Nuprl Lemma : exp_unroll_q
∀[n:ℕ+]. ∀[e:ℚ]. (e ↑ n = (e ↑ n - 1 * e) ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
qrng: <ℚ+*>,
rng_car: |r|,
pi1: fst(t),
rng_times: *,
pi2: snd(t),
infix_ap: x f y,
q-rng-nexp: q-rng-nexp(r;n),
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
true: True,
rev_implies: P ⇐ Q,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
rng_nexp_unroll,
qrng_wf,
crng_subtype_rng,
rationals_wf,
nat_plus_subtype_nat,
subtract_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
qexp-eq-q-rng-nexp,
qmul_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
applyEquality,
sqequalRule,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
because_Cache,
isect_memberFormation_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[e:\mBbbQ{}]. (e \muparrow{} n = (e \muparrow{} n - 1 * e))
Date html generated:
2020_05_20-AM-09_24_39
Last ObjectModification:
2020_01_25-AM-11_48_52
Theory : rationals
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