Nuprl Lemma : member-insert-combine2
∀T:Type. ∀cmp:comparison(T). ∀f:T ⟶ T ⟶ T. ∀x,z:T. ∀v:T List.
((z ∈ insert-combine(cmp;f;x;v))
⇐⇒ (∃l:T List. (l @ [z] ≤ v ∧ (∀y∈l.cmp x y < 0) ∧ cmp x z < 0))
∨ (∃l,l':T List
∃a:T
(((l @ [a / l']) = v ∈ (T List))
∧ (∀y∈l.cmp x y < 0)
∧ ((0 < cmp x a ∧ (z ∈ [x; [a / l']])) ∨ (((cmp x a) = 0 ∈ ℤ) ∧ (z ∈ [f x a / l'])))))
∨ ((z = x ∈ T) ∧ (∀y∈v.cmp x y < 0)))
Proof
Definitions occuring in Statement :
insert-combine: insert-combine(cmp;f;x;l)
,
comparison: comparison(T)
,
iseg: l1 ≤ l2
,
l_all: (∀x∈L.P[x])
,
l_member: (x ∈ l)
,
append: as @ bs
,
cons: [a / b]
,
nil: []
,
list: T List
,
less_than: a < b
,
all: ∀x:A. B[x]
,
exists: ∃x:A. B[x]
,
iff: P
⇐⇒ Q
,
or: P ∨ Q
,
and: P ∧ Q
,
apply: f a
,
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]
,
iff: P
⇐⇒ Q
,
and: P ∧ Q
,
implies: P
⇒ Q
,
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
so_lambda: λ2x.t[x]
,
prop: ℙ
,
or: P ∨ Q
,
exists: ∃x:A. B[x]
,
so_apply: x[s]
,
top: Top
,
comparison: comparison(T)
,
rev_implies: P
⇐ Q
,
uiff: uiff(P;Q)
,
uimplies: b supposing a
,
subtype_rel: A ⊆r B
,
guard: {T}
,
cand: A c∧ B
,
true: True
,
insert-combine: insert-combine(cmp;f;x;l)
,
has-value: (a)↓
,
append: as @ bs
,
so_lambda: so_lambda(x,y,z.t[x; y; z])
,
so_apply: x[s1;s2;s3]
,
not: ¬A
,
false: False
,
decidable: Dec(P)
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
squash: ↓T
,
sq_type: SQType(T)
,
ifthenelse: if b then t else f fi
,
btrue: tt
,
bfalse: ff
,
assert: ↑b
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
cons: [a / b]
,
bnot: ¬bb
,
nequal: a ≠ b ∈ T
,
less_than: a < b
,
nat: ℕ
,
le: A ≤ B
,
subtract: n - m
,
less_than': less_than'(a;b)
,
listp: A List+
Lemmas referenced :
list_induction,
l_member_wf,
insert-combine_wf,
or_wf,
list_wf,
insert-combine-nil,
member_singleton,
equal_wf,
nil_wf,
insert-combine-cons,
cons_wf,
exists_wf,
iseg_wf,
append_wf,
l_all_wf,
less_than_wf,
length_wf,
length-append,
equal-wf-T-base,
cons_member,
l_all_nil_iff,
length_of_nil_lemma,
l_all_wf_nil,
l_all_cons,
length_of_cons_lemma,
comparison_wf,
true_wf,
value-type-has-value,
int-value-type,
eq_int_wf,
list_ind_nil_lemma,
l_all_nil,
assert_wf,
bnot_wf,
not_wf,
lt_int_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
cons_iseg,
nil_iseg,
and_wf,
list_ind_cons_lemma,
squash_wf,
iff_weakening_equal,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_lt_int,
iseg_nil,
null_append,
subtype_rel_list,
top_wf,
null_cons_lemma,
band_wf,
null_wf,
bfalse_wf,
false_wf,
assert_of_band,
assert_of_null,
list-cases,
product_subtype_list,
member_append,
l_all_iff,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
hd_wf,
listp_properties,
cons_neq_nil,
length_wf_nat,
nat_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
reduce_hd_cons_lemma,
tl_wf,
reduce_tl_cons_lemma,
not-equal-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_functionElimination,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
cumulativity,
independent_functionElimination,
addLevel,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
levelHypothesis,
promote_hyp,
rename,
productEquality,
setElimination,
natural_numberEquality,
setEquality,
applyLambdaEquality,
intEquality,
baseClosed,
unionElimination,
orFunctionality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
inrFormation,
callbyvalueReduce,
inlFormation,
dependent_pairFormation,
int_eqEquality,
computeAll,
hyp_replacement,
dependent_set_memberEquality,
imageElimination,
equalityUniverse,
imageMemberEquality,
instantiate,
impliesFunctionality,
equalityElimination,
hypothesis_subsumption,
addEquality,
minusEquality
Latex:
\mforall{}T:Type. \mforall{}cmp:comparison(T). \mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T. \mforall{}x,z:T. \mforall{}v:T List.
((z \mmember{} insert-combine(cmp;f;x;v))
\mLeftarrow{}{}\mRightarrow{} (\mexists{}l:T List. (l @ [z] \mleq{} v \mwedge{} (\mforall{}y\mmember{}l.cmp x y < 0) \mwedge{} cmp x z < 0))
\mvee{} (\mexists{}l,l':T List
\mexists{}a:T
(((l @ [a / l']) = v)
\mwedge{} (\mforall{}y\mmember{}l.cmp x y < 0)
\mwedge{} ((0 < cmp x a \mwedge{} (z \mmember{} [x; [a / l']])) \mvee{} (((cmp x a) = 0) \mwedge{} (z \mmember{} [f x a / l'])))))
\mvee{} ((z = x) \mwedge{} (\mforall{}y\mmember{}v.cmp x y < 0)))
Date html generated:
2017_04_17-AM-08_29_30
Last ObjectModification:
2017_02_27-PM-04_56_17
Theory : list_1
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