Nuprl Lemma : insert-combine-sorted-by
∀T:Type. ∀cmp:comparison(T).
((∀u,x,z:T. (0 < cmp x u ⇒ 0 < cmp u z ⇒ 0 < cmp x z))
⇒ (∀f:T ⟶ T ⟶ T
((∀u,x:T. (((cmp x u) = 0 ∈ ℤ) ⇒ ((cmp u (f x u)) = 0 ∈ ℤ)))
⇒ (∀L:T List. ∀x:T. (sorted-by(λx,y. 0 < cmp x y;L) ⇒ sorted-by(λx,y. 0 < cmp x y;insert-combine(cmp;f;x;L)))\000C))))
Proof
Definitions occuring in Statement :
insert-combine: insert-combine(cmp;f;x;l),
comparison: comparison(T),
sorted-by: sorted-by(R;L),
list: T List,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
comparison: comparison(T),
so_apply: x[s],
insert-combine: insert-combine(cmp;f;x;l),
sorted-by: sorted-by(R;L),
select: L[n],
uimplies: b supposing a,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
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,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
l_all: (∀x∈L.P[x]),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
decidable: Dec(P),
less_than: a < b,
nequal: a ≠ b ∈ T ,
l_exists: (∃x∈L. P[x])
Lemmas referenced :
list_induction,
all_wf,
sorted-by_wf,
l_member_wf,
less_than_wf,
insert-combine_wf,
list_wf,
equal-wf-T-base,
comparison_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,
int_seg_wf,
list_ind_cons_lemma,
value-type-has-value,
int-value-type,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
sorted-by-cons,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
cons_wf,
squash_wf,
true_wf,
iff_weakening_equal,
select_wf,
length_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
intformeq_wf,
int_formula_prop_eq_lemma,
l_all_cons,
minus-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma,
false_wf,
l_all_iff,
member-insert-combine,
and_wf,
select_member,
minus_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality,
because_Cache,
hypothesis,
setElimination,
rename,
natural_numberEquality,
applyEquality,
setEquality,
dependent_functionElimination,
functionExtensionality,
independent_functionElimination,
intEquality,
baseClosed,
universeEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
callbyvalueReduce,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
imageElimination,
imageMemberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}cmp:comparison(T).
((\mforall{}u,x,z:T. (0 < cmp x u {}\mRightarrow{} 0 < cmp u z {}\mRightarrow{} 0 < cmp x z))
{}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T
((\mforall{}u,x:T. (((cmp x u) = 0) {}\mRightarrow{} ((cmp u (f x u)) = 0)))
{}\mRightarrow{} (\mforall{}L:T List. \mforall{}x:T.
(sorted-by(\mlambda{}x,y. 0 < cmp x y;L) {}\mRightarrow{} sorted-by(\mlambda{}x,y. 0 < cmp x y;insert-combine(cmp;f;x;\000CL)))))))
Date html generated:
2017_04_17-AM-08_29_48
Last ObjectModification:
2017_02_27-PM-04_52_53
Theory : list_1
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