Nuprl Lemma : tuple-decomp
∀[n:ℕ]. ∀[F:Top]. (tuple(n;i.F[i]) ~ if (n =z 0) then ⋅ if (n =z 1) then F[0] else <F[0], tuple(n - 1;i.F[i + 1])> fi )
Proof
Definitions occuring in Statement :
tuple: tuple(n;i.F[i]),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
it: ⋅,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
pair: <a, b>,
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
tuple: tuple(n;i.F[i]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
eq_int: (i =z j),
subtract: n - m,
btrue: tt,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
append: as @ bs,
bool: 𝔹,
unit: Unit,
it: ⋅,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
compose: f o g,
so_lambda: λ2x.t[x],
so_apply: x[s],
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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,
less_than_wf,
top_wf,
map_nil_lemma,
list_ind_nil_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
upto_decomp1,
map_cons_lemma,
list_ind_cons_lemma,
null_nil_lemma,
upto_decomp2,
nat_wf,
le_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
null-map,
null-upto,
map-map
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
unionElimination,
instantiate,
cumulativity,
because_Cache,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
equalityElimination,
baseApply,
closedConclusion,
applyEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
impliesFunctionality,
promote_hyp
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[F:Top].
(tuple(n;i.F[i]) \msim{} if (n =\msubz{} 0) then \mcdot{}
if (n =\msubz{} 1) then F[0]
else <F[0], tuple(n - 1;i.F[i + 1])>
fi )
Date html generated:
2017_04_17-AM-09_29_16
Last ObjectModification:
2017_02_27-PM-05_29_27
Theory : tuples
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