Nuprl Lemma : length-interpolate-list
∀[T:Type]
∀[f:T ⟶ T ⟶ T]. ∀[L:T List]. (||interpolate-list(x,y.f[x;y];L)|| = if null(L) then 0 else (2 * ||L||) - 1 fi ∈ ℤ)
supposing value-type(T)
Proof
Definitions occuring in Statement :
interpolate-list: interpolate-list(x,y.f[x; y];L),
length: ||as||,
null: null(as),
list: T List,
value-type: value-type(T),
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
interpolate-list: interpolate-list(x,y.f[x; y];L),
nil: [],
it: ⋅,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
bfalse: ff,
subtract: n - m,
has-value: (a)↓,
uiff: uiff(P;Q)
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,
istype-less_than,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
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,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
null_cons_lemma,
length_of_cons_lemma,
value-type-has-value,
list_wf,
list-value-type,
interpolate-list_wf,
cons_wf,
subtract-is-int-iff,
multiply-is-int-iff,
add-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
false_wf,
istype-nat,
value-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIstype,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
callbyvalueReduce,
pointwiseFunctionality,
Error :isectIsTypeImplies,
Error :functionIsType,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}[f:T {}\mrightarrow{} T {}\mrightarrow{} T]. \mforall{}[L:T List].
(||interpolate-list(x,y.f[x;y];L)|| = if null(L) then 0 else (2 * ||L||) - 1 fi )
supposing value-type(T)
Date html generated:
2019_06_20-PM-01_49_48
Last ObjectModification:
2019_03_06-AM-10_29_47
Theory : list_1
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