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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
implies: P ⇒ Q,
W: W(A;a.B[a]),
param-W: pW,
coW: coW(A;a.B[a]),
coW-wfdd: coW-wfdd(a.B[a];w),
all: ∀x:A. B[x],
squash: ↓T,
nat: ℕ,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
pcw-path: Path,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
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,
pi2: snd(t),
coW-dom: coW-dom(a.B[a];w),
pi1: fst(t),
ext-eq: A ≡ B,
pcw-steprel: StepRel(s1;s2),
spreadn: spread3,
pcw-step-agree: StepAgree(s;p1;w),
cand: A c∧ B,
respects-equality: respects-equality(S;T),
coW-item: coW-item(w;b),
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],
copath-length: copath-length(p),
copath: copath(a.B[a];w),
eq_int: (i =z j),
coPath-at: coPath-at(n;w;p),
copath-at: copath-at(w;p),
pcw-pp-barred: Barred(pp),
pcw-partial: pcw-partial(path;n),
isr: isr(x)
Lemmas referenced :
sq_stable__coW-wfdd,
W-subtype-coW,
istype-nat,
copath_wf,
istype-int,
copath-length_wf,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
istype-le,
copathAgree_wf,
W_wf,
istype-universe,
decidable__int_equal,
nat_wf,
it_wf,
copath-at_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
le_antisymmetry_iff,
not-lt-2,
decidable__lt,
copath-last_wf,
coW_wf,
true_wf,
squash_wf,
coW-dom_wf,
le-add-cancel2,
not-equal-2,
copathAgree-last,
unit_wf2,
subtype_rel_weakening,
coW-ext,
subtype_rel_universe1,
equal_wf,
subtype_rel_self,
iff_weakening_equal,
coW-subtype1,
subtype-respects-equality,
coW-item_wf,
subtype_rel-equal,
pcw-steprel_wf,
minus-minus,
less-iff-le,
subtract_wf,
subtract-add-cancel,
assert_of_band,
iff_weakening_uiff,
iff_transitivity,
member_wf,
iff_functionality_wrt_iff,
btrue_wf,
assert_wf,
band_wf,
iff_imp_equal_bool,
not_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
Error :lambdaEquality_alt,
independent_functionElimination,
setElimination,
rename,
Error :lambdaFormation_alt,
imageElimination,
imageMemberEquality,
baseClosed,
Error :setIsType,
Error :functionIsType,
Error :universeIsType,
Error :equalityIstype,
intEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
Error :dependent_set_memberEquality_alt,
addEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
Error :isect_memberEquality_alt,
minusEquality,
baseApply,
equalityTransitivity,
instantiate,
cumulativity,
universeEquality,
Error :dependent_pairEquality_alt,
functionExtensionality,
promote_hyp,
Error :equalityIsType1,
Error :dependent_pairFormation_alt,
equalityElimination,
Error :inhabitedIsType,
Error :inlEquality_alt,
hyp_replacement,
applyLambdaEquality,
Error :inrEquality_alt,
functionEquality,
productEquality,
hypothesis_subsumption,
Error :unionIsType,
Error :productIsType,
Error :equalityIsType4
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:
2019_06_20-PM-00_57_29
Last ObjectModification:
2019_01_02-PM-01_34_18
Theory : co-recursion-2
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