Nuprl Lemma : dep-accum_wf
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type]. ∀[f:very-dep-fun(A;B;a,b.C[a;b])]. ∀[g:a:A ⟶ b:B ⟶ C[a;b]]. ∀[bs:B List].
(dep-accum(L,b.f[L;b];a,b.g[a;b];bs) ∈ {L:(a:A × b:B × C[a;b]) List|
vdf-eq(A;f;L) ∧ (map(λx.(fst(snd(x)));L) = bs ∈ (B List))} )
Proof
Definitions occuring in Statement :
dep-accum: dep-accum(L,b.f[L; b];a,bb.g[a; bb];bs),
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
vdf-eq: vdf-eq(A;f;L),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
dep-accum: dep-accum(L,b.f[L; b];a,bb.g[a; bb];bs),
cand: A c∧ B,
vdf-eq: vdf-eq(A;f;L),
select: L[n],
nil: [],
it: ⋅,
firstn: firstn(n;as),
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
dep-all: dep-all(n;i.P[i]),
true: True,
map: map(f;as),
list_ind: list_ind,
int_iseg: {i...j},
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
cons: [a / b],
let: let,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
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,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
non_neg_length,
length_wf,
last-decomp2,
subtype_rel_list,
top_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
list_wf,
very-dep-fun_wf,
istype-universe,
eq_int_wf,
equal-wf-T-base,
bool_wf,
assert_wf,
equal-wf-base,
le_wf,
list_accum_nil_lemma,
nil_wf,
length_of_nil_lemma,
stuck-spread,
istype-base,
list_ind_nil_lemma,
vdf-eq_wf,
map_wf,
pi1_wf,
pi2_wf,
bnot_wf,
not_wf,
istype-assert,
istype-void,
list_accum_append,
firstn_wf,
list_accum_cons_lemma,
less_than_wf,
squash_wf,
true_wf,
length_firstn_eq,
subtract-is-int-iff,
false_wf,
iff_weakening_equal,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
very-dep-fun-subtype,
last_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
append_wf,
cons_wf,
implies-vdf-eq-append1,
map_append_sq,
map_cons_lemma,
map_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
productElimination,
unionElimination,
applyEquality,
instantiate,
because_Cache,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
promote_hyp,
hypothesis_subsumption,
imageElimination,
addEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
universeEquality,
baseClosed,
intEquality,
productEquality,
equalityIstype,
sqequalBase,
closedConclusion,
pointwiseFunctionality,
baseApply,
imageMemberEquality,
equalityElimination,
dependent_pairEquality_alt
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} B {}\mrightarrow{} Type]. \mforall{}[f:very-dep-fun(A;B;a,b.C[a;b])]. \mforall{}[g:a:A {}\mrightarrow{} b:B {}\mrightarrow{} C[a;b]].
\mforall{}[bs:B List].
(dep-accum(L,b.f[L;b];a,b.g[a;b];bs) \mmember{} \{L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List|
vdf-eq(A;f;L) \mwedge{} (map(\mlambda{}x.(fst(snd(x)));L) = bs)\} )
Date html generated:
2020_05_19-PM-09_51_45
Last ObjectModification:
2020_03_09-PM-05_55_48
Theory : list_1
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