Step
*
2
2
1
1
1
of Lemma
add-poly-lemma1
1. u : iMonomial()
2. v : iMonomial() List
3. ∀q:iMonomial() List. ∀m:iMonomial().
((∀i:ℕ||v||. ∀j:ℕi. imonomial-less(v[j];v[i]))
⇒ (∀i:ℕ||q||. ∀j:ℕi. imonomial-less(q[j];q[i]))
⇒ (0 < ||v|| ⇒ imonomial-less(m;v[0]))
⇒ (0 < ||q|| ⇒ imonomial-less(m;q[0]))
⇒ 0 < ||add-ipoly(v;q)||
⇒ imonomial-less(m;add-ipoly(v;q)[0]))
4. u1 : iMonomial()
5. v1 : iMonomial() List
6. ∀m:iMonomial()
((∀i:ℕ||[u / v]||. ∀j:ℕi. imonomial-less([u / v][j];[u / v][i]))
⇒ (∀i:ℕ||v1||. ∀j:ℕi. imonomial-less(v1[j];v1[i]))
⇒ (0 < ||[u / v]|| ⇒ imonomial-less(m;[u / v][0]))
⇒ (0 < ||v1|| ⇒ imonomial-less(m;v1[0]))
⇒ 0 < ||add-ipoly([u / v];v1)||
⇒ imonomial-less(m;add-ipoly([u / v];v1)[0]))
7. m : iMonomial()
8. ∀i:ℕ||[u / v]||. ∀j:ℕi. imonomial-less([u / v][j];[u / v][i])
9. ∀i:ℕ||[u1 / v1]||. ∀j:ℕi. imonomial-less([u1 / v1][j];[u1 / v1][i])
10. 0 < ||[u / v]|| ⇒ imonomial-less(m;[u / v][0])
11. 0 < ||[u1 / v1]|| ⇒ imonomial-less(m;[u1 / v1][0])
12. ↑imonomial-le(u;u1)
13. ↑imonomial-le(u1;u)
⊢ 0 < ||eval x = add-ipoly(v;v1) in
let cp,vs = u
in eval c = cp + (fst(u1)) in
if c=0 then x else [<c, vs> / x]||
⇒ imonomial-less(m;eval x = add-ipoly(v;v1) in
let cp,vs = u
in eval c = cp + (fst(u1)) in
if c=0 then x else [<c, vs> / x][0])
BY
{ ((CallByValueReduce 0 THENA Auto)
THEN DVar `u'
THEN DVar `u1'
THEN Reduce 0
THEN (CallByValueReduce 0 THENA Auto)
THEN AutoSplit
THEN All Reduce) }
1
1. u2 : ℤ-o
2. u3 : {vs:ℤ List| sorted(vs)}
3. v : iMonomial() List
4. ∀q:iMonomial() List. ∀m:iMonomial().
((∀i:ℕ||v||. ∀j:ℕi. imonomial-less(v[j];v[i]))
⇒ (∀i:ℕ||q||. ∀j:ℕi. imonomial-less(q[j];q[i]))
⇒ (0 < ||v|| ⇒ imonomial-less(m;v[0]))
⇒ (0 < ||q|| ⇒ imonomial-less(m;q[0]))
⇒ 0 < ||add-ipoly(v;q)||
⇒ imonomial-less(m;add-ipoly(v;q)[0]))
5. u4 : ℤ-o
6. u5 : {vs:ℤ List| sorted(vs)}
7. v1 : iMonomial() List
8. ∀m:iMonomial()
((∀i:ℕ||v|| + 1. ∀j:ℕi. imonomial-less([<u2, u3> / v][j];[<u2, u3> / v][i]))
⇒ (∀i:ℕ||v1||. ∀j:ℕi. imonomial-less(v1[j];v1[i]))
⇒ (0 < ||v|| + 1 ⇒ imonomial-less(m;<u2, u3>))
⇒ (0 < ||v1|| ⇒ imonomial-less(m;v1[0]))
⇒ 0 < ||add-ipoly([<u2, u3> / v];v1)||
⇒ imonomial-less(m;add-ipoly([<u2, u3> / v];v1)[0]))
9. m : iMonomial()
10. ∀i:ℕ||v|| + 1. ∀j:ℕi. imonomial-less([<u2, u3> / v][j];[<u2, u3> / v][i])
11. ∀i:ℕ||v1|| + 1. ∀j:ℕi. imonomial-less([<u4, u5> / v1][j];[<u4, u5> / v1][i])
12. 0 < ||v|| + 1 ⇒ imonomial-less(m;<u2, u3>)
13. 0 < ||v1|| + 1 ⇒ imonomial-less(m;<u4, u5>)
14. ↑imonomial-le(<u2, u3>;<u4, u5>)
15. ↑imonomial-le(<u4, u5>;<u2, u3>)
16. (u2 + u4) = 0 ∈ ℤ
⊢ 0 < ||add-ipoly(v;v1)|| ⇒ imonomial-less(m;add-ipoly(v;v1)[0])
2
1. u2 : ℤ-o
2. u3 : {vs:ℤ List| sorted(vs)}
3. v : iMonomial() List
4. ∀q:iMonomial() List. ∀m:iMonomial().
