Nuprl Lemma : compact-type-corec-lemma0
∀[F:Type ⟶ Type]
(Monotone(T.F T)
⇒ (((⋂n:ℕ. (F^n Top)) ⟶ 𝔹) ⊆r ⋃n:ℕ.((F^n Top) ⟶ 𝔹))
⇒ ((⋂n:ℕ. compact-type2(F^n Top)) ⊆r compact-type2(corec(T.F T))))
Proof
Definitions occuring in Statement :
compact-type2: compact-type2(T),
corec: corec(T.F[T]),
type-monotone: Monotone(T.F[T]),
fun_exp: f^n,
nat: ℕ,
bool: 𝔹,
subtype_rel: A ⊆r B,
tunion: ⋃x:A.B[x],
uall: ∀[x:A]. B[x],
top: Top,
implies: P ⇒ Q,
apply: f a,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
corec: corec(T.F[T]),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
compose: f o g,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
nat: ℕ,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
compact-type2: compact-type2(T),
sq_exists: ∃x:A [B[x]],
tunion: ⋃x:A.B[x],
pi2: snd(t),
p-selector: p-selector(T;x;p)
Lemmas referenced :
type-monotone_wf,
subtype_rel_self,
ext-eq_weakening,
subtype_rel_weakening,
subtype_rel_dep_function,
tunion_wf,
corec_wf,
subtype_rel_transitivity,
type-monotone_fun_exp_top,
subtype_rel_wf,
subtype_rel-equal,
int_seg_wf,
primrec_wf,
fun_exp_wf,
isect_subtype_rel_trivial,
nat_wf,
iff_weakening_equal,
true_wf,
squash_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
le_wf,
fun_exp_unroll,
primrec-unroll,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
top_wf,
fun_exp0_lemma,
primrec0_lemma,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
void_wf,
compact-type2_wf,
sq_exists_wf,
p-selector_wf,
full-omega-unsat,
exists_wf,
equal-wf-T-base,
pi2_wf,
pi1_wf
Rules used in proof :
functionEquality,
isectEquality,
baseClosed,
imageMemberEquality,
universeEquality,
functionExtensionality,
imageElimination,
applyEquality,
cumulativity,
instantiate,
promote_hyp,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
because_Cache,
dependent_set_memberEquality,
unionElimination,
axiomEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
approximateComputation,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
(Monotone(T.F T)
{}\mRightarrow{} (((\mcap{}n:\mBbbN{}. (F\^{}n Top)) {}\mrightarrow{} \mBbbB{}) \msubseteq{}r \mcup{}n:\mBbbN{}.((F\^{}n Top) {}\mrightarrow{} \mBbbB{}))
{}\mRightarrow{} ((\mcap{}n:\mBbbN{}. compact-type2(F\^{}n Top)) \msubseteq{}r compact-type2(corec(T.F T))))
Date html generated:
2018_05_21-PM-06_18_40
Last ObjectModification:
2018_05_16-PM-01_57_03
Theory : basic
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