Nuprl Lemma : rel_star_symmetric
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  (Sym(T;x,y.x R y) 
⇒ Sym(T;x,y.x (R^*) y))
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
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
sym: Sym(T;x,y.E[x; y])
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
prop: ℙ
, 
infix_ap: x f y
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
rel_inverse: R^-1
, 
rel_implies: R1 => R2
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
rel_star: R^*
, 
exists: ∃x:A. B[x]
Lemmas referenced : 
rel_star_wf, 
all_wf, 
rel_star_monotonic, 
rel_inverse_wf, 
rel_inverse_star
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
Error :isect_memberFormation_alt, 
lambdaFormation, 
applyEquality, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
lambdaEquality, 
functionEquality, 
Error :functionIsType, 
Error :universeIsType, 
Error :inhabitedIsType, 
universeEquality, 
dependent_functionElimination, 
independent_functionElimination, 
productElimination, 
independent_pairFormation
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (Sym(T;x,y.x  R  y)  {}\mRightarrow{}  Sym(T;x,y.x  rel\_star(T;  R)  y))
Date html generated:
2019_06_20-PM-00_30_57
Last ObjectModification:
2018_09_26-PM-00_41_51
Theory : relations
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