Nuprl Lemma : mon_for_of_op
∀g:IAbMonoid. ∀A:Type. ∀e,f:A ⟶ |g|. ∀as:A List.
((For{g} x ∈ as. (e[x] * f[x])) = ((For{g} x ∈ as. e[x]) * (For{g} x ∈ as. f[x])) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_for: For{g} x ∈ as. f[x],
list: T List,
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
iabmonoid: IAbMonoid,
grp_op: *,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
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,
iabmonoid: IAbMonoid,
imon: IMonoid,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
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,
mon_for_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
list_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,
mon_for_cons_lemma,
nat_wf,
istype-universe,
grp_car_wf,
iabmonoid_wf,
grp_id_wf,
equal_wf,
squash_wf,
true_wf,
mon_ident,
subtype_rel_self,
iff_weakening_equal,
grp_op_wf,
infix_ap_wf,
mon_for_wf,
mon_assoc,
abmonoid_ac_1,
abmonoid_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIsType1,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
functionIsType,
universeEquality,
imageMemberEquality
Latex:
\mforall{}g:IAbMonoid. \mforall{}A:Type. \mforall{}e,f:A {}\mrightarrow{} |g|. \mforall{}as:A List.
((For\{g\} x \mmember{} as. (e[x] * f[x])) = ((For\{g\} x \mmember{} as. e[x]) * (For\{g\} x \mmember{} as. f[x])))
Date html generated:
2019_10_16-PM-01_02_44
Last ObjectModification:
2018_10_08-PM-00_29_16
Theory : list_2
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