Nuprl Lemma : vdf-eq-firstn
∀A:Type. ∀f:Top. ∀L:(a:Top × b:Top × Top) List. ∀i:ℕ||L||.
(vdf-eq(A;f;firstn(i + 1;L)) ~ x:vdf-eq(A;f;firstn(i;L)) ⋂ let tr = L[i] in
(fst(tr)) = (f firstn(i;L) (fst(snd(tr)))) ∈ A)
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-},
let: let,
top: Top,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
apply: f a,
product: x:A × B[x],
add: n + m,
natural_number: $n,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
let: let,
vdf-eq: vdf-eq(A;f;L),
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_iseg: {i...j},
int_seg: {i..j-},
and: P ∧ Q,
cand: A c∧ B,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
dep-all: dep-all(n;i.P[i]),
less_than': less_than'(a;b),
true: True,
nat: ℕ,
ge: i ≥ j ,
guard: {T},
int_upper: {i...},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
spreadn: spread3,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
int_seg_wf,
length_wf,
top_wf,
list_wf,
istype-top,
istype-universe,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
subtype_rel_sets_simple,
lelt_wf,
le_wf,
istype-less_than,
less_than_wf,
add-subtract-cancel,
length_firstn,
nat_properties,
ge_wf,
istype-int_upper,
subtract-1-ge-0,
int_upper_properties,
istype-nat,
int_seg_subtype_nat,
istype-false,
decidable__lt,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
upper_subtype_upper,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
subtract-add-cancel,
subtype_rel_list,
upper_subtype_nat,
not-lt-2,
zero-add,
add_functionality_wrt_le,
select-firstn,
firstn-firstn,
select_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
productEquality,
hypothesisEquality,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
because_Cache,
productIsType,
applyEquality,
intEquality,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
intWeakElimination,
functionIsTypeImplies,
minusEquality,
multiplyEquality,
closedConclusion,
equalityIstype
Latex:
\mforall{}A:Type. \mforall{}f:Top. \mforall{}L:(a:Top \mtimes{} b:Top \mtimes{} Top) List. \mforall{}i:\mBbbN{}||L||.
(vdf-eq(A;f;firstn(i + 1;L)) \msim{} x:vdf-eq(A;f;firstn(i;L)) \mcap{} let tr = L[i] in
(fst(tr))
= (f firstn(i;L) (fst(snd(tr)))))
Date html generated:
2020_05_19-PM-09_40_36
Last ObjectModification:
2020_03_09-PM-01_10_06
Theory : co-recursion-2
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