Nuprl Lemma : vdf-eq-implies2
∀[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].
(vdf-eq(A;f;L) ⇒ {∀[i:ℕ||L||]. ((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,
implies: P ⇒ Q,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
so_lambda: λ2x y.t[x; y],
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,
prop: ℙ,
guard: {T}
Lemmas referenced :
vdf-eq-implies,
length_wf_nat,
length_wf,
istype-int,
vdf_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
vdf-eq_wf,
list_wf,
very-dep-fun_wf,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
productEquality,
applyEquality,
sqequalRule,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
isectIsType,
universeIsType,
independent_isectElimination,
dependent_functionElimination,
addEquality,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
voidElimination,
functionIsType,
inhabitedIsType,
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].
(vdf-eq(A;f;L) {}\mRightarrow{} \{\mforall{}[i:\mBbbN{}||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))))\})
Date html generated:
2020_05_19-PM-09_40_53
Last ObjectModification:
2020_03_06-PM-01_56_55
Theory : co-recursion-2
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