Step
*
1
1
1
of Lemma
sqle-list_accum
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀a,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[r1;a]] ≤ G[J[r2;a]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum H[y;h] t otherwise if v = Ax then y otherwise ⊥^j - 1
⊥
b1
as] ≤ G[λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum J[y;h] t otherwise if v = Ax then y otherwise ⊥^j\000C - 1
⊥
b2
as]))
11. as : Base@i
12. b1 : Base@i
13. b2 : Base@i
14. F[b1] ≤ G[b2]@i
15. ∀[a,B:Top]. (F[eval x = a in B[x]] ~ eval x = a in F[B[x]])
16. ∀[a,B:Top]. (G[eval x = a in B[x]] ~ eval x = a in G[B[x]])
17. (as)↓@i
18. ∀[a,b,c:Top]. (F[if a is a pair then b otherwise c] ~ if a is a pair then F[b] otherwise F[c])
19. ∀[a,b,c:Top]. (G[if a is a pair then b otherwise c] ~ if a is a pair then G[b] otherwise G[c])
20. ∀[a,b,c:Top]. (F[if a = Ax then b otherwise c] ~ if a = Ax then F[b] otherwise F[c])
21. ∀[a,b,c:Top]. (G[if a = Ax then b otherwise c] ~ if a = Ax then G[b] otherwise G[c])
⊢ if as is a pair then F[let h,t = as
in λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum H[y;h] t
otherwise if v = Ax then y otherwise ⊥^j - 1
⊥
H[b1;h]
t] otherwise if as = Ax then F[b1] otherwise F[⊥]
≤ if as is a pair then G[let h,t = as
in λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum J[y;h] t
otherwise if v = Ax then y otherwise ⊥^j - 1
⊥
J[b2;h]
t] otherwise if as = Ax then G[b2] otherwise G[⊥]
BY
{ (AssumeHasValue
THEN (HasValueD (-1) ORELSE (ExceptionCases (-1) THEN Try (Complete (SameException))))
THEN (HVimplies2 0 [1] THEN (RWO "-1" 0 THENA MemTop) THEN Reduce 0)
THEN Try ((HVimplies2 0 [1] THEN (RWO "-1" 0 THENA MemTop) THEN Reduce 0))
THEN Try ((BackThruSomeHyp THEN Try (MemBase) THEN BackThruSomeHyp THEN Auto))) }
1
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀a,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[r1;a]] ≤ G[J[r2;a]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum H[y;h] t otherwise if v = Ax then y otherwise ⊥^j - 1
⊥
b1
as] ≤ G[λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum J[y;h] t otherwise if v = Ax then y otherwise ⊥^j\000C - 1
⊥
b2
as]))
11. as : Base@i
12. b1 : Base@i
13. b2 : Base@i
14. F[b1] ≤ G[b2]@i
15. ∀[a,B:Top]. (F[eval x = a in B[x]] ~ eval x = a in F[B[x]])
16. ∀[a,B:Top]. (G[eval x = a in B[x]] ~ eval x = a in G[B[x]])
17. (as)↓@i
18. ∀[a,b,c:Top]. (F[if a is a pair then b otherwise c] ~ if a is a pair then F[b] otherwise F[c])
19. ∀[a,b,c:Top]. (G[if a is a pair then b otherwise c] ~ if a is a pair then G[b] otherwise G[c])
20. ∀[a,b,c:Top]. (F[if a = Ax then b otherwise c] ~ if a = Ax then F[b] otherwise F[c])
21. ∀[a,b,c:Top]. (G[if a = Ax then b otherwise c] ~ if a = Ax then G[b] otherwise G[c])
22. (F[⊥])↓
23. (as)↓
24. ∀a,b:Top. (if as is a pair then a otherwise b ~ b)
25. ∀a,b:Top. (if as = Ax then a otherwise b ~ b)
⊢ F[⊥] ≤ G[⊥]
2
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀a,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[r1;a]] ≤ G[J[r2;a]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum H[y;h] t otherwise if v = Ax then y otherwise ⊥^j - 1
⊥
b1
as] ≤ G[λlist_accum,y,L. eval v = L in
