Nuprl Lemma : corec-value-type
∀[F:Type ⟶ Type]
(value-type(corec(T.F[T]))) supposing ((∃n:ℕ. value-type(F^n Top)) and (∀A,B:Type. (A ≡ B ⇒ F[A] ≡ F[B])))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
fun_exp: f^n,
nat: ℕ,
value-type: value-type(T),
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
value-type: value-type(T),
sq_stable: SqStable(P),
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
has-value: (a)↓,
exists: ∃x:A. B[x],
corec: corec(T.F[T]),
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
squash: ↓T,
fun_exp: f^n,
false: False,
ge: i ≥ j ,
guard: {T},
ext-eq: A ≡ B,
and: P ∧ Q,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
compose: f o g,
nequal: a ≠ b ∈ T
Lemmas referenced :
sq_stable__has-value,
corec_wf,
subtype_rel_weakening,
primrec_wf,
top_wf,
int_seg_wf,
fun_exp_wf,
value-type-has-value,
equal_wf,
equal-wf-base,
base_wf,
exists_wf,
nat_wf,
value-type_wf,
all_wf,
ext-eq_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
primrec0_lemma,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_weakening,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-le-2,
not-equal-2,
le_wf,
compose_wf,
ext-eq_weakening,
primrec-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
universeEquality,
cumulativity,
lambdaFormation,
productElimination,
instantiate,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
isect_memberEquality,
axiomSqleEquality,
functionEquality,
intWeakElimination,
voidElimination,
independent_pairEquality,
axiomEquality,
voidEquality,
unionElimination,
independent_pairFormation,
addEquality,
intEquality,
minusEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
dependent_set_memberEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]
(value-type(corec(T.F[T]))) supposing
((\mexists{}n:\mBbbN{}. value-type(F\^{}n Top)) and
(\mforall{}A,B:Type. (A \mequiv{} B {}\mRightarrow{} F[A] \mequiv{} F[B])))
Date html generated:
2017_04_14-AM-07_41_47
Last ObjectModification:
2017_02_27-PM-03_13_26
Theory : co-recursion
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