Nuprl Lemma : rel_plus_idempotent
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y:T. (R+ x y ⇐⇒ R++ x y)
Proof
Definitions occuring in Statement :
rel_plus: R+,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
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,
rev_implies: P ⇐ Q,
rel_plus: R+,
rel_implies: R1 => R2,
infix_ap: x f y,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
cand: A c∧ B,
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
btrue: tt
Lemmas referenced :
rel_plus_trans,
rel_plus_minimal,
le_wf,
false_wf,
infix_ap_wf,
nat_plus_subtype_nat,
rel_exp_wf,
less_than_wf,
rel_plus_monotone,
rel_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
applyEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
introduction,
imageMemberEquality,
baseClosed,
productEquality,
instantiate,
because_Cache,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:T. (R\msupplus{} x y \mLeftarrow{}{}\mRightarrow{} R\msupplus{}\msupplus{} x y)
Date html generated:
2016_05_14-PM-03_55_18
Last ObjectModification:
2016_01_14-PM-11_10_44
Theory : relations2
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