Nuprl Lemma : canonicalizable-iff
∀[T:Type]. (canonicalizable(T) ⇐⇒ ∀t:T. ∃x:Base. (t = x ∈ T))
Proof
Definitions occuring in Statement :
canonicalizable: canonicalizable(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
canonicalizable: canonicalizable(T),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
prop: ℙ,
pi1: fst(t)
Lemmas referenced :
istype-base,
istype-universe,
equal_wf,
iff_weakening_equal,
trivial-equal,
squash_wf,
true_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
universeIsType,
hypothesisEquality,
sqequalRule,
productIsType,
functionIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
equalityIstype,
applyEquality,
sqequalBase,
equalitySymmetry,
instantiate,
sqequalHypSubstitution,
isectElimination,
thin,
universeEquality,
productElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
independent_isectElimination,
independent_functionElimination,
promote_hyp,
rename,
functionExtensionality,
inhabitedIsType
Latex:
\mforall{}[T:Type]. (canonicalizable(T) \mLeftarrow{}{}\mRightarrow{} \mforall{}t:T. \mexists{}x:Base. (t = x))
Date html generated:
2019_10_15-AM-10_20_00
Last ObjectModification:
2019_08_29-AM-10_56_34
Theory : call!by!value_2
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