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Proof step 1 from proof dlls-fold-head (conclusion) (back to tree)

⊦ (r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ↔ (dlls(r, r0, r2, a ' + x) * hP)(h) ∧ True((h(r)).right = r1)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof dlls-fold-head (previous step) (back to tree)

¬((r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ↔ (dlls(r, r0, r2, a ' + x) * hP)(h) ∧ True((h(r)).right = r1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: dlls-rec: ⊦ dlls(r, r0, r1, a ' + x) = _∃(λ r2. r => mknode(r0, a, r2) * dlls(r2, r, r1, x)); 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 3 from proof dlls-fold-head (previous step) (back to tree)

⊦ (r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ↔ (∃ r1. ((r => mknode(r0, a, r1) * dlls(r1, r, r2, x)) * hP)(h)) ∧ True((h(r)).right = r1)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: node, name:elim-mknode, seq: ⊦ (left = nd.left ∧ val = nd.val) ∧ right = nd.right ↔ nd = mknode(left, val, right), subst:[nd, left, val, right] → [h(r), r3, a0, r4]

Additional information

Used simplifier rules: ⊦ (left = nd.left ∧ val = nd.val) ∧ right = nd.right ↔ nd = mknode(left, val, right); (elim) Used terms: h(r), r3, a0, r4

Proof step 4 from proof dlls-fold-head (previous step) (back to tree)

h(r) = mknode(r3, a0, r4) ⊦ (r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ↔ (∃ r1. ((r => mknode(r0, a, r1) * dlls(r1, r, r2, x)) * hP)(h)) ∧ True(r4 = r1)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 5 from proof dlls-fold-head (previous step) (back to tree)

h(r) = mknode(r3, a0, r4), (r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ⊦ (∃ r1. ((r => mknode(r0, a, r1) * dlls(r1, r, r2, x)) * hP)(h)) ∧ True(r4 = r1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ mknode(left, val, left0) = mknode(left1, val0, left2) ↔ left = left1 ∧ val = val0 ∧ left0 = left2; (ws) ⊦ (r => o0 * hP)(h) → h(r) = o0; (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep

Proof step 6 from proof dlls-fold-head (previous step) (back to tree)

(r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ⊦ (∃ r1. ((r => mknode(r0, a, r1) * dlls(r1, r, r2, x)) * hP)(h)) ∧ True(true)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 7 from proof dlls-fold-head (previous step) (back to tree)

(r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ⊦ ∃ r1. ((r => mknode(r0, a, r1) * dlls(r1, r, r2, x)) * hP)(h)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: r1

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: r1

Proof step 8 from proof dlls-fold-head (previous step) (back to tree)

(r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h), ¬ True(true) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: refmap-separation, name: True, seq: ⊦ True(bv) ↔ bv, subst:[bv] → [true], <all paths>

Additional information

Used terms: true

Proof step 9 from proof dlls-fold-head (previous step) (back to tree)

⊦ True(bv) ↔ bv

A spec-lemma

Proof step 10 from proof dlls-fold-head (previous step) (back to tree)

h(r) = mknode(r3, a0, r4), True(r4 = r1), ¬ (r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h), ∃ r1. ((r => mknode(r0, a, r1) * dlls(r1, r, r2, x)) * hP)(h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ mknode(left, val, left0) = mknode(left1, val0, left2) ↔ left = left1 ∧ val = val0 ∧ left0 = left2; (ws) ⊦ (r => o0 * hP)(h) → h(r) = o0; (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification map-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification map-sep

Proof step 11 from proof dlls-fold-head (previous step) (back to tree)

True(false), ((r => mknode(r0, a, r4) * dlls(r4, r, r2, x)) * hP)(h), ¬ (r => mknode(r0, a, r1) * dlls(r1, r, r2, x) * hP)(h) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: refmap-separation, name: True, seq: ⊦ True(bv) ↔ bv, subst:[bv] → [false], <all paths>

Additional information

Used terms: false

Proof step 12 from proof dlls-fold-head (previous step) (back to tree)

⊦ True(bv) ↔ bv

A spec-lemma

Proof Tree for dlls-fold-head (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