Nuprl Lemma : fpf-sub-functionality
∀[A,A':Type].
∀[B:A ⟶ Type]. ∀[C:A' ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[eq':EqDecider(A')]. ∀[f,g:a:A fp-> B[a]].
(f ⊆ g) supposing (f ⊆ g and (∀a:A. (B[a] ⊆r C[a])))
supposing strong-subtype(A;A')
Proof
Definitions occuring in Statement :
fpf-sub: f ⊆ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
guard: {T},
fpf-sub: f ⊆ g,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
prop: ℙ,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
fpf-ap: f(x)
Lemmas referenced :
strong-subtype-implies,
assert_wf,
fpf-dom_wf,
strong-subtype-deq-subtype,
subtype-fpf2,
top_wf,
fpf-sub_witness,
fpf-sub_wf,
all_wf,
subtype_rel_wf,
fpf_wf,
deq_wf,
strong-subtype_wf,
fpf-dom_functionality2,
fpf-ap_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
lambdaFormation,
dependent_functionElimination,
cumulativity,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
lambdaEquality,
functionExtensionality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A,A':Type].
\mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[C:A' {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[eq':EqDecider(A')]. \mforall{}[f,g:a:A fp-> B[a]].
(f \msubseteq{} g) supposing (f \msubseteq{} g and (\mforall{}a:A. (B[a] \msubseteq{}r C[a])))
supposing strong-subtype(A;A')
Date html generated:
2018_05_21-PM-09_18_55
Last ObjectModification:
2018_02_09-AM-10_17_19
Theory : finite!partial!functions
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