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Proof step 1 from proof assign-known-weak (conclusion) (back to tree)

⊦ h1 ≡ h0(r := o1) → ((r => o0 * hP)(h0) ↔ (r => o1 * hP)(h1) ∧ h0(r) = o0 ∧ r ∈ h0)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof assign-known-weak (previous step) (back to tree)

¬((r => o0 * hP)(h0) ↔ (r => o1 * hP)(h1) ∧ h0(r) = o0 ∧ r ∈ h0), h1 ≡ h0(r := o1) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Used simplifier rules: eq: ⊦ h0 ≡ h1 ↔ h0 = h1; ⊦ (r => o0 * hP)(h) → h(r) = o0; (s) ⊦ (r => o * hP)(h) → r ∈ h; (ss) r ∈ h ⊦ (r => o * hP)(h) → ((r => o0 * hP)(h(r := o1)) ↔ o0 = o1); (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 assign-known-weak (previous step) (back to tree)

h0(r) = o0, h1 ≡ h0(r := o1), (r => o1 * hP)(h1), r ∈ h0, ¬ (r => o0 * hP)(h0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: eq: ⊦ h0 ≡ h1 ↔ h0 = h1; sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep map-write: r ∈ h ⊦ (r => o0 * hP)(h(r := o1)) → o0 = o1; (forward) from specification maplet

Proof step 4 from proof assign-known-weak (previous step) (back to tree)

(r => o1 * hP)(h0(r := o1)), r ∈ h0, ¬ (r => h0(r) * hP)(h0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: ex-maplet, name:map-putex, seq:r ∈ h ⊦ (r => o0 * hP)(h(r := o1)) ↔ o0 = o1 ∧ (r.=>_ * hP)(h), subst:[r, o0, h, hP, o1] → [r, o1, h0, hP, o1]; [A] → [A], lhs:(r => o1 * hP)(h0(r := o1)), <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: r, o1, h0, hP, o1

Proof step 5 from proof assign-known-weak (previous step) (back to tree)

r ∈ h ⊦ (r => o0 * hP)(h(r := o1)) ↔ o0 = o1 ∧ (r.=>_ * hP)(h)

A spec-lemma

Proof step 6 from proof assign-known-weak (previous step) (back to tree)

o1 = o1, (r.=>_ * hP)(h0), r ∈ h0, ¬ (r => h0(r) * hP)(h0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ (r.=>_ * hP)(h) → r ∈ h; (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep

Proof step 7 from proof assign-known-weak (previous step) (back to tree)

(r.=>_ * hP)(h0), ¬ (r => h0(r) * hP)(h0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: ex-maplet, name:mapex, seq: ⊦ r.=>_ = _∃(λ o. r => o), subst:[r] → [r]; [A] → [A], lhs:r.=>_, <all paths>

Additional information

Used simplifier rules: ⊦ (_∃ bP * hP)(h) ↔ (∃ b. (bP(b) * hP)(h)); (s) sep-ass: ⊦ (hP0 * hP1) * hP2 = hP0 * hP1 * hP2; from specification heap-sep sep-com: ⊦ hP0 * hP1 = hP1 * hP0; from specification heap-sep ⊦ (r => o0 * hP)(h) → h(r) = o0; (s) Used terms: r

Proof step 8 from proof assign-known-weak (previous step) (back to tree)

⊦ r.=>_ = _∃(λ o. r => o)

A spec-lemma

Proof Tree for assign-known-weak (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