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Proof step 9 from proof rmpos-size (previous step)

n = n1 +1, n ≤ # x0, ∀ m, n0. # x0 = m + 1 ∧ n0 ≤ m → #(x0 -1l n0) = m ⊦ x0 = [], #((a ' + x0) -1l n) = # x0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
n1 < # x0, ∀ m, n0. # x0 = m + 1 ∧ n0 ≤ m → #(x0 -1l n0) = m ⊦ #(x0 -1l n1) + 1 = # x0, x0 = []

Rule argument

Additional information

Used simplifier rules: delpos-rec: ⊦ (a ' + x) -1l (n + 1) = a + (x -1l n);%l: ⊦ # x = 0 ↔ x = []; from specification list-data%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%succ-01: ⊦ m + 1 ≤ n ↔ m < n; from specification nat%len-05: ⊦ #(a ' + x) = (# x)+1; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ a + x = a ' + x; from specification list%succ: ⊦ m +1 = m + 1; from specification nat-basic2