Nuprl Lemma : bpa-norm-equiv
∀p:{2...}. ∀x:basic-padic(p). bpa-equiv(p;x;bpa-norm(p;x))
Proof
Definitions occuring in Statement :
bpa-norm: bpa-norm(p;x),
bpa-equiv: bpa-equiv(p;x;y),
basic-padic: basic-padic(p),
int_upper: {i...},
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
basic-padic: basic-padic(p),
bpa-norm: bpa-norm(p;x),
bpa-equiv: bpa-equiv(p;x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_upper: {i...},
nat: ℕ,
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 ,
top: Top,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
squash: ↓T,
nat_plus: ℕ+,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
true: True,
p-adics: p-adics(p),
int_seg: {i..j-},
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x.t[x],
subtract: n - m,
lelt: i ≤ j < k,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
p-units: p-units(p),
crng: CRng,
rng: Rng,
p-adic-ring: ℤ(p),
ring_p: IsRing(T;plus;zero;neg;times;one),
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
inverse: Inverse(T;op;id;inv),
infix_ap: x f y,
comm: Comm(T;op)
Lemmas referenced :
basic-padic_wf,
int_upper_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
exp0_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
false_wf,
nat_properties,
nequal-le-implies,
zero-add,
le_wf,
int_subtype_base,
p-mul_wf,
squash_wf,
true_wf,
p-adics_wf,
nat_plus_wf,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel,
less_than_wf,
p-int_wf,
int_seg_wf,
exp_wf2,
p-adic-property,
nat_plus_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
nat_plus_subtype_nat,
p-mul-1,
p-shift_wf,
subtype_rel_self,
iff_weakening_equal,
p-shift-mul,
all_wf,
eqmod_wf,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
int_seg_properties,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
p-unitize_wf,
not-equal-2,
p-units_wf,
int_seg_subtype_nat,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
p-adic-ring_wf,
crng_properties,
rng_properties,
exp_add,
subtract-add-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
p-mul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
hypothesis_subsumption,
independent_pairFormation,
dependent_set_memberEquality,
intEquality,
applyEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
addEquality,
approximateComputation,
int_eqEquality,
universeEquality,
functionEquality,
minusEquality,
applyLambdaEquality,
productEquality,
setEquality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}p:\{2...\}. \mforall{}x:basic-padic(p). bpa-equiv(p;x;bpa-norm(p;x))
Date html generated:
2018_05_21-PM-03_25_20
Last ObjectModification:
2018_05_19-AM-08_23_15
Theory : rings_1
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