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Proof step 8 from proof refine-get (previous step)

n = 0, r = rxs, x = [], ((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n0)))(H), # xs < h0, n0 ≤ # xs + # ys, n0 + # xs ≤ h0, ys = [] ∨ # ys = h0 ⊦ ⟪while n < h0 ∧ r ≠ null do {x := x + H[r].val '; r := H[r].nxt; n := n + 1}; r := rys; while n < h0 ∧ r ≠ null do {x := x + H[r].val '; r := H[r].nxt; n := n + 1}⟫ ⟨caterpillar_get#(h0, xs, ys; ; x0)⟩ ((∃ n0. ((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0) ∧ x = x0)

User applied rule invariant right

And obtained the following new goals

Proof step 9:
n2 = 0, r0 = rxs, x0 = [], ((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n0)))(H), # xs < h0, n < h0, n0 + # xs ≤ h0, n0 ≤ # xs + # ys, ∃ x1. ( xs = x + x1 ∧ ((lls(rxs, r, x) * ls(r, x1) * ls(rys, ys) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n = # x) ∧ n = n1, ys = [] ∨ # ys = h0 ⊦ ⟪x := x + H[r].val '; r := H[r].nxt; n := n + 1⟫ ∃ x1. ( xs = x + x1 ∧ ((lls(rxs, r, x) * ls(r, x1) * ls(rys, ys) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n = # x) ∧ h0 - n < h0 - n1, r = null
Proof step 16:
n2 = 0, r0 = rxs, x0 = [], ((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n0)))(H), # xs < h0, n0 + # xs ≤ h0, n0 ≤ # xs + # ys, ∃ x1. xs = x + x1 ∧ ((lls(rxs, r, x) * ls(r, x1) * ls(rys, ys) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n = # x, ys = [] ∨ # ys = h0 ⊦ ⟪r := rys; while n < h0 ∧ r ≠ null do {x := x + H[r].val '; r := H[r].nxt; n := n + 1}⟫ ⟨caterpillar_get#(h0, xs, ys; ; x0)⟩ ((∃ n0. ((ls(rxs, xs) * ls(rys, ys) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0) ∧ x = x0), n < h0 ∧ r ≠ null

Rule argument

inv: ∃ x1. xs = x + x1 ∧ (( lls(rxs, r, x) * ls(r, x1) * ls(rys, ys) * del-list(to_delete, n0)) )(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n = # x fmapos: right 1 bound: h0 - n

Additional information

Used simplifier rules: ⊦ [] + x = x; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ lls(r0, r1, []) = (emp ∧ ⌜r0 = r1⌝ ); (s)% ⊦ ⌜true⌝ = true; (s)% ⊦ (true ∧ P) = P; (s)%and-ass: ⊦ ((P0 ∧ P1) ∧ P2) = (P0 ∧ P1 ∧ P2); from specification heap-sep%and-com: ⊦ (P0 ∧ P1) = (P1 ∧ P0); from specification heap-sep% ⊦ emp * P = P; (s)%sep-ass: ⊦ (P0 * P1) * P2 = P0 * P1 * P2; from specification heap-sep%sep-com: ⊦ P0 * P1 = P1 * P0; from specification heap-sep% ⊦ # [] = 0; (s) Used formulas: ∃ x1. xs = x + x1 ∧ (( lls(rxs, r, x) * ls(r, x1) * ls(rys, ys) * del-list(to_delete, n0)) )(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n = # x, h0 - n = h0 - n