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Proof step 162 from proof validrefs-all-same (previous step)

∀ str, ty, st, st0. validrefs(str, refs0, ty, st) ∧ (∀ r. r ∈ refs0 ∧ r ≠ jvmref → st[r -- _type] = st0[r -- _type] ∧ (is_classtype(st[r -- _type] .type) → (∀ sk. sk ∈ instfields(st[r -- _type] .type .class) → st[r -- sk] = st0[r -- sk])) ∧ ( is_arraytype(st[r -- _type] .type) → st[r -- _length] = st0[r -- _length] ∧ (∀ i. ¬ i < 0 ∧ ¬ st[r -- _length] .intval ≤ i → st[r -- i '] = st0[r -- i ']))) → validrefs(str, refs0, ty, st0), validrefs(str, r + refs0, ty, st), ∀ r0. r0 ∈ r + refs0 ∧ 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 '])) ⊦ validrefs(str, r + refs0, ty, st0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 161:
validrefs(str, r ' + refs0, ty, st), ∀ str0, ty0, st1, st2. validrefs(str0, refs0, ty0, st1) ∧ (∀ r0. r0 ∈ refs0 ∧ 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, refs0, ty0, st2), ∀ r0. (r0 = r ∨ r0 ∈ refs0) ∧ 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 '])) ⊦ validrefs(str, r ' + refs0, ty, st0)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ r ∈ r0 ' + refs ↔ r ≠ r0 ∧ ¬ r ∈ refs], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ r + refs = r ' + refs])]]