Nuprl Lemma : choice-iff-canonicalizable
∀T:Type. (ChoicePrinciple(T) ⇐⇒ ⇃(canonicalizable(T)))
Proof
Definitions occuring in Statement :
choice-principle: ChoicePrinciple(T),
quotient: x,y:A//B[x; y],
canonicalizable: canonicalizable(T),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
true: True,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
choice-principle: ChoicePrinciple(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
true: True,
guard: {T},
pi1: fst(t),
canonicalizable: canonicalizable(T),
squash: ↓T,
subtype_rel: A ⊆r B
Lemmas referenced :
choice-principle_wf,
quotient_wf,
canonicalizable_wf,
true_wf,
equiv_rel_true,
istype-universe,
exists_wf,
base_wf,
equal-wf-T-base,
istype-base,
quotient-member-eq,
equal-wf-base,
implies-quotient-true2,
all_wf,
trivial-quotient-true,
equal_wf,
squash_wf,
subtype_rel_self,
iff_weakening_equal,
all-quotient-true
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
because_Cache,
independent_isectElimination,
instantiate,
universeEquality,
dependent_functionElimination,
closedConclusion,
productElimination,
independent_functionElimination,
pointwiseFunctionalityForEquality,
Error :productIsType,
Error :equalityIsType3,
Error :equalityIsType4,
equalityTransitivity,
equalitySymmetry,
Error :dependent_pairEquality_alt,
axiomEquality,
natural_numberEquality,
Error :functionIsType,
promote_hyp,
applyEquality,
rename,
Error :dependent_pairFormation_alt,
functionExtensionality,
Error :equalityIsType1,
imageElimination,
imageMemberEquality,
baseClosed,
Error :equalityIsType2
Latex:
\mforall{}T:Type. (ChoicePrinciple(T) \mLeftarrow{}{}\mRightarrow{} \00D9(canonicalizable(T)))
Date html generated:
2019_06_20-PM-02_54_40
Last ObjectModification:
2018_10_17-AM-09_27_27
Theory : continuity
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