Nuprl Lemma : can-apply-compose-iff
∀[A,B,C:Type]. ∀[g:A ⟶ (B + Top)]. ∀[f:B ⟶ (C + Top)]. ∀[x:A].
uiff(↑can-apply(f o g;x);(↑can-apply(g;x)) ∧ (↑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),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
top: Top,
and: P ∧ Q,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
top: Top,
implies: P ⇒ Q,
prop: ℙ,
can-apply: can-apply(f;x),
p-compose: f o g,
not: ¬A,
false: False,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
can-apply-compose,
assert_witness,
can-apply_wf,
subtype_rel_union,
top_wf,
do-apply_wf,
assert_wf,
p-compose_wf,
isl_wf,
bool_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
productElimination,
sqequalRule,
independent_pairEquality,
cumulativity,
functionExtensionality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
productEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
unionEquality,
universeEquality,
baseClosed,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_functionElimination
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:A {}\mrightarrow{} (B + Top)]. \mforall{}[f:B {}\mrightarrow{} (C + Top)]. \mforall{}[x:A].
uiff(\muparrow{}can-apply(f o g;x);(\muparrow{}can-apply(g;x)) \mwedge{} (\muparrow{}can-apply(f;do-apply(g;x))))
Date html generated:
2017_10_01-AM-09_13_39
Last ObjectModification:
2017_07_26-PM-04_48_59
Theory : general
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