Nuprl Lemma : Rice-theorem-for-Type_3
∀F:Type ⟶ 𝔹
((∀X,Y:Type. (X ~ Y ⇒ F X = F Y)) ⇒ weak-continuity(𝔹;𝔹) ⇒ ((∀X:Type. (↑(F X))) ∨ (∀X:Type. (¬↑(F X)))))
Proof
Definitions occuring in Statement :
weak-continuity: weak-continuity(T;V),
equipollent: A ~ B,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
false: False,
weak-continuity: weak-continuity(T;V),
decidable: Dec(P),
squash: ↓T,
exists: ∃x:A. B[x],
isl: isl(x),
iff: P ⇐⇒ Q,
and: P ∧ Q,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
not: ¬A,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
sq_type: SQType(T)
Lemmas referenced :
Rice-theorem-for-Type_2,
weak-continuity_wf,
bool_wf,
all_wf,
equipollent_wf,
equal_wf,
nat_wf,
decidable_wf,
equal-wf-T-base,
isl_wf,
not_wf,
squash_wf,
true_wf,
bfalse_wf,
iff_weakening_equal,
equal-wf-base,
btrue_neq_bfalse,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
iff_wf,
lt_int_wf,
iff_imp_equal_bool,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
less_than_wf,
assert_of_lt_int,
assert_wf,
decidable__le,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
le_wf,
eqff_to_assert,
bnot_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
subtype_base_sq,
bool_subtype_base,
assert_functionality_wrt_uiff,
or_wf,
btrue_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
instantiate,
universeEquality,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality,
applyEquality,
functionExtensionality,
unionElimination,
voidElimination,
because_Cache,
rename,
baseClosed,
equalityTransitivity,
equalitySymmetry,
imageElimination,
productElimination,
dependent_pairFormation,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
independent_isectElimination,
setElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
addLevel,
impliesFunctionality,
dependent_set_memberEquality,
addEquality,
equalityElimination,
inlFormation,
orFunctionality,
allFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
inrFormation
Latex:
\mforall{}F:Type {}\mrightarrow{} \mBbbB{}
((\mforall{}X,Y:Type. (X \msim{} Y {}\mRightarrow{} F X = F Y))
{}\mRightarrow{} weak-continuity(\mBbbB{};\mBbbB{})
{}\mRightarrow{} ((\mforall{}X:Type. (\muparrow{}(F X))) \mvee{} (\mforall{}X:Type. (\mneg{}\muparrow{}(F X)))))
Date html generated:
2017_10_01-AM-08_29_47
Last ObjectModification:
2017_07_26-PM-04_24_09
Theory : basic
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