Nuprl Lemma : search_succ
∀[k:ℕ]. ∀[P:ℕk + 1 ⟶ 𝔹].
(search(k + 1;P) = if P 0 then 1 if 0 <z search(k;λi.(P (i + 1))) then search(k;λi.(P (i + 1))) + 1 else 0 fi ∈ ℤ)
Proof
Definitions occuring in Statement :
search: search(k;P),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
iff: P ⇐⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
subtract: n - m,
guard: {T},
sq_type: SQType(T),
assert: ↑b,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_wf,
bool_wf,
nat_wf,
false_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
search_property,
decidable__le,
le_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
decidable__equal_int,
search_wf,
subtract_wf,
itermSubtract_wf,
intformeq_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
add-member-int_seg2,
add-subtract-cancel,
lt_int_wf,
less_than_wf,
le_int_wf,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
subtype_base_sq,
int_subtype_base,
add-commutes,
add-associates,
add-swap,
int_seg_properties,
assert_elim,
bool_subtype_base,
int_seg_subtype,
all_wf,
isect_wf,
not_assert_elim,
btrue_neq_bfalse,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
equalityElimination,
independent_functionElimination,
functionExtensionality,
imageElimination,
instantiate,
cumulativity,
minusEquality,
applyLambdaEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[P:\mBbbN{}k + 1 {}\mrightarrow{} \mBbbB{}].
(search(k + 1;P)
= if P 0 then 1
if 0 <z search(k;\mlambda{}i.(P (i + 1))) then search(k;\mlambda{}i.(P (i + 1))) + 1
else 0
fi )
Date html generated:
2017_04_17-AM-09_52_59
Last ObjectModification:
2017_02_27-PM-05_48_25
Theory : num_thy_1
Home
Index