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Proof step 83 from proof unchanged-junkrefs-level (previous step)

st[r -- i '] = refval(r1), st[r -- _type] .type = mkarraytype(ty1), st[r -- _length] = st1[r -- _length], st1[r -- i '] = refval(r1), st1[r -- _type] = st[r -- _type], 0 < st1[r -- _length] .intval, is_array(str0, r, st), unchangedNorec(r, st, st1), is_array(str, r, st), 0 ≤ i, is_array(str, r, st1), is_valid_ref(str0, r, st1), mkarraytype(ty1) ≤ ty, i < st1[r -- _length] .intval, junkrefs(str, r1 ' + refs, ty1, st), unchanged(str0, r, st, st1), level⊆(str, str0), is_valid_ref(str, r1, st1), st1[r1 -- _type] .type ≤ ty1, ∀ i0. ¬ i0 < 0 ∧ ¬ st1[r -- _length] .intval ≤ i0 → st[r -- i0 '] = st1[r -- i0 '], ∀ r0. (path∃(str0, r, r0, st) ↔ path∃(str0, r, r0, st1)) ∧ (path∃(str0, r, r0, st) → unchangedNorec(r0, st, st1)) ⊦ r = r1, r1 = jvmref, r = jvmref, junkrefs(str, r1 ' + refs, ty1, st1), unchanged(str0, r1, st, st1), path∃(str0, r, r1, st1), st[r1 -- _type] .type ≤ ty1 ∧ is_valid_ref(str0, r1, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 77:
⊦ is_array(str, r, st) ∧ r ≠ r0 → ( path∃(str, r, r0, st) ↔ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ ( st[r -- i '] .refval ≠ jvmref ∧ st[st[r -- i '] .refval -- _type] .type ≤ st[r -- _type] .type .type ∧ path∃(str, st[r -- i '] .refval, r0, st) ∨ st[r -- i '] .refval = jvmref ∧ r0 = jvmref)))
Proof step 82:
is_array(str0, r, st1) ∧ r ≠ r1, st[r -- i '] = refval(r1), st[r -- _type] .type = mkarraytype(ty1), st[r -- _length] = st1[r -- _length], st1[r -- i '] = refval(r1), st1[r -- _type] = st[r -- _type], 0 < st1[r -- _length] .intval, is_array(str0, r, st), unchangedNorec(r, st, st1), is_array(str, r, st), 0 ≤ i, is_array(str, r, st1), is_valid_ref(str0, r, st1), mkarraytype(ty1) ≤ ty, i < st1[r -- _length] .intval, junkrefs(str, r1 ' + refs, ty1, st), unchanged(str0, r, st, st1), level⊆(str, str0), is_valid_ref(str, r1, st1), st1[r1 -- _type] .type ≤ ty1, ∀ i0. ¬ i0 < 0 ∧ ¬ st1[r -- _length] .intval ≤ i0 → st[r -- i0 '] = st1[r -- i0 '], ∀ r0. (path∃(str0, r, r0, st) ↔ path∃(str0, r, r0, st1)) ∧ (path∃(str0, r, r0, st) → unchangedNorec(r0, st, st1)) ⊦ r = r1, r1 = jvmref, r = jvmref, junkrefs(str, r1 ' + refs, ty1, st1), unchanged(str0, r1, st, st1), ∃ i. 0 ≤ i ∧ i < st1[r -- _length] .intval ∧ is_referencevalue(st1[r -- i ']) ∧ ( st1[r -- i '] .refval ≠ jvmref ∧ st1[st1[r -- i '] .refval -- _type] .type ≤ st1[r -- _type] .type .type ∧ path∃(str0, st1[r -- i '] .refval, r1, st1) ∨ st1[r -- i '] .refval = jvmref ∧ r1 = jvmref), st[r1 -- _type] .type ≤ ty1 ∧ is_valid_ref(str0, r1, st)

Rule argument

[x0lemmaarg: spec-name: cyclic instance-name: lemma-name: pathex-array antecedent: succedent: is_array(str, r, st) ∧ r ≠ r0 → ( path∃(str, r, r0, st) ↔ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ ( st[r -- i '] .refval ≠ jvmref ∧ st[st[r -- i '] .refval -- _type] .type ≤ st[r -- _type] .type .type ∧ path∃(str, st[r -- i '] .refval, r0, st) ∨ st[r -- i '] .refval = jvmref ∧ r0 = jvmref))) substlist: [st, r0, r, str] → [st1, r1, r, str0] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 5)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : list, references, a, ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPNAMEDSEQ : junkrefs, , levelsubset-is-array, ⊦ is_array(str, r, st) ∧ level⊆(str, str0) → ( ¬ is_array(str0, r, st) ↔ false)])], [EXTRATERMS : (st1, r1, r, str0)]]