### 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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