Proof step 1 from proof precise-alt-def (conclusion) (back to tree)

⊦ precise(hP) ↔ (∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1)

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 2:
⊦ precise(hP) ↔ (∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0)
Proof step 3:
⊦ (∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0) ↔ (∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1)

Rule argument

spec: <toplevel>, name:precise, seq:⊦ precise(hP) ↔ (∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0), subst:[hP] → [hP], <all paths>

Additional information

Used terms: hP

Proof step 2 from proof precise-alt-def (previous step) (back to tree)

⊦ precise(hP) ↔ (∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0)

Lemma precise

Proof step 3 from proof precise-alt-def (previous step) (back to tree)

⊦ (∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0) ↔ (∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1)

User applied rule case distinction

And obtained the following new goals

Proof step 4:
∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦ ∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1
Proof step 12:
∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1 ⊦ ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0

Rule argument

pos: right 1

Additional information

Proof step 4 from proof precise-alt-def (previous step) (back to tree)

∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦ ∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
h0 ≠ h1, hP(h0), hP(h1), h0 ⊆ h, h1 ⊆ h, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦

Rule argument

Additional information

Proof step 5 from proof precise-alt-def (previous step) (back to tree)

h0 ≠ h1, hP(h0), hP(h1), h0 ⊆ h, h1 ⊆ h, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)
Proof step 7:
h0 ≠ h1, hP(h1), hP(h0), h1 ⊆ h, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0, ∃ h2. h0 ∪ h2 = h ∧ h2 ⊥ h0 ⊦

Rule argument

spec: heap, name:subset-uni, seq: ⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0), subst:[h0, h1] → [h0, h], <all paths>

Additional information

Used terms: h0, h

Proof step 6 from proof precise-alt-def (previous step) (back to tree)

⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)

A spec-lemma

Proof step 7 from proof precise-alt-def (previous step) (back to tree)

h0 ≠ h1, hP(h1), hP(h0), h1 ⊆ h, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0, ∃ h2. h0 ∪ h2 = h ∧ h2 ⊥ h0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
h0 ≠ h1, hP(h0), hP(h1), h1 ⊆ h0 ∪ h2, h2 ⊥ h0, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦

Rule argument

Additional information

Proof step 8 from proof precise-alt-def (previous step) (back to tree)

h0 ≠ h1, hP(h0), hP(h1), h1 ⊆ h0 ∪ h2, h2 ⊥ h0, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)
Proof step 10:
h0 ≠ h1, hP(h1), hP(h0), h2 ⊥ h0, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0, ∃ h. h1 ∪ h = h0 ∪ h2 ∧ h ⊥ h1 ⊦

Rule argument

spec: heap, name:subset-uni, seq: ⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0), subst:[h0, h1] → [h1, h0 ∪ h2], <all paths>

Additional information

Used terms: h1, h0 ∪ h2

Proof step 9 from proof precise-alt-def (previous step) (back to tree)

⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)

A spec-lemma

Proof step 10 from proof precise-alt-def (previous step) (back to tree)

h0 ≠ h1, hP(h1), hP(h0), h2 ⊥ h0, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0, ∃ h. h1 ∪ h = h0 ∪ h2 ∧ h ⊥ h1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
h1 ∪ h = h0 ∪ h2, h0 ≠ h1, hP(h0), hP(h1), h ⊥ h1, h2 ⊥ h0, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦

Rule argument

Additional information

Proof step 11 from proof precise-alt-def (previous step) (back to tree)

h1 ∪ h = h0 ∪ h2, h0 ≠ h1, hP(h0), hP(h1), h ⊥ h1, h2 ⊥ h0, ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: h0, h1, h2, h abort: F discard: T computed: T

Additional information

Used simplifier rules: ⊦ h0 ⊥ h1 → h1 ⊥ h0; (ss) Used terms: h0, h1, h2, h

Proof step 12 from proof precise-alt-def (previous step) (back to tree)

∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1 ⊦ ∀ h, h0, h1, h2. hP(h) ∧ hP(h0) ∧ h ⊥ h1 ∧ h0 ⊥ h2 ∧ h ∪ h1 = h0 ∪ h2 → h = h0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2, ∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1 ⊦

