Nuprl Lemma : rel_star_iff2
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y:T.  (x (R^*) y 
⇐⇒ (∃z:T. ((x R z) ∧ (z (R^*) y))) ∨ (x = y ∈ T))
Proof
Definitions occuring in Statement : 
rel_star: R^*
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
infix_ap: x f y
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
rel_star: R^*
, 
infix_ap: x f y
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
exists: ∃x:A. B[x]
, 
member: t ∈ T
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
rev_implies: P 
⇐ Q
, 
or: P ∨ Q
, 
cand: A c∧ B
, 
nat: ℕ
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
false: False
, 
not: ¬A
, 
top: Top
, 
guard: {T}
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
rel_exp: R^n
, 
ifthenelse: if b then t else f fi 
, 
eq_int: (i =z j)
, 
btrue: tt
Lemmas referenced : 
false_wf, 
infix_ap_wf, 
less_than_wf, 
add-subtract-cancel, 
decidable__lt, 
int_term_value_add_lemma, 
itermAdd_wf, 
le_wf, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_subtract_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
intformless_wf, 
itermVar_wf, 
itermSubtract_wf, 
itermConstant_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
decidable__le, 
nat_properties, 
subtract_wf, 
rel_exp_iff2, 
equal_wf, 
and_wf, 
or_wf, 
rel_exp_wf, 
nat_wf, 
exists_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
lambdaFormation, 
independent_pairFormation, 
sqequalHypSubstitution, 
productElimination, 
thin, 
cut, 
lemma_by_obid, 
isectElimination, 
hypothesis, 
lambdaEquality, 
applyEquality, 
hypothesisEquality, 
unionElimination, 
functionEquality, 
cumulativity, 
universeEquality, 
dependent_functionElimination, 
independent_functionElimination, 
inlFormation, 
dependent_pairFormation, 
dependent_set_memberEquality, 
setElimination, 
rename, 
natural_numberEquality, 
independent_isectElimination, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll, 
inrFormation, 
addEquality, 
because_Cache, 
productEquality, 
instantiate
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}x,y:T.    (x  (R\^{}*)  y  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}z:T.  ((x  R  z)  \mwedge{}  (z  (R\^{}*)  y)))  \mvee{}  (x  =  y))
Date html generated:
2016_05_14-PM-03_52_43
Last ObjectModification:
2016_01_14-PM-11_11_06
Theory : relations2
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