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Proof step 12 from proof shift-03 (previous step)

a ∈ y, a ∈ y -1l a0 → (perm(x1 + a ' + x0, y -1l a0) ↔ perm(x1 + x0, (y -1l a0) -1l a)) ⊦ a0 ∈ y ∧ perm(x1 + a ' + x0, y -1l a0) ↔ a0 ∈ y -1l a ∧ perm(x1 + x0, (y -1l a0) -1l a), a ∈ y -1l a0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
perm(x1 + a0 ' + x0, y -1l a0), perm(y -1l a0, x1 + a0 ' + x0), a0 ∈ y ⊦ a0 ∈ y -1l a0

Rule argument

Additional information

Used simplifier rules: sym-01: ⊦ perm(x, y) → perm(y, x);%trans: ⊦ perm(x, y) ∧ perm(y, z) → perm(x, z);%app: ⊦ perm(x + y, x + z) ↔ perm(y, z);%ref: ⊦ perm(x, x);%a: ⊦ (x + y) + z = x + y + z; from specification list%in-05: ⊦ ¬ a ∈ y -1l b ∧ a ∈ y → a = b; (forward) from specification list-del% ⊦ ¬ a ∈ x → x -1l a = x; (s)