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Proof step 58 from proof validrefs-subset-01 (previous step)

st[r0 -- i '] = refval(r), validrefs(str, r ' + refs, st[r0 -- _type] .type .type, st), i < st[r0 -- _length] .intval, st[r0 -- _type] .type ≤ ty, st ⊆ st0, is_array(str, r0, st), st0[r0 -- _type] .type ≤ ty, 0 < st[r0 -- _length] .intval, 0 ≤ i, ∀ str0, ty0, st1, st2. validrefs(str0, r ' + refs, ty0, st1) ∧ st1 ⊆ st2 ∧ ¬ junkref(str0, r, st1) → validrefs(str0, r ' + refs, ty0, st2) ⊦ r0 = jvmref, junkref(str, r0, st), ∃ i0. ¬ i0 < 0 ∧ ¬ st0[r0 -- _length] .intval ≤ i0 ∧ is_referencevalue(st0[r0 -- i0 ']) ∧ r = st0[r0 -- i0 '] .refval ∧ validrefs(str, st0[r0 -- i0 '] .refval ' + refs, st0[r0 -- _type] .type .type, st0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ rk ∈ st ∧ st ⊆ st1 → st1[rk] = st[rk]
Proof step 57:
r0 -- _type ∈ st ∧ st ⊆ st0, st[r0 -- i '] = refval(r), validrefs(str, r ' + refs, st[r0 -- _type] .type .type, st), i < st[r0 -- _length] .intval, st[r0 -- _type] .type ≤ ty, st ⊆ st0, is_array(str, r0, st), st[r0 -- _type] .type ≤ ty, 0 < st[r0 -- _length] .intval, 0 ≤ i, ∀ str0, ty0, st1, st2. validrefs(str0, r ' + refs, ty0, st1) ∧ st1 ⊆ st2 ∧ ¬ junkref(str0, r, st1) → validrefs(str0, r ' + refs, ty0, st2) ⊦ r0 = jvmref, junkref(str, r0, st), ∃ i0. ¬ i0 < 0 ∧ ¬ st0[r0 -- _length] .intval ≤ i0 ∧ is_referencevalue(st0[r0 -- i0 ']) ∧ r = st0[r0 -- i0 '] .refval ∧ validrefs(str, st0[r0 -- i0 '] .refval ' + refs, st[r0 -- _type] .type .type, st0)

Rule argument

[speclemmaarg: ,rk ∈ st ∧ st ⊆ st1 → st1[rk] = st[rk] ,[rk, st, st1] → [r0 -- _type, st, st0] ,F ,store ,javastore-ostorea ,superset-elem-eq ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ ¬ rk ∈ st → st[rk] = ∅[rk]], [CSIMPSEQ : ⊦ ∅[rk] = noval], [CSIMPSEQ : ⊦ ¬ r -- _type ∈ st → ¬ is_array(str, r, st)])], [EXTRATERMS : (r0 -- _type, st, st0)]]