[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 17 from proof multi-refin (previous step)

# refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), r0 ' + refs2) = r ' + refs1, rk ∈ st, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ ty, acc, ints, i, st, rk, refs0. # refs0 < # (r0 + refs2) → ( true → ( rk ∈ st ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → rk ∈ addmultiarrays(refs0, i, acc, ty, ints, st))) ⊦ 0 = i0, (~1) = i2, rk ∈ addmultiarrays (r0 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2), i2, r ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r ' + refs1, i1, [], ty, i1 ' + ints1, st)), 0 ∈ (i1 ' + ints1) .butlast

User applied rule apply induction

And obtained the following new goals

Proof step 11:
# refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), r0 ' + refs2) = r ' + refs1, rk ∈ st, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ ty, acc, ints, i, st, rk, refs0. # refs0 < # (r0 + refs2) → ( true → ( rk ∈ st ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → rk ∈ addmultiarrays(refs0, i, acc, ty, ints, st))) ⊦ # (r0 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2)) < # (r0 + refs2), 0 = i0, (~1) = i2, rk ∈ addmultiarrays (r0 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2), i2, r ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r ' + refs1, i1, [], ty, i1 ' + ints1, st)), 0 ∈ (i1 ' + ints1) .butlast
Proof step 16:
rk ∈ addmultiarrays(r ' + refs1, i1, [], ty, i1 ' + ints1, st) ∧ # (r0 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2)) = i→n(countrefs(i2 ' + (i0 ' + i1 ' + ints1) .rest)) ∧ i0 ' + i1 ' + ints1 ≠ [] ∧ 0 ≤ i2 ∧ posints(i0 ' + i1 ' + ints1) ∧ ¬ 0 ∈ (i0 ' + i1 ' + ints1) .butlast ∧ i2 ≤ (i0 ' + i1 ' + ints1) .first → rk ∈ addmultiarrays (r0 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2), i2, r ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r ' + refs1, i1, [], ty, i1 ' + ints1, st)), # refs2 = i→n(countrefs(i1 ' + ints1)) * i→n(i2) + i→n(countrefs(i1 ' + ints1)), lastn(i→n(countrefs(i1 ' + ints1)), r0 ' + refs2) = r ' + refs1, rk ∈ st, posints(ints1), 0 ≤ i1, 0 < i0, i2 < i0, 0 ≤ i2, i→n(countrefs(i1 ' + ints1)) ≤ 1 + # refs2, ∀ ty, acc, ints, i, st, rk, refs0. # refs0 < # (r0 + refs2) → ( true → ( rk ∈ st ∧ # refs0 = i→n(countrefs(i ' + ints .rest)) ∧ ints ≠ [] ∧ 0 ≤ i ∧ posints(ints) ∧ ¬ 0 ∈ ints .butlast ∧ i ≤ ints .first → rk ∈ addmultiarrays(refs0, i, acc, ty, ints, st))) ⊦ 0 = i0, (~1) = i2, rk ∈ addmultiarrays (r0 ' + firstn(i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2), i2, r ' + acc, ty, i0 ' + i1 ' + ints1, addmultiarrays(r ' + refs1, i1, [], ty, i1 ' + ints1, st)), 0 ∈ (i1 ' + ints1) .butlast

Rule argument

[substlist-arg: [ty, acc, ints, i, st, rk, refs0] → [ty, r ' + acc, i0 ' + i1 ' + ints1, i2, addmultiarrays (r ' + refs1, i1, [], ty, i1 ' + ints1, st), rk, r0 ' + firstn (i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2)] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (ty, r ' + acc, i0 ' + i1 ' + ints1, i2, addmultiarrays (r ' + refs1, i1, [], ty, i1 ' + ints1, st), rk, r0 ' + firstn (i→n(countrefs(i1 ' + ints1)) * i→n(i2), refs2))]]