Nuprl Lemma : fpf-cap_functionality_wrt_sub
∀[A:Type]. ∀[d1,d2,d3,d4:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f,g:a:A fp-> B[a]]. ∀[x:A]. ∀[z:B[x]].
(f(x)?z = g(x)?z ∈ B[x]) supposing ((↑x ∈ dom(f)) and f ⊆ g)
Proof
Definitions occuring in Statement :
fpf-sub: f ⊆ g,
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
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 :
bfalse: ff,
ifthenelse: if b then t else f fi ,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
top: Top,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
fpf-cap: f(x)?z,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
pi2: snd(t),
fpf-ap: f(x),
cand: A c∧ B,
guard: {T},
fpf-sub: f ⊆ g,
false: False,
not: ¬A
Lemmas referenced :
equal_wf,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
not_wf,
bnot_wf,
equal-wf-T-base,
bool_wf,
deq_wf,
fpf_wf,
fpf-sub_wf,
top_wf,
subtype-fpf2,
fpf-dom_wf,
assert_wf,
fpf-dom_functionality2
Rules used in proof :
dependent_functionElimination,
independent_functionElimination,
productElimination,
equalityElimination,
unionElimination,
baseClosed,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
because_Cache,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaFormation,
independent_isectElimination,
functionExtensionality,
lambdaEquality,
sqequalRule,
applyEquality,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
universeIsType,
Error :memTop,
lambdaFormation_alt,
inhabitedIsType,
lambdaEquality_alt
Latex:
\mforall{}[A:Type]. \mforall{}[d1,d2,d3,d4:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f,g:a:A fp-> B[a]]. \mforall{}[x:A]. \mforall{}[z:B[x]].
(f(x)?z = g(x)?z) supposing ((\muparrow{}x \mmember{} dom(f)) and f \msubseteq{} g)
Date html generated:
2020_05_20-AM-09_02_24
Last ObjectModification:
2020_01_25-AM-11_42_15
Theory : finite!partial!functions
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