Nuprl Lemma : mon_for_eq_itop
∀g:IMonoid. ∀A:Type. ∀as:A List. ∀f:A ⟶ |g|. ((For{g} x ∈ as. f[x]) = (Π 0 ≤ i < ||as||. f[as[i]]) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_for: For{g} x ∈ as. f[x],
select: L[n],
length: ||as||,
list: T List,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
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,
mon_for: For{g} x ∈ as. f[x],
for: For{T,op,id} x ∈ as. f[x],
mon_reduce: mon_reduce,
squash: ↓T,
prop: ℙ,
tlambda: λx:T. b[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ
Lemmas referenced :
grp_car_wf,
list_wf,
imon_wf,
equal_wf,
squash_wf,
true_wf,
mon_reduce_eq_itop,
map_wf,
mon_itop_wf,
length_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
iff_weakening_equal,
map-length,
length_wf_nat,
and_wf,
nat_wf,
less_than_wf,
lelt_wf,
map_length,
map_select
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
functionEquality,
cumulativity,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
universeEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
functionExtensionality,
natural_numberEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_set_memberEquality,
addLevel,
hyp_replacement,
applyLambdaEquality,
levelHypothesis
Latex:
\mforall{}g:IMonoid. \mforall{}A:Type. \mforall{}as:A List. \mforall{}f:A {}\mrightarrow{} |g|.
((For\{g\} x \mmember{} as. f[x]) = (\mPi{} 0 \mleq{} i < ||as||. f[as[i]]))
Date html generated:
2017_10_01-AM-09_55_12
Last ObjectModification:
2017_03_03-PM-00_50_20
Theory : list_2
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