Nuprl Lemma : map-upto-length
∀[T:Type]. ∀[L:T List]. ∀[f:ℕ||L|| ⟶ T]. L = map(f;upto(||L||)) ∈ (T List) supposing ∀i:ℕ||L||. ((f i) = L[i] ∈ T)
Proof
Definitions occuring in Statement :
upto: upto(n),
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
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,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
ge: i ≥ j ,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
nat: ℕ,
less_than': less_than'(a;b),
btrue: tt,
cons: [a / b],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
nat_plus: ℕ+,
compose: f o g
Lemmas referenced :
list_induction,
uall_wf,
int_seg_wf,
length_wf,
isect_wf,
all_wf,
equal_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
list_wf,
map_wf,
upto_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
map_nil_lemma,
nil_wf,
equal-wf-T-base,
length_of_cons_lemma,
cons_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
squash_wf,
true_wf,
iff_weakening_equal,
select_cons_tl,
le_wf,
less_than_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel,
upto_decomp,
add_nat_wf,
length_wf_nat,
false_wf,
nat_wf,
nat_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
map_append_sq,
map_cons_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
add_nat_plus,
nat_plus_wf,
nat_plus_properties,
map-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
natural_numberEquality,
cumulativity,
hypothesis,
because_Cache,
applyEquality,
functionExtensionality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
baseClosed,
lambdaFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
universeEquality,
dependent_set_memberEquality,
imageMemberEquality,
productEquality,
minusEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[f:\mBbbN{}||L|| {}\mrightarrow{} T].
L = map(f;upto(||L||)) supposing \mforall{}i:\mBbbN{}||L||. ((f i) = L[i])
Date html generated:
2018_05_21-PM-07_37_11
Last ObjectModification:
2017_07_26-PM-05_11_06
Theory : general
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