Nuprl Lemma : llex-le-lin-order
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
((∀a:A. (¬<[a;a]))
⇒ Trans(A;a,b.<[a;b])
⇒ (∀a,b:A. (<[a;b] ∨ (a = b ∈ A) ∨ <[b;a]))
⇒ Linorder(A List;as,bs.as llex-le(A;a,b.<[a;b]) bs))
Proof
Definitions occuring in Statement :
llex-le: llex-le(A;a,b.<[a; b]),
list: T List,
linorder: Linorder(T;x,y.R[x; y]),
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
linorder: Linorder(T;x,y.R[x; y]),
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
connex: Connex(T;x,y.R[x; y]),
llex-le: llex-le(A;a,b.<[a; b]),
infix_ap: x f y,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
llex-le-order,
all_wf,
or_wf,
equal_wf,
trans_wf,
not_wf,
list_wf,
llex-linear,
llex_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
independent_pairFormation,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
because_Cache,
universeEquality,
dependent_functionElimination,
unionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
((\mforall{}a:A. (\mneg{}<[a;a]))
{}\mRightarrow{} Trans(A;a,b.<[a;b])
{}\mRightarrow{} (\mforall{}a,b:A. (<[a;b] \mvee{} (a = b) \mvee{} <[b;a]))
{}\mRightarrow{} Linorder(A List;as,bs.as llex-le(A;a,b.<[a;b]) bs))
Date html generated:
2017_02_20-AM-10_55_52
Last ObjectModification:
2017_02_02-PM-09_43_50
Theory : general
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