Nuprl Lemma : equal-padics
∀[p:ℤ]. ∀[x,y:padic(p)]. uiff(x = y ∈ padic(p);x = y ∈ basic-padic(p))
Proof
Definitions occuring in Statement :
padic: padic(p),
basic-padic: basic-padic(p),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
padic: padic(p),
basic-padic: basic-padic(p),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi2: snd(t),
pi1: fst(t),
nat: ℕ,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
not: ¬A,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
eq_int: (i =z j),
int_upper: {i...},
p-units: p-units(p),
p-adics: p-adics(p),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
int_seg: {i..j-}
Lemmas referenced :
equal_wf,
padic_wf,
basic-padic_wf,
padic_subtype_basic-padic,
pi2_wf,
nat_wf,
p-adics_wf,
pi1_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__equal_int,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
false_wf,
ifthenelse_wf,
p-units_wf,
upper_subtype_nat,
nequal-le-implies,
zero-add,
not_wf,
equal-wf-T-base,
less_than_wf,
int_seg_wf,
exp_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
because_Cache,
intEquality,
productElimination,
applyLambdaEquality,
lambdaEquality,
promote_hyp,
instantiate,
cumulativity,
independent_isectElimination,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
unionElimination,
lambdaFormation,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
voidElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality,
dependent_pairEquality,
dependent_set_memberEquality,
universeEquality,
hypothesis_subsumption,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[p:\mBbbZ{}]. \mforall{}[x,y:padic(p)]. uiff(x = y;x = y)
Date html generated:
2018_05_21-PM-03_26_12
Last ObjectModification:
2018_05_19-AM-08_23_10
Theory : rings_1
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