{ 
[B:Type]. [n:
]. [m:n + 1]. [q:m + 1]. [A:n 
Type].
[lst:k:{q..n
} (A k)]. [r:m]. [a:A r].
[f:funtype(n - m;
x.(A (x + m));B)].
((apply_gen(n;x.if (x =
r) then a else lst x fi ) m f)
= (apply_gen(n;lst) m f)) }
{ Proof }
Definitions occuring in Statement :
apply_gen: apply_gen(n;lst),
eq_int: (i = j),
ifthenelse: if b then t else f fi ,
int_seg: {i..j},
nat: ,
uall: [x:A]. B[x],
apply: f a,
lambda: x.A[x],
function: x:A B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t,
funtype: funtype(n;A;T)
Definitions :
Auto: Error :Auto,
exists: 
x:A. B[x],
nat: ,
equal: s = t,
int: 
,
subtract: n - m,
D: Error :D,
CollapseTHEN: Error :CollapseTHEN,
CollapseTHENA: Error :CollapseTHENA,
int_seg: {i..j},
apply: f a,
function: x:A B[x],
member: t 
T,
isect: 
x:A. B[x],
uall: [x:A]. B[x],
universe: Type,
funtype: funtype(n;A;T),
apply_gen: apply_gen(n;lst),
lambda: x.A[x],
ifthenelse: if b then t else f fi ,
eq_int: (i = j),
axiom: Ax,
all: x:A. B[x],
subtype: S 
T,
rationals: 
,
real: 
,
set: {x:A| B[x]} ,
le: A 
B,
not: 
A,
false: False,
implies: P 
Q,
void: Void,
less_than: a < b,
add: n + m,
minus: -n,
lelt: i j < k,
and: P 
Q,
prop: 
,
subtype_rel: A r B,
uiff: uiff(P;Q),
product: x:A 
B[x],
uimplies: b supposing a,
ge: i 
j ,
natural_number: $n,
strong-subtype: strong-subtype(A;B),
grp_car: |g|,
limited-type: LimitedType,
sq_type: SQType(T),
guard: {T},
squash: 
T,
true: True,
Unfold: Error :Unfold,
sqequal: s ~ t,
btrue: tt,
MaAuto: Error :MaAuto,
primrec: primrec(n;b;c),
ycomb: Y,
top: Top,
bool: 
,
bfalse: ff,
Try: Error :Try,
Complete: Error :Complete,
fpf: a:A fp-> B[a],
pair: <a, b>,
eclass: EClass(A[eo; e]),
append: as @ bs,
qabs: |r|,
nil: [],
filter: filter(P;l),
bnot: 
b,
assert: 
b,
iff: P 
Q,
bor: p 
q,
band: p q,
bimplies: p q,
es-ble: e loc e',
es-bless: e <loc e',
es-eq-E: e = e',
eq_lnk: a = b,
eq_id: a = b,
name_eq: name_eq(x;y),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
deq-disjoint: deq-disjoint(eq;as;bs),
deq-member: deq-member(eq;x;L),
q_le: q_le(r;s),
q_less: q_less(r;s),
qeq: qeq(r;s),
eq_atom: eq_atom$n(x;y),
eq_type: eq_type(T;T'),
b-exists: (i<n.P[i])_b,
bl-exists: (xL.P[x])_b,
bl-all: (xL.P[x])_b,
dcdr-to-bool: [d],
infix_ap: x f y,
grp_blt: a < b,
set_blt: a < b,
null: null(as),
eq_atom: x =a y,
rev_implies: P Q,
or: P Q,
union: left + right,
decide: case b of inl(x) => s[x] | inr(y) => t[y]
Lemmas :
primrec_wf,
eqtt_to_assert,
bool_cases,
rev_implies_wf,
iff_wf,
assert_of_eq_int,
iff_weakening_uiff,
iff_functionality_wrt_iff,
assert_wf,
iff_imp_equal_bool,
bfalse_wf,
eq_int_eq_true,
bool_wf,
bool_subtype_base,
eq_int_wf,
ycomb-unroll,
squash_wf,
true_wf,
nat_ind_tp,
int_subtype_base,
subtype_base_sq,
int_seg_properties,
funtype_wf,
int_seg_wf,
nat_wf,
member_wf,
false_wf,
le_wf,
not_wf,
nat_properties,
ge_wf
\mforall{}[B:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n + 1]. \mforall{}[q:\mBbbN{}m + 1]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[lst:k:\{q..n\msupminus{}\} {}\mrightarrow{} (A k)]. \mforall{}[r:\mBbbN{}m].
\mforall{}[a:A r]. \mforall{}[f:funtype(n - m;\mlambda{}x.(A (x + m));B)].
((apply\_gen(n;\mlambda{}x.if (x =\msubz{} r) then a else lst x fi ) m f) = (apply\_gen(n;lst) m f))
Date html generated:
2011_08_17-PM-06_03_55
Last ObjectModification:
2011_05_31-PM-09_23_40
Home
Index