Proof step 1 from proof dom-restrict (conclusion) (back to tree)

⊦ s ⊆ dom st → dom(st | s) = s

System applied rule normalize

And obtained the following new goals

Proof step 2:
dom(st | s) ≠ s, s ⊆ dom st ⊦

Rule argument

Additional information

Proof step 2 from proof dom-restrict (previous step) (back to tree)

dom(st | s) ≠ s, s ⊆ dom st ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
dom st \ dom st \ s ≠ s, s ⊆ dom st ⊦

Rule argument

Additional information

Used simplifier rules: restrict: ⊦ st | s = st \ (dom st \ s); dom-rem: ⊦ dom(st \ s) = dom st \ s;

Proof step 3 from proof dom-restrict (previous step) (back to tree)

dom st \ dom st \ s ≠ s, s ⊆ dom st ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2)
Proof step 5:
s ⊆ dom st, ∃ a. ¬(a ∈ dom st \ dom st \ s ↔ a ∈ s) ⊦

Rule argument

spec: polyset-basic, name:Extension, seq: ⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2), subst:[s2, s1] → [s, dom st \ dom st \ s], <all paths>

Additional information

Used terms: s, dom st \ dom st \ s

Proof step 4 from proof dom-restrict (previous step) (back to tree)

⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2)

A spec-lemma

Proof step 5 from proof dom-restrict (previous step) (back to tree)

s ⊆ dom st, ∃ a. ¬(a ∈ dom st \ dom st \ s ↔ a ∈ s) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
¬(a ∈ dom st \ dom st \ s ↔ a ∈ s), s ⊆ dom st ⊦

Rule argument

Additional information

Proof step 6 from proof dom-restrict (previous step) (back to tree)

¬(a ∈ dom st \ dom st \ s ↔ a ∈ s), s ⊆ dom st ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ a ∈ s1 \ s2 ↔ a ∈ s1 ∧ ¬ a ∈ s2

Rule argument

spec: polyset-union, name:Difference, seq: ⊦ a ∈ s1 \ s2 ↔ a ∈ s1 ∧ ¬ a ∈ s2, subst:[s2, s1, a] → [dom st \ s, dom st, a], <all paths>

Additional information

Used simplifier rules: Dom: ⊦ a ∈ dom st ↔ a ∈ st; sub-in: ⊦ a ∈ s ∧ s ⊆ s0 → a ∈ s0; (forward) from specification polyset-basic ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification polyset-basic ⊦ a ∈ s1 \ s2 ↔ a ∈ s1 ∧ ¬ a ∈ s2; (ws) Used terms: dom st \ s, dom st, a

Proof step 7 from proof dom-restrict (previous step) (back to tree)

⊦ a ∈ s1 \ s2 ↔ a ∈ s1 ∧ ¬ a ∈ s2

A spec-lemma

Proof Tree for dom-restrict (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.