Nuprl Lemma : binary-search_wf
∀a:ℤ. ∀b:{a + 1...}. ∀f:{a..b + 1-} ⟶ 𝔹.
binary-search(f;a;b) ∈ {x:{a..b-}| (¬↑(f x)) ∧ (↑(f (x + 1)))}
supposing ↓∃x:{a..b-}. ((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))
Proof
Definitions occuring in Statement :
binary-search: binary-search(f;a;b),
int_upper: {i...},
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
int_upper: {i...},
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
so_apply: x[s],
binary-search: binary-search(f;a;b),
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
cand: A c∧ B,
uiff: uiff(P;Q),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
subtype_rel: A ⊆r B,
less_than: a < b,
nat_plus: ℕ+,
less_than': less_than'(a;b),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
squash_wf,
exists_wf,
int_seg_wf,
all_wf,
not_wf,
assert_wf,
int_seg_properties,
int_upper_properties,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
lelt_wf,
bool_wf,
le_wf,
subtract_wf,
int_upper_wf,
subtract-1-ge-0,
decidable__equal_int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__le,
nat_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
add-member-int_seg2,
add-subtract-cancel,
subtype_base_sq,
int_subtype_base,
nequal_wf,
rem_bounds_1,
itermMultiply_wf,
int_term_value_mul_lemma,
mul_cancel_in_lt,
div_rem_sum2,
value-type-has-value,
int-value-type,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
subtype_rel_function,
int_seg_subtype,
le_reflexive,
add-is-int-iff,
subtype_rel_self,
subtype_rel_sets,
subtype_rel_set,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
zero-add,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productEquality,
addEquality,
applyEquality,
functionExtensionality,
Error :dependent_set_memberEquality_alt,
productElimination,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isect_memberFormation_alt,
Error :functionIsType,
unionElimination,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
imageElimination,
Error :productIsType,
instantiate,
cumulativity,
intEquality,
Error :equalityIsType4,
multiplyEquality,
remainderEquality,
divideEquality,
imageMemberEquality,
baseClosed,
Error :equalityIsType1,
callbyvalueReduce,
equalityElimination,
promote_hyp,
baseApply,
closedConclusion,
minusEquality,
functionEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
dependent_set_memberEquality,
lambdaFormation
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}b:\{a + 1...\}. \mforall{}f:\{a..b + 1\msupminus{}\} {}\mrightarrow{} \mBbbB{}.
binary-search(f;a;b) \mmember{} \{x:\{a..b\msupminus{}\}| (\mneg{}\muparrow{}(f x)) \mwedge{} (\muparrow{}(f (x + 1)))\}
supposing \mdownarrow{}\mexists{}x:\{a..b\msupminus{}\}. ((\mforall{}y:\{a..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))) \mwedge{} (\mforall{}z:\{x + 1..b + 1\msupminus{}\}. (\muparrow{}(f z))))
Date html generated:
2019_06_20-PM-01_16_17
Last ObjectModification:
2018_10_03-PM-11_02_41
Theory : int_2
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