Nuprl Lemma : xxorder_split
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(order(T;R) ⇒ (∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀a,b:T. (R a b ⇐⇒ ((R\) a b) ∨ (a = b ∈ T))))
Proof
Definitions occuring in Statement :
s_part: E\,
xxorder: order(T;R),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s1;s2],
strict_part: strict_part(x,y.R[x; y];a;b),
order: Order(T;x,y.R[x; y]),
s_part: E\,
xxorder: order(T;R),
xxanti_sym: anti_sym(T;R),
xxtrans: trans(T;E),
xxrefl: refl(T;E)
Lemmas referenced :
order_split
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
hypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(order(T;R) {}\mRightarrow{} (\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}a,b:T. (R a b \mLeftarrow{}{}\mRightarrow{} ((R\mbackslash{}) a b) \mvee{} (a = b))))
Date html generated:
2016_05_15-PM-00_01_40
Last ObjectModification:
2015_12_26-PM-11_26_07
Theory : gen_algebra_1
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