Nuprl Lemma : id-fun-subtype
∀[A,B:Type]. id-fun(B) ⊆r id-fun(A) supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
id-fun: id-fun(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
guard: {T},
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
id-fun: id-fun(T),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x]
Lemmas referenced :
strong-subtype-implies,
id-fun_wf,
strong-subtype_wf,
equal_wf,
set_wf,
subtype_rel_sets,
exists_wf,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
promote_hyp,
productElimination,
cumulativity,
sqequalRule,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionExtensionality,
dependent_set_memberEquality,
applyEquality,
lambdaFormation,
dependent_functionElimination,
independent_isectElimination,
setElimination,
rename,
setEquality,
dependent_pairFormation
Latex:
\mforall{}[A,B:Type]. id-fun(B) \msubseteq{}r id-fun(A) supposing strong-subtype(A;B)
Date html generated:
2017_04_14-AM-07_37_00
Last ObjectModification:
2017_02_27-PM-03_09_46
Theory : subtype_1
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