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Proof step 17 from proof gc-heap-free-h (previous step)

lpt[n] = mklp(n0, n1, LP_GROUP_NODES, n2), lpt[m3] = mklp(m, m4, lpf, m5), lpt[n, mklp(m0, m1, LP_FREE, m2)][m3] = mklp(m6, m7, lpf0, m8), gc-heap(lpt[n, mklp(m0, m1, LP_FREE, m2)], nx, m3) = gc-heap(lpt, nx, m3) -- key-index(n0, n), m3 < # lpt, n < # lpt, n ≤ m3 ⊦ (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ (¬ m3 ∈ nx ∧ lpf0 = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅) = (gc-heap(lpt, nx, m3) -- key-index(n0, n)) ∪ ((¬ m3 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅) -- key-index(n0, n)), n ∈ nx

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: gc-heap-write-above seq: ⊦ n < # lpt ∧ m ≤ n → gc-heap(lpt[n, lp], nx, m) = gc-heap(lpt, nx, m) subst: [m, nx, lp, n, lpt] → [m3, nx, mklp(m0, m1, LP_FREE, m2), n, lpt] flags: <all paths>

Additional information

Used simplifier rules: ⊦ gchs -- ki = gchs ↔ ¬ ki ∈ gchs; (s)% ⊦ ¬ ki ∈ gchs → gchs -- ki = gchs; (s)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < # lpt → lpt[m, lp][m] = lp; (s)%⊦ mklp(m, m0, lpf, m1) = mklp(m2, m3, lpf0, m4) ↔ m = m2 ∧ m0 = m3 ∧ lpf = lpf0 ∧ m1 = m4; (ws)% ⊦ m ≤ m; (s)% ⊦ LP_FREE ≠ LP_GROUP_NODES; (s)% ⊦ gchs ∪ ∅ = gchs; (s)%una: ⊦ (gchs ∪ gchs0) ∪ gchs1 = gchs ∪ gchs0 ∪ gchs1; from specification set-union[gc-heap-rename-gc-heap-base-keyindex-set]%unc: ⊦ gchs ∪ gchs0 = gchs0 ∪ gchs; from specification set-union[gc-heap-rename-gc-heap-base-keyindex-set]% ⊦ gchs ∪ gchs0 = gchs0 ↔ gchs ⊆ gchs0; (s)% ⊦ (gchs ++ ki) -- ki = gchs -- ki; (s)% ⊦ ∅ -- ki = ∅; (s)% ⊦ ∅ ⊆ gchs; (s)% ⊦ m ≠ n → lpt[n, lp][m] = lpt[m]; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ (gchs -- ki) ∪ gchs0 = gchs ∪ gchs0 ↔ ki ∈ gchs0 ∨ ¬ ki ∈ gchs; (ws)% ⊦ ki ∈ gchs -- ki0 ↔ ki ≠ ki0 ∧ ki ∈ gchs; (s)%ne-01: ⊦ ki ∈ gchs → gchs ≠ ∅; (forward) from specification set-basic[gc-heap-rename-gc-heap-base-keyindex-set]% ⊦ ∅ ≠ gchs ++ ki; (s)% ⊦ ki ∈ ∅ ++ ki0 ↔ ki = ki0; (s)%⊦ key-index(natkey, m) = key-index(natkey0, m0) ↔ natkey = natkey0 ∧ m = m0; (ws) Used terms: m3, nx, mklp(m0, m1, LP_FREE, m2), n, lpt