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Proof step 9 from proof valid-group-nodes-shorter-prefix (previous step)

valid-group-nodes(prefixn(n0, gndx + gnd ')) < valid-group-nodes(prefixn(n, gndx + gnd ')), valid-group-nodes(prefixn(n0, gndx + gnd ')) < # prefixn(n, gndx + gnd '), n ≤ n0, ∀ n0, n1. valid-group-nodes(prefixn(n0, gndx)) < valid-group-nodes(prefixn(n1, gndx)) ∧ valid-group-nodes(prefixn(n0, gndx)) < # prefixn(n1, gndx) ∧ n1 ≤ n0 → gndx = [] ∨ n1 = 0 ⊦ n = 0, n0 ≤ # gndx

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
# gndx < n0, (gnd.end ⊃ # gndx + 1;valid-group-nodes(gndx)) < valid-group-nodes(prefixn(n, gndx + gnd ')), (gnd.end ⊃ # gndx + 1;valid-group-nodes(gndx)) < # prefixn(n, gndx + gnd '), n ≤ n0, ∀ n0, n1. valid-group-nodes(prefixn(n0, gndx)) < valid-group-nodes(prefixn(n1, gndx)) ∧ valid-group-nodes(prefixn(n0, gndx)) < # prefixn(n1, gndx) ∧ n1 ≤ n0 → gndx = [] ∨ n1 = 0 ⊦ n = 0

Rule argument

Additional information

Used simplifier rules: valid-group-nodes-rec: ⊦ valid-group-nodes(gndx + gnd ') = (gnd.end ⊃ # gndx + 1;valid-group-nodes(gndx));%valid-group-nodes-ls: ⊦ n < valid-group-nodes(gndx) → n < # gndx;% ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≤ n ↔ ¬ n < m; (ss)%# gndx ≤ m ⊦ prefixn(m, gndx) = gndx; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]% ⊦ # gnd ' = 0 +1; (s)% ⊦ #(gndx + gndx0) = # gndx + # gndx0; (s)