Nuprl Lemma : rel_star_closure
∀[T:Type]. ∀[R,R2:T ⟶ T ⟶ ℙ].
  (Trans(T)(R2[_1;_2]) 
⇒ (∀x,y:T.  ((x R y) 
⇒ (x R2 y))) 
⇒ (∀x,y:T.  ((x (R^*) y) 
⇒ ((x R2 y) ∨ (x = y ∈ T)))))
Proof
Definitions occuring in Statement : 
rel_star: R^*
, 
trans: Trans(T;x,y.E[x; y])
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
infix_ap: x f y
, 
so_apply: x[s1;s2]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
bfalse: ff
, 
it: ⋅
, 
unit: Unit
, 
bool: 𝔹
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s]
, 
true: True
, 
top: Top
, 
subtype_rel: A ⊆r B
, 
subtract: n - m
, 
uiff: uiff(P;Q)
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
decidable: Dec(P)
, 
so_lambda: λ2x.t[x]
, 
uimplies: b supposing a
, 
not: ¬A
, 
false: False
, 
less_than': less_than'(a;b)
, 
and: P ∧ Q
, 
le: A ≤ B
, 
nat: ℕ
, 
prop: ℙ
, 
member: t ∈ T
, 
btrue: tt
, 
eq_int: (i =z j)
, 
ifthenelse: if b then t else f fi 
, 
rel_exp: R^n
, 
or: P ∨ Q
, 
guard: {T}
, 
exists: ∃x:A. B[x]
, 
infix_ap: x f y
, 
rel_star: R^*
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
uall: ∀[x:A]. B[x]
, 
trans: Trans(T;x,y.E[x; y])
Lemmas referenced : 
assert_of_bnot, 
eqff_to_assert, 
iff_weakening_uiff, 
iff_transitivity, 
assert_of_eq_int, 
eqtt_to_assert, 
uiff_transitivity, 
exists_wf, 
not_wf, 
bnot_wf, 
less_than_irreflexivity, 
le_weakening, 
less_than_transitivity1, 
assert_wf, 
int_subtype_base, 
equal-wf-base, 
bool_wf, 
eq_int_wf, 
trans_wf, 
rel_star_wf, 
nat_wf, 
primrec-wf2, 
less_than_wf, 
set_wf, 
equal_wf, 
or_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-le-2, 
decidable__le, 
subtract_wf, 
all_wf, 
le_weakening2, 
le_wf, 
false_wf, 
rel_exp_wf, 
infix_ap_wf
Rules used in proof : 
impliesFunctionality, 
equalityElimination, 
productEquality, 
equalitySymmetry, 
equalityTransitivity, 
baseClosed, 
closedConclusion, 
baseApply, 
minusEquality, 
intEquality, 
voidEquality, 
isect_memberEquality, 
addEquality, 
voidElimination, 
unionElimination, 
functionEquality, 
lambdaEquality, 
independent_isectElimination, 
dependent_functionElimination, 
independent_functionElimination, 
setElimination, 
rename, 
independent_pairFormation, 
natural_numberEquality, 
dependent_set_memberEquality, 
universeEquality, 
because_Cache, 
isectElimination, 
extract_by_obid, 
introduction, 
instantiate, 
cumulativity, 
hypothesisEquality, 
functionExtensionality, 
applyEquality, 
inrFormation, 
hypothesis, 
cut, 
thin, 
productElimination, 
sqequalRule, 
sqequalHypSubstitution, 
lambdaFormation, 
isect_memberFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
inlFormation, 
applyLambdaEquality, 
hyp_replacement
Latex:
\mforall{}[T:Type].  \mforall{}[R,R2:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    (Trans(T)(R2[$_{1}$;$_{2}$])
    {}\mRightarrow{}  (\mforall{}x,y:T.    ((x  R  y)  {}\mRightarrow{}  (x  R2  y)))
    {}\mRightarrow{}  (\mforall{}x,y:T.    ((x  rel\_star(T;  R)  y)  {}\mRightarrow{}  ((x  R2  y)  \mvee{}  (x  =  y)))))
Date html generated:
2019_06_20-PM-00_30_44
Last ObjectModification:
2018_08_03-PM-05_27_20
Theory : relations
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