Proof step 1 from proof mapex-put (conclusion) (back to tree)

r ∈ h ⊦ (r.=>_ * hP)(h(r := o)) ↔ (r => o * hP)(h(r := o))

System applied rule normalize

And obtained the following new goals

Proof step 2:
¬((r.=>_ * hP)(h(r := o)) ↔ (r => o * hP)(h(r := o))), r ∈ h ⊦

Rule argument

Additional information

Proof step 2 from proof mapex-put (previous step) (back to tree)

¬((r.=>_ * hP)(h(r := o)) ↔ (r => o * hP)(h(r := o))), r ∈ h ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
r ∈ h ⊦ (∃ b. (r => b * hP)(h(r := o))) ↔ (r => o * hP)(h(r := o))

Rule argument

Additional information

Used simplifier rules: mapex: ⊦ r.=>_ = _∃(λ o. r => o); 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)

Proof step 3 from proof mapex-put (previous step) (back to tree)

r ∈ h ⊦ (∃ b. (r => b * hP)(h(r := o))) ↔ (r => o * hP)(h(r := o))

User applied rule case distinction

And obtained the following new goals

Proof step 4:
r ∈ h, (r => o * hP)(h(r := o)) ⊦ ∃ b. (r => b * hP)(h(r := o))

Rule argument

pos: right 1

Additional information

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

Proof step 4 from proof mapex-put (previous step) (back to tree)

r ∈ h, (r => o * hP)(h(r := o)) ⊦ ∃ b. (r => b * hP)(h(r := o))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
(r => o * hP)(h(r := o)), r ∈ h ⊦ ∃ b. (r => b * hP)(h(r := o))

Rule argument

Additional information

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

Proof step 5 from proof mapex-put (previous step) (back to tree)

(r => o * hP)(h(r := o)), r ∈ h ⊦ ∃ b. (r => b * hP)(h(r := o))

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: o

Additional information

Used terms: o

Proof Tree for mapex-put (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.