Nuprl Lemma : mon_reduce_eq_itop
∀g:IMonoid. ∀as:|g| List. ((Π as) = (Π 0 ≤ i < ||as||. as[i]) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_reduce: mon_reduce,
select: L[n],
length: ||as||,
list: T List,
all: ∀x:A. B[x],
natural_number: $n,
equal: s = t ∈ T,
mon_itop: Π lb ≤ i < ub. E[i],
imon: IMonoid,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
imon: IMonoid,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
mon_reduce: mon_reduce,
select: L[n],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
nat_plus: ℕ+,
uiff: uiff(P;Q),
subtract: n - m,
cand: A c∧ B
Lemmas referenced :
list_wf,
grp_car_wf,
imon_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
nat_wf,
grp_id_wf,
int_seg_properties,
int_seg_wf,
reduce_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mon_itop_unroll_base,
subtype_rel_self,
iff_weakening_equal,
reduce_cons_lemma,
length_of_cons_lemma,
grp_op_wf,
mon_reduce_wf,
mon_itop_unroll_lo,
length_wf,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
false_wf,
select_wf,
cons_wf,
non_neg_length,
mon_itop_wf,
select_cons_tl,
mon_itop_shift,
add-associates,
zero-add,
add-commutes,
add-swap,
minus-minus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIsType1,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
universeEquality,
imageMemberEquality,
addEquality,
pointwiseFunctionality,
functionIsType,
minusEquality,
hyp_replacement,
productEquality
Latex:
\mforall{}g:IMonoid. \mforall{}as:|g| List. ((\mPi{} as) = (\mPi{} 0 \mleq{} i < ||as||. as[i]))
Date html generated:
2019_10_16-PM-01_02_17
Last ObjectModification:
2018_10_08-AM-11_46_15
Theory : list_2
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