Nuprl Lemma : insert-by-sorted-by
∀[T:Type]
∀eq,r:T ⟶ T ⟶ 𝔹.
Linorder(T;a,b.↑(r a b))
⇒ (∀x:T. ∀L:T List. (sorted-by(λx,y. (↑(r x y));L) ⇒ sorted-by(λx,y. (↑(r x y));insert-by(eq;r;x;L))))
supposing ∀a,b:T. (↑(eq a b) ⇐⇒ a = b ∈ T)
Proof
Definitions occuring in Statement :
insert-by: insert-by(eq;r;x;l),
sorted-by: sorted-by(R;L),
list: T List,
linorder: Linorder(T;x,y.R[x; y]),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
insert-by: insert-by(eq;r;x;l),
sorted-by: sorted-by(R;L),
select: L[n],
nil: [],
it: ⋅,
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
l_all: (∀x∈L.P[x]),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
linorder: Linorder(T;x,y.R[x; y]),
order: Order(T;x,y.R[x; y]),
trans: Trans(T;x,y.E[x; y]),
connex: Connex(T;x,y.R[x; y])
Lemmas referenced :
assert_wf,
assert_witness,
equal_wf,
list_induction,
sorted-by_wf,
l_member_wf,
insert-by_wf,
list_wf,
linorder_wf,
all_wf,
iff_wf,
bool_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
list_ind_nil_lemma,
length_of_cons_lemma,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
select_wf,
cons_wf,
nil_wf,
int_seg_wf,
list_ind_cons_lemma,
equal-wf-T-base,
bnot_wf,
not_wf,
sorted-by-cons,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
l_all_cons,
length_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
l_all_iff,
member-insert-by
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
independent_functionElimination,
rename,
because_Cache,
functionEquality,
setElimination,
setEquality,
universeEquality,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
imageElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq,r:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}.
Linorder(T;a,b.\muparrow{}(r a b))
{}\mRightarrow{} (\mforall{}x:T. \mforall{}L:T List.
(sorted-by(\mlambda{}x,y. (\muparrow{}(r x y));L) {}\mRightarrow{} sorted-by(\mlambda{}x,y. (\muparrow{}(r x y));insert-by(eq;r;x;L))))
supposing \mforall{}a,b:T. (\muparrow{}(eq a b) \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2017_04_17-AM-08_32_00
Last ObjectModification:
2017_02_27-PM-04_52_30
Theory : list_1
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