Nuprl Lemma : fpf-all-single
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]. ∀[P:x:A ⟶ B[x] ⟶ ℙ]. ∀x:A. ∀v:B[x]. (∀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],
so_apply: x[s],
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,
so_apply: x[s],
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],
subtype_rel: A ⊆r B,
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,
subtype_rel_self,
subtype_rel_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,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
functionEquality,
universeEquality,
isectElimination,
independent_pairFormation,
independent_functionElimination,
inlFormation,
because_Cache,
addLevel,
orFunctionality,
productElimination,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
unionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
levelHypothesis,
promote_hyp
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:x:A {}\mrightarrow{} B[x] {}\mrightarrow{} \mBbbP{}].
\mforall{}x:A. \mforall{}v:B[x]. (\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_05
Last ObjectModification:
2018_02_09-AM-10_24_42
Theory : finite!partial!functions
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