Nuprl Lemma : cube-mod-9
∀n:ℤ. ((n^3 ≡ 0 mod 9) ∨ (n^3 ≡ 1 mod 9) ∨ (n^3 ≡ (-1) mod 9))
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
exp: i^n,
all: ∀x:A. B[x],
or: P ∨ Q,
minus: -n,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
absval: |i|,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
modulus: a mod n,
remainder: n rem m,
nat_plus: ℕ+,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m
Lemmas referenced :
rem-eqmod,
subtype_base_sq,
int_subtype_base,
nequal_wf,
exp_functionality_wrt_eqmod,
remainder_wfa,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
exp_wf2,
eqmod_wf,
eqmod_functionality_wrt_eqmod,
eqmod_inversion,
eqmod_weakening,
rem_bounds_absval,
nat_wf,
set_subtype_base,
le_wf,
istype-false,
absval_pos,
absval_strict_ubound,
intformand_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
istype-less_than,
int_seg_properties,
decidable__lt,
modulus-equal-iff-eqmod,
decidable__equal_int,
int_seg_subtype_special,
int_seg_cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
Error :equalityIstype,
Error :inhabitedIsType,
baseClosed,
sqequalBase,
Error :universeIsType,
because_Cache,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
sqequalRule,
Error :unionIsType,
minusEquality,
Error :inlFormation_alt,
productElimination,
Error :inrFormation_alt,
independent_pairFormation,
imageElimination,
int_eqEquality,
Error :productIsType,
setElimination,
rename,
hypothesis_subsumption
Latex:
\mforall{}n:\mBbbZ{}. ((n\^{}3 \mequiv{} 0 mod 9) \mvee{} (n\^{}3 \mequiv{} 1 mod 9) \mvee{} (n\^{}3 \mequiv{} (-1) mod 9))
Date html generated:
2019_06_20-PM-02_26_09
Last ObjectModification:
2019_03_18-AM-11_43_17
Theory : num_thy_1
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