Nuprl Lemma : l-ordered-cons
∀[T:Type]
∀x:T. ∀L:T List.
∀[R:T ⟶ T ⟶ ℙ]. (l-ordered(T;x,y.R[x;y];[x / L]) ⇐⇒ l-ordered(T;x,y.R[x;y];L) ∧ (∀y:T. ((y ∈ L) ⇒ R[x;y])))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
or: P ∨ Q,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
l_member_wf,
l-ordered_wf,
cons_wf,
all_wf,
list_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
l-ordered-append,
nil_wf,
cons_member,
l-ordered-single,
member_singleton,
and_wf,
equal_wf
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,
productElimination,
productEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
because_Cache,
inlFormation,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
setElimination,
rename,
setEquality
Latex:
\mforall{}[T:Type]
\mforall{}x:T. \mforall{}L:T List.
\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]
(l-ordered(T;x,y.R[x;y];[x / L]) \mLeftarrow{}{}\mRightarrow{} l-ordered(T;x,y.R[x;y];L) \mwedge{} (\mforall{}y:T. ((y \mmember{} L) {}\mRightarrow{} R[x;y])))
Date html generated:
2016_10_25-AM-10_56_43
Last ObjectModification:
2016_07_12-AM-07_03_49
Theory : general
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