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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