Nuprl Lemma : ulinorder_lt_neg

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  ((∀x,y:T.  Dec(R[x;y])) `` UniformLinorder(T;x,y.R[x;y]) `` (∀[a,b:T].  uiff(¬strict_part(x,y.R[x;y];a;b);R[b;a])))`

Proof

Definitions occuring in Statement :  ulinorder: `UniformLinorder(T;x,y.R[x; y])` strict_part: `strict_part(x,y.R[x; y];a;b)` decidable: `Dec(P)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` ulinorder: `UniformLinorder(T;x,y.R[x; y])` and: `P ∧ Q` connex: `Connex(T;x,y.R[x; y])` uorder: `UniformOrder(T;x,y.R[x; y])` uanti_sym: `UniformlyAntiSym(T;x,y.R[x; y])` utrans: `UniformlyTrans(T;x,y.E[x; y])` urefl: `UniformlyRefl(T;x,y.E[x; y])` strict_part: `strict_part(x,y.R[x; y];a;b)` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` uimplies: `b supposing a` not: `¬A` false: `False` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` cand: `A c∧ B`
Lemmas referenced :  ulinorder_wf all_wf decidable_wf subtype_rel_self not_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  lambdaFormation sqequalHypSubstitution productElimination thin Error :inhabitedIsType,  hypothesisEquality Error :universeIsType,  cut introduction extract_by_obid isectElimination sqequalRule lambdaEquality applyEquality hypothesis Error :functionIsType,  universeEquality independent_pairFormation dependent_functionElimination voidElimination productEquality instantiate rename because_Cache independent_functionElimination independent_isectElimination unionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}x,y:T.    Dec(R[x;y]))
{}\mRightarrow{}  UniformLinorder(T;x,y.R[x;y])
{}\mRightarrow{}  (\mforall{}[a,b:T].    uiff(\mneg{}strict\_part(x,y.R[x;y];a;b);R[b;a])))

Date html generated: 2019_06_20-PM-00_30_05
Last ObjectModification: 2018_09_26-PM-00_06_21

Theory : rel_1

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