Nuprl Lemma : compact-type-corec-lemma1
∀[F:Type ⟶ Type]
(Monotone(T.F T)
⇒ (((⋂n:ℕ. (F^n Top)) ⟶ 𝔹) ⊆r ⋃n:ℕ.((F^n Top) ⟶ 𝔹))
⇒ (∀[T:Type]. (compact-type2(T) ⇒ compact-type2(F T)))
⇒ 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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
compact-type2: compact-type2(T),
sq_exists: ∃x:{A| B[x]},
p-selector: p-selector(T;x;p),
squash: ↓T,
true: True,
nat_plus: ℕ+
Lemmas referenced :
compact-type-corec-lemma0,
uall_wf,
compact-type2_wf,
subtype_rel_wf,
nat_wf,
fun_exp_wf,
top_wf,
bool_wf,
tunion_wf,
type-monotone_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
fun_exp0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
void_wf,
exists_wf,
equal-wf-T-base,
p-selector_wf,
equal_wf,
squash_wf,
true_wf,
fun_exp_unroll_1,
le_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
rename,
instantiate,
universeEquality,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality,
applyEquality,
functionExtensionality,
isectEquality,
because_Cache,
isect_memberEquality,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_set_memberEquality,
productElimination,
baseClosed,
addLevel,
hyp_replacement,
imageElimination,
equalityUniverse,
levelHypothesis,
imageMemberEquality
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{} (\mforall{}[T:Type]. (compact-type2(T) {}\mRightarrow{} compact-type2(F T)))
{}\mRightarrow{} compact-type2(corec(T.F T)))
Date html generated:
2016_10_25-AM-10_14_00
Last ObjectModification:
2016_07_12-AM-06_24_28
Theory : basic
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