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Proof step 6 from proof rmpos-size (previous step)

n ≤ # x0, ∀ m, n. # x0 = m + 1 ∧ n ≤ m → #(x0 -1l n) = m ∨ x0 = [] ⊦ #((a ' + x0) -1l n) = # x0

Heuristic cut applied rule cut formula

And obtained the following new goals

Proof step 7:
n ≤ # x0, ∀ m, n. # x0 = m + 1 ∧ n ≤ m → #(x0 -1l n) = m ∨ x0 = [] ⊦ x0 = [], #((a ' + x0) -1l n) = # x0

Rule argument

fma: x0 = []

Additional information

Used simplifier rules: delnr-one: ⊦ a ' -1l 0 = [];%app: ⊦ x + [] = x; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%size-01: ⊦ # [] = 0; from specification list-data%l: ⊦ # x = 0 ↔ x = []; from specification list-data%lez-01: ⊦ m ≤ 0 ↔ m = 0; from specification nat%oz: ⊦ m + 1 ≠ 0; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2 Used formulas: x0 = []