Nuprl Lemma : param-W-ext
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P].
pW ≡ λp.(a:A[p] × (b:B[p;a] ⟶ (pW C[p;a;b])))
Proof
Definitions occuring in Statement :
param-W: pW,
ext-family: F ≡ G,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
apply: f a,
lambda: λx.A[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,
ext-family: F ≡ G,
all: ∀x:A. B[x],
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
param-W: pW,
implies: P ⇒ Q,
squash: ↓T,
pcw-path: Path,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
exists: ∃x:A. B[x],
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pcw-step-agree: StepAgree(s;p1;w),
spreadn: spread3,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
pi1: fst(t),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
int_upper: {i...},
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
subtract: n - m,
top: Top,
true: True,
eq_int: (i =z j),
cand: A c∧ B,
pcw-steprel: StepRel(s1;s2),
nequal: a ≠ b ∈ T ,
pcw-pp-barred: Barred(pp),
pcw-partial: pcw-partial(path;n),
isr: isr(x),
pW-sup: pW-sup(a;f)
Lemmas referenced :
param-co-W_wf,
param-W_wf,
param-co-W-ext,
pcw-step-agree_wf,
istype-void,
le_wf,
pcw-path_wf,
squash_wf,
exists_wf,
nat_wf,
pcw-pp-barred_wf,
pcw-partial_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
unit_wf2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
zero-add,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
istype-int,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
pcw-steprel_wf,
add-zero,
le_antisymmetry_iff,
int_subtype_base,
general_arith_equation1,
not-equal-2,
less-iff-le,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
decidable__lt,
not-lt-2,
minus-zero,
le-add-cancel-alt,
pW-sup_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
Error :lambdaEquality_alt,
Error :universeIsType,
applyEquality,
cumulativity,
functionExtensionality,
because_Cache,
Error :inhabitedIsType,
hypothesis,
Error :productIsType,
Error :functionIsType,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
Error :isect_memberEquality_alt,
universeEquality,
setElimination,
rename,
hypothesis_subsumption,
Error :dependent_pairEquality_alt,
Error :functionExtensionality_alt,
Error :dependent_set_memberEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
unionElimination,
equalityElimination,
independent_isectElimination,
Error :inlEquality_alt,
Error :unionIsType,
Error :dependent_pairFormation_alt,
promote_hyp,
instantiate,
voidElimination,
addEquality,
minusEquality,
intEquality,
hyp_replacement,
applyLambdaEquality,
Error :equalityIsType4
Latex:
\mforall{}[P:Type]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type]. \mforall{}[C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P].
pW \mequiv{} \mlambda{}p.(a:A[p] \mtimes{} (b:B[p;a] {}\mrightarrow{} (pW C[p;a;b])))
Date html generated:
2019_06_20-PM-00_35_51
Last ObjectModification:
2018_10_02-AM-10_40_38
Theory : co-recursion
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