Step
*
2
1
2
2
2
of Lemma
mul_poly-sq
1. u : iMonomial()
2. v : iMonomial() List
3. ∀[q:iMonomial() List]. (mul_ipoly(v;q) ~ mul-ipoly(v;q))
4. u1 : iMonomial()
5. v1 : iMonomial() List
6. ∀p,q:iMonomial() List. (add_ipoly(p;q) ∈ iMonomial() List)
7. ∀[f:(iMonomial() List) ⟶ iMonomial() ⟶ (iMonomial() List)]
cbv_list_accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v) ~ accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing value-type(iMonomial() List)
8. ∀[f:(iMonomial() List) ⟶ iMonomial() ⟶ (iMonomial() List)]
eager-accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v) ~ accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing valueall-type(iMonomial() List)
⊢ accumulate (with value sofar and list item m):
add_ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1])) ~ accumulate (with value sofar and list item m):
add-ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
BY
{ Assert ⌜accumulate (with value sofar and list item m):
add_ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
= accumulate (with value sofar and list item m):
add-ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
∈ (iMonomial() List)⌝⋅ }
1
.....assertion.....
1. u : iMonomial()
2. v : iMonomial() List
3. ∀[q:iMonomial() List]. (mul_ipoly(v;q) ~ mul-ipoly(v;q))
4. u1 : iMonomial()
5. v1 : iMonomial() List
6. ∀p,q:iMonomial() List. (add_ipoly(p;q) ∈ iMonomial() List)
7. ∀[f:(iMonomial() List) ⟶ iMonomial() ⟶ (iMonomial() List)]
cbv_list_accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v) ~ accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing value-type(iMonomial() List)
8. ∀[f:(iMonomial() List) ⟶ iMonomial() ⟶ (iMonomial() List)]
eager-accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v) ~ accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing valueall-type(iMonomial() List)
⊢ accumulate (with value sofar and list item m):
add_ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
= accumulate (with value sofar and list item m):
add-ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
∈ (iMonomial() List)
2
1. u : iMonomial()
2. v : iMonomial() List
3. ∀[q:iMonomial() List]. (mul_ipoly(v;q) ~ mul-ipoly(v;q))
4. u1 : iMonomial()
5. v1 : iMonomial() List
6. ∀p,q:iMonomial() List. (add_ipoly(p;q) ∈ iMonomial() List)
7. ∀[f:(iMonomial() List) ⟶ iMonomial() ⟶ (iMonomial() List)]
cbv_list_accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v) ~ accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing value-type(iMonomial() List)
8. ∀[f:(iMonomial() List) ⟶ iMonomial() ⟶ (iMonomial() List)]
eager-accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v) ~ accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing valueall-type(iMonomial() List)
9. accumulate (with value sofar and list item m):
add_ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
= accumulate (with value sofar and list item m):
add-ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
∈ (iMonomial() List)
⊢ accumulate (with value sofar and list item m):
add_ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1])) ~ accumulate (with value sofar and list item m):
add-ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
Latex:
Latex:
1. u : iMonomial()
2. v : iMonomial() List
3. \mforall{}[q:iMonomial() List]. (mul\_ipoly(v;q) \msim{} mul-ipoly(v;q))
4. u1 : iMonomial()
5. v1 : iMonomial() List
6. \mforall{}p,q:iMonomial() List. (add\_ipoly(p;q) \mmember{} iMonomial() List)
7. \mforall{}[f:(iMonomial() List) {}\mrightarrow{} iMonomial() {}\mrightarrow{} (iMonomial() List)]
cbv\_list\_accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v)
\msim{} accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing value-type(iMonomial() List)
8. \mforall{}[f:(iMonomial() List) {}\mrightarrow{} iMonomial() {}\mrightarrow{} (iMonomial() List)]
eager-accum(x,a.f[x;a];mul-mono-poly(u;[u1 / v1]);v)
\msim{} accumulate (with value x and list item a):
f[x;a]
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
supposing valueall-type(iMonomial() List)
\mvdash{} accumulate (with value sofar and list item m):
add\_ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1])) \msim{} accumulate (with value sofar and list item m):
add-ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
By
Latex:
Assert \mkleeneopen{}accumulate (with value sofar and list item m):
add\_ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))
= accumulate (with value sofar and list item m):
add-ipoly(sofar;mul-mono-poly(m;[u1 / v1]))
over list:
v
with starting value:
mul-mono-poly(u;[u1 / v1]))\mkleeneclose{}\mcdot{}
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