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Proof step 9 from proof size-rmdup-del (previous step)

# rmdup(a0 ' + x0) = m + 1, ∀ a. a ∈ x0 → # rmdup(x0 -l a) = # rmdup(x0) - 1 ⊦ # rmdup(x0 -l a0) = m

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
# rmdup(a0 ' + x0) = m + 1, # rmdup(x0) = m0 + 1, ∀ a. a ∈ x0 → # rmdup(x0 -l a) = m0 ⊦ # rmdup(x0 -l a0) = m, # rmdup(x0) = 0

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [# rmdup(x0), m0]

Additional information

Used simplifier rules: del-e: ⊦ [] -l a = [];% ⊦ # x = 0 ↔ x = []; (s)% ⊦ rmdup(x) = [] ↔ x = []; (s)% ⊦ x + [] = x; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ rmdup(a ') = a '; (s)% ⊦ # a ' = 0 +1; (s)% ⊦ 1 = m + 1 ↔ m = 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ rmdup([]) = []; (s)% ⊦ # [] = 0; (s)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # rmdup(x0), m0