Nuprl Lemma : has-value-is-list-approx-is-type
∀[T:Type]. ∀[t:colist(T)]. ∀[n:ℕ].
((λis-list,t. eval u = t in if u is a pair then is-list (snd(u)) otherwise if u = Ax then tt otherwise ⊥^n ⊥ t)↓ ∈ Typ\000Ce)
Proof
Definitions occuring in Statement :
colist: colist(T),
fun_exp: f^n,
nat: ℕ,
has-value: (a)↓,
callbyvalue: callbyvalue,
bottom: ⊥,
btrue: tt,
uall: ∀[x:A]. B[x],
pi2: snd(t),
ispair: if z is a pair then a otherwise b,
isaxiom: if z = Ax then a otherwise b,
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
compose: f o g,
has-value: (a)↓,
pi2: snd(t),
ext-eq: A ≡ B
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
fun_exp0_lemma,
strictness-apply,
has-value_wf_base,
subtract-1-ge-0,
fun_exp_unroll,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
value-type-has-value,
colist_wf,
colist-value-type,
co-list-cases,
unit_subtype_colist,
sqle_wf_base,
subtype_rel_transitivity,
b-union_wf,
unit_wf2,
subtype_rel_b-union-right,
colist-ext,
istype-nat,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
baseClosed,
because_Cache,
dependent_set_memberEquality_alt,
unionElimination,
equalityElimination,
productElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
callbyvalueReduce,
hypothesis_subsumption,
productEquality,
isectIsTypeImplies,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[t:colist(T)]. \mforall{}[n:\mBbbN{}].
((\mlambda{}is-list,t. eval u = t in
if u is a pair then is-list (snd(u)) otherwise if u = Ax then tt otherwise \mbot{}\^{}n
\mbot{}
t)\mdownarrow{} \mmember{} Type)
Date html generated:
2019_10_16-AM-11_38_22
Last ObjectModification:
2019_06_26-PM-05_07_14
Theory : eval!all
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