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Proof step 455 from proof hasSharing-put-index-refval (previous step)

(st[r2 -- i ', refval(r3)][(st[r2 -- i ', refval(r3)][r -- sks]) .refval -- (i ' ' + sks2)]) = refval(r1), st[r2 -- _type] .type = mkarraytype(ty), (st[r2 -- i ', refval(r3)][r3 -- sks3]) = refval(r1), st[r -- sks1] = refval(r2), st[r -- sks] = refval(r2), 0 < st[r2 -- _length] .intval, validref(str, r3, ty, st), disjoint(str, r ' + r3 ', st), fits(r, sks1, st), fits(r3, sks3, st[r2 -- i ', refval(r3)]), i < st[r2 -- _length] .intval, 0 ≤ i, is_referencevalue(st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks3)]), fits(r, sks, st), fits(r3, sks2, st[r2 -- i ', refval(r3)]), r3 = jvmref → is_reftype(ty) ⊦ sks1 = [], sks + i ' ' + sks2 = sks1 + i ' ' + sks3, r1 = jvmref, r = r3, r2 = jvmref, sks = [], hasSharing(r, st), hasSharing(r3, st), junkref(str, r, st), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .refval ≠ r2), ∃ sks0, sks2. sks1 = sks0 + i ' ' + sks2 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks0 → st[r -- sks4] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r -- sks4] .refval ≠ r2), ∃ sks0, sks4. sks = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 340:
⊦ ¬ (∃ sks0, sks1. sks = sks0 + (sk + sks1) ∧ st[r -- sks0] .refval = r0 ∧ (∀ sks2. sks2 + sk ⊑ sks0 → st[r -- sks2] .refval ≠ r0)) → (st[r0 -- sk, val][r -- sks]) = st[r -- sks]
Proof step 344:
¬ ¬ (∃ sks0, sks1. sks = sks0 + (i ' + sks1) ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks2. sks2 + i ' ⊑ sks0 → st[r -- sks2] .refval ≠ r2)), (st[r2 -- i ', refval(r3)][(st[r2 -- i ', refval(r3)][r -- sks]) .refval -- (i ' ' + sks2)]) = refval(r1), st[r2 -- _type] .type = mkarraytype(ty), (st[r2 -- i ', refval(r3)][r3 -- sks3]) = refval(r1), st[r -- sks1] = refval(r2), st[r -- sks] = refval(r2), 0 < st[r2 -- _length] .intval, validref(str, r3, ty, st), disjoint(str, r ' + r3 ', st), fits(r, sks1, st), fits(r3, sks3, st[r2 -- i ', refval(r3)]), i < st[r2 -- _length] .intval, 0 ≤ i, is_referencevalue(st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks3)]), fits(r, sks, st), fits(r3, sks2, st[r2 -- i ', refval(r3)]), r3 = jvmref → is_reftype(ty) ⊦ sks1 = [], sks + i ' ' + sks2 = sks1 + i ' ' + sks3, r1 = jvmref, r = r3, r2 = jvmref, sks = [], hasSharing(r, st), hasSharing(r3, st), junkref(str, r, st), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .refval ≠ r2), ∃ sks0, sks2. sks1 = sks0 + i ' ' + sks2 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks0 → st[r -- sks4] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r -- sks4] .refval ≠ r2), ∃ sks0, sks4. sks = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval)
Proof step 454:
¬ (∃ sks0, sks1. sks = sks0 + (i ' + sks1) ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks2. sks2 + i ' ⊑ sks0 → st[r -- sks2] .refval ≠ r2)), (st[r2 -- i ', refval(r3)][st[r -- sks] .refval -- (i ' ' + sks2)]) = refval(r1), st[r2 -- _type] .type = mkarraytype(ty), (st[r2 -- i ', refval(r3)][r3 -- sks3]) = refval(r1), st[r -- sks1] = refval(r2), st[r -- sks] = refval(r2), 0 < st[r2 -- _length] .intval, validref(str, r3, ty, st), disjoint(str, r ' + r3 ', st), fits(r, sks1, st), fits(r3, sks3, st[r2 -- i ', refval(r3)]), i < st[r2 -- _length] .intval, 0 ≤ i, is_referencevalue(st[r2 -- i ', refval(r3)][r2 -- (i ' ' + sks3)]), fits(r, sks, st), fits(r3, sks2, st[r2 -- i ', refval(r3)]), r3 = jvmref → is_reftype(ty) ⊦ sks1 = [], sks + i ' ' + sks2 = sks1 + i ' ' + sks3, r1 = jvmref, r = r3, r2 = jvmref, sks = [], hasSharing(r, st), hasSharing(r3, st), junkref(str, r, st), ∃ sks2. ¬ (sks2 + i ' ' ⊑ sks1 → st[r -- sks2] .refval ≠ r2), ∃ sks0, sks2. sks1 = sks0 + i ' ' + sks2 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks4. sks4 + i ' ' ⊑ sks0 → st[r -- sks4] .refval ≠ st[r -- sks0] .refval), ∃ sks4. ¬ (sks4 + i ' ' ⊑ sks → st[r -- sks4] .refval ≠ r2), ∃ sks0, sks4. sks = sks0 + i ' ' + sks4 ∧ st[r -- sks0] .refval = r2 ∧ (∀ sks5. sks5 + i ' ' ⊑ sks0 → st[r -- sks5] .refval ≠ st[r -- sks0] .refval)

Rule argument

[x0lemmaarg: spec-name: storefuns instance-name: lemma-name: lookup-put-other antecedent: succedent: ¬ (∃ sks0, sks1. sks = sks0 + (sk + sks1) ∧ st[r -- sks0] .refval = r0 ∧ (∀ sks2. sks2 + sk ⊑ sks0 → st[r -- sks2] .refval ≠ r0)) → (st[r0 -- sk, val][r -- sks]) = st[r -- sks] substlist: [sks, r, val, sk, r0, st] → [sks, r, refval(r3), i ', r2, st] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (sks, r, refval(r3), i ', r2, st)]]