Proof step 1 from proof ll-trav2-end (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  h[r2] = a × null, r2 ≠ null ⊦ ((ll-trav(r0, r1, r2, x) * hP))(h) ↔ x ≠ [] ∧ (r1 = null ⊃ r0 = r2 ∧ ((lls(r0, null, x) * hP))(h);∃ y, b. x = (y + b) + a ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h))

Rule argument

Additional information

Proof step 2 from proof ll-trav2-end (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  h[r2] = a × null, r2 ≠ null ⊦ ((ll-trav(r0, r1, r2, x) * hP))(h) ↔ x ≠ [] ∧ (r1 = null ⊃ r0 = r2 ∧ ((lls(r2, null, x) * hP))(h) ;∃ y, b. x = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h) )

Rule argument

Additional information

Used simplifier rules: ⊦ x + a = x + a '; (s)

Proof step 3 from proof ll-trav2-end (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 4:

  h[r2] = a × null, r2 ≠ null, ((ll-trav(r0, r1, r2, x) * hP))(h) ⊦ x ≠ [] ∧ (r1 = null ⊃ r0 = r2 ∧ ((lls(r2, null, x) * hP))(h) ;∃ y, b. x = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h) )

• Proof step 13:

  h[r2] = a × null, r2 ≠ null, x ≠ [], ¬ ((ll-trav(r0, r1, r2, x) * hP))(h), (r1 = null ⊃ r0 = r2 ∧ ((lls(r2, null, x) * hP))(h);∃ y, b. x = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h)) ⊦

Rule argument

Additional information

Proof step 4 from proof ll-trav2-end (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 5:

  h[r2] = a × null, x = [], r2 ≠ null, ((ll-trav(r0, r1, r2, x) * hP))(h) ⊦

• Proof step 8:

  h[r2] = a × null, r2 ≠ null, ((ll-trav(r0, r1, r2, x) * hP))(h) ⊦ (r1 = null ⊃ r0 = r2 ∧ ((lls(r2, null, x) * hP))(h);∃ y, b. x = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h))

Rule argument

Additional information

Proof step 5 from proof ll-trav2-end (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  h[r2] = a × null, r2 ≠ null, ((ll-trav(r0, r1, r2, []) * hP))(h) ⊦

Rule argument

Additional information

Proof step 6 from proof ll-trav2-end (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 7:

  ⊦ r1 ≠ null → (((ll-trav(r0, r1, r2, x) * hP))(h) ↔ (∃ x0, x1, a. x = x0 + (a + x1) ∧ ((lls(r0, r1, x0) * r1 => a × r2 * lls(r2, null, x1) * hP))(h)))

Rule argument

Additional information

Used simplifier rules: ll-trav2-null-def: ⊦ ((ll-trav(r0, null, r2, x) * hP))(h) ↔ r2 = r0 ∧ ((lls(r0, null, x) * hP))(h); lls-base: ⊦ lls(r0, r1, []) = (emp ∧ ⌜r0 = r1⌝ ); ⊦ a + x = a ' + x; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (as) ⊦ a ' + x ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (⌜bv⌝ ∧ hP0) * hP1 = (⌜bv⌝ ∧ hP0 * hP1); (s) ⊦ emp * hP = hP; (s) ⊦ ((hP0 ∧ hP1))(h) ↔ hP0(h) ∧ hP1(h); (s) ⊦ ⌜false⌝ = false; (s) ⊦ ¬ false(h); (s) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification heap-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep Used terms: r2, h, [], hP, r0, r1

Proof step 7 from proof ll-trav2-end (previous step) (back to tree)

Lemma ll-trav2-nonnull-def

Proof step 8 from proof ll-trav2-end (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 9:

  h[r2] = a × null, r1 ≠ null, r2 ≠ null, ((ll-trav(r0, r1, r2, x) * hP))(h) ⊦ ∃ y, b. x = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h)

Rule argument

Additional information

Used simplifier rules: ll-trav2-null-def: ⊦ ((ll-trav(r0, null, r2, x) * hP))(h) ↔ r2 = r0 ∧ ((lls(r0, null, x) * hP))(h); sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep

Proof step 9 from proof ll-trav2-end (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  h[r2] = a × null, r1 ≠ null, r2 ≠ null, ((lls(r0, r1, x0) * r1 => a0 × r2 * lls(r2, null, x1) * hP))(h) ⊦ ∃ y, b. x0 + a0 ' + x1 = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h)

Rule argument

Additional information

Used simplifier rules: ll-trav2-nonnull-def: ⊦ r1 ≠ null → ( ((ll-trav(r0, r1, r2, x) * hP))(h) ↔ (∃ x0, x1, a. x = x0 + (a + x1) ∧ ((lls(r0, r1, x0) * r1 => a × r2 * lls(r2, null, x1) * hP))(h))); sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

