Nuprl Lemma : very-dep-fun-subtype-domain
∀[A,B1,B2:Type]. ∀[C:A ⟶ B2 ⟶ Type].
very-dep-fun(A;B2;a,b.C[a;b]) ⊆r very-dep-fun(A;B1;a,b.C[a;b]) supposing B1 ⊆r B2
Proof
Definitions occuring in Statement :
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
vdf: vdf(A;B;a,b.C[a; b];n),
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
istype: istype(T),
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
subtype_rel_wf,
vdf_wf,
istype-int,
isect_subtype,
decidable__le,
istype-le,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_formula_prop_wf,
ge_wf,
istype-less_than,
subtract-1-ge-0,
subtype_rel_dep_function,
list_wf,
equal-wf-base,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
subtype_rel_sets,
subtype_rel_set,
subtype_rel_list,
subtype_rel_product,
vdf-wf+,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
lt_int_wf,
dep-isect-subtype2,
length_wf,
subtract-add-cancel,
istype-universe,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
axiomEquality,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
because_Cache,
intEquality,
lambdaEquality_alt,
applyEquality,
lambdaFormation_alt,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_set_memberEquality_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setElimination,
rename,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
functionIsTypeImplies,
setEquality,
productEquality,
baseClosed,
functionEquality,
setIsType,
sqequalBase,
productElimination,
productIsType,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
functionExtensionality,
universeEquality
Latex:
\mforall{}[A,B1,B2:Type]. \mforall{}[C:A {}\mrightarrow{} B2 {}\mrightarrow{} Type].
very-dep-fun(A;B2;a,b.C[a;b]) \msubseteq{}r very-dep-fun(A;B1;a,b.C[a;b]) supposing B1 \msubseteq{}r B2
Date html generated:
2020_05_19-PM-09_40_25
Last ObjectModification:
2020_03_10-PM-05_33_48
Theory : co-recursion-2
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