Nuprl Lemma : sub-corec-family
∀[P:Type]. ∀[H:(P ⟶ Type) ⟶ P ⟶ Type].
corec-family(H) ⊆ H corec-family(H) supposing type-family-continuous{i:l}(P;H)
Proof
Definitions occuring in Statement :
corec-family: corec-family(H),
type-family-continuous: type-family-continuous{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 :
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
sub-family: F ⊆ G,
so_apply: x[s],
so_lambda: λ2x.t[x],
corec-family: corec-family(H),
type-family-continuous: type-family-continuous{i:l}(P;H),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
isect-family: ⋂a:A. F[a],
top: Top,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
fun_exp_add1,
istype-void,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-int,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
istype-le,
fun_exp_wf,
top_wf,
subtype_rel_isect,
nat_wf,
sub-family_transitivity,
isect-family_wf,
type-family-continuous_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
because_Cache,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
independent_isectElimination,
sqequalRule,
universeEquality,
hypothesisEquality,
cumulativity,
functionEquality,
lemma_by_obid,
instantiate,
applyEquality,
lambdaEquality,
thin,
isectElimination,
hypothesis,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
isectEquality,
lambdaFormation,
extract_by_obid,
Error :isect_memberEquality_alt,
voidElimination,
Error :lambdaEquality_alt,
Error :dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_pairFormation,
Error :lambdaFormation_alt,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
Error :isectIsType,
Error :inhabitedIsType,
Error :universeIsType,
closedConclusion
Latex:
\mforall{}[P:Type]. \mforall{}[H:(P {}\mrightarrow{} Type) {}\mrightarrow{} P {}\mrightarrow{} Type].
corec-family(H) \msubseteq{} H corec-family(H) supposing type-family-continuous\{i:l\}(P;H)
Date html generated:
2019_06_20-PM-00_35_27
Last ObjectModification:
2018_11_20-PM-00_25_09
Theory : co-recursion
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