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Proof step 14 from proof linearity (previous step)

path(r ' + x1, H), H[(r ' + x1).last].nxt ∈ H, reachable(r, r1, H), path(r ' + x1 + H[(r ' + x1).last].nxt ', H), r1 ∈ H, ∀ x0, r2, H0, r1. # x0 < #(x1 + H[(r ' + x1).last].nxt ') → path(r2 ' + x0, H0) ∧ reachable(r2, r1, H0) ∧ (r2 ' + x0).last ∈ H0 ∧ r1 ∈ H0 → reachable((r2 ' + x0).last, r1, H0) ∨ reachable(r1, (r2 ' + x0).last, H0) ⊦ H[(r ' + x1).last].nxt = null, reachable(r1, H[(r ' + x1).last].nxt, H), reachable(H[(r ' + x1).last].nxt, r1, H)

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [x0, r2, H0, r1] → [x1, r, H, r1]

Additional information

Used simplifier rules: path-butlast: ⊦ path(r ' + x + r0 ', H) → (¬ path(r ' + x, H) ↔ false);%path-prelast: ⊦ path(r ' + x + r0 ', H) → (¬ (r ' + x).last ∈ H ↔ false);%reachable-notnullinH: ⊦ reachable(r, r1, H) → (¬ r1 ∈ H ↔ false);%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%len: ⊦ #(x0 + x1) = # x0 + # x1; from specification list[reflist]%len-02: ⊦ # r ' = 0 +1; from specification list[reflist]%s: ⊦ m < n + 1 ↔ ¬ n < m; from specification nat-basic2%ref-1: ⊦ ¬ n < n; from specification nat-basic1%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2 Used terms: x1, r, H, r1