Step
*
1
1
of Lemma
sqle-list_ind
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀x,y,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[x;y;r1]] ≤ G[J[x;y;r2]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j - 1
⊥
as] ≤ G[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
as]))
11. as : Base@i
12. b1 : Base@i
13. b2 : Base@i
14. F[b1] ≤ G[b2]@i
⊢ F[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j
⊥
as] ≤ G[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j
⊥
as]
BY
{ ((RWO "fun_exp_unroll_1" 0 THENA Auto)
THEN Reduce 0
THEN ((InstLemma `lifting-strict-callbyvalue`
[⌜so_lambda(x,y,z,w.F[x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWW "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-callbyvalue`
[⌜so_lambda(x,y,z,w.G[x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWW "-1" 0 THENA Auto)
)
THEN UseCbvSqle
THEN ((InstLemma `lifting-strict-ispair`
[⌜so_lambda(x,y,z,w.F[x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWO "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-ispair`
[⌜so_lambda(x,y,z,w.G[x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWO "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-isaxiom`
[⌜so_lambda(x,y,z,w.F[x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWO "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-isaxiom`
[⌜so_lambda(x,y,z,w.G[x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM ((RWO "-1" 0 THENA Auto)
THEN 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)))
)) }
1
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀x,y,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[x;y;r1]] ≤ G[J[x;y;r2]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j - 1
⊥
as] ≤ G[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
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. 0 ≤ 0@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[H[fst(as);snd(as);λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j\000C - 1
⊥
(snd(as))]])↓
23. 0 ≤ 0
24. as ~ <fst(as), snd(as)>
⊢ F[H[fst(as);snd(as);λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j - \000C1
⊥
(snd(as))]] ≤ G[J[fst(as);snd(as);λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t]
otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
(snd(as))]]
2
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀x,y,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[x;y;r1]] ≤ G[J[x;y;r2]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j - 1
⊥
as] ≤ G[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
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[⊥]
3
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀x,y,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[x;y;r1]] ≤ G[J[x;y;r2]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j - 1
⊥
as] ≤ G[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
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. 0 ≤ 0@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[H[fst(as);snd(as);λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t]
otherwise if v = Ax then b1 otherwise ⊥^j - 1
⊥
(snd(as))]])
23. 0 ≤ 0
24. as ~ <fst(as), snd(as)>
⊢ F[H[fst(as);snd(as);λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j - \000C1
⊥
(snd(as))]] ≤ G[J[fst(as);snd(as);λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t]
otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
(snd(as))]]
4
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. strict1(λx.G[x])
5. H : Base
6. J : Base
7. ∀x,y,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[x;y;r1]] ≤ G[J[x;y;r2]]))
8. j : ℤ
9. 0 < j
10. ∀as,b1,b2:Base.
((F[b1] ≤ G[b2])
⇒ (F[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list_ind t] otherwise if v = Ax then b1 otherwise ⊥^j - 1
⊥
as] ≤ G[λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
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{}x,y,r1,r2:Base. ((F[r1] \mleq{} G[r2]) {}\mRightarrow{} (F[H[x;y;r1]] \mleq{} G[J[x;y;r2]]))
8. j : \mBbbZ{}
9. 0 < j
10. \mforall{}as,b1,b2:Base.
((F[b1] \mleq{} G[b2])
{}\mRightarrow{} (F[\mlambda{}list$_{ind}$,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list$_{ind}$ t]
otherwise if v = Ax then b1 otherwise \mbot{}\^{}j - 1
\mbot{}
as] \mleq{} G[\mlambda{}list$_{ind}$,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list$_{ind}$ t]
otherwise if v = Ax then b2 otherwise \mbot{}\^{}j - 1
\mbot{}
as]))
11. as : Base@i
12. b1 : Base@i
13. b2 : Base@i
14. F[b1] \mleq{} G[b2]@i
\mvdash{} F[\mlambda{}list$_{ind}$,L. eval v = L in
if v is a pair then let h,t = v
in H[h;t;list$_{ind}$ t] otherwise if v = Ax\000C then b1 otherwise \mbot{}\^{}j
\mbot{}
as] \mleq{} G[\mlambda{}list$_{ind}$,L. eval v = L in
if v is a pair then let h,t = v
in J[h;t;list$_{ind}$ t]
otherwise if v = Ax then b2 otherwise \mbot{}\^{}j
\mbot{}
as]
By
Latex:
((RWO "fun\_exp\_unroll\_1" 0 THENA Auto)
THEN Reduce 0
THEN ((InstLemma `lifting-strict-callbyvalue`
[\mkleeneopen{}so\_lambda(x,y,z,w.F[x])\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWW "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-callbyvalue`
[\mkleeneopen{}so\_lambda(x,y,z,w.G[x])\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWW "-1" 0 THENA Auto)
)
THEN UseCbvSqle
THEN ((InstLemma `lifting-strict-ispair`
[\mkleeneopen{}so\_lambda(x,y,z,w.F[x])\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWO "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-ispair`
[\mkleeneopen{}so\_lambda(x,y,z,w.G[x])\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWO "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-isaxiom`
[\mkleeneopen{}so\_lambda(x,y,z,w.F[x])\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWO "-1" 0 THENA Auto)
)
THEN ((InstLemma `lifting-strict-isaxiom`
[\mkleeneopen{}so\_lambda(x,y,z,w.G[x])\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM ((RWO "-1" 0 THENA Auto)
THEN 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)))
))
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