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

⊦ h0 ⊥ h1 ∧ dll(r, r0, x)(h0) → ((dll(r, r0, 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 dll-union (previous step) (back to tree)

¬((dll(r, r0, y) * hP)(h0 ∪ h1) ↔ x = y ∧ hP(h1)), dll(r, r0, x)(h0), h0 ⊥ h1 ⊦

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: (dll(r, r0, y) * hP)(h0 ∪ h1)

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 map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep Used formulas: (dll(r, r0, y) * hP)(h0 ∪ h1)

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

¬((dll(r, r0, y) * hP)(h0 ∪ h1) ↔ x = y ∧ hP(h1)), dll(r, r0, x)(h0), (dll(r, r0, y) * hP)(h0 ∪ h1), h0 ⊥ h1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

(dll(r, r0, y) * hP)(h0 ∪ h1), dll(r, r0, x)(h0), h0 ⊥ h1 ⊦ x = y ∧ hP(h1)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: y

Additional information

Used simplifier rules: dll-base: ⊦ dll(r, r0, []) = (emp ∧ ⌜r = null⌝ ); dll-prev-null: ⊦ dll(null, r, x) = (emp ∧ ⌜x = []⌝ ); List[A] freely generated by [], +; ⊦ (⌜bv⌝ ∧ hP0) * hP1 = (⌜bv⌝ ∧ hP0 * hP1); (s) ⊦ emp * hP = hP; (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-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 map-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep ⊦ ∅ ⊥ h; (s) ⊦ ∅ ∪ h = h; (s)

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

dll(r, r0, x)(h0), (dll(r, r0, a + y0) * hP)(h0 ∪ h1), h0 ⊥ h1, ∀ r, r0, hP, h0, h1, x. (dll(r, r0, y0) * hP)(h0 ∪ h1) ∧ dll(r, r0, 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: dll-rec: ⊦ dll(r, r0, a ' + x) = _∃(λ r1. r => mknode(r0, a, r1) * dll(r1, r, x)); ⊦ a + x = a ' + x; (s) sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep ⊦ (_∃ bP * hP)(h) ↔ (∃ b. (bP(b) * hP)(h)); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

dll(r, r0, x)(h0), h0 ⊥ h1, ((r => mknode(r0, a, r1) * dll(r1, r, y0)) * hP)(h0 ∪ h1), ∀ r, r0, hP, h0, h1, x. (dll(r, r0, y0) * hP)(h0 ∪ h1) ∧ dll(r, r0, 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: dll-base: ⊦ dll(r, r0, []) = (emp ∧ ⌜r = null⌝ ); List[A] 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 map-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification map-sep ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (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 map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep ⊦ ¬ false(h); (s)

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

x = a0 + x0, dll(r, r0, x)(h0), h0 ⊥ h1, ((r => mknode(r0, a, r1) * dll(r1, r, y0)) * hP)(h0 ∪ h1), ∀ r, r0, hP, h0, h1, x. (dll(r, r0, y0) * hP)(h0 ∪ h1) ∧ dll(r, r0, 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: dll-rec: ⊦ dll(r, r0, a ' + x) = _∃(λ r1. r => mknode(r0, a, r1) * dll(r1, r, x)); ⊦ a + x = a ' + x; (s) ⊦ (_∃ bP)(h) ↔ (∃ b. bP(b)(h)); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

h0 ⊥ h1, ((r => mknode(r0, a, r1) * dll(r1, r, y0)) * hP)(h0 ∪ h1), (r => mknode(r0, a0, r2) * dll(r2, r, x0))(h0), ∀ r, r0, hP, h0, h1, x. (dll(r, r0, y0) * hP)(h0 ∪ h1) ∧ dll(r, r0, 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: refmap-maplet, name:map-del, seq: ⊦ (r => o * hP)(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h(r) = o ∧ r ≠ null, subst:[h, r, o, hP] → [h0 ∪ h1, r, mknode(r0, a, r1), dll(r1, r, y0) * hP]; [A] → [Node[A]], lhs:((r => mknode(r0, a, r1) * dll(r1, r, y0)) * hP)(h0 ∪ h1), <all paths>

Additional information

Used simplifier rules: sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep Used terms: h0 ∪ h1, r, mknode(r0, a, r1), dll(r1, r, y0) * hP

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

⊦ (r => o * hP)(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h(r) = o ∧ r ≠ null

