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
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