Nuprl Lemma : exp_add
∀[n,m:ℕ]. ∀[i:ℤ]. (i^n + m = (i^n * i^m) ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
exp: i^n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
nequal: a ≠ b ∈ T ,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
nat_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
exp0_lemma,
zero-add,
one-mul,
exp_wf2,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
bnot_wf,
not_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
mul-associates,
squash_wf,
true_wf,
le_wf,
iff_weakening_equal,
general_arith_equation1,
primrec-unroll,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
because_Cache,
unionElimination,
addEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
cumulativity,
applyEquality,
imageElimination,
universeEquality,
multiplyEquality,
dependent_set_memberEquality,
imageMemberEquality,
impliesFunctionality
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[i:\mBbbZ{}]. (i\^{}n + m = (i\^{}n * i\^{}m))
Date html generated:
2017_04_14-AM-09_22_19
Last ObjectModification:
2017_02_27-PM-03_58_05
Theory : int_2
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