A spec-lemma

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

(h0 ∪ h1)(r) = mknode(r0, a, r1), r ≠ null, (dll(r1, r, y0) * hP)((h0 ∪ h1) -- r), h0 ⊥ h1, (r => mknode(r0, a0, r2) * dll(r2, r, x0))(h0), r ∈ h0 ∪ h1, ∀ r, r0, hP, h0, h1, x. (dll(r, r0, y0) * hP)(h0 ∪ h1) ∧ dll(r, r0, 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: refmap-maplet, name:map-del, seq: ⊦ (r => o * hP)(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h(r) = o ∧ r ≠ null, subst:[h, r, o, hP] → [h0, r, mknode(r0, a0, r2), dll(r2, r, x0)]; [A] → [Node[A]], lhs:(r => mknode(r0, a0, r2) * dll(r2, r, x0))(h0), <all paths>

Additional information

Used terms: h0, r, mknode(r0, a0, r2), dll(r2, r, x0)

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

⊦ (r => o * hP)(h) ↔ hP(h -- r) ∧ r ∈ h ∧ h(r) = o ∧ r ≠ null

A spec-lemma

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

h0(r) = mknode(r0, a0, r2), (h0 ∪ h1)(r) = mknode(r0, a, r1), r ≠ null, dll(r2, r, x0)(h0 -- r), (dll(r1, r, y0) * hP)((h0 ∪ h1) -- r), h0 ⊥ h1, r ∈ h0 ∪ h1, r ∈ h0, ∀ r, r0, hP, h0, h1, x. (dll(r, r0, y0) * hP)(h0 ∪ h1) ∧ dll(r, r0, x)(h0) ∧ h0 ⊥ h1 → x = y0 ∧ hP(h1) ⊦ a0 ' + x0 = a ' + y0 ∧ hP(h1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: r2, r, hP, h0 -- r, h1 -- r, x0 abort: F discard: T computed: F

Additional information

Used terms: r2, r, hP, h0 -- r, h1 -- r, x0

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

(h0 ∪ h1)(r) = mknode(r0, a, r1), h0(r) = mknode(r0, a0, r2), r ≠ null, (dll(r1, r, y0) * hP)((h0 ∪ h1) -- r), dll(r2, r, x0)(h0 -- r), h0 ⊥ h1, r ∈ h0, r ∈ h0 ∪ h1, (dll(r2, r, y0) * hP)((h0 -- r) ∪ (h1 -- r)) ∧ dll(r2, r, x0)(h0 -- r) ∧ h0 -- r ⊥ h1 -- r → x0 = y0 ∧ hP(h1 -- r) ⊦ 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 = a0 ' + y ↔ a = a0 ∧ x = y; (ws) ⊦ mknode(left, val, left0) = mknode(left1, val0, left2) ↔ left = left1 ∧ val = val0 ∧ left0 = left2; (ws) ⊦ h0 ⊥ h1 → h0 -- r ⊥ h1 -- r; (ss) ⊦ h0 ⊥ h1 → (h0 ∪ h1) -- r = (h0 -- r) ∪ (h1 -- r); (s) ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); (s) r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r)); (s)

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

h0(r) = mknode(r0, a0, r2), r ≠ null, (dll(r2, r, y0) * hP)((h0 -- r) ∪ (h1 -- r)), dll(r2, r, y0)(h0 -- r), h0 ⊥ h1, hP(h1 -- r), r ∈ h0, ¬ hP(h1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: heapfuns-map, name:disj, seq: ⊦ h0 ⊥ h1 ↔ (∀ r. ¬ r ∈ h0 ∨ ¬ r ∈ h1), subst:[h0, h1] → [h0, h1]; [A, B] → [Ref[Node[A]], Node[A]], lhs:h0 ⊥ h1, <all paths>

Additional information

Used terms: h0, h1

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

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

A spec-lemma

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

h0(r) = mknode(r0, a0, r2), r ≠ null, dll(r2, r, y0)(h0 -- r), (dll(r2, r, y0) * hP)((h0 -- r) ∪ (h1 -- r)), hP(h1 -- r), r ∈ h0, ¬ hP(h1), ∀ r. ¬ r ∈ h0 ∨ ¬ r ∈ h1 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: r abort: F discard: T computed: T

Additional information

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

Proof Tree for dll-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