Step
*
1
1
1
1
3
1
1
of Lemma
extend-half-cube-face
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. ∀i:ℕk. (↑Inhabited(a i))
6. 0 < dim(b)
7. ∀i:ℕk. a i ≤ b i
8. ∀i:ℕk. (↑is-half-interval(b i;c i))
9. dim(a) = (dim(b) - 1) ∈ ℤ
10. i : ℕk
11. dim(c i) = 1 ∈ ℤ
12. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
13. (a i) = [fst((b i))] ∈ ℚInterval
14. (fst((b i))) = qavg(fst((c i));snd((c i))) ∈ ℚ
15. (snd((b i))) = (snd((c i))) ∈ ℚ
⊢ ∃!b':ℚCube(k). ((∀i:ℕk. a i ≤ b' i) ∧ (↑is-half-cube(k;b';c)) ∧ (¬(b' = b ∈ ℚCube(k))))
BY
{ ((Assert λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ∈ ℚCube(k) BY
(Unfold `rational-cube` 0 THEN Auto))
THEN D 0 With ⌜λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ⌝
THEN Auto
THEN Reduce 0) }
1
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. ∀i:ℕk. (↑Inhabited(a i))
6. 0 < dim(b)
7. ∀i:ℕk. a i ≤ b i
8. ∀i:ℕk. (↑is-half-interval(b i;c i))
9. dim(a) = (dim(b) - 1) ∈ ℤ
10. i : ℕk
11. dim(c i) = 1 ∈ ℤ
12. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
13. (a i) = [fst((b i))] ∈ ℚInterval
14. (fst((b i))) = qavg(fst((c i));snd((c i))) ∈ ℚ
15. (snd((b i))) = (snd((c i))) ∈ ℚ
16. λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ∈ ℚCube(k)
17. i@0 : ℕk
⊢ a i@0 ≤ if (i@0 =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b i@0 fi
2
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. ∀i:ℕk. (↑Inhabited(a i))
6. 0 < dim(b)
7. ∀i:ℕk. a i ≤ b i
8. ∀i:ℕk. (↑is-half-interval(b i;c i))
9. dim(a) = (dim(b) - 1) ∈ ℤ
10. i : ℕk
11. dim(c i) = 1 ∈ ℤ
12. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
13. (a i) = [fst((b i))] ∈ ℚInterval
14. (fst((b i))) = qavg(fst((c i));snd((c i))) ∈ ℚ
15. (snd((b i))) = (snd((c i))) ∈ ℚ
16. λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ∈ ℚCube(k)
17. ∀i@0:ℕk. a i@0 ≤ (λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ) i@0
⊢ ↑is-half-cube(k;λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ;c)
3
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. ∀i:ℕk. (↑Inhabited(a i))
6. 0 < dim(b)
7. ∀i:ℕk. a i ≤ b i
8. ∀i:ℕk. (↑is-half-interval(b i;c i))
9. dim(a) = (dim(b) - 1) ∈ ℤ
10. i : ℕk
11. dim(c i) = 1 ∈ ℤ
12. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
13. (a i) = [fst((b i))] ∈ ℚInterval
14. (fst((b i))) = qavg(fst((c i));snd((c i))) ∈ ℚ
15. (snd((b i))) = (snd((c i))) ∈ ℚ
16. λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ∈ ℚCube(k)
17. ∀i@0:ℕk. a i@0 ≤ (λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ) i@0
18. ↑is-half-cube(k;λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ;c)
⊢ ¬((λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ) = b ∈ ℚCube(k))
4
1. k : ℕ
2. a : ℚCube(k)
3. b : ℚCube(k)
4. c : ℚCube(k)
5. ∀i:ℕk. (↑Inhabited(a i))
6. 0 < dim(b)
7. ∀i:ℕk. a i ≤ b i
8. ∀i:ℕk. (↑is-half-interval(b i;c i))
9. dim(a) = (dim(b) - 1) ∈ ℤ
10. i : ℕk
11. dim(c i) = 1 ∈ ℤ
12. ∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((a j) = (b j) ∈ ℚInterval))
13. (a i) = [fst((b i))] ∈ ℚInterval
14. (fst((b i))) = qavg(fst((c i));snd((c i))) ∈ ℚ
15. (snd((b i))) = (snd((c i))) ∈ ℚ
16. λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ∈ ℚCube(k)
17. ∀i@0:ℕk. a i@0 ≤ (λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ) i@0
18. ↑is-half-cube(k;λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ;c)
19. ¬((λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ) = b ∈ ℚCube(k))
20. y : ℚCube(k)
21. ∀i:ℕk. a i ≤ y i
22. ↑is-half-cube(k;y;c)
23. ¬(y = b ∈ ℚCube(k))
⊢ y = (λj.if (j =z i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi ) ∈ ℚCube(k)
Latex:
Latex:
1. k : \mBbbN{}
2. a : \mBbbQ{}Cube(k)
3. b : \mBbbQ{}Cube(k)
4. c : \mBbbQ{}Cube(k)
5. \mforall{}i:\mBbbN{}k. (\muparrow{}Inhabited(a i))
6. 0 < dim(b)
7. \mforall{}i:\mBbbN{}k. a i \mleq{} b i
8. \mforall{}i:\mBbbN{}k. (\muparrow{}is-half-interval(b i;c i))
9. dim(a) = (dim(b) - 1)
10. i : \mBbbN{}k
11. dim(c i) = 1
12. \mforall{}j:\mBbbN{}k. ((\mneg{}(j = i)) {}\mRightarrow{} ((a j) = (b j)))
13. (a i) = [fst((b i))]
14. (fst((b i))) = qavg(fst((c i));snd((c i)))
15. (snd((b i))) = (snd((c i)))
\mvdash{} \mexists{}!b':\mBbbQ{}Cube(k). ((\mforall{}i:\mBbbN{}k. a i \mleq{} b' i) \mwedge{} (\muparrow{}is-half-cube(k;b';c)) \mwedge{} (\mneg{}(b' = b)))
By
Latex:
((Assert \mlambda{}j.if (j =\msubz{} i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi \mmember{} \mBbbQ{}Cube(k) BY
(Unfold `rational-cube` 0 THEN Auto))
THEN D 0 With \mkleeneopen{}\mlambda{}j.if (j =\msubz{} i) then <fst((c i)), qavg(fst((c i));snd((c i)))> else b j fi \mkleeneclose{}
THEN Auto
THEN Reduce 0)
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