Nuprl Lemma : do-apply-compose'
∀[A,B,C:Type]. ∀[g:A ⟶ (B + Top)]. ∀[f:A ⟶ B ⟶ C]. ∀[x:A].
do-apply(f o' g;x) ~ f x do-apply(g;x) supposing ↑can-apply(f o' 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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
do-apply: do-apply(f;x),
p-compose': f o' g,
can-apply: can-apply(f;x),
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,
assert: ↑b,
prop: ℙ,
bfalse: ff,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top
Lemmas referenced :
top_wf,
true_wf,
false_wf,
equal_wf,
assert_wf,
can-apply_wf,
p-compose'_wf,
subtype_rel_dep_function
Rules used in proof :
cut,
thin,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
unionEquality,
introduction,
extract_by_obid,
hypothesis,
lambdaFormation,
unionElimination,
sqequalHypSubstitution,
voidElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidEquality,
functionEquality,
universeEquality,
isect_memberFormation,
sqequalAxiom
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[g:A {}\mrightarrow{} (B + Top)]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} C]. \mforall{}[x:A].
do-apply(f o' g;x) \msim{} f x do-apply(g;x) supposing \muparrow{}can-apply(f o' g;x)
Date html generated:
2017_10_01-AM-09_13_52
Last ObjectModification:
2017_07_26-PM-04_49_08
Theory : general
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