if v is a pair then let h,t = v
in list_accum J[y;h] t otherwise if v = Ax then y otherwise ⊥^j\000C - 1
⊥
b2
as]))
11. as : Base@i
12. b1 : Base@i
13. b2 : Base@i
14. F[b1] ≤ G[b2]@i
15. ∀[a,B:Top]. (F[eval x = a in B[x]] ~ eval x = a in F[B[x]])
16. ∀[a,B:Top]. (G[eval x = a in B[x]] ~ eval x = a in G[B[x]])
17. (as)↓@i
18. ∀[a,b,c:Top]. (F[if a is a pair then b otherwise c] ~ if a is a pair then F[b] otherwise F[c])
19. ∀[a,b,c:Top]. (G[if a is a pair then b otherwise c] ~ if a is a pair then G[b] otherwise G[c])
20. ∀[a,b,c:Top]. (F[if a = Ax then b otherwise c] ~ if a = Ax then F[b] otherwise F[c])
21. ∀[a,b,c:Top]. (G[if a = Ax then b otherwise c] ~ if a = Ax then G[b] otherwise G[c])
22. is-exception(F[⊥])
23. (as)↓
24. ∀a,b:Top. (if as is a pair then a otherwise b ~ b)
25. ∀a,b:Top. (if as = Ax then a otherwise b ~ b)
⊢ F[⊥] ≤ G[⊥]
Latex:
Latex:
1. F : Base
2. strict1(\mlambda{}x.F[x])
3. G : Base
4. strict1(\mlambda{}x.G[x])
5. H : Base
6. J : Base
7. \mforall{}a,r1,r2:Base. ((F[r1] \mleq{} G[r2]) {}\mRightarrow{} (F[H[r1;a]] \mleq{} G[J[r2;a]]))
8. j : \mBbbZ{}
9. 0 < j
10. \mforall{}as,b1,b2:Base.
((F[b1] \mleq{} G[b2])
{}\mRightarrow{} (F[\mlambda{}list$_{accum}$,y,L. eval v = L in
if v is a pair then let h,t = v
in list$_{accum}$ H[y;h] t
otherwise if v = Ax then y otherwise \mbot{}\^{}j - 1
\mbot{}
b1
as] \mleq{} G[\mlambda{}list$_{accum}$,y,L. eval v = L in
if v is a pair then let h,t = v
in list$_{accum}$ J[y;h]\000C t
otherwise if v = Ax then y otherwise \mbot{}\^{}j - 1
\mbot{}
b2
as]))
11. as : Base@i
12. b1 : Base@i
13. b2 : Base@i
14. F[b1] \mleq{} G[b2]@i
15. \mforall{}[a,B:Top]. (F[eval x = a in B[x]] \msim{} eval x = a in F[B[x]])
16. \mforall{}[a,B:Top]. (G[eval x = a in B[x]] \msim{} eval x = a in G[B[x]])
17. (as)\mdownarrow{}@i
18. \mforall{}[a,b,c:Top]. (F[if a is a pair then b otherwise c] \msim{} if a is a pair then F[b] otherwise F[c])
19. \mforall{}[a,b,c:Top]. (G[if a is a pair then b otherwise c] \msim{} if a is a pair then G[b] otherwise G[c])
20. \mforall{}[a,b,c:Top]. (F[if a = Ax then b otherwise c] \msim{} if a = Ax then F[b] otherwise F[c])
21. \mforall{}[a,b,c:Top]. (G[if a = Ax then b otherwise c] \msim{} if a = Ax then G[b] otherwise G[c])
\mvdash{} if as is a pair then F[let h,t = as
in \mlambda{}list$_{accum}$,y,L. eval v = L in
if v is a pair then let h,t = v
in list$_{accum}\mbackslash{}ff2\000C4 H[y;h] t
otherwise if v = Ax then y otherwise \mbot{}\^{}j - 1
\mbot{}
H[b1;h]
t] otherwise if as = Ax then F[b1] otherwise F[\mbot{}]
\mleq{} if as is a pair then G[let h,t = as
in \mlambda{}list$_{accum}$,y,L. eval v = L in
if v is a pair then let h,t = v
in list$_{accum}\mbackslash{}f\000Cf24 J[y;h] t
otherwise if v = Ax then y otherwise \mbot{}\^{}j - 1
\mbot{}
J[b2;h]
t] otherwise if as = Ax then G[b2] otherwise G[\mbot{}]
By
Latex:
(AssumeHasValue
THEN (HasValueD (-1) ORELSE (ExceptionCases (-1) THEN Try (Complete (SameException))))
THEN (HVimplies2 0 [1] THEN (RWO "-1" 0 THENA MemTop) THEN Reduce 0)
THEN Try ((HVimplies2 0 [1] THEN (RWO "-1" 0 THENA MemTop) THEN Reduce 0))
THEN Try ((BackThruSomeHyp THEN Try (MemBase) THEN BackThruSomeHyp THEN Auto)))
Home
Index