Rule argument

Additional information

Proof step 13 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2, ∀ h, h0, h1. h0 ⊆ h ∧ h1 ⊆ h ∧ hP(h0) ∧ hP(h1) → h0 = h1 ⊦

User applied rule all left

And obtained the following new goals

Proof step 14:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, h ⊆ h0 ∪ h2 ∧ h0 ⊆ h0 ∪ h2 ∧ hP(h) ∧ hP(h0) → h = h0 ⊦

Rule argument

pos: left 7 terms: h0 ∪ h2, h, h0 abort: F discard: T computed: F

Additional information

Used terms: h0 ∪ h2, h, h0

Proof step 14 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, h ⊆ h0 ∪ h2 ∧ h0 ⊆ h0 ∪ h2 ∧ hP(h) ∧ hP(h0) → h = h0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2 ⊦ h ⊆ h0 ∪ h2 ∧ h0 ⊆ h0 ∪ h2

Rule argument

Additional information

Proof step 15 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2 ⊦ h ⊆ h0 ∪ h2 ∧ h0 ⊆ h0 ∪ h2

User applied rule case distinction

And obtained the following new goals

Proof step 16:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ¬ h ⊆ h0 ∪ h2 ⊦
Proof step 20:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ¬ h0 ⊆ h0 ∪ h2 ⊦

Rule argument

pos: right 1

Additional information

Proof step 16 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ¬ h ⊆ h0 ∪ h2 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)
Proof step 18:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2, ∀ h3. ¬(h ∪ h3 = h0 ∪ h2 ∧ h3 ⊥ h) ⊦

Rule argument

spec: heap, name:subset-uni, seq: ⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0), subst:[h0, h1] → [h, h0 ∪ h2], <all paths>

Additional information

Used terms: h, h0 ∪ h2

Proof step 17 from proof precise-alt-def (previous step) (back to tree)

⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)

A spec-lemma

Proof step 18 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2, ∀ h3. ¬(h ∪ h3 = h0 ∪ h2 ∧ h3 ⊥ h) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ∀ h3. h ∪ h3 ≠ h0 ∪ h2 ∨ ¬ h3 ⊥ h ⊦

Rule argument

Additional information

Proof step 19 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ∀ h3. h ∪ h3 ≠ h0 ∪ h2 ∨ ¬ h3 ⊥ h ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: h1

Additional information

Used simplifier rules: ⊦ h0 ⊥ h1 → h1 ⊥ h0; (ss) Used terms: h1

Proof step 20 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ¬ h0 ⊆ h0 ∪ h2 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)
Proof step 22:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2, ∀ h. ¬(h0 ∪ h = h0 ∪ h2 ∧ h ⊥ h0) ⊦

Rule argument

spec: heap, name:subset-uni, seq: ⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0), subst:[h0, h1] → [h0, h0 ∪ h2], <all paths>

Additional information

Used terms: h0, h0 ∪ h2

Proof step 21 from proof precise-alt-def (previous step) (back to tree)

⊦ h0 ⊆ h1 ↔ (∃ h. h0 ∪ h = h1 ∧ h ⊥ h0)

A spec-lemma

Proof step 22 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h), hP(h0), h ⊥ h1, h0 ⊥ h2, ∀ h. ¬(h0 ∪ h = h0 ∪ h2 ∧ h ⊥ h0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 23:
h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ∀ h3. h0 ∪ h3 ≠ h0 ∪ h2 ∨ ¬ h3 ⊥ h0 ⊦

Rule argument

Additional information

Proof step 23 from proof precise-alt-def (previous step) (back to tree)

h ∪ h1 = h0 ∪ h2, h ≠ h0, hP(h0), hP(h), h0 ⊥ h2, h ⊥ h1, ∀ h3. h0 ∪ h3 ≠ h0 ∪ h2 ∨ ¬ h3 ⊥ h0 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: h2

Additional information

Used simplifier rules: ⊦ h0 ⊥ h1 → h1 ⊥ h0; (ss) Used terms: h2

Proof Tree for precise-alt-def (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.