Nuprl Lemma : list_decomp_member
∀[T:Type]. ∀L:T List. ∀i:ℕ||L||. ∃as,bs:T List. (L = (as @ [L[i]] @ bs) ∈ (T List))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
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,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
int_iseg: {i...j},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
firstn_wf,
nth_tl_wf,
nth_tl_decomp_eq,
int_seg_subtype_nat,
length_wf,
false_wf,
int_seg_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
equal_wf,
list_wf,
append_wf,
cons_wf,
select_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
nil_wf,
exists_wf,
int_seg_wf,
subtype_rel_sets,
lelt_wf,
le_wf,
squash_wf,
true_wf,
iff_weakening_equal,
append_firstn_lastn
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
setElimination,
rename,
hypothesis,
addEquality,
natural_numberEquality,
because_Cache,
applyEquality,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
productElimination,
dependent_functionElimination,
unionElimination,
imageElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
universeEquality,
productEquality,
setEquality,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}i:\mBbbN{}||L||. \mexists{}as,bs:T List. (L = (as @ [L[i]] @ bs))
Date html generated:
2017_04_14-AM-09_25_33
Last ObjectModification:
2017_02_27-PM-03_59_47
Theory : list_1
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