Nuprl Lemma : llex-le-order
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
((∀a:A. (¬<[a;a])) ⇒ Trans(A;a,b.<[a;b]) ⇒ Order(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,
order: Order(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,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
order: Order(T;x,y.R[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
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],
llex-le: llex-le(A;a,b.<[a; b]),
infix_ap: x f y,
guard: {T},
or: P ∨ Q,
trans: Trans(T;x,y.E[x; y]),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
anti_sym: AntiSym(T;x,y.R[x; y]),
not: ¬A,
false: False
Lemmas referenced :
list_wf,
trans_wf,
all_wf,
not_wf,
llex_wf,
or_wf,
equal_wf,
llex_transitivity,
squash_wf,
true_wf,
iff_weakening_equal,
llex-le_wf,
llex-irreflexive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
inrFormation,
unionElimination,
independent_functionElimination,
dependent_functionElimination,
inlFormation,
rename,
because_Cache,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
voidElimination
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
((\mforall{}a:A. (\mneg{}<[a;a])) {}\mRightarrow{} Trans(A;a,b.<[a;b]) {}\mRightarrow{} Order(A List;as,bs.as llex-le(A;a,b.<[a;b]) bs))
Date html generated:
2017_02_20-AM-10_55_44
Last ObjectModification:
2017_02_02-PM-09_39_29
Theory : general
Home
Index