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of Lemma
assert-rat-term-eq
1. ∀r:rat_term(). let p,q = rat_term_to_ipolys(r) in r ≡ ipolynomial-term(p)/ipolynomial-term(q)
2. r1 : rat_term()
3. r2 : rat_term()
4. v4 : iPolynomial()
5. v5 : iPolynomial()
6. rat_term_to_ipolys(r1) = <v4, v5> ∈ (iPolynomial() × iPolynomial())
7. v2 : iPolynomial()
8. v3 : iPolynomial()
9. rat_term_to_ipolys(r2) = <v2, v3> ∈ (iPolynomial() × iPolynomial())
10. ↑null(add-ipoly(mul-ipoly(v4;v3);mul-ipoly(minus-poly(v2);v5)))
11. f : ℤ ⟶ ℝ
12. P1 : ℙ
13. v8 : ⋂:P1. ℝ
14. rat_term_to_real(f;r2) = <P1, v8> ∈ (P:ℙ × ℝ supposing P)
15. P : ℙ
16. v7 : ⋂:P. ℝ
17. rat_term_to_real(f;r1) = <P, v7> ∈ (P:ℙ × ℝ supposing P)
18. P
19. P1
20. real_term_value(f;ipolynomial-term(v5)) ≠ r0
21. v7 = (real_term_value(f;ipolynomial-term(v4))/real_term_value(f;ipolynomial-term(v5)))
22. real_term_value(f;ipolynomial-term(v3)) ≠ r0
23. v8 = (real_term_value(f;ipolynomial-term(v2))/real_term_value(f;ipolynomial-term(v3)))
24. v : ℝ
25. real_term_value(f;ipolynomial-term(v3)) = v ∈ ℝ
26. v9 : ℝ
27. real_term_value(f;ipolynomial-term(v4)) = v9 ∈ ℝ
28. v10 : ℝ
29. real_term_value(f;ipolynomial-term(v2)) = v10 ∈ ℝ
30. v11 : ℝ
31. real_term_value(f;ipolynomial-term(v5)) = v11 ∈ ℝ
32. v12 : ℝ
33. real_term_value(f;ipolynomial-term(mul-ipoly(v4;v3))) = v12 ∈ ℝ
34. v13 : ℝ
35. real_term_value(f;ipolynomial-term(mul-ipoly(minus-poly(v2);v5))) = v13 ∈ ℝ
36. v14 : ℝ
37. real_term_value(f;ipolynomial-term(add-ipoly(mul-ipoly(v4;v3);mul-ipoly(minus-poly(v2);v5)))) = v14 ∈ ℝ
⊢ (v12 = (v9 * v))
⇒ (v13 = (real_term_value(f;ipolynomial-term(minus-poly(v2))) * v11))
⇒ (v14 = (v12 + v13))
⇒ (real_term_value(f;ipolynomial-term(minus-poly(v2))) = -(v10))
⇒ (v14 = r0)
⇒ ((v * v9) = (v10 * v11))
BY
{ (GenConclTerm ⌜real_term_value(f;ipolynomial-term(minus-poly(v2)))⌝ ⋅ THENA Auto) }
1
1. ∀r:rat_term(). let p,q = rat_term_to_ipolys(r) in r ≡ ipolynomial-term(p)/ipolynomial-term(q)
2. r1 : rat_term()
3. r2 : rat_term()
4. v4 : iPolynomial()
5. v5 : iPolynomial()
6. rat_term_to_ipolys(r1) = <v4, v5> ∈ (iPolynomial() × iPolynomial())
7. v2 : iPolynomial()
8. v3 : iPolynomial()
9. rat_term_to_ipolys(r2) = <v2, v3> ∈ (iPolynomial() × iPolynomial())
10. ↑null(add-ipoly(mul-ipoly(v4;v3);mul-ipoly(minus-poly(v2);v5)))
11. f : ℤ ⟶ ℝ
12. P1 : ℙ
13. v8 : ⋂:P1. ℝ
14. rat_term_to_real(f;r2) = <P1, v8> ∈ (P:ℙ × ℝ supposing P)
15. P : ℙ
16. v7 : ⋂:P. ℝ
17. rat_term_to_real(f;r1) = <P, v7> ∈ (P:ℙ × ℝ supposing P)
18. P
19. P1
20. real_term_value(f;ipolynomial-term(v5)) ≠ r0
21. v7 = (real_term_value(f;ipolynomial-term(v4))/real_term_value(f;ipolynomial-term(v5)))
22. real_term_value(f;ipolynomial-term(v3)) ≠ r0
23. v8 = (real_term_value(f;ipolynomial-term(v2))/real_term_value(f;ipolynomial-term(v3)))
