Nuprl Lemma : exp-zero
∀[n:ℕ+]. (0^n = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
exp: i^n,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
subtract: n - m,
nat: ℕ,
decidable: Dec(P)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
exp1,
iff_weakening_equal,
nat_plus_properties,
equal-wf-base,
int_subtype_base,
nat_plus_wf,
primrec-wf-nat-plus,
equal-wf-T-base,
exp_wf2,
nat_plus_subtype_nat,
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
satisfiable-full-omega-tt,
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,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
itermAdd_wf,
int_term_value_add_lemma,
add-associates,
zero-mul,
primrec_wf,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
le_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
intEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
lambdaFormation,
rename,
setElimination,
baseApply,
closedConclusion,
addEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
independent_pairFormation,
computeAll,
promote_hyp,
instantiate,
cumulativity,
minusEquality,
dependent_set_memberEquality,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. (0\^{}n = 0)
Date html generated:
2017_04_17-AM-09_44_52
Last ObjectModification:
2017_02_27-PM-05_38_58
Theory : num_thy_1
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