Nuprl Lemma : very-dep-fun-subtype
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type]. ∀[f:very-dep-fun(A;B;a,b.C[a;b])].
(f ∈ {L:(a:A × b:B × C[a;b]) List| vdf-eq(A;f;L)} ⟶ B ⟶ 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),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
vdf: vdf(A;B;a,b.C[a; b];n),
not: ¬A,
implies: P ⇒ Q,
false: False,
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
cons: [a / b],
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
decidable: Dec(P),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B
Lemmas referenced :
list_wf,
vdf-eq_wf,
very-dep-fun_wf,
istype-universe,
length_wf,
istype-int,
vdf_wf,
lt_int_wf,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
istype-less_than,
istype-assert,
istype-void,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
le_weakening2,
non_neg_length,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
rename,
setElimination,
thin,
sqequalHypSubstitution,
hypothesis,
setEquality,
extract_by_obid,
isectElimination,
productEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality_alt,
universeIsType,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
instantiate,
universeEquality,
isectIsType,
natural_numberEquality,
because_Cache,
unionElimination,
cumulativity,
independent_isectElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
lambdaFormation_alt,
promote_hyp,
hypothesis_subsumption,
Error :memTop,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
dependent_set_memberEquality_alt,
equalityIstype,
intEquality,
baseClosed,
sqequalBase,
dependentIntersectionEqElimination,
productIsType
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} B {}\mrightarrow{} Type]. \mforall{}[f:very-dep-fun(A;B;a,b.C[a;b])].
(f \mmember{} \{L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List| vdf-eq(A;f;L)\} {}\mrightarrow{} B {}\mrightarrow{} A)
Date html generated:
2020_05_19-PM-09_41_01
Last ObjectModification:
2020_03_05-AM-11_40_26
Theory : co-recursion-2
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