Nuprl Lemma : mon_reduce_append
∀g:IMonoid. ∀as,bs:|g| List. ((Π as @ bs) = ((Π as) * (Π bs)) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_reduce: mon_reduce,
append: as @ bs,
list: T List,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
imon: IMonoid,
grp_op: *,
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,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
mon_reduce: mon_reduce,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
infix_ap: x f y
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,
list_ind_nil_lemma,
reduce_nil_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mon_reduce_wf,
mon_ident,
subtype_rel_self,
iff_weakening_equal,
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,
list_ind_cons_lemma,
reduce_cons_lemma,
grp_op_wf,
append_wf,
mon_assoc,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
inhabitedIsType,
hypothesisEquality,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
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,
axiomEquality,
functionIsTypeImplies,
because_Cache,
unionElimination,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
imageMemberEquality,
baseClosed,
instantiate,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
dependent_set_memberEquality_alt,
applyLambdaEquality,
equalityIsType4,
baseApply,
closedConclusion,
intEquality
Latex:
\mforall{}g:IMonoid. \mforall{}as,bs:|g| List. ((\mPi{} as @ bs) = ((\mPi{} as) * (\mPi{} bs)))
Date html generated:
2019_10_16-PM-01_02_13
Last ObjectModification:
2018_10_08-AM-11_48_46
Theory : list_2
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