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Proof step 68 from proof validrefs-put-badclassfield (previous step)

is_array(str, r, st), st[r0 -- _type] .type = mkclasstype(class), st[r -- _type] .type ≤ ty, validrefs(str, r ' + r1 ' + refs, ty, st[r0 -- fs ', val]), ∀ str0, ty0, st0, fs0, r2, val0. ¬ validrefs(str0, r1 ' + refs, ty0, st0) ∧ ¬ junkref(str0, r1, st0) ∧ fs0 ' ≠ _type ∧ is_classtype(st0[r2 -- _type] .type) ∧ ¬ fs0 ' ∈ instfields(st0[r2 -- _type] .type .class) → ¬ validrefs(str0, r1 ' + refs, ty0, st0[r2 -- fs0 ', val0]), is_obj(str, r, st[r0 -- fs ', val]) ∨ is_array(str, r, st[r0 -- fs ', val]) ⊦ r = jvmref, fs ' = _type, validrefs(str, r ' + r1 ' + refs, ty, st), junkref(str, r, st), fs ' ∈ instfields(class), r ∈ (r1 ' + refs) .butlast, dups((r1 ' + refs) .butlast)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ 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 48:
¬ (is_array(str, r, st[r0 -- fs ', val]) ∧ (st[r0 -- fs ', val][r -- _type]) .type ≤ ty), is_array(str, r, st), st[r0 -- _type] .type = mkclasstype(class), st[r -- _type] .type ≤ ty, validrefs(str, r ' + r1 ' + refs, ty, st[r0 -- fs ', val]), ∀ str0, ty0, st0, fs0, r2, val0. ¬ validrefs(str0, r1 ' + refs, ty0, st0) ∧ ¬ junkref(str0, r1, st0) ∧ fs0 ' ≠ _type ∧ is_classtype(st0[r2 -- _type] .type) ∧ ¬ fs0 ' ∈ instfields(st0[r2 -- _type] .type .class) → ¬ validrefs(str0, r1 ' + refs, ty0, st0[r2 -- fs0 ', val0]), is_obj(str, r, st[r0 -- fs ', val]) ∨ is_array(str, r, st[r0 -- fs ', val]) ⊦ r = jvmref, fs ' = _type, validrefs(str, r ' + r1 ' + refs, ty, st), junkref(str, r, st), fs ' ∈ instfields(class), r ∈ (r1 ' + refs) .butlast, dups((r1 ' + refs) .butlast)
Proof step 67:
is_array(str, r, st[r0 -- fs ', val]) ∧ (st[r0 -- fs ', val][r -- _type]) .type ≤ ty, is_array(str, r, st), st[r0 -- _type] .type = mkclasstype(class), st[r -- _type] .type ≤ ty, r1 ' + refs = [] ∨ (∃ i. 0 ≤ i ∧ i < (st[r0 -- fs ', val][r -- _length]) .intval ∧ is_referencevalue(st[r0 -- fs ', val][r -- i ']) ∧ (r1 ' + refs) .first = (st[r0 -- fs ', val][r -- i ']) .refval ∧ validrefs(str, r1 ' + refs, (st[r0 -- fs ', val][r -- _type]) .type .type, st[r0 -- fs ', val])), ∀ str0, ty0, st0, fs0, r2, val0. ¬ validrefs(str0, r1 ' + refs, ty0, st0) ∧ ¬ junkref(str0, r1, st0) ∧ fs0 ' ≠ _type ∧ is_classtype(st0[r2 -- _type] .type) ∧ ¬ fs0 ' ∈ instfields(st0[r2 -- _type] .type .class) → ¬ validrefs(str0, r1 ' + refs, ty0, st0[r2 -- fs0 ', val0]), is_obj(str, r, st[r0 -- fs ', val]) ∨ is_array(str, r, st[r0 -- fs ', val]) ⊦ r = jvmref, fs ' = _type, validrefs(str, r ' + r1 ' + refs, ty, st), junkref(str, r, st), fs ' ∈ instfields(class), r ∈ (r1 ' + refs) .butlast, dups((r1 ' + refs) .butlast)

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st[r0 -- fs ', val], ty, r1 ' + refs, r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (st[r0 -- fs ', val], ty, r1 ' + refs, r, str)]]