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Proof step 7 from proof perm-ins (previous step)

perm(x + ins≤(a, y), ins≤(a, y) + x) ⊦ perm((x + ins≤(a, y)) -1l a, z) ↔ perm(x + y, z), perm(x + y, (x + ins≤(a, y)) -1l a), perm((ins≤(a, y) + x) -1l a, x + y)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ a ∈ x → (x + y) -1l a = (x -1l a) + y

Rule argument

spec: list-del inst: name: app seq: ⊦ a ∈ x → (x + y) -1l a = (x -1l a) + y subst: [a, y, x] → [a, x, ins≤(a, y)] flags: rewrite

Additional information

Used simplifier rules: insperm-01: ⊦ perm(ins≤(a, x), ins≤(a, y)) ↔ perm(x, y);%in-04: ⊦ a ∈ ins≤(a, x);%in-04: ⊦ a ∈ ins≤(a, x);%insperm-01: ⊦ perm(ins≤(a, x), ins≤(a, y)) ↔ perm(x, y);%del1-02: ⊦ ins≤(a, x) -1l a = x;%app-02: ⊦ perm(x + y, z + x) ↔ perm(y, z); from specification list-perm%ref: ⊦ perm(x, x); from specification list-perm%a: ⊦ (x + y) + z = x + y + z; from specification list%app-01: ⊦ ¬ a ∈ x → (x + y) -1l a = x + (y -1l a); from specification list-del Used terms: a, x, ins≤(a, y)