Nuprl Lemma : member-insert-combine
∀T:Type. ∀cmp:comparison(T). ∀f:T ⟶ T ⟶ T. ∀x,z:T. ∀v:T List.
((z ∈ insert-combine(cmp;f;x;v)) ⇒ ((z ∈ v) ∨ (z = x ∈ T) ∨ (∃y∈v. ((cmp x y) = 0 ∈ ℤ) ∧ (z = (f x y) ∈ T))))
Proof
Definitions occuring in Statement :
insert-combine: insert-combine(cmp;f;x;l),
comparison: comparison(T),
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
all: ∀x:A. B[x],
implies: 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],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
comparison: comparison(T),
so_apply: x[s],
or: P ∨ Q,
insert-combine: insert-combine(cmp;f;x;l),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
guard: {T},
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
has-value: (a)↓,
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
subtract: n - m,
ge: i ≥ j
Lemmas referenced :
list_induction,
l_member_wf,
insert-combine_wf,
or_wf,
equal_wf,
l_exists_wf,
equal-wf-T-base,
list_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
comparison_wf,
l_exists_wf_nil,
and_wf,
nil_wf,
member_singleton,
cons_wf,
value-type-has-value,
int-value-type,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
cons_member,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
length_wf,
select-cons-hd,
select_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
select-cons-tl,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesisEquality,
dependent_functionElimination,
functionExtensionality,
applyEquality,
hypothesis,
setElimination,
rename,
productEquality,
intEquality,
baseClosed,
setEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
inrFormation,
inlFormation,
addLevel,
impliesFunctionality,
productElimination,
callbyvalueReduce,
independent_isectElimination,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
computeAll,
addEquality,
imageElimination
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)) {}\mRightarrow{} ((z \mmember{} v) \mvee{} (z = x) \mvee{} (\mexists{}y\mmember{}v. ((cmp x y) = 0) \mwedge{} (z = (f x y)))))
Date html generated:
2017_04_17-AM-08_29_04
Last ObjectModification:
2017_02_27-PM-04_50_55
Theory : list_1
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