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Proof step 94 from proof pathex-put-pathex-array-refval (previous step)

st[r0 -- _type] .type = mkarraytype(ty), list2set(refs1) ∩ list2set(refs) = ∅, refs1 + r0 ' = r1 ' + refs0, is_valid_ref(str, r0, st[r0 -- i ', refval(r2)]), validrefs(str, r0 ' + refs, mkarraytype(ty), st[r0 -- i ', refval(r2)]), validrefs(str, r ' + refs1 + r0 ' + refs, st[r -- _type] .type, st[r0 -- i ', refval(r2)]), path∃(str, r, r0, st), i < st[r0 -- _length] .intval, validref(str, r2, ty, st), 0 < st[r0 -- _length] .intval, 0 ≤ i, validrefs(str, refs1 + r0 ' + refs, st[r1 -- _type] .type, st[r0 -- i ', refval(r2)]), r2 = jvmref → is_reftype(ty) ⊦ r = (r0 ' + refs) .last, r0 = jvmref, r = r0, r0 ∈ refs, r0 ∈ refs1, dups(refs), dups(refs1), path∃(str, r, (r0 ' + refs) .last, st), path∃(str, r2, (r0 ' + refs) .last, st), r ∈ refs1, r ∈ refs, ¬ r1 ∈ refs1 ∧ r1 ≠ r0, r0 ≠ r1 ∧ ¬ r0 ∈ refs0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ 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 93:
is_array(str, r0, st[r0 -- i ', refval(r2)]) ∧ (st[r0 -- i ', refval(r2)][r0 -- _type]) .type ≤ mkarraytype(ty), st[r0 -- _type] .type = mkarraytype(ty), list2set(refs1) ∩ list2set(refs) = ∅, refs1 + r0 ' = r1 ' + refs0, is_valid_ref(str, r0, st[r0 -- i ', refval(r2)]), refs = [] ∨ (∃ i0. 0 ≤ i0 ∧ i0 < (st[r0 -- i ', refval(r2)][r0 -- _length]) .intval ∧ is_referencevalue(st[r0 -- i ', refval(r2)][r0 -- i0 ']) ∧ refs .first = (st[r0 -- i ', refval(r2)][r0 -- i0 ']) .refval ∧ validrefs(str, refs, (st[r0 -- i ', refval(r2)][r0 -- _type]) .type .type, st[r0 -- i ', refval(r2)])), validrefs(str, r ' + refs1 + r0 ' + refs, st[r -- _type] .type, st[r0 -- i ', refval(r2)]), path∃(str, r, r0, st), i < st[r0 -- _length] .intval, validref(str, r2, ty, st), 0 < st[r0 -- _length] .intval, 0 ≤ i, validrefs(str, refs1 + r0 ' + refs, st[r1 -- _type] .type, st[r0 -- i ', refval(r2)]), r2 = jvmref → is_reftype(ty) ⊦ r = (r0 ' + refs) .last, r0 = jvmref, r = r0, r0 ∈ refs, r0 ∈ refs1, dups(refs), dups(refs1), path∃(str, r, (r0 ' + refs) .last, st), path∃(str, r2, (r0 ' + refs) .last, st), r ∈ refs1, r ∈ refs, ¬ r1 ∈ refs1 ∧ r1 ≠ r0, r0 ≠ r1 ∧ ¬ r0 ∈ refs0

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: validrefs-array antecedent: succedent: 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))) substlist: [st, ty, refs, r, str] → [st[r0 -- i ', refval(r2)], mkarraytype(ty), refs, r0, str] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((0, 0, 4)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : javastore, , type-at, ⊦ (st[r -- i ', val] [r0 -- _type ] ) = st[r0 -- _type]], [CSIMPNAMEDSEQ : addfuns, , is_valid_ref-01, ⊦ ¬ is_array(str, r, st) → ( is_valid_ref(str, r, st) ↔ is_obj(str, r, st))], [CSIMPNAMEDSEQ : addfuns, , is_obj-put-index, ⊦ is_obj (str, r, st[r0 -- i ', val]) ↔ is_obj(str, r, st)], [CSIMPNAMEDSEQ : addfuns, , is_obj-type-02, ⊦ st[r -- _type] .type = mkarraytype(ty) → ¬ is_obj(str, r, st)], [CSIMPNAMEDSEQ : addfuns, , is_valid_ref-isarray, ⊦ is_array(str, r, st) → is_valid_ref(str, r, st)], [CSIMPNAMEDSEQ : typefuns, , subarray1, ⊦ mkarraytype(ty) ≤ mkarraytype(ty0) ↔ ty ≤ ty0], [CSIMPNAMEDSEQ : typefuns, , ref, ⊦ ty ≤ ty])], [EXTRATERMS : (st[r0 -- i ', refval(r2)], mkarraytype(ty), refs, r0, str)]]