Nuprl Lemma : reduce-permutation
∀[A:Type]. ∀[f:A ⟶ A ⟶ A]. ∀[e:A].
(∀[as,bs:A List]. reduce(f;e;as) = reduce(f;e;bs) ∈ A supposing permutation(A;as;bs)) supposing
(Comm(A;λx,y. f[x;y]) and
Assoc(A;λx,y. f[x;y]))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
reduce: reduce(f;k;as),
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s1;s2],
true: True,
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
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]
Lemmas referenced :
reduce-as-combine-list,
permutation_wf,
list_wf,
comm_wf,
assoc_wf,
combine-list-permutation,
cons_wf,
length_of_cons_lemma,
non_neg_length,
decidable__lt,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
nil_wf,
permutation_weakening,
append_functionality_wrt_permutation,
list_ind_cons_lemma,
list_ind_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
cumulativity,
sqequalRule,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
natural_numberEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
addEquality,
unionElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A]. \mforall{}[e:A].
(\mforall{}[as,bs:A List]. reduce(f;e;as) = reduce(f;e;bs) supposing permutation(A;as;bs)) supposing
(Comm(A;\mlambda{}x,y. f[x;y]) and
Assoc(A;\mlambda{}x,y. f[x;y]))
Date html generated:
2017_04_17-AM-08_23_25
Last ObjectModification:
2017_02_27-PM-04_45_15
Theory : list_1
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