Nuprl Lemma : array_subtype
∀[Val1,Val2:Type]. ∀[m,n:ℕ].
(array{i:l}(Val1;m) ⊆r array{i:l}(Val2;n)) supposing ((n ≤ m) and ((Val1 ⊆r Val2) ∧ (Val2 ⊆r Val1)))
Proof
Definitions occuring in Statement :
array: array{i:l}(Val;n),
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
array: array{i:l}(Val;n),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nequal: a ≠ b ∈ T ,
less_than: a < b
Lemmas referenced :
subtype_rel_product,
int_seg_wf,
uall_wf,
equal_wf,
eq_int_wf,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
int_seg_properties,
subtype_rel_self,
le_wf,
subtype_rel_wf,
nat_wf,
equal_subtype,
equal_functionality_wrt_subtype_rel2,
subtype_rel_isect_general,
subtype_rel_isect-2,
eqtt_to_assert,
assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
instantiate,
extract_by_obid,
isectElimination,
universeEquality,
lambdaEquality,
productEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
cumulativity,
hypothesisEquality,
functionExtensionality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
axiomEquality,
isectEquality
Latex:
\mforall{}[Val1,Val2:Type]. \mforall{}[m,n:\mBbbN{}].
(array\{i:l\}(Val1;m) \msubseteq{}r array\{i:l\}(Val2;n)) supposing
((n \mleq{} m) and
((Val1 \msubseteq{}r Val2) \mwedge{} (Val2 \msubseteq{}r Val1)))
Date html generated:
2017_10_01-AM-08_43_46
Last ObjectModification:
2017_07_26-PM-04_29_56
Theory : monads
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