Nuprl Lemma : vdf-eq-firstn-general
∀A:Type. ∀f:Top. ∀L:(a:Top × b:Top × Top) List. ∀i:ℕ||L|| + 1.
(vdf-eq(A;f;firstn(i;L)) ~ if (i =z 0)
then True
else x:vdf-eq(A;f;firstn(i - 1;L)) ⋂ let tr = L[i - 1] in
(fst(tr)) = (f firstn(i - 1;L) (fst(snd(tr)))) ∈ A
fi )
Proof
Definitions occuring in Statement :
vdf-eq: vdf-eq(A;f;L),
firstn: firstn(n;as),
select: L[n],
length: ||as||,
list: T List,
dep-isect: x:A ⋂ B[x],
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
let: let,
top: Top,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
true: True,
apply: f a,
product: x:A × B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
vdf-eq: vdf-eq(A;f;L),
select: L[n],
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
dep-all: dep-all(n;i.P[i]),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ
Lemmas referenced :
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
first0,
subtype_rel_list,
top_wf,
length_of_nil_lemma,
stuck-spread,
istype-base,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
vdf-eq-firstn,
subtract_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
add-is-int-iff,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
false_wf,
istype-le,
istype-less_than,
subtract-add-cancel,
int_seg_wf,
list_wf,
istype-top,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
sqequalRule,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
productEquality,
lambdaEquality_alt,
Error :memTop,
productIsType,
baseClosed,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
imageElimination,
independent_pairFormation,
approximateComputation,
int_eqEquality,
universeIsType,
pointwiseFunctionality,
baseApply,
closedConclusion,
addEquality,
universeEquality
Latex:
\mforall{}A:Type. \mforall{}f:Top. \mforall{}L:(a:Top \mtimes{} b:Top \mtimes{} Top) List. \mforall{}i:\mBbbN{}||L|| + 1.
(vdf-eq(A;f;firstn(i;L)) \msim{} if (i =\msubz{} 0)
then True
else x:vdf-eq(A;f;firstn(i - 1;L)) \mcap{} let tr = L[i - 1] in
(fst(tr)) = (f firstn(i - 1;L) (fst(snd(tr))))
fi )
Date html generated:
2020_05_19-PM-09_40_39
Last ObjectModification:
2020_03_09-PM-01_11_37
Theory : co-recursion-2
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