Nuprl Lemma : sorted-by-append1
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x:T. ∀L:T List. (sorted-by(R;L @ [x]) ⇐⇒ sorted-by(R;L) ∧ (∀z∈L.R z x))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
l_all: (∀x∈L.P[x]),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
istype: istype(T),
top: Top,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
guard: {T}
Lemmas referenced :
sorted-by-reverse,
sorted-by-cons,
reverse-append,
cons_wf,
nil_wf,
append_wf,
sorted-by_wf,
subtype_rel_dep_function,
l_member_wf,
l_all_wf,
list_wf,
istype-universe,
reverse-cons,
istype-void,
reverse_nil_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
reverse_wf,
l_all_iff,
member-reverse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
sqequalRule,
hypothesis,
productElimination,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
promote_hyp,
Error :universeIsType,
instantiate,
cumulativity,
functionEquality,
universeEquality,
setEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
Error :setIsType,
independent_isectElimination,
Error :productIsType,
Error :functionIsType,
Error :isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}x:T. \mforall{}L:T List. (sorted-by(R;L @ [x]) \mLeftarrow{}{}\mRightarrow{} sorted-by(R;L) \mwedge{} (\mforall{}z\mmember{}L.R z x))
Date html generated:
2019_06_20-PM-01_45_05
Last ObjectModification:
2018_10_06-PM-11_56_11
Theory : list_1
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