Nuprl Lemma : combine-list-permutation
∀[A:Type]. ∀[f:A ⟶ A ⟶ A].
(∀[as,bs:A List].
(combine-list(x,y.f[x;y];as) = combine-list(x,y.f[x;y];bs) ∈ A) supposing
(permutation(A;as;bs) and
0 < ||as||)) 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),
combine-list: combine-list(x,y.f[x; y];L),
length: ||as||,
list: T List,
comm: Comm(T;op),
assoc: Assoc(T;op),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
lambda: λx.A[x],
function: x:A ⟶ B[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,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
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,
not: ¬A,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
permutation_wf,
less_than_wf,
length_wf,
comm_wf,
assoc_wf,
permutation-invariant,
equal_wf,
combine-list_wf,
list_wf,
append_wf,
cons_wf,
nil_wf,
length_of_cons_lemma,
non_neg_length,
decidable__lt,
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,
permutation-length,
intformeq_wf,
int_formula_prop_eq_lemma,
permutation_transitivity,
list_ind_cons_lemma,
list_ind_nil_lemma,
permutation-rotate,
length-append,
length_of_nil_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
false_wf,
squash_wf,
true_wf,
combine-list-append,
iff_weakening_equal,
permutation_weakening,
append_functionality_wrt_permutation,
combine-list-flip,
trivial-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
independent_isectElimination,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
voidElimination,
voidEquality,
addEquality,
unionElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
setElimination,
rename,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
imageElimination,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A].
(\mforall{}[as,bs:A List].
(combine-list(x,y.f[x;y];as) = combine-list(x,y.f[x;y];bs)) supposing
(permutation(A;as;bs) and
0 < ||as||)) 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_22_58
Last ObjectModification:
2017_02_27-PM-04_44_41
Theory : list_1
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