[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 18 from proof refine-get (previous step)

n = # x, r = rys, xs = x, (((lls(rxs, null, x) * ls(rys, ys)) * del-list(to_delete, n0)))(H), ((ls(rxs, x) * ls(rys, ys) * del-list(to_delete, n0)))(H), # x < h0, n0 ≤ # x + # ys, n0 + # x ≤ h0, ys = [] ∨ # ys = h0 ⊦ ⟪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 19:
n2 = # x0, r0 = rys, xs = x0, ((ls(rxs, x0) * ls(rys, ys) * del-list(to_delete, n0)))(H), (((lls(rxs, null, x0) * ls(rys, ys)) * del-list(to_delete, n0)))(H), # x0 < h0, n < h0, n0 + # x0 ≤ h0, n0 ≤ # x0 + # ys, ∃ x1, y1. ( x = xs + x1 ∧ ys = x1 + y1 ∧ ((ls(rxs, xs) * lls(rys, r, x1) * ls(r, y1) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n ≤ h0 ∧ n = # x) ∧ n = n1, ys = [] ∨ # ys = h0 ⊦ ⟪x := x + H[r].val '; r := H[r].nxt; n := n + 1⟫ ∃ x1, y1. ( x = xs + x1 ∧ ys = x1 + y1 ∧ ((ls(rxs, xs) * lls(rys, r, x1) * ls(r, y1) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n ≤ h0 ∧ n = # x) ∧ h0 - n < h0 - n1, r = null
Proof step 26:
n2 = # x0, r0 = rys, xs = x0, ((ls(rxs, x0) * ls(rys, ys) * del-list(to_delete, n0)))(H), (((lls(rxs, null, x0) * ls(rys, ys)) * del-list(to_delete, n0)))(H), # x0 < h0, n0 + # x0 ≤ h0, n0 ≤ # x0 + # ys, ∃ x1, y1. x = xs + x1 ∧ ys = x1 + y1 ∧ ((ls(rxs, xs) * lls(rys, r, x1) * ls(r, y1) * del-list(to_delete, n0)))(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n ≤ h0 ∧ n = # x, ys = [] ∨ # ys = h0 ⊦ ⟨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, y1. x = xs + x1 ∧ ys = x1 + y1 ∧ (( ls(rxs, xs) * lls(rys, r, x1) * ls(r, y1) * del-list(to_delete, n0)) )(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n ≤ h0 ∧ n = # x fmapos: right 1 bound: h0 - n

Additional information

Used simplifier rules: ⊦ x = x + y ↔ y = []; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ [] + x = x; (s)% ⊦ 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% ⊦ m < n → m ≤ n; (s) Used formulas: ∃ x1, y1. x = xs + x1 ∧ ys = x1 + y1 ∧ (( ls(rxs, xs) * lls(rys, r, x1) * ls(r, y1) * del-list(to_delete, n0)) )(H) ∧ n0 ≤ # xs + # ys ∧ n0 + # xs ≤ h0 ∧ n ≤ h0 ∧ n = # x, h0 - n = h0 - n