Nuprl Lemma : corec-sub-family
∀[P:Type]. ∀[H:(P ⟶ Type) ⟶ P ⟶ Type]. H corec-family(H) ⊆ corec-family(H) supposing family-monotone{i:l}(P;H)
Proof
Definitions occuring in Statement :
corec-family: corec-family(H),
family-monotone: family-monotone{i:l}(P;H),
sub-family: F ⊆ G,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
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,
corec-family: corec-family(H),
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
sub-family: F ⊆ G,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
compose: f o g,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
family-monotone: family-monotone{i:l}(P;H),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
true: True,
isect-family: ⋂a:A. F[a]
Lemmas referenced :
sub-isect-family,
corec-family_wf,
nat_wf,
fun_exp_wf,
top_wf,
fun_exp_unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
subtract_wf,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
family-monotone_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality,
hypothesis,
lambdaEquality,
instantiate,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
because_Cache,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
minusEquality,
isectEquality,
axiomEquality
Latex:
\mforall{}[P:Type]. \mforall{}[H:(P {}\mrightarrow{} Type) {}\mrightarrow{} P {}\mrightarrow{} Type].
H corec-family(H) \msubseteq{} corec-family(H) supposing family-monotone\{i:l\}(P;H)
Date html generated:
2017_04_14-AM-07_41_54
Last ObjectModification:
2017_02_27-PM-03_13_42
Theory : co-recursion
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