Nuprl Lemma : list-decomp-nat
∀[T:Type]. ∀L:T List. ∀i:ℕ||L|| + 1. ∃K,J:T List. ((L = (K @ J) ∈ (T List)) ∧ (||K|| = i ∈ ℤ))
Proof
Definitions occuring in Statement :
length: ||as||,
append: as @ bs,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
top: Top,
so_apply: x[s],
implies: P ⇒ Q,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
int_seg_wf,
length_wf,
exists_wf,
list_wf,
equal_wf,
append_wf,
length-append,
length_of_nil_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
int_seg_subtype,
false_wf,
int_seg_cases,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
length_of_cons_lemma,
nil_wf,
cons_wf,
list_ind_nil_lemma,
equal-wf-base-T,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
decidable__lt,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
list_ind_cons_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
append_back_nil,
equal-wf-base,
equal-wf-T-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
addEquality,
cumulativity,
hypothesis,
because_Cache,
productEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
unionElimination,
instantiate,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
independent_pairFormation,
productElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
baseClosed,
dependent_set_memberEquality,
pointwiseFunctionality,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
applyEquality,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}i:\mBbbN{}||L|| + 1. \mexists{}K,J:T List. ((L = (K @ J)) \mwedge{} (||K|| = i))
Date html generated:
2017_04_17-AM-08_45_02
Last ObjectModification:
2017_02_27-PM-05_04_53
Theory : list_1
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