Nuprl Lemma : Rice-theorem-for-Type
∀F:Type ⟶ 𝔹. ((∀X,Y:Type. (X ~ Y ⇒ F X = F Y)) ⇒ ((F = (λT.tt) ∈ (Type ⟶ 𝔹)) ∨ (F = (λT.ff) ∈ (Type ⟶ 𝔹))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
bfalse: ff,
btrue: tt,
bool: 𝔹,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
compose: f o g,
false: False,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
equal-wf-base-T,
equal-wf-T-base,
btrue_neq_bfalse,
assert_elim,
false_wf,
assert_wf,
iff_imp_equal_bool,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
nat-inf_wf,
compose_wf,
nat-inf-limit,
nat-inf-infinity_wf,
nat2inf_wf,
nat_wf,
nat-inf-attach,
bfalse_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
btrue_wf,
eqtt_to_assert,
bool_wf,
equal_wf,
equipollent_wf,
all_wf
Rules used in proof :
inrFormation,
functionExtensionality,
inlFormation,
voidEquality,
independent_pairFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
rename,
voidElimination,
independent_functionElimination,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
because_Cache,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
applyEquality,
hypothesis,
hypothesisEquality,
functionEquality,
cumulativity,
lambdaEquality,
sqequalRule,
universeEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}F:Type {}\mrightarrow{} \mBbbB{}. ((\mforall{}X,Y:Type. (X \msim{} Y {}\mRightarrow{} F X = F Y)) {}\mRightarrow{} ((F = (\mlambda{}T.tt)) \mvee{} (F = (\mlambda{}T.ff))))
Date html generated:
2018_07_29-AM-09_29_31
Last ObjectModification:
2018_07_27-PM-04_34_49
Theory : basic
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