Nuprl Lemma : reduce-as-combine-list
∀[A:Type]. ∀[f:A ⟶ A ⟶ A].
(∀[L:A List]. ∀[z:A]. (reduce(f;z;L) = combine-list(x,y.f[x;y];[z / L]) ∈ A)) 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),
reduce: reduce(f;k;as),
cons: [a / b],
list: T List,
comm: Comm(T;op),
assoc: Assoc(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s1;s2],
combine-list: combine-list(x,y.f[x; y];L),
comm: Comm(T;op),
assoc: Assoc(T;op),
infix_ap: x f y,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
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,
decidable: Dec(P),
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
list-cases,
reduce_nil_lemma,
list_accum_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
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,
le_wf,
reduce_cons_lemma,
list_accum_cons_lemma,
equal_wf,
squash_wf,
true_wf,
reduce_wf,
subtype_rel_self,
iff_weakening_equal,
istype-nat,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
isect_memberFormation_alt,
lambdaFormation_alt,
isectElimination,
hypothesisEquality,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
because_Cache,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIstype,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
imageMemberEquality,
isectIsType,
functionIsType,
universeEquality,
hyp_replacement
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A].
(\mforall{}[L:A List]. \mforall{}[z:A]. (reduce(f;z;L) = combine-list(x,y.f[x;y];[z / L]))) supposing
(Comm(A;\mlambda{}x,y. f[x;y]) and
Assoc(A;\mlambda{}x,y. f[x;y]))
Date html generated:
2020_05_19-PM-09_45_07
Last ObjectModification:
2019_12_31-PM-00_13_15
Theory : list_1
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