Nuprl Lemma : search_wf
∀[k:ℕ]. ∀[P:ℕk ⟶ 𝔹]. (search(k;P) ∈ ℕk + 1)
Proof
Definitions occuring in Statement :
search: search(k;P),
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
search: search(k;P),
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,
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,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtract: n - m
Lemmas referenced :
primrec_wf,
int_seg_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,
lt_int_wf,
bool_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_of_lt_int,
less_than_wf,
eqtt_to_assert,
add-member-int_seg2,
int_seg_properties,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-subtract-cancel,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
applyEquality,
functionExtensionality,
axiomEquality,
functionEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[P:\mBbbN{}k {}\mrightarrow{} \mBbbB{}]. (search(k;P) \mmember{} \mBbbN{}k + 1)
Date html generated:
2017_04_17-AM-09_52_19
Last ObjectModification:
2017_02_27-PM-05_46_42
Theory : num_thy_1
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