Nuprl Lemma : combine-antecedent-surjections
es:EO
[A,B,P,Q:E 
].
((e:E. Dec(P e))
(e:E. Dec(A e))
(e:E. Dec(B e))
(f:{e:E| A e} {e:E| P e} . g:{e:E| B e} {e:E| Q e} .
(P 
f A
Q g B
(
h:{e:E| (A e) 
(B e)} {e:E| (P e) (Q e)}
(
e.((P e) (Q e)) h e.((A e) (B e))
(e:{e:E| (A e) (B e)} . ((A e) ((h e) = (f e))))
(e:{e:E| (A e) (B e)} . (h e) = (g e) supposing 
(A e))))))) supposing
((e:E. (((P e) (Q e)))) and
(e:E. (((A e) (B e)))))
Proof not projected
Definitions occuring in Statement :
antecedent-surjection: Q f P,
es-E: E,
event_ordering: EO,
decidable: Dec(P),
uimplies: b supposing a,
uall: [x:A]. B[x],
prop: ,
all: x:A. B[x],
exists: x:A. B[x],
not: A,
implies: P Q,
or: P Q,
and: P Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: x.A[x],
function: x:A B[x],
equal: s = t
Definitions :
so_lambda: 
x.t[x],
cand: A c B,
false: False,
member: t 
T,
implies: P Q,
and: P Q,
not: A,
uimplies: b supposing a,
prop: ,
uall: [x:A]. B[x],
all: x:A. B[x],
exists: x:A. B[x],
rev_implies: P 
Q,
true: True,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
assert: 
b,
iff: P Q,
guard: {T},
or: P Q,
antecedent-surjection: Q f P,
antecedent-function: Q f P,
squash: 
T,
so_apply: x[s],
decidable: Dec(P),
unit: Unit,
bool: 
,
uiff: uiff(P;Q),
it: 
Lemmas :
event_ordering_wf,
not_wf,
decidable_wf,
all_wf,
antecedent-surjection_wf,
es-E_wf,
assert_wf,
iff_wf,
bfalse_wf,
btrue_wf,
false_wf,
true_wf,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
iff_weakening_uiff,
bnot_wf,
subtype_rel_self,
subtype_rel_sets,
bool_wf,
equal_wf,
or_wf,
isect_wf,
es-causl_wf,
and_wf,
squash_wf
\mforall{}es:EO
\mforall{}[A,B,P,Q:E {}\mrightarrow{} \mBbbP{}].
((\mforall{}e:E. Dec(P e))
{}\mRightarrow{} (\mforall{}e:E. Dec(A e))
{}\mRightarrow{} (\mforall{}e:E. Dec(B e))
{}\mRightarrow{} (\mforall{}f:\{e:E| A e\} {}\mrightarrow{} \{e:E| P e\} . \mforall{}g:\{e:E| B e\} {}\mrightarrow{} \{e:E| Q e\} .
(P \mleftarrow{}\mleftarrow{}{} f{}{} A
{}\mRightarrow{} Q \mleftarrow{}\mleftarrow{}{} g{}{} B
{}\mRightarrow{} (\mexists{}h:\{e:E| (A e) \mvee{} (B e)\} {}\mrightarrow{} \{e:E| (P e) \mvee{} (Q e)\}
(\mlambda{}e.((P e) \mvee{} (Q e)) \mleftarrow{}\mleftarrow{}{} h{}{} \mlambda{}e.((A e) \mvee{} (B e))
\mwedge{} (\mforall{}e:\{e:E| (A e) \mvee{} (B e)\} . ((A e) {}\mRightarrow{} ((h e) = (f e))))
\mwedge{} (\mforall{}e:\{e:E| (A e) \mvee{} (B e)\} . (h e) = (g e) supposing \mneg{}(A e))))))) supposing
((\mforall{}e:E. (\mneg{}((P e) \mwedge{} (Q e)))) and
(\mforall{}e:E. (\mneg{}((A e) \mwedge{} (B e)))))
Date html generated:
2012_01_23-PM-12_15_06
Last ObjectModification:
2011_12_13-PM-12_50_05
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