Nuprl Lemma : p-digit_wf
∀[p:ℕ+]. ∀[a:p-adics(p)]. ∀[n:ℕ+]. (p-digit(p;a;n) ∈ ℕp)
Proof
Definitions occuring in Statement :
p-digit: p-digit(p;a;n),
p-adics: p-adics(p),
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
p-adics: p-adics(p),
all: ∀x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
p-digit: p-digit(p;a;n),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
eqmod: a ≡ b mod m,
divides: b | a,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
top: Top,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
nat: ℕ,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
p-adics_wf,
nat_plus_wf,
int_seg_wf,
subtype_rel_self,
exp1,
exp-positive,
set_subtype_base,
less_than_wf,
subtract_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
istype-less_than,
subtract-add-cancel,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
le_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
exp-fastexp,
equal-wf-base,
nequal_wf,
subtype_rel_sets,
exp_wf3,
div-cancel2,
p-adic-bounds,
eqmod_wf,
exp_wf2,
istype-le,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
exp_wf_nat_plus,
mul_cancel_in_le,
mul_cancel_in_lt,
mul-commutes,
exp_step,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
lambdaEquality,
baseClosed,
imageMemberEquality,
independent_pairFormation,
dependent_set_memberEquality,
applyEquality,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
voidElimination,
productElimination,
promote_hyp,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
equalityElimination,
lambdaFormation,
setEquality,
functionIsType,
addEquality,
productIsType,
closedConclusion,
baseApply,
pointwiseFunctionality,
multiplyEquality,
applyLambdaEquality
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. \mforall{}[a:p-adics(p)]. \mforall{}[n:\mBbbN{}\msupplus{}]. (p-digit(p;a;n) \mmember{} \mBbbN{}p)
Date html generated:
2020_05_19-PM-10_08_18
Last ObjectModification:
2020_01_08-PM-06_00_08
Theory : rings_1
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