Nuprl Lemma : pa-sep-or
∀[p:{2...}]. ∀x,y:basic-padic(p). (pa-sep(p;x;y) ⇒ (∀z:basic-padic(p). (pa-sep(p;z;x) ∨ pa-sep(p;z;y))))
Proof
Definitions occuring in Statement :
pa-sep: pa-sep(p;x;y),
basic-padic: basic-padic(p),
int_upper: {i...},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
basic-padic: basic-padic(p),
pa-sep: pa-sep(p;x;y),
or: P ∨ Q,
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
guard: {T},
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
nat_plus: ℕ+,
int_upper: {i...},
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
decidable__equal_int,
decidable__int_equal,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
not_wf,
equal_wf,
p-sep_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
or_wf,
basic-padic_wf,
pa-sep_wf,
int_upper_wf,
p-sep-or
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
unionElimination,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
inlFormation,
equalityTransitivity,
equalitySymmetry,
inrFormation,
isectElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality,
applyEquality,
because_Cache
Latex:
\mforall{}[p:\{2...\}]
\mforall{}x,y:basic-padic(p). (pa-sep(p;x;y) {}\mRightarrow{} (\mforall{}z:basic-padic(p). (pa-sep(p;z;x) \mvee{} pa-sep(p;z;y))))
Date html generated:
2018_05_21-PM-03_28_31
Last ObjectModification:
2018_05_19-AM-08_24_41
Theory : rings_1
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