Nuprl Lemma : fpf-all-empty
∀[A:Type]. ∀eq,P:Top. (∀y∈dom(⊗). w=⊗(y) ⇒ P[y;w] ⇐⇒ True)
Proof
Definitions occuring in Statement :
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
fpf-empty: ⊗,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
true: True,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
fpf-empty: ⊗,
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
member: t ∈ T,
fpf-dom: x ∈ dom(f),
pi1: fst(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
true: True,
false: False,
rev_implies: P ⇐ Q
Lemmas referenced :
fpf_ap_pair_lemma,
deq_member_nil_lemma,
istype-top,
istype-universe,
istype-void,
istype-true
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
inhabitedIsType,
hypothesisEquality,
instantiate,
isectElimination,
universeEquality,
independent_pairFormation,
natural_numberEquality,
functionIsType,
universeIsType,
voidElimination,
because_Cache
Latex:
\mforall{}[A:Type]. \mforall{}eq,P:Top. (\mforall{}y\mmember{}dom(\motimes{}). w=\motimes{}(y) {}\mRightarrow{} P[y;w] \mLeftarrow{}{}\mRightarrow{} True)
Date html generated:
2020_05_20-AM-09_03_22
Last ObjectModification:
2020_01_28-PM-03_38_58
Theory : finite!partial!functions
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