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

⊦ a ∈ st ∧ a ∈ s → (st | s)[a] = st[a]

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

(st | s)[a] ≠ st[a], a ∈ st, a ∈ s ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict: ⊦ st | s = st \ (dom st \ s); size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic

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

# s ≠ 0, (st \ (dom st \ s))[a] ≠ st[a], s ≠ ∅, a ∈ st, a ∈ s ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:Diff-get, seq: ⊦ ¬ a ∈ s → (st \ s)[a] = st[a], subst:[a, s, st] → [a, dom st \ s, st], <all paths>

Additional information

Used simplifier rules: Diff-mems-get: ⊦ a ∈ s → (st \ s)[a] = ∅[a]; Dom: ⊦ a ∈ dom st ↔ a ∈ st; ⊦ a ∈ s1 \ s2 ↔ a ∈ s1 ∧ ¬ a ∈ s2; (ws) Used terms: a, dom st \ s, st

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

⊦ ¬ a ∈ s → (st \ s)[a] = st[a]

Lemma Diff-get

Proof Tree for restrict-at (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