Nuprl Lemma : vdf-eq-firstn-implies
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type]. ∀[f:very-dep-fun(A;B;a,b.C[a;b])]. ∀[L:(a:A × b:B × C[a;b]) List]. ∀[j:ℕ||L||].
(vdf-eq(A;f;firstn(j;L)) ⇒ {∀[i:ℕj]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))) ∈ A)})
Proof
Definitions occuring in Statement :
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
vdf-eq: vdf-eq(A;f;L),
firstn: firstn(n;as),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
implies: P ⇒ Q,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
int_iseg: {i...j},
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
le: A ≤ B,
so_lambda: λ2x y.t[x; y],
less_than': less_than'(a;b)
Lemmas referenced :
vdf-eq-implies2,
firstn_wf,
length_firstn,
subtype_rel_sets_simple,
lelt_wf,
length_wf,
le_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_wf,
vdf-eq_wf,
list_wf,
very-dep-fun_wf,
istype-universe,
subtype_rel_list,
top_wf,
int_seg_subtype_nat,
istype-false,
select-firstn,
firstn-firstn
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaFormation_alt,
productEquality,
applyEquality,
setElimination,
rename,
productElimination,
imageElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
inhabitedIsType,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
universeIsType,
voidElimination,
productIsType,
because_Cache,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} B {}\mrightarrow{} Type]. \mforall{}[f:very-dep-fun(A;B;a,b.C[a;b])]. \mforall{}[L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List].
\mforall{}[j:\mBbbN{}||L||].
(vdf-eq(A;f;firstn(j;L)) {}\mRightarrow{} \{\mforall{}[i:\mBbbN{}j]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))))\})
Date html generated:
2020_05_19-PM-09_40_55
Last ObjectModification:
2020_03_06-PM-03_48_10
Theory : co-recursion-2
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