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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