Nuprl Lemma : combine-list-as-reduce
∀[A:Type]. ∀[f:A ⟶ A ⟶ A].
(∀[L:A List]
combine-list(x,y.f[x;y];L) = outl(reduce(λx,y. case y of inl(z) => inl f[x;z] | inr(z) => inl x;inr ⋅ ;L)) ∈ A
supposing 0 < ||L||) supposing
(Comm(A;λx,y. f[x;y]) and
Assoc(A;λx,y. f[x;y]))
Proof
Definitions occuring in Statement :
combine-list: combine-list(x,y.f[x; y];L),
length: ||as||,
reduce: reduce(f;k;as),
list: T List,
comm: Comm(T;op),
assoc: Assoc(T;op),
outl: outl(x),
less_than: a < b,
it: ⋅,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
lambda: λx.A[x],
function: x:A ⟶ B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
cons: [a / b],
top: Top,
bfalse: ff,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
isl: isl(x),
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
outl: outl(x),
le: A ≤ B,
combine-list: combine-list(x,y.f[x; y];L),
comm: Comm(T;op),
infix_ap: x f y
Lemmas referenced :
last_lemma,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
length_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
int_subtype_base,
reduce_nil_lemma,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
decidable__equal_int,
reduce_cons_lemma,
bool_wf,
bool_subtype_base,
reduce_wf,
unit_wf2,
it_wf,
btrue_wf,
list_wf,
comm_wf,
assoc_wf,
last_wf,
append_wf,
cons_wf,
nil_wf,
length-append,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
outl_wf,
assert_of_tt,
squash_wf,
true_wf,
combine-list-append,
subtype_rel_self,
iff_weakening_equal,
list_ind_cons_lemma,
list_ind_nil_lemma,
combine-list_wf,
non_neg_length,
assert_wf,
isl_wf,
list_induction,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
list_accum_nil_lemma,
list_accum_cons_lemma,
list_accum_wf,
combine-list-cons,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
hypothesis,
unionElimination,
sqequalRule,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
axiomSqEquality,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
cumulativity,
unionEquality,
inlEquality,
inrEquality,
axiomEquality,
functionEquality,
universeEquality,
hyp_replacement,
imageMemberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionExtensionality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A].
(\mforall{}[L:A List]
combine-list(x,y.f[x;y];L)
= outl(reduce(\mlambda{}x,y. case y of inl(z) => inl f[x;z] | inr(z) => inl x;inr \mcdot{} ;L))
supposing 0 < ||L||) supposing
(Comm(A;\mlambda{}x,y. f[x;y]) and
Assoc(A;\mlambda{}x,y. f[x;y]))
Date html generated:
2019_06_20-PM-01_30_30
Last ObjectModification:
2018_08_21-PM-01_55_59
Theory : list_1
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