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Proof step 4 from proof transaction-prefix-is-transaction-prefix (previous step)

# gndx = # ndx, gndx = gnd + gndx0, gndx.last.end, gnd.start, n < # ndx, ∀ n0. (n0 ≠ 0 ∧ n0 < # ndx → ¬ gndx[n0].start) ∧ (n0 + 1 < # ndx → ¬ gndx[n0].end), ∀ n. n < # ndx → gndx[n].nd = ndx[n] ⊦ gndx = [], ndx = [], ( prefixn(n, gndx) = [] ∨ prefixn(n, gndx).head.start ∧ (n0 ≠ 0 ∧ n0 < n → ¬ prefixn(n, gndx)[n0].start) ∧ (n0 < n → ¬ prefixn(n, gndx)[n0].end)) ∧ (n1 < n → prefixn(n, gndx)[n1].nd = ndx[n1])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
# gndx0 + 1 = # ndx, (gnd ' + gndx0).last.end, gnd.start, n < # ndx, n ≤ # gndx0, ∀ n2. (n2 ≠ 0 ∧ n2 < # ndx → ¬ (gnd ' + gndx0)[n2].start) ∧ (n2 + 1 < # ndx → ¬ (gnd ' + gndx0)[n2].end), ∀ n2. n2 < # ndx → (gnd ' + gndx0)[n2].nd = ndx[n2] ⊦ ndx = [], ( prefixn(n, gnd ' + gndx0) = [] ∨ prefixn(n, gnd ' + gndx0).head.start ∧ (n0 ≠ 0 ∧ n0 < n → ¬ prefixn(n, gnd ' + gndx0)[n0].start) ∧ (n0 < n → ¬ prefixn(n, gnd ' + gndx0)[n0].end)) ∧ (n1 < n → prefixn(n, gnd ' + gndx0)[n1].nd = ndx[n1])

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ gnd ' + gndx ≠ []; (s)%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]% ⊦ m +1 = m + 1; (s)% ⊦ #(gnd ' + gndx) = (# gndx)+1; (s)% ⊦ gnd + gndx = gnd ' + gndx; (s)