Nuprl Lemma : div_is_zero
∀[n:{2...}]. ∀[i:ℤ]. i ÷ n ~ 0 supposing |i| < n
Proof
Definitions occuring in Statement :
absval: |i|,
int_upper: {i...},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
nat: ℕ,
int_upper: {i...},
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_lower: {...i},
gt: i > j,
ge: i ≥ j ,
cand: A c∧ B,
less_than: a < b,
uiff: uiff(P;Q)
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
istype-less_than,
absval_wf,
istype-int,
istype-int_upper,
div_rem_sum,
subtype_rel_sets_simple,
le_wf,
nequal_wf,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-le,
decidable__le,
rem_bounds_1,
int_upper_properties,
decidable__lt,
intformnot_wf,
intformless_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
nat_wf,
set_subtype_base,
absval-non-neg,
absval_pos,
equal_wf,
squash_wf,
true_wf,
istype-universe,
quotient-is-zero,
upper_subtype_nat,
istype-false,
subtype_rel_self,
iff_weakening_equal,
rem_bounds_2,
absval_neg,
itermMinus_wf,
int_term_value_minus_lemma,
mul_preserves_le,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomSqEquality,
hypothesisEquality,
applyEquality,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :inhabitedIsType,
sqequalRule,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
natural_numberEquality,
Error :lambdaFormation_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
Error :universeIsType,
Error :equalityIstype,
baseClosed,
sqequalBase,
because_Cache,
unionElimination,
Error :dependent_set_memberEquality_alt,
productElimination,
imageElimination,
universeEquality,
imageMemberEquality,
minusEquality,
divideEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[n:\{2...\}]. \mforall{}[i:\mBbbZ{}]. i \mdiv{} n \msim{} 0 supposing |i| < n
Date html generated:
2019_06_20-PM-01_18_50
Last ObjectModification:
2019_02_12-PM-00_26_19
Theory : int_2
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