Nuprl Lemma : rnexp-req
∀[k:ℕ]. ∀[x:ℝ]. (x^k = if (k =z 0) then r1 else x * x^k - 1 fi )
Proof
Definitions occuring in Statement :
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
rnexp: x^k1,
eq_int: (i =z j),
subtract: n - m,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
real: ℝ,
reg-seq-nexp: reg-seq-nexp(x;k),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nequal: a ≠ b ∈ T ,
true: True,
fastexp: i^n,
efficient-exp-ext,
less_than: a < b,
squash: ↓T,
bdd-diff: bdd-diff(f;g),
int-to-real: r(n),
reg-seq-mul: reg-seq-mul(x;y),
int_nzero: ℤ-o,
absval: |i|,
respects-equality: respects-equality(S;T),
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B
Lemmas referenced :
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
req_weakening,
int-to-real_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nat_properties,
nequal-le-implies,
zero-add,
istype-le,
req-iff-bdd-diff,
rnexp_wf,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rmul_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
req_witness,
bool_wf,
real_wf,
istype-nat,
reg-seq-mul_wf,
value-type-has-value,
int_upper_wf,
set-value-type,
le_wf,
int-value-type,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_plus_wf,
absval_wf,
istype-int_upper,
canon-bnd_wf,
bdd-diff_functionality,
bdd-diff_weakening,
rmul-bdd-diff-reg-seq-mul,
set_subtype_base,
decidable__equal_int,
accelerate_wf,
fastexp_wf,
istype-less_than,
reg-seq-nexp_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
accelerate-bdd-diff,
nat_plus_properties,
exp-fastexp,
exp0_lemma,
itermMultiply_wf,
int_term_value_mul_lemma,
div-one,
squash_wf,
true_wf,
exp1,
subtype_rel_self,
iff_weakening_equal,
div-cancel,
nequal_wf,
minus-one-mul,
add-mul-special,
zero-mul,
upper_subtype_upper,
divide_wf,
exp_wf4,
subtype_rel_set,
nat_wf,
exp_wf_nat_plus,
add_nat_plus,
multiply_nat_wf,
subtract_nat_wf,
add_nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
respects-equality-sets,
regular-int-seq_wf,
respects-equality-trivial,
reg-seq-mul_functionality_wrt_bdd-diff,
bdd-diff_inversion,
bdd-diff_wf,
canonical-bound_wf,
add-is-int-iff,
false_wf,
mul_cancel_in_le,
absval_nat_plus,
less_than_wf,
absval_mul,
left_mul_subtract_distrib,
left_mul_add_distrib,
div_rem_sum2,
rem_bounds_absval,
exp_step,
mul_nzero,
exp_wf3,
add-commutes,
exp_wf2,
add-swap,
add-associates,
sq_stable__less_than,
mul-associates,
minus-add,
minus-minus,
le_functionality,
le_weakening,
int-triangle-inequality,
nat_plus_inc_int_nzero,
add-zero,
absval_sym,
sq_stable__all,
sq_stable__le,
le_witness_for_triv,
mul_preserves_le,
add_functionality_wrt_le,
exp-positive,
mul-swap,
add_functionality_wrt_eq,
absval_pos,
mul-commutes,
nat_plus_subtype_nat,
exp-positive-stronger,
mul-distributes,
one-mul,
multiply-is-int-iff,
multiply_functionality_wrt_le,
absval_unfold,
lt_int_wf,
assert_of_lt_int,
istype-top,
iff_weakening_uiff,
assert_wf,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
sqequalRule,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
voidElimination,
hypothesis_subsumption,
independent_pairFormation,
dependent_set_memberEquality_alt,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
universeIsType,
closedConclusion,
isectIsTypeImplies,
applyEquality,
callbyvalueReduce,
setEquality,
functionEquality,
multiplyEquality,
addEquality,
divideEquality,
baseClosed,
sqequalBase,
imageMemberEquality,
functionIsType,
imageElimination,
universeEquality,
minusEquality,
applyLambdaEquality,
setIsType,
pointwiseFunctionality,
baseApply,
remainderEquality,
functionIsTypeImplies,
lessCases,
axiomSqEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbR{}]. (x\^{}k = if (k =\msubz{} 0) then r1 else x * x\^{}k - 1 fi )
Date html generated:
2019_10_29-AM-09_34_33
Last ObjectModification:
2019_01_31-PM-09_59_48
Theory : reals
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