24. v : ℝ
25. real_term_value(f;ipolynomial-term(v3)) = v ∈ ℝ
26. v9 : ℝ
27. real_term_value(f;ipolynomial-term(v4)) = v9 ∈ ℝ
28. v10 : ℝ
29. real_term_value(f;ipolynomial-term(v2)) = v10 ∈ ℝ
30. v11 : ℝ
31. real_term_value(f;ipolynomial-term(v5)) = v11 ∈ ℝ
32. v12 : ℝ
33. real_term_value(f;ipolynomial-term(mul-ipoly(v4;v3))) = v12 ∈ ℝ
34. v13 : ℝ
35. real_term_value(f;ipolynomial-term(mul-ipoly(minus-poly(v2);v5))) = v13 ∈ ℝ
36. v14 : ℝ
37. real_term_value(f;ipolynomial-term(add-ipoly(mul-ipoly(v4;v3);mul-ipoly(minus-poly(v2);v5)))) = v14 ∈ ℝ
38. v15 : ℝ
39. real_term_value(f;ipolynomial-term(minus-poly(v2))) = v15 ∈ ℝ
⊢ (v12 = (v9 * v))
⇒ (v13 = (v15 * v11))
⇒ (v14 = (v12 + v13))
⇒ (v15 = -(v10))
⇒ (v14 = r0)
⇒ ((v * v9) = (v10 * v11))
Latex:
Latex:
1. \mforall{}r:rat\_term(). let p,q = rat\_term\_to\_ipolys(r) in r \mequiv{} ipolynomial-term(p)/ipolynomial-term(q)
2. r1 : rat\_term()
3. r2 : rat\_term()
4. v4 : iPolynomial()
5. v5 : iPolynomial()
6. rat\_term\_to\_ipolys(r1) = <v4, v5>
7. v2 : iPolynomial()
8. v3 : iPolynomial()
9. rat\_term\_to\_ipolys(r2) = <v2, v3>
10. \muparrow{}null(add-ipoly(mul-ipoly(v4;v3);mul-ipoly(minus-poly(v2);v5)))
11. f : \mBbbZ{} {}\mrightarrow{} \mBbbR{}
12. P1 : \mBbbP{}
13. v8 : \mcap{}:P1. \mBbbR{}
14. rat\_term\_to\_real(f;r2) = <P1, v8>
15. P : \mBbbP{}
16. v7 : \mcap{}:P. \mBbbR{}
17. rat\_term\_to\_real(f;r1) = <P, v7>
18. P
19. P1
20. real\_term\_value(f;ipolynomial-term(v5)) \mneq{} r0
21. v7 = (real\_term\_value(f;ipolynomial-term(v4))/real\_term\_value(f;ipolynomial-term(v5)))
22. real\_term\_value(f;ipolynomial-term(v3)) \mneq{} r0
23. v8 = (real\_term\_value(f;ipolynomial-term(v2))/real\_term\_value(f;ipolynomial-term(v3)))
24. v : \mBbbR{}
25. real\_term\_value(f;ipolynomial-term(v3)) = v
26. v9 : \mBbbR{}
27. real\_term\_value(f;ipolynomial-term(v4)) = v9
28. v10 : \mBbbR{}
29. real\_term\_value(f;ipolynomial-term(v2)) = v10
30. v11 : \mBbbR{}
31. real\_term\_value(f;ipolynomial-term(v5)) = v11
32. v12 : \mBbbR{}
33. real\_term\_value(f;ipolynomial-term(mul-ipoly(v4;v3))) = v12
34. v13 : \mBbbR{}
35. real\_term\_value(f;ipolynomial-term(mul-ipoly(minus-poly(v2);v5))) = v13
36. v14 : \mBbbR{}
37. real\_term\_value(f;ipolynomial-term(add-ipoly(mul-ipoly(v4;v3);mul-ipoly(minus-poly(v2);v5))))
= v14
\mvdash{} (v12 = (v9 * v))
{}\mRightarrow{} (v13 = (real\_term\_value(f;ipolynomial-term(minus-poly(v2))) * v11))
{}\mRightarrow{} (v14 = (v12 + v13))
{}\mRightarrow{} (real\_term\_value(f;ipolynomial-term(minus-poly(v2))) = -(v10))
{}\mRightarrow{} (v14 = r0)
{}\mRightarrow{} ((v * v9) = (v10 * v11))
By
Latex:
(GenConclTerm \mkleeneopen{}real\_term\_value(f;ipolynomial-term(minus-poly(v2)))\mkleeneclose{} \mcdot{} THENA Auto)
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