Nuprl Lemma : rel_inverse_star
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y:T. (x R^*^-1 y ⇐⇒ x (R^-1^*) y)
Proof
Definitions occuring in Statement :
rel_inverse: R^-1,
rel_star: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
rev_implies: P ⇐ Q,
rel_star: R^*,
rel_inverse: R^-1,
exists: ∃x:A. B[x]
Lemmas referenced :
rel_inverse_wf,
rel_star_wf,
rel_exp_wf,
rel_inverse_exp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
applyEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
universeEquality,
sqequalRule,
productElimination,
dependent_pairFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:T. (x rel\_star(T; R)\^{}-1 y \mLeftarrow{}{}\mRightarrow{} x rel\_star(T; R\^{}-1) y)
Date html generated:
2019_06_20-PM-00_30_55
Last ObjectModification:
2018_09_26-PM-00_41_50
Theory : relations
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