0 ⊥ h1, h0 -- r ⊥ h1 -- r → x0 = y0 ∧ hP(h1 -- r) ⊦ x0 = y0 ∧ hP(h1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: heap[cell-heap], name:disj, seq: ⊦ h0 ⊥ h1 ↔ (∀ r. ¬ r ∈ h0 ∨ ¬ r ∈ h1), subst:[h1, h0] → [h1 -- r, h0 -- r], <all paths>

Additional information

Used terms: h1 -- r, h0 -- r

Proof step 15 from proof ls-union (previous step) (back to tree)

⊦ h0 ⊥ h1 ↔ (∀ r. ¬ r ∈ h0 ∨ ¬ r ∈ h1)

A spec-lemma

Proof step 16 from proof ls-union (previous step) (back to tree)

h0[r] = a0 × r0, ((hP * ls(r0, y0)))((h0 -- r) ∪ (h1 -- r)), ls(r0, x0)(h0 -- r), r ∈ h0, h0 ⊥ h1, (∀ r0. ¬ r0 ∈ h0 -- r ∨ ¬ r0 ∈ h1 -- r) → x0 = y0 ∧ hP(h1 -- r) ⊦ x0 = y0 ∧ hP(h1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ h0 ∧ h0 ⊥ h1 → (r ∈ h1 ↔ false); (s) ⊦ r ∈ h -- r0 ↔ r ≠ r0 ∧ r ∈ h; (s)

Proof step 17 from proof ls-union (previous step) (back to tree)

h0[r] = a0 × r0, ((hP * ls(r0, y0)))((h0 -- r) ∪ (h1 -- r)), hP(h1 -- r), ls(r0, y0)(h0 -- r), r ∈ h0, h0 ⊥ h1, ¬ hP(h1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: heap[cell-heap], name:disj, seq: ⊦ h0 ⊥ h1 ↔ (∀ r. ¬ r ∈ h0 ∨ ¬ r ∈ h1), subst:[h1, h0] → [h1, h0], <all paths>

Additional information

Used terms: h1, h0

Proof step 18 from proof ls-union (previous step) (back to tree)

⊦ h0 ⊥ h1 ↔ (∀ r. ¬ r ∈ h0 ∨ ¬ r ∈ h1)

A spec-lemma

Proof step 19 from proof ls-union (previous step) (back to tree)

h0[r] = a0 × r0, ((hP * ls(r0, y0)))((h0 -- r) ∪ (h1 -- r)), hP(h1 -- r), ls(r0, y0)(h0 -- r), r ∈ h0, ¬ hP(h1), ∀ r. ¬ r ∈ h0 ∨ ¬ r ∈ h1 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: r abort: F discard: T computed: T

Additional information

Used simplifier rules: ⊦ ¬ r ∈ h → h -- r = h; (s) Used terms: r

Proof Tree for ls-union (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