Nuprl Lemma : absval_div_nat
∀[n:ℕ+]. ∀[i:ℤ]. (|i| ÷ n ~ |i ÷ n|)
Proof
Definitions occuring in Statement :
absval: |i|,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
prop: ℙ,
squash: ↓T,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_lower: {...i},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
decidable__le,
nat_plus_wf,
div_bounds_1,
le_wf,
equal_wf,
squash_wf,
true_wf,
absval_pos,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
divide_wf,
subtype_rel_self,
iff_weakening_equal,
div_bounds_2,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
absval_unfold2,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
minus_functionality_wrt_eq,
div_2_to_1,
minus_minus_cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
sqequalRule,
isect_memberEquality,
because_Cache,
dependent_set_memberEquality,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
divideEquality,
setElimination,
rename,
lambdaFormation,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
baseClosed,
imageMemberEquality,
productElimination,
minusEquality,
equalityElimination,
lessCases,
promote_hyp
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[i:\mBbbZ{}]. (|i| \mdiv{} n \msim{} |i \mdiv{} n|)
Date html generated:
2018_05_21-PM-00_30_20
Last ObjectModification:
2018_05_15-PM-05_47_32
Theory : int_2
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