Nuprl Lemma : deq_subtype2
∀[T:Type]. (EqDecider(T) ⊆r (∀x,y:T. Dec(x = y ∈ T)))
Proof
Definitions occuring in Statement :
deq: EqDecider(T),
decidable: Dec(P),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
decidable: Dec(P),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False
Lemmas referenced :
deq_wf,
bool_wf,
eqtt_to_assert,
it_wf,
equal_subtype,
equal-wf-base,
equal_wf,
false_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :universeIsType,
universeEquality,
sqequalRule,
functionExtensionality,
rename,
because_Cache,
setElimination,
dependent_functionElimination,
applyEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
inlEquality,
intEquality,
natural_numberEquality,
independent_functionElimination,
baseClosed,
functionEquality,
cumulativity,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
inrEquality
Latex:
\mforall{}[T:Type]. (EqDecider(T) \msubseteq{}r (\mforall{}x,y:T. Dec(x = y)))
Date html generated:
2019_06_20-PM-00_31_51
Last ObjectModification:
2018_09_26-PM-00_54_53
Theory : equality!deciders
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