Nuprl Lemma : mklist-add
∀[T:Type]. ∀[n,m:ℕ]. ∀[f:ℕn + m ⟶ T]. (mklist(n + m;f) = (mklist(n;f) @ mklist(m;λi.(f (n + i)))) ∈ (T List))
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
append: as @ bs,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
lelt: i ≤ j < k,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
subtract: n - m
Lemmas referenced :
int_seg_wf,
add-member-int_seg1,
nat_properties,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
lelt_wf,
list_extensionality,
mklist_wf,
le_wf,
append_wf,
mklist_length,
length-append,
nat_wf,
less_than_wf,
length_wf,
equal_wf,
squash_wf,
true_wf,
select_wf,
select_append_front,
iff_weakening_equal,
mklist_select,
select_append_back,
subtype_base_sq,
int_subtype_base,
minus-one-mul,
add-commutes,
add-swap,
add-mul-special,
zero-mul,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
cumulativity,
functionEquality,
axiomEquality,
universeEquality,
lambdaFormation,
addLevel,
levelHypothesis,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_functionElimination,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:\mBbbN{}n + m {}\mrightarrow{} T]. (mklist(n + m;f) = (mklist(n;f) @ mklist(m;\mlambda{}i.(f (n + i)))))
Date html generated:
2017_04_17-AM-07_42_02
Last ObjectModification:
2017_02_27-PM-04_15_41
Theory : list_1
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