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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-cap: f(x)?z,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
fpf-sub: f ⊆ g,
guard: {T},
cand: A c∧ B,
fpf-ap: f(x),
pi2: snd(t),
not: ¬A,
false: False
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:
2016_05_16-AM-11_07_58
Last ObjectModification:
2015_12_29-AM-09_16_15
Theory : event-ordering
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