Nuprl Lemma : implies-k-1-continuous
∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ Type].
((∀[A,B:ℕk ⟶ Type]. F[A] ⊆r F[B] supposing A ⊆ B)
⇒ (∀j:ℕk. ∀Z:ℕk ⟶ Type. Continuous(X.F[λi.if (i =z j) then X else Z i fi ]))
⇒ k-1-continuous{i:l}(k;T.F[T]))
Proof
Definitions occuring in Statement :
k-1-continuous: k-1-continuous{i:l}(k;T.F[T]),
k-subtype: A ⊆ B,
type-continuous: Continuous(T.F[T]),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
k-1-continuous: k-1-continuous{i:l}(k;T.F[T]),
k-intersection: ⋂n. X[n],
all: ∀x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
lelt: i ≤ j < k,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
type-continuous: Continuous(T.F[T]),
squash: ↓T,
nequal: a ≠ b ∈ T ,
true: True,
cand: A c∧ B,
k-subtype: A ⊆ B
Lemmas referenced :
k-subtype_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
int_seg_wf,
istype-universe,
type-continuous_wf,
ifthenelse_wf,
eq_int_wf,
subtype_rel_wf,
istype-nat,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
istype-less_than,
subtract-1-ge-0,
subtype_rel_isect-2,
nat_wf,
lt_int_wf,
subtype_rel-equal,
eqtt_to_assert,
assert_of_lt_int,
int_seg_properties,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
squash_wf,
true_wf,
assert_of_eq_int,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
subtype_rel_self,
subtype_rel_isect_general,
imax_wf,
imax_nat,
decidable__equal_int,
subtype_rel_transitivity,
le_wf,
imax_unfold,
iff_weakening_equal,
le_int_wf,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalRule,
Error :functionIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
Error :dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
axiomEquality,
Error :functionIsTypeImplies,
because_Cache,
instantiate,
universeEquality,
Error :isectIsType,
Error :isectIsTypeImplies,
intWeakElimination,
isectEquality,
Error :functionExtensionality_alt,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
promote_hyp,
cumulativity,
Error :productIsType,
hyp_replacement,
imageElimination,
intEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[F:(\mBbbN{}k {}\mrightarrow{} Type) {}\mrightarrow{} Type].
((\mforall{}[A,B:\mBbbN{}k {}\mrightarrow{} Type]. F[A] \msubseteq{}r F[B] supposing A \msubseteq{} B)
{}\mRightarrow{} (\mforall{}j:\mBbbN{}k. \mforall{}Z:\mBbbN{}k {}\mrightarrow{} Type. Continuous(X.F[\mlambda{}i.if (i =\msubz{} j) then X else Z i fi ]))
{}\mRightarrow{} k-1-continuous\{i:l\}(k;T.F[T]))
Date html generated:
2019_06_20-PM-01_13_05
Last ObjectModification:
2019_01_02-PM-03_59_13
Theory : co-recursion-2
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