Nuprl Lemma : fpf-compatible-single-iff
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[x:A]. ∀[v:B[x]].
uiff(f || x : v;v = f(x) ∈ B[x] supposing ↑x ∈ dom(f))
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-compatible: f || g,
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
fpf-compatible: f || g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
top: Top,
cand: A c∧ B,
eqof: eqof(d),
squash: ↓T,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fpf-compatible_wf,
fpf-single_wf,
fpf_ap_single_lemma,
istype-assert,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf-ap_wf,
fpf_wf,
deq_wf,
istype-universe,
fpf-single-dom,
fpf-single-dom-sq,
safe-assert-deq,
equal_wf,
squash_wf,
true_wf,
assert_wf,
subtype_rel-equal,
subtype_rel_self,
iff_weakening_equal,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
because_Cache,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
extract_by_obid,
lambdaEquality_alt,
applyEquality,
instantiate,
lambdaFormation_alt,
dependent_functionElimination,
Error :memTop,
productElimination,
productIsType,
independent_isectElimination,
functionIsTypeImplies,
isectIsType,
equalityIstype,
independent_pairEquality,
functionIsType,
universeEquality,
equalitySymmetry,
voidEquality,
voidElimination,
isect_memberEquality,
independent_functionElimination,
imageElimination,
equalityTransitivity,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
applyLambdaEquality,
setElimination,
rename,
natural_numberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[x:A]. \mforall{}[v:B[x]].
uiff(f || x : v;v = f(x) supposing \muparrow{}x \mmember{} dom(f))
Date html generated:
2020_05_20-AM-09_03_02
Last ObjectModification:
2020_01_04-PM-11_11_37
Theory : finite!partial!functions
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