Nuprl Lemma : fpf-all-single-decl
∀[A:Type]. ∀eq:EqDecider(A). ∀[P:x:A ⟶ Type ⟶ ℙ]. ∀x:A. ∀[v:Type]. (∀y∈dom(x : v). w=x : v(y) 
⇒  P[y;w] 
⇐⇒ P[x;v])
Proof
Definitions occuring in Statement : 
fpf-all: ∀x∈dom(f). v=f(x) 
⇒  P[x; v]
, 
fpf-single: x : v
, 
deq: EqDecider(T)
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s1;s2]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
fpf-all: ∀x∈dom(f). v=f(x) 
⇒  P[x; v]
, 
fpf-single: x : v
, 
fpf-ap: f(x)
, 
fpf-dom: x ∈ dom(f)
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
member: t ∈ T
, 
top: Top
, 
prop: ℙ
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
rev_implies: P 
⇐ Q
, 
eqof: eqof(d)
, 
so_lambda: λ2x.t[x]
, 
deq: EqDecider(T)
, 
so_apply: x[s1;s2]
, 
so_apply: x[s]
, 
false: False
Lemmas referenced : 
deq_member_cons_lemma, 
deq_member_nil_lemma, 
deq_wf, 
false_wf, 
iff_transitivity, 
assert_wf, 
bor_wf, 
eqof_wf, 
bfalse_wf, 
or_wf, 
equal_wf, 
iff_weakening_uiff, 
assert_of_bor, 
safe-assert-deq, 
member_wf, 
all_wf, 
and_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
sqequalRule, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
hypothesis, 
universeEquality, 
hypothesisEquality, 
functionEquality, 
cumulativity, 
isectElimination, 
independent_pairFormation, 
independent_functionElimination, 
inlFormation, 
because_Cache, 
addLevel, 
applyEquality, 
orFunctionality, 
productElimination, 
independent_isectElimination, 
lambdaEquality, 
setElimination, 
rename, 
functionExtensionality, 
unionElimination, 
hyp_replacement, 
equalitySymmetry, 
dependent_set_memberEquality, 
applyLambdaEquality, 
levelHypothesis, 
promote_hyp
Latex:
\mforall{}[A:Type]
    \mforall{}eq:EqDecider(A)
        \mforall{}[P:x:A  {}\mrightarrow{}  Type  {}\mrightarrow{}  \mBbbP{}].  \mforall{}x:A.  \mforall{}[v:Type].  (\mforall{}y\mmember{}dom(x  :  v).  w=x  :  v(y)  {}\mRightarrow{}    P[y;w]  \mLeftarrow{}{}\mRightarrow{}  P[x;v])
Date html generated:
2018_05_21-PM-09_30_09
Last ObjectModification:
2018_02_09-AM-10_24_45
Theory : finite!partial!functions
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