Nuprl Lemma : order_split

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  (Order(T;x,y.R[x;y])`
`  `` (∀x,y:T.  Dec(x = y ∈ T))`
`  `` (∀a,b:T.  (R[a;b] `⇐⇒` strict_part(x,y.R[x;y];a;b) ∨ (a = b ∈ T))))`

Proof

Definitions occuring in Statement :  order: `Order(T;x,y.R[x; y])` strict_part: `strict_part(x,y.R[x; y];a;b)` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` or: `P ∨ Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  strict_part: `strict_part(x,y.R[x; y];a;b)` uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` order: `Order(T;x,y.R[x; y])` anti_sym: `AntiSym(T;x,y.R[x; y])` trans: `Trans(T;x,y.E[x; y])` refl: `Refl(T;x,y.E[x; y])` member: `t ∈ T` prop: `ℙ` so_apply: `x[s1;s2]` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]` decidable: `Dec(P)` or: `P ∨ Q` cand: `A c∧ B` not: `¬A` false: `False` guard: `{T}`
Lemmas referenced :  or_wf subtype_rel_self not_wf equal_wf all_wf decidable_wf order_wf and_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin applyEquality hypothesisEquality cut introduction extract_by_obid isectElimination productEquality hypothesis instantiate universeEquality lambdaEquality Error :functionIsType,  Error :universeIsType,  Error :inhabitedIsType,  dependent_functionElimination unionElimination inrFormation inlFormation independent_functionElimination hyp_replacement equalitySymmetry dependent_set_memberEquality applyLambdaEquality setElimination rename

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(Order(T;x,y.R[x;y])
{}\mRightarrow{}  (\mforall{}x,y:T.    Dec(x  =  y))
{}\mRightarrow{}  (\mforall{}a,b:T.    (R[a;b]  \mLeftarrow{}{}\mRightarrow{}  strict\_part(x,y.R[x;y];a;b)  \mvee{}  (a  =  b))))

Date html generated: 2019_06_20-PM-00_29_53
Last ObjectModification: 2018_09_26-PM-00_04_55

Theory : rel_1

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