Nuprl Lemma : bigrel-iff
∀[T:Type]
  ∀F:(T ⟶ T ⟶ ℙ) ⟶ T ⟶ T ⟶ ℙ
    (rel-monotone{i:l}(T;R.F[R]) 
⇒ rel-continuous{i:l}(T;R.F[R]) 
⇒ (∨R.F[R] => F[∨R.F[R]] ∧ F[∨R.F[R]] => ∨R.F[R]))
Proof
Definitions occuring in Statement : 
bigrel: ∨R.F[R]
, 
rel-continuous: rel-continuous{i:l}(T;R.F[R])
, 
rel-monotone: rel-monotone{i:l}(T;R.F[R])
, 
rel_implies: R1 => R2
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
member: t ∈ T
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
rel-continuous: rel-continuous{i:l}(T;R.F[R])
, 
nat: ℕ
, 
bigrel: ∨R.F[R]
, 
rel_implies: R1 => R2
, 
isect-rel: isect-rel(T;i.R[i])
, 
infix_ap: x f y
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
iff: P 
⇐⇒ Q
, 
not: ¬A
, 
rev_implies: P 
⇐ Q
, 
false: False
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
subtract: n - m
, 
subtype_rel: A ⊆r B
, 
top: Top
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
true: True
, 
guard: {T}
, 
sq_type: SQType(T)
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
bfalse: ff
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
exists: ∃x:A. B[x]
, 
bnot: ¬bb
, 
assert: ↑b
, 
ge: i ≥ j 
, 
int_upper: {i...}
, 
rel-monotone: rel-monotone{i:l}(T;R.F[R])
Lemmas referenced : 
rel-continuous_wf, 
rel-monotone_wf, 
primrec_wf, 
true_wf, 
int_seg_wf, 
nat_wf, 
decidable__le, 
false_wf, 
not-le-2, 
sq_stable__le, 
condition-implies-le, 
minus-add, 
minus-one-mul, 
zero-add, 
minus-one-mul-top, 
add-associates, 
add-swap, 
add-commutes, 
add_functionality_wrt_le, 
add-zero, 
le-add-cancel, 
le_wf, 
primrec-unroll, 
infix_ap_wf, 
isect-rel_wf, 
eq_int_wf, 
le_antisymmetry_iff, 
assert_wf, 
bnot_wf, 
not_wf, 
equal-wf-T-base, 
add-subtract-cancel, 
bool_cases, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
eqtt_to_assert, 
assert_of_eq_int, 
eqff_to_assert, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_bnot, 
equal_wf, 
bool_cases_sqequal, 
assert-bnot, 
neg_assert_of_eq_int, 
int_upper_subtype_nat, 
nat_properties, 
nequal-le-implies, 
bigrel_wf, 
subtract_wf, 
minus-minus
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
independent_pairFormation, 
hypothesis, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
cumulativity, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
functionEquality, 
universeEquality, 
dependent_functionElimination, 
instantiate, 
because_Cache, 
natural_numberEquality, 
setElimination, 
rename, 
independent_functionElimination, 
dependent_set_memberEquality, 
addEquality, 
unionElimination, 
voidElimination, 
productElimination, 
independent_isectElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
isect_memberEquality, 
voidEquality, 
intEquality, 
minusEquality, 
equalityTransitivity, 
equalitySymmetry, 
impliesFunctionality, 
equalityElimination, 
dependent_pairFormation, 
promote_hyp, 
hypothesis_subsumption
Latex:
\mforall{}[T:Type]
    \mforall{}F:(T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{})  {}\mrightarrow{}  T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}
        (rel-monotone\{i:l\}(T;R.F[R])
        {}\mRightarrow{}  rel-continuous\{i:l\}(T;R.F[R])
        {}\mRightarrow{}  (\mvee{}R.F[R]  =>  F[\mvee{}R.F[R]]  \mwedge{}  F[\mvee{}R.F[R]]  =>  \mvee{}R.F[R]))
Date html generated:
2017_04_14-AM-07_38_49
Last ObjectModification:
2017_02_27-PM-03_10_41
Theory : relations
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