Nuprl Lemma : firstn_decomp2
∀[T:Type]. ∀[j:ℕ]. ∀[l:T List]. (firstn(j - 1;l) @ [l[j - 1]] ~ firstn(j;l)) supposing ((j ≤ ||l||) and 0 < j)
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
select: L[n],
length: ||as||,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
subtract: n - m,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
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,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
subtract: n - m,
le: A ≤ B,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
lt_int: i <z j,
select: L[n],
cons: [a / b],
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
append: as @ bs,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bnot: ¬bb,
assert: ↑b
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,
le_wf,
length_wf,
list_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
list_decomp,
decidable__lt,
intformeq_wf,
int_formula_prop_eq_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
first0,
tl_wf,
subtype_rel_list,
top_wf,
squash_wf,
true_wf,
length_tl,
iff_weakening_equal,
lt_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
eqtt_to_assert,
assert_of_lt_int,
select-cons-tl,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
le_int_wf,
bnot_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
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,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageElimination,
productElimination,
because_Cache,
unionElimination,
universeEquality,
instantiate,
applyEquality,
imageMemberEquality,
baseClosed,
baseApply,
closedConclusion,
equalityElimination,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[j:\mBbbN{}]. \mforall{}[l:T List].
(firstn(j - 1;l) @ [l[j - 1]] \msim{} firstn(j;l)) supposing ((j \mleq{} ||l||) and 0 < j)
Date html generated:
2017_04_17-AM-07_52_08
Last ObjectModification:
2017_02_27-PM-04_25_09
Theory : list_1
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