Nuprl Lemma : regularize_wf
∀[k:ℕ+]. ∀[f:ℕ+ ⟶ ℤ]. (regularize(k;f) ∈ ℕ+ ⟶ ℤ)
Proof
Definitions occuring in Statement :
regularize: regularize(k;f),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
regularize: regularize(k;f),
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
decidable: Dec(P),
not: ¬A,
regular-upto: regular-upto(k;n;f),
top: Top,
true: True,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
absval: |i|,
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
regular-upto_wf,
nat_plus_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
nat_plus_subtype_nat,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_wf,
assert_wf,
assert_of_bnot,
bnot_wf,
exists_wf,
nat_wf,
mu-property,
mu_wf,
uall_wf,
isect_wf,
less_than_wf,
decidable__equal_int,
int_subtype_base,
bdd_all_zero_lemma,
assert-bdd-all,
false_wf,
le_wf,
le_int_wf,
absval_wf,
subtract_wf,
decidable__lt,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
int_seg_wf,
bdd-all_wf,
all_wf,
assert_of_le_int,
int_seg_properties,
nat_properties,
nat_plus_properties,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
int_seg_cases,
int_seg_subtype,
intformand_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
decidable__le,
value-type-has-value,
int-value-type,
set-value-type,
seq-min-upper_wf,
mul_nzero,
subtype_rel_sets,
nequal_wf,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
functionExtensionality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
functionEquality,
intEquality,
addLevel,
existsFunctionality,
productEquality,
setElimination,
rename,
natural_numberEquality,
isect_memberEquality,
voidEquality,
allFunctionality,
dependent_set_memberEquality,
independent_pairFormation,
multiplyEquality,
addEquality,
minusEquality,
levelHypothesis,
allLevelFunctionality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality,
hypothesis_subsumption,
callbyvalueReduce,
divideEquality,
setEquality
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. (regularize(k;f) \mmember{} \mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{})
Date html generated:
2017_10_03-AM-09_07_30
Last ObjectModification:
2017_09_11-PM-01_40_53
Theory : reals
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