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Proof step 8 from proof put-del1-in (previous step)

m0 < # x0, ∀ m, a. x0[m] ≠ a ∧ m < # x0 → (a ∈ x0[m, a] -1l a ↔ a ∈ x0) ⊦ x0[m0] = a, a ∈ (a0 ' + x0[m0, a]) -1l a ↔ a = a0 ∨ a ∈ x0, x0 = []

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 9:
m0 < # x0, ∀ m, a. x0[m] ≠ a ∧ m < # x0 → (a ∈ x0[m, a] -1l a ↔ a ∈ x0) ⊦ a = a0, x0[m0] = a, a ∈ (a0 ' + x0[m0, a]) -1l a ↔ a = a0 ∨ a ∈ x0, x0 = []

Rule argument

fma: a = a0

Additional information

Used simplifier rules: del1-n: ⊦ a ≠ b → (b ' + x) -1l a = b ' + (x -1l a);%del1-in: ⊦ a ≠ b → (a ∈ x -1l b ↔ a ∈ x);%put-in: ⊦ a ≠ x[m] ∧ m < # x → (a ∈ x[m, b] ↔ a = b ∨ a ∈ x);%del1-y: ⊦ (a ' + x) -1l a = x;%put-in-02: ⊦ m < # x → a ∈ x[m, a];%a: ⊦ (x + y) + z = x + y + z; from specification list%in-01: ⊦ a ≠ b → (a ∈ b ' + x ↔ a ∈ x); from specification list Used formulas: a = a0