Nuprl Lemma : rel_star_symmetric_2
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y:T.  (Sym(T;x,y.x R y) 
⇒ (x (R^*) y) 
⇒ (y (R^*) x))
Proof
Definitions occuring in Statement : 
rel_star: R^*
, 
sym: Sym(T;x,y.E[x; y])
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
infix_ap: x f y
, 
all: ∀x:A. B[x]
, 
implies: 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
, 
member: t ∈ T
, 
prop: ℙ
, 
infix_ap: x f y
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
sym: Sym(T;x,y.E[x; y])
, 
guard: {T}
Lemmas referenced : 
rel_star_symmetric, 
rel_star_wf, 
sym_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
because_Cache, 
hypothesisEquality, 
independent_functionElimination, 
hypothesis, 
applyEquality, 
sqequalRule, 
lambdaEquality, 
Error :functionIsType, 
Error :universeIsType, 
Error :inhabitedIsType, 
universeEquality, 
dependent_functionElimination
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}x,y:T.    (Sym(T;x,y.x  R  y)  {}\mRightarrow{}  (x  (R\^{}*)  y)  {}\mRightarrow{}  (y  (R\^{}*)  x))
Date html generated:
2019_06_20-PM-00_30_58
Last ObjectModification:
2018_09_26-PM-00_44_09
Theory : relations
Home
Index