Nuprl Lemma : qexp-minus-one
∀[n:ℕ]. (-1 ↑ n = if (n rem 2 =z 0) then 1 else -1 fi ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
rationals: ℚ,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
remainder: n rem m,
minus: -n,
natural_number: $n,
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,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
cand: A c∧ B,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t)
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,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
rationals_wf,
int-subtype-rationals,
equal_wf,
squash_wf,
true_wf,
qexp-zero,
iff_weakening_equal,
exp_unroll_q,
eq_int_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
equal-wf-T-base,
qmul_wf,
rem_addition,
le_wf,
false_wf,
rem_bounds_1,
le_weakening2,
decidable__equal_int,
subtract-add-cancel,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
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,
unionElimination,
minusEquality,
applyEquality,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality,
remainderEquality,
addLevel,
instantiate,
cumulativity,
equalityElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
baseApply,
closedConclusion,
inlFormation,
productEquality,
inrFormation
Latex:
\mforall{}[n:\mBbbN{}]. (-1 \muparrow{} n = if (n rem 2 =\msubz{} 0) then 1 else -1 fi )
Date html generated:
2018_05_22-AM-00_00_40
Last ObjectModification:
2017_07_26-PM-06_49_31
Theory : rationals
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