Nuprl Lemma : combine-list-cons
∀[A:Type]. ∀[f:A ⟶ A ⟶ A].
∀[L:A List]. ∀[a:A]. (combine-list(x,y.f[x;y];[a / L]) = f[a;combine-list(x,y.f[x;y];L)] ∈ A) supposing 0 < ||L||
supposing Assoc(A;λx,y. f[x;y])
Proof
Definitions occuring in Statement :
combine-list: combine-list(x,y.f[x; y];L),
length: ||as||,
cons: [a / b],
list: T List,
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,
all: ∀x:A. B[x],
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
cons: [a / b],
top: Top,
combine-list: combine-list(x,y.f[x; y];L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assoc: Assoc(T;op),
infix_ap: x f y
Lemmas referenced :
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
list_accum_cons_lemma,
less_than_wf,
length_wf,
list_wf,
assoc_wf,
list_induction,
all_wf,
equal_wf,
list_accum_wf,
list_accum_nil_lemma,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
axiomEquality,
because_Cache,
natural_numberEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
independent_functionElimination,
lambdaFormation,
rename,
imageMemberEquality,
baseClosed,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} A {}\mrightarrow{} A].
\mforall{}[L:A List]
\mforall{}[a:A]. (combine-list(x,y.f[x;y];[a / L]) = f[a;combine-list(x,y.f[x;y];L)])
supposing 0 < ||L||
supposing Assoc(A;\mlambda{}x,y. f[x;y])
Date html generated:
2017_04_14-AM-08_41_36
Last ObjectModification:
2017_02_27-PM-03_31_29
Theory : list_0
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