Nuprl Lemma : q-geometric-series
∀[a:ℚ]. ∀[n:ℕ]. (Σ0 ≤ i < n. a ↑ i = if qeq(a;1) then n else (1 - a ↑ n/1 - a) fi ∈ ℚ)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qsum: Σa ≤ j < b. E[j],
qsub: r - s,
qdiv: (r/s),
rationals: ℚ,
qeq: qeq(r;s),
nat: ℕ,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
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 ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_apply: x[s],
less_than': less_than'(a;b),
le: A ≤ B,
so_lambda: λ2x.t[x],
nat: ℕ,
prop: ℙ,
true: True,
squash: ↓T,
guard: {T},
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
qadd: r + s,
qsub: r - s,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sum_of_geometric_prog_q,
istype-nat,
rationals_wf,
qeq_wf2,
int-subtype-rationals,
equal-wf-T-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
istype-void,
uiff_transitivity,
eqtt_to_assert,
assert-qeq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
le_wf,
subtype_rel_set,
int_seg_wf,
false_wf,
int_seg_subtype_nat,
qexp_wf,
qsum_wf,
equal_wf,
squash_wf,
true_wf,
qexp-one,
iff_weakening_equal,
qmul_one_qrng,
qsum-const,
qadd_wf,
qmul_wf,
qinv_inv_q,
istype-universe,
mon_ident_q,
subtype_rel_self,
qadd_assoc,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_var_lemma,
int_formula_prop_wf,
istype-le,
qdiv_wf,
qmul-preserves-eq,
qmul-qdiv-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
because_Cache,
applyEquality,
baseClosed,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
functionIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
voidElimination,
dependent_functionElimination,
intEquality,
lambdaFormation,
lambdaEquality,
rename,
setElimination,
applyLambdaEquality,
hyp_replacement,
imageElimination,
universeEquality,
functionEquality,
imageMemberEquality,
minusEquality,
closedConclusion,
lambdaEquality_alt,
instantiate,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop
Latex:
\mforall{}[a:\mBbbQ{}]. \mforall{}[n:\mBbbN{}]. (\mSigma{}0 \mleq{} i < n. a \muparrow{} i = if qeq(a;1) then n else (1 - a \muparrow{} n/1 - a) fi )
Date html generated:
2020_05_20-AM-09_26_14
Last ObjectModification:
2020_02_26-AM-09_59_18
Theory : rationals
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