Nuprl Lemma : rel_star_transitivity
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[x,y,z:T]. ((x (R^*) y) ⇒ (y (R^*) z) ⇒ (x (R^*) z))
Proof
Definitions occuring in Statement :
rel_star: R^*,
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 :
rel_star: R^*,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
add_nat_wf,
nat_wf,
sq_stable__le,
equal_wf,
le_wf,
rel_exp_wf,
exists_wf,
rel_exp_add
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
cut,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
applyEquality,
lambdaEquality,
Error :inhabitedIsType,
Error :universeIsType,
Error :functionIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[x,y,z:T].
((x (R\^{}*) y) {}\mRightarrow{} (y (R\^{}*) z) {}\mRightarrow{} (x (R\^{}*) z))
Date html generated:
2019_06_20-PM-00_30_38
Last ObjectModification:
2018_09_26-PM-00_50_36
Theory : relations
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