Nuprl Lemma : cnv-taba-property
∀[A,B:Type].
∀xs:A List. ∀ys:B List. ((||xs|| ≤ ||ys||) ⇒ (cnv-taba(xs;ys) = zip(xs;rev(firstn(||xs||;ys))) ∈ ((A × B) List)))
Proof
Definitions occuring in Statement :
cnv-taba: cnv-taba(xs;ys),
zip: zip(as;bs),
firstn: firstn(n;as),
length: ||as||,
reverse: rev(as),
list: T List,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cnv-taba: cnv-taba(xs;ys),
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
firstn: firstn(n;as),
int_seg: {i..j-},
lelt: i ≤ j < k,
append: as @ bs,
true: True,
int_iseg: {i...j},
cand: A c∧ B,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
tl: tl(l),
pi2: snd(t),
subtract: n - m,
reverse: rev(as),
rev-append: rev(as) + bs,
list_accum: list_accum,
pi1: fst(t)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
list-cases,
length_of_nil_lemma,
list_ind_nil_lemma,
zip_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
length_of_cons_lemma,
list_ind_cons_lemma,
nat_wf,
list_wf,
length_wf,
nil_wf,
add-is-int-iff,
false_wf,
le_int_wf,
uiff_transitivity,
equal-wf-T-base,
bool_wf,
assert_wf,
eqtt_to_assert,
assert_of_le_int,
non_neg_length,
lt_int_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
bnot_of_lt_int,
add-subtract-cancel,
append_firstn_lastn_sq,
subtype_rel_list,
top_wf,
istype-universe,
decidable__lt,
nth_tl_wf,
firstn_wf,
reverse_nil_lemma,
reverse-cons,
reduce_tl_cons_lemma,
zip_cons_nil_lemma,
reduce_tl_nil_lemma,
first0,
zip_cons_cons_lemma,
cons_wf,
length_firstn,
length_wf_nat,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
firstn-append,
nth_tl-append,
length_nth_tl,
reverse-append,
zip_wf,
append_wf,
reverse_wf,
pi1_wf_top,
equal_wf,
squash_wf,
true_wf,
length_of_null_list,
subtype_rel_self,
iff_weakening_equal,
spread_wf,
subtype_rel-equal,
list_ind_wf,
tl_wf,
subtype_rel_product,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectElimination,
universeIsType,
universeEquality,
lambdaFormation_alt,
extract_by_obid,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIsType1,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
cumulativity,
productEquality,
independent_pairEquality,
lambdaFormation,
pointwiseFunctionality,
addEquality,
equalityElimination,
hyp_replacement,
productIsType,
imageMemberEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}xs:A List. \mforall{}ys:B List. ((||xs|| \mleq{} ||ys||) {}\mRightarrow{} (cnv-taba(xs;ys) = zip(xs;rev(firstn(||xs||;ys)))))
Date html generated:
2019_10_15-AM-11_34_58
Last ObjectModification:
2018_10_10-PM-01_59_05
Theory : general
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