Nuprl Lemma : p-adic-non-decreasing
∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ+. ∀i:ℕ+n + 1. ((a i) ≤ (a n))
Proof
Definitions occuring in Statement :
p-adics: p-adics(p),
int_seg: {i..j-},
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
p-adics: p-adics(p),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
subtract: n - m
Lemmas referenced :
int_seg_wf,
p-adics_wf,
nat_plus_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
le_witness_for_triv,
subtract-1-ge-0,
istype-nat,
add-zero,
int_seg_properties,
nat_plus_properties,
decidable__le,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
intformnot_wf,
int_formula_prop_not_lemma,
subtract_wf,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
le_functionality,
le_weakening,
p-adic-bounds,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtract-is-int-iff,
false_wf,
istype-le,
minus-one-mul,
add-associates,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
inhabitedIsType,
intWeakElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
applyEquality,
dependent_set_memberEquality_alt,
unionElimination,
because_Cache,
instantiate,
cumulativity,
intEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}a:p-adics(p). \mforall{}n:\mBbbN{}\msupplus{}. \mforall{}i:\mBbbN{}\msupplus{}n + 1. ((a i) \mleq{} (a n))
Date html generated:
2019_10_15-AM-10_34_32
Last ObjectModification:
2018_12_08-PM-00_23_49
Theory : rings_1
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