Nuprl Lemma : can-apply-compose-sq
∀[A,B,C:Type]. ∀[g:A ⟶ (B + Top)]. ∀[f:B ⟶ (C + Top)]. ∀[x:A].
(can-apply(f o g;x) ~ can-apply(g;x) ∧b can-apply(f;do-apply(g;x)))
Proof
Definitions occuring in Statement :
p-compose: f o g,
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
band: p ∧b q,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
p-compose: f o g,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
isl: isl(x),
outl: outl(x),
ifthenelse: if b then t else f fi ,
btrue: tt,
band: p ∧b q,
bfalse: ff,
uall: ∀[x:A]. B[x],
prop: ℙ
Lemmas referenced :
top_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
thin,
unionEquality,
introduction,
extract_by_obid,
hypothesis,
lambdaFormation,
unionElimination,
sqequalHypSubstitution,
isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
functionEquality,
universeEquality,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:A {}\mrightarrow{} (B + Top)]. \mforall{}[f:B {}\mrightarrow{} (C + Top)]. \mforall{}[x:A].
(can-apply(f o g;x) \msim{} can-apply(g;x) \mwedge{}\msubb{} can-apply(f;do-apply(g;x)))
Date html generated:
2017_10_01-AM-09_13_30
Last ObjectModification:
2017_07_26-PM-04_48_53
Theory : general
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