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Proof step 13 from proof bin-heap-inv-cleanup-lpt (previous step)

2exp-size(# kar), heap-order(kar, m), no-dups-idx(kar, m), m ≤ # kar ⊦ (∀ m0. m0 < m → kar[m0].idx < # lpt ∧ lpt.map(λ lp. (lp.ref-size = 0 ∧ lp.flags ≠ LP_FREE ⊃ mklp(0, 0, LP_FREE, lp.gcheapidx);lp))[kar[m0].idx].gcheapidx = m0) ↔ valid-idx(kar, lpt, m)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ valid-idx(kar, lpt, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # lpt ∧ lpt[kar[m].idx].gcheapidx = m)
Proof step 15:
2exp-size(# kar), heap-order(kar, m), no-dups-idx(kar, m), m ≤ # kar ⊦ (∀ m0. m0 < m → kar[m0].idx < # lpt ∧ lpt.map(λ lp. (lp.ref-size = 0 ∧ lp.flags ≠ LP_FREE ⊃ mklp(0, 0, LP_FREE, lp.gcheapidx);lp))[kar[m0].idx].gcheapidx = m0) ↔ m ≤ # kar ∧ (∀ m0. m0 < m → kar[m0].idx < # lpt ∧ lpt[kar[m0].idx].gcheapidx = m0)

Rule argument

spec: bin-min-heap-inv, inst: gc-heap-rename-gc-heap-base name: valid-idx seq: ⊦ valid-idx(kar, lpt, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # lpt ∧ lpt[kar[m].idx].gcheapidx = m) subst: [n, lpt, kar] → [m, lpt, kar] flags: <all paths>

Additional information

Used terms: m, lpt, kar