Nuprl Lemma : firstn_last_mklist
∀[T:Type]. ∀[F:ℕ ⟶ T]. ∀n:ℕ+. (mklist(n;F) = (firstn(n - 1;mklist(n;F)) @ [last(mklist(n;F))]) ∈ (T List))
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
firstn: firstn(n;as),
last: last(L),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
nat: ℕ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
ge: i ≥ j ,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
mklist_length,
nat_plus_subtype_nat,
nat_plus_properties,
decidable__equal_int,
subtract_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__le,
intformand_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
le_wf,
subtype_base_sq,
nat_wf,
set_subtype_base,
int_subtype_base,
nat_plus_wf,
non_null_iff_length,
mklist_wf,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_list,
top_wf,
nat_properties,
decidable__lt,
length_wf,
subtract-is-int-iff,
list_wf,
equal_wf,
squash_wf,
true_wf,
firstn_last,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
applyEquality,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
because_Cache,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
instantiate,
cumulativity,
independent_functionElimination,
axiomEquality,
functionEquality,
universeEquality,
productElimination,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[F:\mBbbN{} {}\mrightarrow{} T]. \mforall{}n:\mBbbN{}\msupplus{}. (mklist(n;F) = (firstn(n - 1;mklist(n;F)) @ [last(mklist(n;F))]))
Date html generated:
2017_04_17-AM-07_59_40
Last ObjectModification:
2017_02_27-PM-04_30_28
Theory : list_1
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