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Proof step 68 from proof validrefs-put-notin-01 (previous step)

r ∈ st, okval(st[r -- mkfs(class, ty1, fieldvar) '], ty1), r -- mkfs(class, ty1, fieldvar) ' ∈ st, okval(val, ty1), is_obj(str, r1, st) ∨ is_array(str, r1, st), st[r1 -- _type] .type ≤ ty, ∀ r2, fs0, val0, st0, str0, r3, ty0. ¬ r2 ∈ acc ∧ fs0 ' ≠ _type ∧ fs0 ' ≠ _length ∧ okval(val0, fs0 .type) ∧ r2 -- fs0 ' ∈ st0 ∧ okval(st0[r2 -- fs0 '], fs0 .type) → (validrefs(str0, acc + r3 ', ty0, st0[r2 -- fs0 ', val0]) ↔ validrefs(str0, acc + r3 ', ty0, st0)) ⊦ r1 = jvmref, r = r1, validrefs(str, (r1 ' + acc) + r0 ', ty, st[r -- mkfs(class, ty1, fieldvar) ', val]) ↔ validrefs(str, (r1 ' + acc) + r0 ', ty, st), mkfs(class, ty1, fieldvar) ' = _type, mkfs(class, ty1, fieldvar) ' = _length, r ∈ acc

User applied rule case distinction

And obtained the following new goals

Proof step 44:
is_obj(str, r1, st), r ∈ st, okval(st[r -- mkfs(class, ty1, fieldvar) '], ty1), r -- mkfs(class, ty1, fieldvar) ' ∈ st, okval(val, ty1), st[r1 -- _type] .type ≤ ty, ∀ r2, fs0, val0, st0, str0, r3, ty0. ¬ r2 ∈ acc ∧ fs0 ' ≠ _type ∧ fs0 ' ≠ _length ∧ okval(val0, fs0 .type) ∧ r2 -- fs0 ' ∈ st0 ∧ okval(st0[r2 -- fs0 '], fs0 .type) → (validrefs(str0, acc + r3 ', ty0, st0[r2 -- fs0 ', val0]) ↔ validrefs(str0, acc + r3 ', ty0, st0)) ⊦ r1 = jvmref, r = r1, validrefs(str, (r1 ' + acc) + r0 ', ty, st[r -- mkfs(class, ty1, fieldvar) ', val]) ↔ validrefs(str, (r1 ' + acc) + r0 ', ty, st), mkfs(class, ty1, fieldvar) ' = _type, mkfs(class, ty1, fieldvar) ' = _length, r ∈ acc
Proof step 67:
is_array(str, r1, st), r ∈ st, okval(st[r -- mkfs(class, ty1, fieldvar) '], ty1), r -- mkfs(class, ty1, fieldvar) ' ∈ st, okval(val, ty1), st[r1 -- _type] .type ≤ ty, ∀ r2, fs0, val0, st0, str0, r3, ty0. ¬ r2 ∈ acc ∧ fs0 ' ≠ _type ∧ fs0 ' ≠ _length ∧ okval(val0, fs0 .type) ∧ r2 -- fs0 ' ∈ st0 ∧ okval(st0[r2 -- fs0 '], fs0 .type) → (validrefs(str0, acc + r3 ', ty0, st0[r2 -- fs0 ', val0]) ↔ validrefs(str0, acc + r3 ', ty0, st0)) ⊦ r1 = jvmref, r = r1, validrefs(str, (r1 ' + acc) + r0 ', ty, st[r -- mkfs(class, ty1, fieldvar) ', val]) ↔ validrefs(str, (r1 ' + acc) + r0 ', ty, st), mkfs(class, ty1, fieldvar) ' = _type, mkfs(class, ty1, fieldvar) ' = _length, r ∈ acc

Rule argument

[CASEDARG : pos: left 5, ()]

Additional information

[[CONCRETESIMPRULES : ()]]