Nuprl Lemma : search_property
∀k:ℕ. ∀P:ℕk ⟶ 𝔹.
((∃i:ℕk. (↑(P i)) ⇐⇒ 0 < search(k;P))
∧ (↑(P (search(k;P) - 1))) ∧ (∀j:ℕk. ¬↑(P j) supposing j < search(k;P) - 1) supposing 0 < search(k;P))
Proof
Definitions occuring in Statement :
search: search(k;P),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
guard: {T},
ge: i ≥ j ,
le: A ≤ B,
search: search(k;P),
subtract: n - m,
cand: A c∧ B,
less_than': less_than'(a;b),
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
true: True
Lemmas referenced :
all_wf,
int_seg_wf,
subtract_wf,
bool_wf,
iff_wf,
exists_wf,
assert_wf,
less_than_wf,
search_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
decidable__lt,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
isect_wf,
not_wf,
set_wf,
primrec-wf2,
nat_properties,
nat_wf,
primrec0_lemma,
assert_witness,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
decidable__exists_int_seg,
decidable__assert,
subtype_base_sq,
int_subtype_base,
primrec-unroll,
eq_int_wf,
equal-wf-base,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
uiff_transitivity,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
subtract-add-cancel,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
natural_numberEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
because_Cache,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
isectEquality,
productElimination,
addEquality,
imageElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
isect_memberFormation,
independent_pairEquality,
instantiate,
promote_hyp,
cumulativity,
baseApply,
closedConclusion,
baseClosed,
equalityElimination,
impliesFunctionality
Latex:
\mforall{}k:\mBbbN{}. \mforall{}P:\mBbbN{}k {}\mrightarrow{} \mBbbB{}.
((\mexists{}i:\mBbbN{}k. (\muparrow{}(P i)) \mLeftarrow{}{}\mRightarrow{} 0 < search(k;P))
\mwedge{} (\muparrow{}(P (search(k;P) - 1))) \mwedge{} (\mforall{}j:\mBbbN{}k. \mneg{}\muparrow{}(P j) supposing j < search(k;P) - 1)
supposing 0 < search(k;P))
Date html generated:
2017_04_17-AM-09_52_40
Last ObjectModification:
2017_02_27-PM-05_48_12
Theory : num_thy_1
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