Proof step 10 from proof ll-trav2-end (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 11:

  h[r2] = a × null, x1 = a1 + x, r1 ≠ null, r2 ≠ null, ((lls(r0, r1, x0) * r1 => a0 × r2 * lls(r2, null, x1) * hP))(h) ⊦ ∃ y, b. x0 + a0 ' + x1 = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h)

Rule argument

Additional information

Used simplifier rules: lls-base: ⊦ lls(r0, r1, []) = (emp ∧ ⌜r0 = r1⌝ ); list freely generated by [], +;; ⊦ ⌜false⌝ = false; (s) ⊦ (false ∧ hP) = false; (s) and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification heap-sep and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification heap-sep ⊦ 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 11 from proof ll-trav2-end (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  r1 ≠ null, r2 ≠ null, ((((r2 => a × null * lls(r0, r1, x0)) * r1 => a0 × r2) * hP))(h) ⊦ ∃ y, b. x0 + a0 ' = y + b ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h)

Rule argument

Additional information

Used simplifier rules: lls-rec: ⊦ lls(r, r1, a ' + x) = _∃(λ r0. r => a × r0 * lls(r0, r1, x)); lls-null: ⊦ lls(null, r, x) = (emp ∧ ⌜r = null⌝ ∧ ⌜x = []⌝ ); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + y = z + y ↔ x = z; (s) ⊦ x + [] = x; (s) and-com: ⊦ (hP0 ∧ hP1) = (hP1 ∧ hP0); from specification heap-sep and-ass: ⊦ ((hP0 ∧ hP1) ∧ hP2) = (hP0 ∧ hP1 ∧ hP2); from specification heap-sep ⊦ ⌜bv⌝ (h) ↔ bv; (s) ⊦ ((hP0 ∧ hP1))(h) ↔ hP0(h) ∧ hP1(h); (s) sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep ⊦ (⌜bv⌝ ∧ hP0) * hP1 = (⌜bv⌝ ∧ hP0 * hP1); (s) ⊦ emp * hP = hP; (s) ⊦ (true ∧ hP) = hP; (s) ⊦ ⌜true⌝ = true; (s) ⊦ a + x = a ' + x; (s) ⊦ a × r = a0 × r0 ↔ a = a0 ∧ r = r0; (ws) ⊦ ((r => o0 * hP))(h) → (h[r] = o1 ↔ o0 = o1); (s) ⊦ ((_∃ rP * hP))(h) ↔ (∃ r. ((rP(r) * hP))(h)); (s)

Proof step 12 from proof ll-trav2-end (previous step) (back to tree)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

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: x0, a0

Proof step 13 from proof ll-trav2-end (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  h[r2] = a × null, r2 ≠ null, x ≠ [], ¬ ((ll-trav(r0, r1, r2, x) * hP))(h), (r1 = null ⊃ r0 = r2 ∧ ((lls(r2, null, x) * hP))(h);x = (y + b ') + a ' ∧ ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h)) ⊦

Rule argument

Additional information

Proof step 14 from proof ll-trav2-end (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 15:

  h[r2] = a × null, x = (y + b ') + a ', r1 ≠ null, r2 ≠ null, x ≠ [], ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h), ¬ ((ll-trav(r0, r1, r2, x) * hP))(h) ⊦

Rule argument

Additional information

Used simplifier rules: ll-trav2-null-def: ⊦ ((ll-trav(r0, null, r2, x) * hP))(h) ↔ r2 = r0 ∧ ((lls(r0, null, x) * hP))(h); sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep

Proof step 15 from proof ll-trav2-end (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  r1 ≠ null, r2 ≠ null, ((lls(r0, r1, y) * r1 => b × r2 * r2 => a × null * hP))(h) ⊦ ∃ x0, x1, a0. (y + b ') + a ' = x0 + a0 ' + x1 ∧ ((lls(r0, r1, x0) * r1 => a0 × r2 * lls(r2, null, x1) * hP))(h)

Rule argument

Additional information

Used simplifier rules: ll-trav2-nonnull-def: ⊦ r1 ≠ null → ( ((ll-trav(r0, r1, r2, x) * hP))(h) ↔ (∃ x0, x1, a. x = x0 + (a + x1) ∧ ((lls(r0, r1, x0) * r1 => a × r2 * lls(r2, null, x1) * hP))(h))); sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s) ⊦ x + a ' ≠ []; (s) ⊦ ((r => o0 * hP))(h) → (h[r] = o1 ↔ o0 = o1); (s)

Proof step 16 from proof ll-trav2-end (previous step) (back to tree)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

Additional information

Used simplifier rules: lls-one: ⊦ lls(r0, r1, a ') = r0 => a × r1; a: ⊦ (x + y) + z = x + y + z; from specification list sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep Used terms: y, a ', b