Nuprl Lemma : W-wfdd
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:W(A;a.B[a])]. coW-wfdd(a.B[a];w)
Proof
Definitions occuring in Statement :
coW-wfdd: coW-wfdd(a.B[a];w),
W: W(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
isr: isr(x),
pcw-pp-barred: Barred(pp),
pcw-partial: pcw-partial(path;n),
eq_int: (i =z j),
coPath-at: coPath-at(n;w;p),
copath-at: copath-at(w;p),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
coW-item: coW-item(w;b),
cand: A c∧ B,
pcw-step-agree: StepAgree(s;p1;w),
spreadn: spread3,
pcw-steprel: StepRel(s1;s2),
ext-eq: A ≡ B,
pi1: fst(t),
coW-dom: coW-dom(a.B[a];w),
pi2: snd(t),
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
exposed-bfalse: exposed-bfalse,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pcw-path: Path,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
prop: ℙ,
squash: ↓T,
all: ∀x:A. B[x],
coW-wfdd: coW-wfdd(a.B[a];w),
coW: coW(A;a.B[a]),
param-W: pW,
W: W(A;a.B[a]),
implies: P ⇒ Q,
sq_stable: SqStable(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
not_wf,
iff_wf,
assert_of_band,
iff_weakening_uiff,
assert_wf,
iff_transitivity,
member_wf,
btrue_wf,
band_wf,
iff_imp_equal_bool,
subtract-add-cancel,
minus-minus,
less-iff-le,
subtract_wf,
int_subtype_base,
equal-wf-T-base,
pcw-steprel_wf,
subtype_rel-equal,
coW-item_wf,
iff_weakening_equal,
subtype_rel_self,
subtype_rel_weakening,
coW-ext,
unit_wf2,
copathAgree-last,
le-add-cancel2,
not-equal-2,
pi1_wf,
true_wf,
squash_wf,
coW-dom_wf,
coW_wf,
le_antisymmetry_iff,
not-lt-2,
decidable__lt,
copath-last_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
copath-at_wf,
it_wf,
decidable__int_equal,
W_wf,
copathAgree_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
copath-length_wf,
equal_wf,
all_wf,
copath_wf,
nat_wf,
set_wf,
W-subtype-coW,
sq_stable__coW-wfdd
Rules used in proof :
impliesFunctionality,
addLevel,
unionEquality,
inrEquality,
hypothesis_subsumption,
applyLambdaEquality,
hyp_replacement,
productEquality,
inlEquality,
promote_hyp,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
dependent_pairEquality,
universeEquality,
instantiate,
minusEquality,
voidEquality,
isect_memberEquality,
independent_isectElimination,
productElimination,
voidElimination,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
addEquality,
dependent_set_memberEquality,
intEquality,
because_Cache,
functionExtensionality,
cumulativity,
functionEquality,
baseClosed,
imageMemberEquality,
imageElimination,
lambdaFormation,
rename,
setElimination,
independent_functionElimination,
lambdaEquality,
sqequalRule,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:W(A;a.B[a])]. coW-wfdd(a.B[a];w)
Date html generated:
2018_07_25-PM-01_42_13
Last ObjectModification:
2018_07_24-PM-00_02_46
Theory : co-recursion
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