Nuprl Lemma : listfun_mklist
∀A,B:Type. ∀G:(A List) ⟶ (B List).
((∀L:A List. (||L|| = ||G L|| ∈ ℕ))
⇒ (∀L1,L2:A List. ∀i:ℕ.
((i ≤ ||L1||) ⇒ (i ≤ ||L2||) ⇒ (∀j:ℕi. (L1[j] = L2[j] ∈ A)) ⇒ (∀j:ℕi. (G L1[j] = G L2[j] ∈ B))))
⇒ (∀f:ℕ ⟶ A. ∀x:ℕ. ((G mklist(x;f)) = mklist(x;λn.G mklist(n + 1;f)[n]) ∈ (B List))))
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
less_than: a < b
Lemmas referenced :
mklist_wf,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
mklist_length,
nat_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,
all_wf,
list_wf,
length_wf,
int_seg_wf,
equal_wf,
select_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
squash_wf,
true_wf,
iff_weakening_equal,
length_wf_nat,
list_extensionality,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
itermAdd_wf,
int_term_value_add_lemma,
mklist_select,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
instantiate,
independent_isectElimination,
intEquality,
lambdaEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyLambdaEquality,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
functionEquality,
productElimination,
functionExtensionality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
addEquality,
comment
Latex:
\mforall{}A,B:Type. \mforall{}G:(A List) {}\mrightarrow{} (B List).
((\mforall{}L:A List. (||L|| = ||G L||))
{}\mRightarrow{} (\mforall{}L1,L2:A List. \mforall{}i:\mBbbN{}.
((i \mleq{} ||L1||) {}\mRightarrow{} (i \mleq{} ||L2||) {}\mRightarrow{} (\mforall{}j:\mBbbN{}i. (L1[j] = L2[j])) {}\mRightarrow{} (\mforall{}j:\mBbbN{}i. (G L1[j] = G L2[j]))))
{}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} A. \mforall{}x:\mBbbN{}. ((G mklist(x;f)) = mklist(x;\mlambda{}n.G mklist(n + 1;f)[n]))))
Date html generated:
2017_04_17-AM-07_42_31
Last ObjectModification:
2017_02_27-PM-04_15_51
Theory : list_1
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