Step
*
1
1
of Lemma
sqle-list_accum-list_ind
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. ∀z:Base. strict1(λx.G[z;x])
5. H : Base
6. J : Base
7. ∀a,b,c:Base. (G[H[b;a];c] ≤ G[b;J[a;c]])
8. b2 : Base@i
9. ∀x:Base. (F[x] ≤ G[x;b2])
10. j : ℤ
11. 0 < j
12. ∀as,x:Base.
(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
⊥
x
as] ≤ G[x;λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
as])
13. as : Base@i
14. x : Base@i
⊢ 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
⊥
x
as] ≤ G[x;λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j
⊥
as]
BY
{ (RenameVar `b1' (-1)
THEN (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[b1;x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THEN (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[b1;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[b1;x])⌝;⌜⋅⌝;⌜⋅⌝;⌜⋅⌝]⋅
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THENM (RWO "-1" 0 THENA Auto)
)) }
1
1. F : Base
2. strict1(λx.F[x])
3. G : Base
4. ∀z:Base. strict1(λx.G[z;x])
5. H : Base
6. J : Base
7. ∀a,b,c:Base. (G[H[b;a];c] ≤ G[b;J[a;c]])
8. b2 : Base@i
9. ∀x:Base. (F[x] ≤ G[x;b2])
10. j : ℤ
11. 0 < j
12. ∀as,x:Base.
(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
⊥
x
as] ≤ G[x;λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;list_ind t] otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
as])
13. as : Base@i
14. b1 : Base@i
15. ∀[a,B:Top]. (F[eval x = a in B[x]] ~ eval x = a in F[B[x]])
16. ∀[a,B:Top]. (G[b1;eval x = a in B[x]] ~ eval x = a in G[b1;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[b1;if a is a pair then b otherwise c] ~ if a is a pair then G[b1;b] otherwise G[b1;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[b1;if a = Ax then b otherwise c] ~ if a = Ax then G[b1;b] otherwise G[b1;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[b1;let h,t = as
in J[h;λlist_ind,L. eval v = L in
if v is a pair then let h,t = v
in J[h;list_ind t]
otherwise if v = Ax then b2 otherwise ⊥^j - 1
⊥
t]] otherwise if as = Ax then G[b1;b2] otherwise G[b1;⊥]
Latex:
Latex:
1. F : Base
2. strict1(\mlambda{}x.F[x])
3. G : Base
4. \mforall{}z:Base. strict1(\mlambda{}x.G[z;x])
5. H : Base
6. J : Base
7. \mforall{}a,b,c:Base. (G[H[b;a];c] \mleq{} G[b;J[a;c]])
8. b2 : Base@i
9. \mforall{}x:Base. (F[x] \mleq{} G[x;b2])
10. j : \mBbbZ{}
11. 0 < j
12. \mforall{}as,x:Base.
(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{}
x
as] \mleq{} G[x;\mlambda{}list$_{ind}$,L. eval v = L in
if v is a pair then let h,t = v
in J[h;list$_{ind}$ t]
otherwise if v = Ax then b2 otherwise \mbot{}\^{}j - 1
\mbot{}
as])
13. as : Base@i
14. x : Base@i
\mvdash{} 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 \000Cv = Ax then y otherwise \mbot{}\^{}j
\mbot{}
x
as] \mleq{} G[x;\mlambda{}list$_{ind}$,L. eval v = L in
if v is a pair then let h,t = v
in J[h;list$_{ind}$ t]
otherwise if v = Ax then b2 otherwise \mbot{}\^{}j
\mbot{}
as]
By
Latex:
(RenameVar `b1' (-1)
THEN (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[b1;x])\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{};\mkleeneopen{}\mcdot{}\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `strict1-strict4` THEN Auto)
)
THEN (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[b1;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[b1;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)
))
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