Nuprl Lemma : mk_deq_wf
∀[T:Type]. ∀[p:∀x,y:T. Dec(x = y ∈ T)]. (mk_deq(p) ∈ EqDecider(T))
Proof
Definitions occuring in Statement :
mk_deq: mk_deq(p),
deq: EqDecider(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mk_deq: mk_deq(p),
deq: EqDecider(T),
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
isl: isl(x),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
false: False,
not: ¬A,
bfalse: ff,
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B
Lemmas referenced :
istype-universe,
decidable_wf,
equal_wf,
assert_wf,
all_wf,
true_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
Error :inhabitedIsType,
Error :universeIsType,
Error :isect_memberEquality_alt,
universeEquality,
Error :dependent_set_memberEquality_alt,
Error :lambdaEquality_alt,
because_Cache,
Error :lambdaFormation_alt,
unionElimination,
Error :equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
Error :productIsType,
applyEquality,
voidElimination,
natural_numberEquality,
independent_pairFormation,
lemma_by_obid,
lambdaEquality,
lambdaFormation
Latex:
\mforall{}[T:Type]. \mforall{}[p:\mforall{}x,y:T. Dec(x = y)]. (mk\_deq(p) \mmember{} EqDecider(T))
Date html generated:
2019_06_20-PM-00_31_53
Last ObjectModification:
2018_10_06-AM-11_20_18
Theory : equality!deciders
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