Nuprl Lemma : bpa-equiv-equiv
∀[p:{2...}]. EquivRel(basic-padic(p);x,y.bpa-equiv(p;x;y))
Proof
Definitions occuring in Statement :
bpa-equiv: bpa-equiv(p;x;y),
basic-padic: basic-padic(p),
equiv_rel: EquivRel(T;x,y.E[x; y]),
int_upper: {i...},
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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),
and: P ∧ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
basic-padic: basic-padic(p),
bpa-equiv: bpa-equiv(p;x;y),
nat_plus: ℕ+,
int_upper: {i...},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
squash: ↓T,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
p-adics: p-adics(p),
so_lambda: λ2x.t[x],
subtract: n - m,
int_seg: {i..j-},
so_apply: x[s]
Lemmas referenced :
p-adic-ring_wf,
crng_properties,
rng_properties,
p-mul_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
p-int_wf,
exp_wf2,
basic-padic_wf,
bpa-equiv_wf,
int_upper_wf,
squash_wf,
true_wf,
p-adics_wf,
nat_plus_wf,
p-adic-property,
nat_plus_properties,
nat_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,
le_wf,
p-mul-comm,
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_wf,
equal_wf,
subtype_rel_self,
iff_weakening_equal,
p-mul-int-cancelation-1
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
productElimination,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
imageElimination,
imageMemberEquality,
baseClosed,
equalityUniverse,
levelHypothesis,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
minusEquality,
universeEquality,
instantiate
Latex:
\mforall{}[p:\{2...\}]. EquivRel(basic-padic(p);x,y.bpa-equiv(p;x;y))
Date html generated:
2018_05_21-PM-03_24_49
Last ObjectModification:
2018_05_19-AM-08_22_45
Theory : rings_1
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