((∀i:ℕ||v||. ∀j:ℕi. imonomial-less(v[j];v[i]))
⇒ (∀i:ℕ||q||. ∀j:ℕi. imonomial-less(q[j];q[i]))
⇒ (0 < ||v|| ⇒ imonomial-less(m;v[0]))
⇒ (0 < ||q|| ⇒ imonomial-less(m;q[0]))
⇒ 0 < ||add-ipoly(v;q)||
⇒ imonomial-less(m;add-ipoly(v;q)[0]))
5. u4 : ℤ-o
6. u2 + u4 ≠ 0
7. u5 : {vs:ℤ List| sorted(vs)}
8. v1 : iMonomial() List
9. ∀m:iMonomial()
((∀i:ℕ||v|| + 1. ∀j:ℕi. imonomial-less([<u2, u3> / v][j];[<u2, u3> / v][i]))
⇒ (∀i:ℕ||v1||. ∀j:ℕi. imonomial-less(v1[j];v1[i]))
⇒ (0 < ||v|| + 1 ⇒ imonomial-less(m;<u2, u3>))
⇒ (0 < ||v1|| ⇒ imonomial-less(m;v1[0]))
⇒ 0 < ||add-ipoly([<u2, u3> / v];v1)||
⇒ imonomial-less(m;add-ipoly([<u2, u3> / v];v1)[0]))
10. m : iMonomial()
11. ∀i:ℕ||v|| + 1. ∀j:ℕi. imonomial-less([<u2, u3> / v][j];[<u2, u3> / v][i])
12. ∀i:ℕ||v1|| + 1. ∀j:ℕi. imonomial-less([<u4, u5> / v1][j];[<u4, u5> / v1][i])
13. 0 < ||v|| + 1 ⇒ imonomial-less(m;<u2, u3>)
14. 0 < ||v1|| + 1 ⇒ imonomial-less(m;<u4, u5>)
15. ↑imonomial-le(<u2, u3>;<u4, u5>)
16. ↑imonomial-le(<u4, u5>;<u2, u3>)
⊢ 0 < ||add-ipoly(v;v1)|| + 1 ⇒ imonomial-less(m;<u2 + u4, u3>)
Latex:
Latex:
1. u : iMonomial()
2. v : iMonomial() List
3. \mforall{}q:iMonomial() List. \mforall{}m:iMonomial().
((\mforall{}i:\mBbbN{}||v||. \mforall{}j:\mBbbN{}i. imonomial-less(v[j];v[i]))
{}\mRightarrow{} (\mforall{}i:\mBbbN{}||q||. \mforall{}j:\mBbbN{}i. imonomial-less(q[j];q[i]))
{}\mRightarrow{} (0 < ||v|| {}\mRightarrow{} imonomial-less(m;v[0]))
{}\mRightarrow{} (0 < ||q|| {}\mRightarrow{} imonomial-less(m;q[0]))
{}\mRightarrow{} 0 < ||add-ipoly(v;q)||
{}\mRightarrow{} imonomial-less(m;add-ipoly(v;q)[0]))
4. u1 : iMonomial()
5. v1 : iMonomial() List
6. \mforall{}m:iMonomial()
((\mforall{}i:\mBbbN{}||[u / v]||. \mforall{}j:\mBbbN{}i. imonomial-less([u / v][j];[u / v][i]))
{}\mRightarrow{} (\mforall{}i:\mBbbN{}||v1||. \mforall{}j:\mBbbN{}i. imonomial-less(v1[j];v1[i]))
{}\mRightarrow{} (0 < ||[u / v]|| {}\mRightarrow{} imonomial-less(m;[u / v][0]))
{}\mRightarrow{} (0 < ||v1|| {}\mRightarrow{} imonomial-less(m;v1[0]))
{}\mRightarrow{} 0 < ||add-ipoly([u / v];v1)||
{}\mRightarrow{} imonomial-less(m;add-ipoly([u / v];v1)[0]))
7. m : iMonomial()
8. \mforall{}i:\mBbbN{}||[u / v]||. \mforall{}j:\mBbbN{}i. imonomial-less([u / v][j];[u / v][i])
9. \mforall{}i:\mBbbN{}||[u1 / v1]||. \mforall{}j:\mBbbN{}i. imonomial-less([u1 / v1][j];[u1 / v1][i])
10. 0 < ||[u / v]|| {}\mRightarrow{} imonomial-less(m;[u / v][0])
11. 0 < ||[u1 / v1]|| {}\mRightarrow{} imonomial-less(m;[u1 / v1][0])
12. \muparrow{}imonomial-le(u;u1)
13. \muparrow{}imonomial-le(u1;u)
\mvdash{} 0 < ||eval x = add-ipoly(v;v1) in
let cp,vs = u
in eval c = cp + (fst(u1)) in
if c=0 then x else [<c, vs> / x]||
{}\mRightarrow{} imonomial-less(m;eval x = add-ipoly(v;v1) in
let cp,vs = u
in eval c = cp + (fst(u1)) in
if c=0 then x else [<c, vs> / x][0])
By
Latex:
((CallByValueReduce 0 THENA Auto)
THEN DVar `u'
THEN DVar `u1'
THEN Reduce 0
THEN (CallByValueReduce 0 THENA Auto)
THEN AutoSplit
THEN All Reduce)
Home
Index