Step
*
1
1
2
1
2
1
1
of Lemma
closures-meet
.....assertion.....
1. [P] : ℝ ⟶ ℙ
2. [Q] : ℝ ⟶ ℙ
3. a0 : ℝ
4. b0 : ℝ
5. P a0
6. Q b0
7. a0 ≤ b0
8. c : ℝ
9. r0 ≤ c
10. c < r1
11. ∀a,b:ℝ.
(((P a) ∧ (Q b) ∧ (a ≤ b))
⇒ (∃a',b':ℝ. ((P a') ∧ (Q b') ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))))
12. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
∃abp':a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = abp' in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
13. f : abp:(a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))) ⟶ (a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b)))
14. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = f abp in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
15. p0 : (P a0) ∧ (Q b0) ∧ (a0 ≤ b0)
⊢ ∃ab:ℕ ⟶ (ℝ × ℝ)
∀n:ℕ
let a,b = ab[n]
in let a',b' = ab[n + 1]
in (P a) ∧ (Q b) ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
BY
{ InstConcl [⌜λn.let a,b,p = primrec(n;<a0, b0, p0>;λx,y. (f y)) in
<a, b>⌝]⋅ }
1
.....wf.....
1. P : ℝ ⟶ ℙ
2. Q : ℝ ⟶ ℙ
3. a0 : ℝ
4. b0 : ℝ
5. P a0
6. Q b0
7. a0 ≤ b0
8. c : ℝ
9. r0 ≤ c
10. c < r1
11. ∀a,b:ℝ.
(((P a) ∧ (Q b) ∧ (a ≤ b))
⇒ (∃a',b':ℝ. ((P a') ∧ (Q b') ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))))
12. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
∃abp':a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = abp' in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
13. f : abp:(a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))) ⟶ (a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b)))
14. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = f abp in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
15. p0 : (P a0) ∧ (Q b0) ∧ (a0 ≤ b0)
⊢ λn.let a,b,p = primrec(n;<a0, b0, p0>;λx,y. (f y)) in
<a, b> ∈ ℕ ⟶ (ℝ × ℝ)
2
1. [P] : ℝ ⟶ ℙ
2. [Q] : ℝ ⟶ ℙ
3. a0 : ℝ
4. b0 : ℝ
5. P a0
6. Q b0
7. a0 ≤ b0
8. c : ℝ
9. r0 ≤ c
10. c < r1
11. ∀a,b:ℝ.
(((P a) ∧ (Q b) ∧ (a ≤ b))
⇒ (∃a',b':ℝ. ((P a') ∧ (Q b') ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))))
12. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
∃abp':a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = abp' in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
13. f : abp:(a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))) ⟶ (a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b)))
14. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = f abp in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
15. p0 : (P a0) ∧ (Q b0) ∧ (a0 ≤ b0)
⊢ ∀n:ℕ
let a,b = λn.let a,b,p = primrec(n;<a0, b0, p0>;λx,y. (f y)) in
<a, b>[n]
in let a',b' = λn.let a,b,p = primrec(n;<a0, b0, p0>;λx,y. (f y)) in
<a, b>[n + 1]
in (P a) ∧ (Q b) ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
3
.....wf.....
1. P : ℝ ⟶ ℙ
2. Q : ℝ ⟶ ℙ
3. a0 : ℝ
4. b0 : ℝ
5. P a0
6. Q b0
7. a0 ≤ b0
8. c : ℝ
9. r0 ≤ c
10. c < r1
11. ∀a,b:ℝ.
(((P a) ∧ (Q b) ∧ (a ≤ b))
⇒ (∃a',b':ℝ. ((P a') ∧ (Q b') ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))))
12. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
∃abp':a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = abp' in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
13. f : abp:(a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))) ⟶ (a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b)))
14. ∀abp:a:ℝ × b:ℝ × ((P a) ∧ (Q b) ∧ (a ≤ b))
let a,b,p = abp in
let a',b',p' = f abp in
(a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))
15. p0 : (P a0) ∧ (Q b0) ∧ (a0 ≤ b0)
16. ab : ℕ ⟶ (ℝ × ℝ)
⊢ istype(∀n:ℕ
let a,b = ab[n]
in let a',b' = ab[n + 1]
in (P a) ∧ (Q b) ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))
Latex:
Latex:
.....assertion.....
1. [P] : \mBbbR{} {}\mrightarrow{} \mBbbP{}
2. [Q] : \mBbbR{} {}\mrightarrow{} \mBbbP{}
3. a0 : \mBbbR{}
4. b0 : \mBbbR{}
5. P a0
6. Q b0
7. a0 \mleq{} b0
8. c : \mBbbR{}
9. r0 \mleq{} c
10. c < r1
11. \mforall{}a,b:\mBbbR{}.
(((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b))
{}\mRightarrow{} (\mexists{}a',b':\mBbbR{}
((P a') \mwedge{} (Q b') \mwedge{} (a \mleq{} a') \mwedge{} (a' \mleq{} b') \mwedge{} (b' \mleq{} b) \mwedge{} ((b' - a') \mleq{} ((b - a) * c)))))
12. \mforall{}abp:a:\mBbbR{} \mtimes{} b:\mBbbR{} \mtimes{} ((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b))
\mexists{}abp':a:\mBbbR{} \mtimes{} b:\mBbbR{} \mtimes{} ((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b))
let a,b,p = abp in
let a',b',p' = abp' in
(a \mleq{} a') \mwedge{} (a' \mleq{} b') \mwedge{} (b' \mleq{} b) \mwedge{} ((b' - a') \mleq{} ((b - a) * c))
13. f : abp:(a:\mBbbR{} \mtimes{} b:\mBbbR{} \mtimes{} ((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b))) {}\mrightarrow{} (a:\mBbbR{} \mtimes{} b:\mBbbR{} \mtimes{} ((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b)))
14. \mforall{}abp:a:\mBbbR{} \mtimes{} b:\mBbbR{} \mtimes{} ((P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} b))
let a,b,p = abp in
let a',b',p' = f abp in
(a \mleq{} a') \mwedge{} (a' \mleq{} b') \mwedge{} (b' \mleq{} b) \mwedge{} ((b' - a') \mleq{} ((b - a) * c))
15. p0 : (P a0) \mwedge{} (Q b0) \mwedge{} (a0 \mleq{} b0)
\mvdash{} \mexists{}ab:\mBbbN{} {}\mrightarrow{} (\mBbbR{} \mtimes{} \mBbbR{})
\mforall{}n:\mBbbN{}
let a,b = ab[n]
in let a',b' = ab[n + 1]
in (P a) \mwedge{} (Q b) \mwedge{} (a \mleq{} a') \mwedge{} (a' \mleq{} b') \mwedge{} (b' \mleq{} b) \mwedge{} ((b' - a') \mleq{} ((b - a) * c))
By
Latex:
InstConcl [\mkleeneopen{}\mlambda{}n.let a,b,p = primrec(n;<a0, b0, p0>\mlambda{}x,y. (f y)) in
<a, b>\mkleeneclose{}]\mcdot{}
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