Nuprl Lemma : W-path-lemma
∀A:Type. ∀B:A ⟶ Type. ∀x:W(A;a.B[a]). ∀alpha:ℕ ⟶ cw-step(A;a.B[a]).
((∀n:ℕ. (W-rel(A;a.B[a];x) n alpha (alpha n))) ⇒ (alpha ∈ Path))
Proof
Definitions occuring in Statement :
W-rel: W-rel(A;a.B[a];w),
W: W(A;a.B[a]),
cw-step: cw-step(A;a.B[a]),
pcw-path: Path,
nat: ℕ,
it: ⋅,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
pcw-path: Path,
cw-step: cw-step(A;a.B[a]),
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],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
top: Top,
true: True,
prop: ℙ,
W-rel: W-rel(A;a.B[a];w),
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pi2: snd(t),
isl: isl(x),
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
W: W(A;a.B[a])
Lemmas referenced :
nat_wf,
W-rel_wf,
istype-universe,
subtype_rel_function,
cw-step_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-void,
subtype_rel_self,
W_wf,
pcw-step_wf,
unit_wf2,
it_wf,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-int,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
add-subtract-cancel,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
assert_wf,
btrue_wf,
bfalse_wf,
pcw-steprel_wf,
le_weakening2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
less_than_wf,
not-lt-2,
pcw-step-agree_wf,
subtype_rel-equal,
param-W_wf,
param-co-W_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
introduction,
extract_by_obid,
applyEquality,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
independent_pairFormation,
Error :inhabitedIsType,
universeEquality,
Error :dependent_set_memberEquality_alt,
functionExtensionality,
dependent_functionElimination,
addEquality,
unionElimination,
voidElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
Error :isect_memberEquality_alt,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :productIsType,
Error :equalityIsType1,
Error :dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type. \mforall{}x:W(A;a.B[a]). \mforall{}alpha:\mBbbN{} {}\mrightarrow{} cw-step(A;a.B[a]).
((\mforall{}n:\mBbbN{}. (W-rel(A;a.B[a];x) n alpha (alpha n))) {}\mRightarrow{} (alpha \mmember{} Path))
Date html generated:
2019_06_20-PM-00_36_24
Last ObjectModification:
2018_10_06-AM-11_20_32
Theory : co-recursion
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