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

⊦ dom(st0 ⊕ st1) = dom st0 ∪ dom st1

System applied rule normalize

And obtained the following new goals

Proof step 2:
dom(st0 ⊕ st1) ≠ dom st0 ∪ dom st1 ⊦

Rule argument

Additional information

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

dom(st0 ⊕ st1) ≠ dom st0 ∪ dom st1 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2)
Proof step 4:
∃ a. ¬(a ∈ dom(st0 ⊕ st1) ↔ a ∈ dom st0 ∪ dom st1) ⊦

Rule argument

spec: set-basic, name:Extension, seq: ⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2), subst:[s2, s1] → [dom st0 ∪ dom st1, dom(st0 ⊕ st1)], <all paths>

Additional information

Used terms: dom st0 ∪ dom st1, dom(st0 ⊕ st1)

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

⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2)

A spec-lemma

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

∃ a. ¬(a ∈ dom(st0 ⊕ st1) ↔ a ∈ dom st0 ∪ dom st1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
¬(a ∈ st0 ⊕ st1 ↔ a ∈ dom st0 ∪ dom st1) ⊦

Rule argument

Additional information

Used simplifier rules: Dom: ⊦ a ∈ dom st ↔ a ∈ st;

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

¬(a ∈ st0 ⊕ st1 ↔ a ∈ dom st0 ∪ dom st1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ a ∈ s1 ∪ s2 ↔ a ∈ s1 ∨ a ∈ s2

Rule argument

spec: set-union, name:Union, seq: ⊦ a ∈ s1 ∪ s2 ↔ a ∈ s1 ∨ a ∈ s2, subst:[s2, s1, a] → [dom st1, dom st0, a], <all paths>

Additional information

Used simplifier rules: Dom: ⊦ a ∈ dom st ↔ a ∈ st; oplus-l-in: ⊦ a ∈ st0 ⊕ st1 ↔ a ∈ st0 ∨ a ∈ st1; Used terms: dom st1, dom st0, a

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

⊦ a ∈ s1 ∪ s2 ↔ a ∈ s1 ∨ a ∈ s2

A spec-lemma

Proof Tree for dom-union-l (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.