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Proof step 34 from proof validrefs-addarray-junkref (previous step)

is_array(str, r1, addarray(r, ty, vals, st)), validrefs(str, r1 ' + r2 ' + refs, ty0, addarray(r, ty, vals, st)), newref(r, st), st[r1 -- _type] .type ≤ ty0, ∀ str0, ty1, r0, ty2, vals0, st0. validrefs(str0, r2 ' + refs, ty1, addarray(r0, ty2, vals0, st0)) ∧ newref(r0, st0) ∧ (r0 = r2 ∨ r0 ∈ refs) → junkref(str0, r2, st0) ⊦ r1 = jvmref, r = jvmref, r = r1, junkref(str, r1, st), r ≠ r2 ∧ ¬ r ∈ refs

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ refs .first = st[r -- i '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 33:
is_array(str, r1, addarray(r, ty, vals, st)) ∧ (addarray(r, ty, vals, st)[r1 -- _type]) .type ≤ ty0, is_array(str, r1, addarray(r, ty, vals, st)), r2 ' + refs = [] ∨ (∃ i. 0 ≤ i ∧ i < (addarray(r, ty, vals, st)[r1 -- _length]) .intval ∧ is_referencevalue(addarray(r, ty, vals, st)[r1 -- i ']) ∧ (r2 ' + refs) .first = (addarray(r, ty, vals, st)[r1 -- i ']) .refval ∧ validrefs(str, r2 ' + refs, (addarray(r, ty, vals, st)[r1 -- _type]) .type .type, addarray(r, ty, vals, st))), newref(r, st), st[r1 -- _type] .type ≤ ty0, ∀ str0, ty1, r0, ty2, vals0, st0. validrefs(str0, r2 ' + refs, ty1, addarray(r0, ty2, vals0, st0)) ∧ newref(r0, st0) ∧ (r0 = r2 ∨ r0 ∈ refs) → junkref(str0, r2, st0) ⊦ r1 = jvmref, r = jvmref, r = r1, junkref(str, r1, st), r ≠ r2 ∧ ¬ r ∈ refs

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [addarray(r, ty, vals, st), ty0, r2 ' + refs, r1, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r ≠ r0 → ( is_array (str, r, addarray(r0, ty0, vals, st)) ↔ is_array(str, r, st))], [CSIMPSEQ : ⊦ r ≠ r0 → (addarray(r, ty, vals, st) [r0 -- sk ] ) = st[r0 -- sk]])], [EXTRATERMS : (addarray(r, ty, vals, st), ty0, r2 ' + refs, r1, str)]]