Nuprl Lemma : firstn_nth_tl_decomp
∀[T:Type]. ∀[L:T List]. ∀[i:ℕ||L||]. (L ~ firstn(i;L) @ [L[i]] @ nth_tl(1 + i;L))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
select: L[n],
length: ||as||,
nth_tl: nth_tl(n;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
list_wf,
int_seg_wf,
lelt_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
top_wf,
subtype_rel_list,
append_firstn_lastn_sq,
int_seg_properties,
false_wf,
length_wf,
int_seg_subtype_nat,
nth_tl_decomp,
list_ind_nil_lemma,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
setElimination,
rename,
productElimination,
lambdaEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
cumulativity,
sqequalAxiom,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[i:\mBbbN{}||L||]. (L \msim{} firstn(i;L) @ [L[i]] @ nth\_tl(1 + i;L))
Date html generated:
2016_05_14-PM-02_07_57
Last ObjectModification:
2016_01_15-AM-08_03_15
Theory : list_1
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