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Proof step 145 from proof validrefs-all-same (previous step)

st[r -- _length] = st0[r -- _length], st[r -- _type] = st0[r -- _type], st[r -- i '] = refval(r1), 0 < st0[r -- _length] .intval, is_arraytype(st0[r -- _type] .type), 0 ≤ i, is_array(str, r, st0), is_array(str, r, st), st0[r -- _type] .type ≤ ty, i < st0[r -- _length] .intval, validrefs(str, r1 ' + refs, st0[r -- _type] .type .type, st), ∀ i. ¬ i < 0 ∧ ¬ st0[r -- _length] .intval ≤ i → st[r -- i '] = st0[r -- i '], ∀ str0, ty0, st1, st2. validrefs(str0, r1 ' + refs, ty0, st1) ∧ (∀ r0. (r0 = r1 ∨ r0 ∈ refs) ∧ r0 ≠ jvmref → st1[r0 -- _type] = st2[r0 -- _type] ∧ (is_classtype(st1[r0 -- _type] .type) → (∀ sk. sk ∈ instfields(st1[r0 -- _type] .type .class) → st1[r0 -- sk] = st2[r0 -- sk])) ∧ ( is_arraytype(st1[r0 -- _type] .type) → st1[r0 -- _length] = st2[r0 -- _length] ∧ (∀ i. 0 ≤ i ∧ i < st1[r0 -- _length] .intval → st1[r0 -- i '] = st2[r0 -- i ']))) → validrefs(str0, r1 ' + refs, ty0, st2), ∀ r0. (r0 = r ∨ r0 = r1 ∨ r0 ∈ refs) ∧ r0 ≠ jvmref → st[r0 -- _type] = st0[r0 -- _type] ∧ (is_classtype(st[r0 -- _type] .type) → (∀ sk. sk ∈ instfields(st[r0 -- _type] .type .class) → st[r0 -- sk] = st0[r0 -- sk])) ∧ ( is_arraytype(st[r0 -- _type] .type) → st[r0 -- _length] = st0[r0 -- _length] ∧ (∀ i. ¬ i < 0 ∧ ¬ st[r0 -- _length] .intval ≤ i → st[r0 -- i '] = st0[r0 -- i '])) ⊦ r = jvmref, is_referencevalue(st0[r -- i ']) ∧ r1 = st0[r -- i '] .refval ∧ validrefs(str, st0[r -- i '] .refval ' + refs, st0[r -- _type] .type .type, st0)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 144:
st0[r -- _type] .type = mkarraytype(ty1), st[r -- _length] = st0[r -- _length], st[r -- _type] = st0[r -- _type], st[r -- i '] = refval(r1), 0 < st0[r -- _length] .intval, is_arraytype(st0[r -- _type] .type), 0 ≤ i, is_array(str, r, st0), is_array(str, r, st), st0[r -- _type] .type ≤ ty, i < st0[r -- _length] .intval, validrefs(str, r1 ' + refs, ty1, st), ∀ i. ¬ i < 0 ∧ ¬ st0[r -- _length] .intval ≤ i → st[r -- i '] = st0[r -- i '], ∀ str0, ty0, st1, st2. validrefs(str0, r1 ' + refs, ty0, st1) ∧ (∀ r0. (r0 = r1 ∨ r0 ∈ refs) ∧ r0 ≠ jvmref → st1[r0 -- _type] = st2[r0 -- _type] ∧ (is_classtype(st1[r0 -- _type] .type) → (∀ sk. sk ∈ instfields(st1[r0 -- _type] .type .class) → st1[r0 -- sk] = st2[r0 -- sk])) ∧ ( is_arraytype(st1[r0 -- _type] .type) → st1[r0 -- _length] = st2[r0 -- _length] ∧ (∀ i. 0 ≤ i ∧ i < st1[r0 -- _length] .intval → st1[r0 -- i '] = st2[r0 -- i ']))) → validrefs(str0, r1 ' + refs, ty0, st2), ∀ r0. (r0 = r ∨ r0 = r1 ∨ r0 ∈ refs) ∧ r0 ≠ jvmref → st[r0 -- _type] = st0[r0 -- _type] ∧ (is_classtype(st[r0 -- _type] .type) → (∀ sk. sk ∈ instfields(st[r0 -- _type] .type .class) → st[r0 -- sk] = st0[r0 -- sk])) ∧ ( is_arraytype(st[r0 -- _type] .type) → st[r0 -- _length] = st0[r0 -- _length] ∧ (∀ i. ¬ i < 0 ∧ ¬ st[r0 -- _length] .intval ≤ i → st[r0 -- i '] = st0[r0 -- i '])) ⊦ r = jvmref, is_referencevalue(st0[r -- i ']) ∧ r1 = st0[r -- i '] .refval ∧ validrefs(str, st0[r -- i '] .refval ' + refs, ty1, st0)

Rule argument

[x0lemmaarg: spec-name: javatype instance-name: lemma-name: elim-01 antecedent: succedent: is_arraytype(ty) → (ty0 = ty .type ↔ ty = mkarraytype(ty0)) substlist: [ty, ty0] → [st0[r -- _type] .type, ty1] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: T replace all?: T paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPELIM : ⊦ is_arraytype(ty) → ( ty0 = ty .type ↔ ty = mkarraytype(ty0))])], [EXTRATERMS : (st0[r -- _type] .type, ty1)]]