Nuprl Lemma : p-shift_wf
∀[p:ℕ+]. ∀[a:p-adics(p)]. ∀[k:ℕ+]. p-shift(p;a;k) ∈ p-adics(p) supposing (a k) = 0 ∈ ℤ
Proof
Definitions occuring in Statement :
p-shift: p-shift(p;a;k),
p-adics: p-adics(p),
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
p-adics: p-adics(p),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat_plus: ℕ+,
p-shift: p-shift(p;a;k),
all: ∀x:A. B[x],
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
sq_type: SQType(T),
guard: {T},
eqmod: a ≡ b mod m,
divides: b | a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
lelt: i ≤ j < k,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
equal-wf-T-base,
int_seg_wf,
exp_wf2,
nat_plus_subtype_nat,
nat_plus_wf,
p-adics_wf,
p-adic-property,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
subtype_base_sq,
int_subtype_base,
set_subtype_base,
lelt_wf,
decidable__equal_int,
decidable__lt,
less_than_wf,
multiply-is-int-iff,
subtract-is-int-iff,
intformeq_wf,
itermMultiply_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
false_wf,
exp_wf_nat_plus,
int_seg_properties,
div-cancel2,
exp_wf3,
subtype_rel_sets,
nequal_wf,
equal-wf-base,
equal_wf,
exp_add,
mul_cancel_in_le,
mul_cancel_in_lt,
subtract_wf,
all_wf,
eqmod_wf,
not-lt-2,
less-iff-le,
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,
exp_step,
mul_nzero,
squash_wf,
true_wf,
add_functionality_wrt_eq,
subtype_rel_self,
iff_weakening_equal,
div-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
intEquality,
applyEquality,
setElimination,
rename,
hypothesisEquality,
lambdaEquality,
natural_numberEquality,
baseClosed,
isect_memberEquality,
because_Cache,
functionExtensionality,
dependent_functionElimination,
dependent_set_memberEquality,
addEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
instantiate,
cumulativity,
productElimination,
universeEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
applyLambdaEquality,
setEquality,
lambdaFormation,
imageElimination,
multiplyEquality,
minusEquality,
divideEquality,
imageMemberEquality
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. \mforall{}[a:p-adics(p)]. \mforall{}[k:\mBbbN{}\msupplus{}]. p-shift(p;a;k) \mmember{} p-adics(p) supposing (a k) = 0
Date html generated:
2018_05_21-PM-03_21_46
Last ObjectModification:
2018_05_19-AM-08_20_29
Theory : rings_1
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