Nuprl Lemma : fix_wf_corec1
∀[F,H:Type ⟶ Type].
∀[G:⋂T:{T:Type| (F[T] ⊆r T) ∧ (corec(T.F[T]) ⊆r T)} . (H[T] ⟶ H[F[T]]) ⋂ Top ⟶ H[Top]]. (fix(G) ∈ H[corec(T.F[T])])
supposing Continuous(T.H[T]) ∧ Monotone(T.F[T])
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
type-monotone: Monotone(T.F[T]),
type-continuous: Continuous(T.F[T]),
isect2: T1 ⋂ T2,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
fix: fix(F),
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
corec: corec(T.F[T]),
subtype_rel: A ⊆r B,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
prop: ℙ,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
top: Top,
cand: A c∧ B,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
subtract: n - m,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
type-monotone: Monotone(T.F[T]),
squash: ↓T,
type-continuous: Continuous(T.F[T])
Lemmas referenced :
isect2_wf,
subtype_rel_wf,
corec_wf,
top_wf,
type-continuous_wf,
type-monotone_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
primrec0_lemma,
istype-void,
subtract-1-ge-0,
nat_wf,
isect2_decomp,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le_weakening,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
primrec-unroll,
set_wf,
isect2_subtype_rel3,
int_seg_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-equal-2,
not-le-2,
false_wf,
decidable__le,
subtract_wf,
primrec_wf,
le_weakening2,
not-ge-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
productElimination,
thin,
instantiate,
extract_by_obid,
isectElimination,
isectEquality,
setEquality,
universeEquality,
productEquality,
functionExtensionality,
hypothesisEquality,
because_Cache,
cumulativity,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
setElimination,
rename,
functionEquality,
Error :isect_memberEquality_alt,
Error :productIsType,
Error :functionIsType,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
Error :equalityIsType1,
independent_pairFormation,
lambdaEquality,
promote_hyp,
dependent_pairFormation,
equalityElimination,
unionElimination,
lambdaFormation,
voidEquality,
isect_memberEquality,
inlFormation,
minusEquality,
intEquality,
addEquality,
dependent_set_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality
Latex:
\mforall{}[F,H:Type {}\mrightarrow{} Type].
\mforall{}[G:\mcap{}T:\{T:Type| (F[T] \msubseteq{}r T) \mwedge{} (corec(T.F[T]) \msubseteq{}r T)\} . (H[T] {}\mrightarrow{} H[F[T]]) \mcap{} Top {}\mrightarrow{} H[Top]]
(fix(G) \mmember{} H[corec(T.F[T])])
supposing Continuous(T.H[T]) \mwedge{} Monotone(T.F[T])
Date html generated:
2019_06_20-PM-00_36_46
Last ObjectModification:
2018_10_01-AM-10_05_17
Theory : co-recursion
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