[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 18 from proof putarray-addarray-one (previous step)

¬ (i < 0 ∨ n→i(# (refval(jvmref) ' + vals)) ≤ i), r0 -- i ' ∈ addarray(r0, ty, n→i(# vals) + 1, st) ↔ r0 -- i ' ∈ addarray(r0, ty, refval(jvmref) ' + vals, st), is_reftype(ty), i ≤ 0 ∨ n→i(# vals) < i ⊦ (addarray(r0, ty, n→i(# vals) + 1, st)[r0 -- i ']) = (addarray(r0, ty, refval(jvmref) ' + vals, st)[r0 -- i ']), ¬ i ≤ 0 ∧ ¬ n→i(# vals) < i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
⊦ r -- j ' ∈ addarray(r, ty, vals, st) ↔ r -- j ' ∈ st ∨ 0 ≤ j ∧ j < n→i(# vals)
Proof step 13:
⊦ (addarray(r, ty, val ' + vals, st)[r -- 0 ']) = val
Proof step 14:
⊦ r ∈ addarray(r, ty0, vals, st)
Proof step 15:
⊦ r ∈ addarray(r, ty0, i, st)
Proof step 17:
r0 -- 0 ' ∈ addarray(r0, ty, refval(jvmref) ' + vals, st), r0 -- 0 ' ∈ addarray(r0, ty, n→i(# vals) + 1, st), is_reftype(ty) ⊦ (addarray(r0, ty, n→i(# vals) + 1, st)[r0 -- 0 ']) = refval(jvmref)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ 0 ≤ n→i(n)], [CSIMPFORWARD : ⊦ r -- sk ∈ st → r ∈ st], [CSIMPFORWARD : ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ i ≤ j → (j ≤ i ↔ i = j)], [CSIMPSEQ : ⊦ i < j → (¬ ¬ j ≤ i ↔ false)], [CSIMPSEQ : ⊦ i + 1 ≤ j ↔ i < j], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i < j + 1 ↔ i ≤ j], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ n→i(m + n) = n→i(m) + n→i(n)], [CSIMPSEQ : ⊦ (vals + vals0) + vals1 = vals + vals0 + vals1], [CSIMPSEQ : ⊦ # val ' = 0 +1], [CSIMPSEQ : ⊦ # (vals + vals0) = # vals + # vals0], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ ∅[rk] = noval], [CSIMPSEQ : ⊦ ¬ rk ∈ st → st[rk] = ∅[rk]])]]