Nuprl Lemma : fix_wf_corec2'
∀[F,H:Type ⟶ Type].
  ∀[G:⋂T:{T:Type| 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])
Proof
Definitions occuring in Statement : 
corec: corec(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]
, 
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
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
false: False
, 
ge: i ≥ j 
, 
guard: {T}
, 
top: Top
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
le: A ≤ B
, 
not: ¬A
, 
less_than': less_than'(a;b)
, 
true: True
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
assert: ↑b
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
decidable: Dec(P)
, 
subtract: n - m
, 
type-continuous: Continuous(T.F[T])
Lemmas referenced : 
isect2_wf, 
subtype_rel_wf, 
corec_wf, 
subtype_rel_universe1, 
istype-universe, 
top_wf, 
type-continuous_wf, 
nat_properties, 
less_than_transitivity1, 
less_than_irreflexivity, 
ge_wf, 
istype-less_than, 
primrec0_lemma, 
istype-void, 
subtract-1-ge-0, 
istype-nat, 
isect2_decomp, 
primrec-unroll, 
lt_int_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
less-iff-le, 
add_functionality_wrt_le, 
add-associates, 
add-zero, 
add-commutes, 
le-add-cancel2, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
iff_weakening_uiff, 
assert_wf, 
less_than_wf, 
isect2_subtype_rel3, 
primrec_wf, 
subtract_wf, 
decidable__le, 
istype-false, 
not-le-2, 
condition-implies-le, 
minus-one-mul, 
zero-add, 
minus-one-mul-top, 
minus-add, 
minus-minus, 
add-swap, 
le-add-cancel, 
istype-le, 
int_seg_wf, 
nat_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
introduction, 
cut, 
applyEquality, 
hypothesis, 
sqequalHypSubstitution, 
sqequalRule, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
Error :universeIsType, 
thin, 
instantiate, 
extract_by_obid, 
isectElimination, 
isectEquality, 
setEquality, 
closedConclusion, 
universeEquality, 
Error :lambdaEquality_alt, 
hypothesisEquality, 
Error :inhabitedIsType, 
because_Cache, 
Error :lambdaFormation_alt, 
setElimination, 
rename, 
functionEquality, 
cumulativity, 
Error :isect_memberEquality_alt, 
Error :isectIsTypeImplies, 
Error :functionIsType, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
independent_functionElimination, 
voidElimination, 
dependent_functionElimination, 
Error :functionIsTypeImplies, 
Error :equalityIsType1, 
lambdaEquality, 
productElimination, 
independent_pairFormation, 
unionElimination, 
equalityElimination, 
addEquality, 
Error :dependent_pairFormation_alt, 
Error :equalityIstype, 
promote_hyp, 
Error :inlFormation_alt, 
Error :isectIsType, 
Error :setIsType, 
Error :dependent_set_memberEquality_alt, 
minusEquality, 
functionExtensionality
Latex:
\mforall{}[F,H:Type  {}\mrightarrow{}  Type].
    \mforall{}[G:\mcap{}T:\{T:Type|  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])
Date html generated:
2019_06_20-PM-00_36_51
Last ObjectModification:
2018_11_28-AM-11_40_49
Theory : co-recursion
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