Nuprl Lemma : pa-sep-irrefl
∀[p:{2...}]. ∀x:basic-padic(p). (¬pa-sep(p;x;x))
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],
not: ¬A,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
basic-padic: basic-padic(p),
pa-sep: pa-sep(p;x;y),
or: P ∨ Q,
nat: ℕ,
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
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 :
nat_properties,
int_upper_properties,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
pa-sep_wf,
basic-padic_wf,
int_upper_wf,
p-sep-irrefl,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
productElimination,
unionElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
because_Cache,
dependent_set_memberEquality,
independent_pairFormation,
applyEquality
Latex:
\mforall{}[p:\{2...\}]. \mforall{}x:basic-padic(p). (\mneg{}pa-sep(p;x;x))
Date html generated:
2018_05_21-PM-03_28_36
Last ObjectModification:
2018_05_19-AM-08_24_36
Theory : rings_1
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