Nuprl Lemma : subtype_rel_dep_function_iff
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[C:Type]. ∀[D:C ⟶ Type].
(uiff(∀[a:C]. (B[a] ⊆r D[a]);(a:A ⟶ B[a]) ⊆r (c:C ⟶ D[c]))) supposing
((∀x,y:A. Dec(x = y ∈ A)) and
(a:A ⟶ B[a]) and
(C ⊆r A))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uiff: uiff(P;Q),
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,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
bfalse: ff,
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
squash: ↓T,
guard: {T},
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
subtype_rel_dep_function,
istype-universe,
subtype_rel_wf,
decidable_wf,
equal_wf,
subtype_rel_self,
true_wf,
false_wf,
assert_wf,
equal-wf-T-base,
bool_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
subtype_rel-equal,
squash_wf,
iff_weakening_equal,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
hypothesis,
independent_isectElimination,
Error :lambdaFormation_alt,
axiomEquality,
Error :isectIsType,
Error :universeIsType,
because_Cache,
Error :isect_memberEquality_alt,
functionEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
Error :inhabitedIsType,
universeEquality,
rename,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
unionElimination,
natural_numberEquality,
voidElimination,
Error :productIsType,
baseClosed,
equalityElimination,
instantiate,
imageElimination,
imageMemberEquality,
promote_hyp,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[C:Type]. \mforall{}[D:C {}\mrightarrow{} Type].
(uiff(\mforall{}[a:C]. (B[a] \msubseteq{}r D[a]);(a:A {}\mrightarrow{} B[a]) \msubseteq{}r (c:C {}\mrightarrow{} D[c]))) supposing
((\mforall{}x,y:A. Dec(x = y)) and
(a:A {}\mrightarrow{} B[a]) and
(C \msubseteq{}r A))
Date html generated:
2019_06_20-PM-00_27_37
Last ObjectModification:
2018_10_06-AM-11_20_16
Theory : subtype_1
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