Nuprl Lemma : fpf-sub_transitivity
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f,g,h:a:A fp-> B[a]]. (f ⊆ h) supposing (g ⊆ h and f ⊆ g)
Proof
Definitions occuring in Statement :
fpf-sub: f ⊆ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fpf-sub: f ⊆ g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
prop: ℙ
Lemmas referenced :
assert_elim,
fpf-dom_wf,
subtype-fpf2,
top_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert_wf,
fpf-sub_witness,
fpf-sub_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
productElimination,
lemma_by_obid,
isectElimination,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
independent_pairFormation,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g,h:a:A fp-> B[a]].
(f \msubseteq{} h) supposing (g \msubseteq{} h and f \msubseteq{} g)
Date html generated:
2018_05_21-PM-09_19_06
Last ObjectModification:
2018_02_09-AM-10_17_25
Theory : finite!partial!functions
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