Nuprl Lemma : compose-fpf-dom
∀[A:Type]. ∀[B:A ⟶ Type].
∀f:x:A fp-> B[x]
∀[C:Type]
∀a:A ⟶ (C?). ∀b:C ⟶ A. ∀y:C.
((y ∈ fpf-domain(compose-fpf(a;b;f))) ⇐⇒ ∃x:A. ((x ∈ fpf-domain(f)) ∧ ((↑isl(a x)) c∧ (y = outl(a x) ∈ C))))
Proof
Definitions occuring in Statement :
compose-fpf: compose-fpf(a;b;f),
fpf-domain: fpf-domain(f),
fpf: a:A fp-> B[a],
l_member: (x ∈ l),
outl: outl(x),
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
cand: A c∧ B,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
fpf: a:A fp-> B[a],
fpf-domain: fpf-domain(f),
compose-fpf: compose-fpf(a;b;f),
pi1: fst(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
cand: A c∧ B,
outl: outl(x),
uimplies: b supposing a,
isl: isl(x),
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
unit_wf2,
fpf_wf,
exists_wf,
l_member_wf,
assert_wf,
isl_wf,
equal_wf,
assert_elim,
and_wf,
bfalse_wf,
btrue_neq_bfalse,
member_map_filter,
outl_wf,
mapfilter_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
functionEquality,
cumulativity,
unionEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
universeEquality,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
productElimination,
independent_pairFormation,
productEquality,
because_Cache,
unionElimination,
independent_isectElimination,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
applyLambdaEquality,
setElimination,
rename,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
addLevel,
impliesFunctionality,
setEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}f:x:A fp-> B[x]
\mforall{}[C:Type]
\mforall{}a:A {}\mrightarrow{} (C?). \mforall{}b:C {}\mrightarrow{} A. \mforall{}y:C.
((y \mmember{} fpf-domain(compose-fpf(a;b;f)))
\mLeftarrow{}{}\mRightarrow{} \mexists{}x:A. ((x \mmember{} fpf-domain(f)) \mwedge{} ((\muparrow{}isl(a x)) c\mwedge{} (y = outl(a x)))))
Date html generated:
2018_05_21-PM-09_27_58
Last ObjectModification:
2018_02_09-AM-10_23_25
Theory : finite!partial!functions
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