Proof of Nuprl Lemma : derivative-rinv

Step * 1 1 1 of Lemma derivative-rinv

1. I : Interval
2. f : I ⟶ℝ
3. g : I ⟶ℝ
4. ∀x,y:{t:ℝ| t ∈ I} .  ((x = y) ⇒ (g[x] = g[y]))
5. d(f[x])/dx = λx.g[x] on I
6. k : ℕ⁺
7. n : {n:ℕ⁺| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
8. ∀a,b:{x:ℝ| x ∈ i-approx(I;n)} .  ((a = b) ⇒ (f[a] = f[b]))
9. i-approx(I;n) ⊆ I
10. ∀x:ℝ. ((x ∈ I) ⇒ f[x] ≠ r0)
11. ∃c:ℝ [((r0 < c) ∧ (∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ (c ≤ |f[x]|))))]
12. (r1/f[x]) continuous for x ∈ i-approx(I;n)
13. mcg : g[x] continuous for x ∈ i-approx(I;n)
14. ∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ f[x] ≠ r0)
15. ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|g[x]| ≤ a)
⊢ ∃M:ℕ⁺. ∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ ((|g[x]| ≤ r(M)) ∧ (|(r1/f[x])| ≤ r(M))))
BY
{ (RenameVar `mcfinv' (-4)⋅
   THEN (Assert ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|(r1/f[x])| ≤ a) BY

               ((InstLemma `Inorm-bound` [⌜i-approx(I;n)⌝;⌜λ₂x.(r1/f[x])⌝;⌜mcfinv⌝]⋅ THENA Auto)

                THEN (InstLemma `Inorm_wf` [⌜i-approx(I;n)⌝;⌜λ₂x.(r1/f[x])⌝;⌜mcfinv⌝]⋅ THENA Auto)

                THEN With ⌜||(r1/f[x])||_i-approx(I;n)⌝ (D 0)⋅
                THEN Auto))
   )⋅ }

1
1. I : Interval
2. f : I ⟶ℝ
3. g : I ⟶ℝ
4. ∀x,y:{t:ℝ| t ∈ I} .  ((x = y) ⇒ (g[x] = g[y]))
5. d(f[x])/dx = λx.g[x] on I
6. k : ℕ⁺
7. n : {n:ℕ⁺| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
8. ∀a,b:{x:ℝ| x ∈ i-approx(I;n)} .  ((a = b) ⇒ (f[a] = f[b]))
9. i-approx(I;n) ⊆ I
10. ∀x:ℝ. ((x ∈ I) ⇒ f[x] ≠ r0)
11. ∃c:ℝ [((r0 < c) ∧ (∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ (c ≤ |f[x]|))))]
12. mcfinv : (r1/f[x]) continuous for x ∈ i-approx(I;n)
13. mcg : g[x] continuous for x ∈ i-approx(I;n)
14. ∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ f[x] ≠ r0)
15. ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|g[x]| ≤ a)
16. ∃a:ℝ. ∀[x:{r:ℝ| r ∈ i-approx(I;n)} ]. (|(r1/f[x])| ≤ a)
⊢ ∃M:ℕ⁺. ∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ ((|g[x]| ≤ r(M)) ∧ (|(r1/f[x])| ≤ r(M))))

Latex:

Latex:
1. I : Interval 2. f : I {}\mrightarrow{}\mBbbR{} 3. g : I {}\mrightarrow{}\mBbbR{} 4. \mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])) 5. d(f[x])/dx = \mlambda{}x.g[x] on I 6. k : \mBbbN{}\msupplus{} 7. n : \{n:\mBbbN{}\msupplus{}| icompact(i-approx(I;n)) \mwedge{} iproper(i-approx(I;n))\} 8. \mforall{}a,b:\{x:\mBbbR{}| x \mmember{} i-approx(I;n)\} . ((a = b) {}\mRightarrow{} (f[a] = f[b])) 9. i-approx(I;n) \msubseteq{} I 10. \mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} f[x] \mneq{} r0) 11. \mexists{}c:\mBbbR{} [((r0 < c) \mwedge{} (\mforall{}x:\mBbbR{}. ((x \mmember{} i-approx(I;n)) {}\mRightarrow{} (c \mleq{} |f[x]|))))] 12. (r1/f[x]) continuous for x \mmember{} i-approx(I;n) 13. mcg : g[x] continuous for x \mmember{} i-approx(I;n) 14. \mforall{}x:\mBbbR{}. ((x \mmember{} i-approx(I;n)) {}\mRightarrow{} f[x] \mneq{} r0) 15. \mexists{}a:\mBbbR{}. \mforall{}[x:\{r:\mBbbR{}| r \mmember{} i-approx(I;n)\} ]. (|g[x]| \mleq{} a) \mvdash{} \mexists{}M:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbR{}. ((x \mmember{} i-approx(I;n)) {}\mRightarrow{} ((|g[x]| \mleq{} r(M)) \mwedge{} (|(r1/f[x])| \mleq{} r(M)))) By Latex: (RenameVar `mcfinv' (-4)\mcdot{} THEN (Assert \mexists{}a:\mBbbR{}. \mforall{}[x:\{r:\mBbbR{}| r \mmember{} i-approx(I;n)\} ]. (|(r1/f[x])| \mleq{} a) BY ((InstLemma `Inorm-bound` [\mkleeneopen{}i-approx(I;n)\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}x.(r1/f[x])\mkleeneclose{};\mkleeneopen{}mcfinv\mkleeneclose{}]\mcdot{} THENA Auto) THEN (InstLemma `Inorm\_wf` [\mkleeneopen{}i-approx(I;n)\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}x.(r1/f[x])\mkleeneclose{};\mkleeneopen{}mcfinv\mkleeneclose{}]\mcdot{} THENA Auto) THEN With \mkleeneopen{}||(r1/f[x])||\_i-approx(I;n)\mkleeneclose{} (D 0)\mcdot{} THEN Auto)) )\mcdot{} Home Index