Nuprl Lemma : l-ordered-equality
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(∀as,bs:T List.
(l-ordered(T;x,y.R[x;y];as)
⇒ l-ordered(T;x,y.R[x;y];bs)
⇒ (as = bs ∈ (T List) ⇐⇒ ∀x:T. ((x ∈ as) ⇐⇒ (x ∈ bs))))) supposing
((∀x,y:T. (R[x;y] ⇒ (¬R[y;x]))) and
(∀x:T. (¬R[x;x])))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
l-ordered: l-ordered(T;x,y.R[x; y];L),
or: P ∨ Q,
guard: {T}
Lemmas referenced :
no_repeats-before-equality,
l-ordered-no_repeats,
l_member_wf,
l-ordered_wf,
list_wf,
subtype_rel_self,
istype-void,
istype-universe,
l_before_wf,
l_tricotomy,
l_before_member2,
l_before_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
lambdaFormation_alt,
extract_by_obid,
isectElimination,
independent_isectElimination,
because_Cache,
hypothesis,
independent_pairFormation,
productElimination,
independent_functionElimination,
equalityIstype,
functionIsType,
productIsType,
universeIsType,
applyEquality,
instantiate,
universeEquality,
unionElimination,
hyp_replacement,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(\mforall{}as,bs:T List.
(l-ordered(T;x,y.R[x;y];as)
{}\mRightarrow{} l-ordered(T;x,y.R[x;y];bs)
{}\mRightarrow{} (as = bs \mLeftarrow{}{}\mRightarrow{} \mforall{}x:T. ((x \mmember{} as) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} bs))))) supposing
((\mforall{}x,y:T. (R[x;y] {}\mRightarrow{} (\mneg{}R[y;x]))) and
(\mforall{}x:T. (\mneg{}R[x;x])))
Date html generated:
2020_05_20-AM-08_09_34
Last ObjectModification:
2020_01_17-AM-10_30_59
Theory : general
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