Nuprl Lemma : coW-is-W
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. w ∈ W(A;a.B[a]) supposing coW-wfdd(a.B[a];w)
Proof
Definitions occuring in Statement :
coW-wfdd: coW-wfdd(a.B[a];w),
coW: coW(A;a.B[a]),
W: W(A;a.B[a]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
exists: ∃x:A. B[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
pcw-path: Path,
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],
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
coW: coW(A;a.B[a]),
param-W: pW,
W: W(A;a.B[a]),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
top: Top,
subtract: n - m,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
coW-wfdd: coW-wfdd(a.B[a];w),
copath-length: copath-length(p),
pi1: fst(t),
pcw-path-coPath: pcw-path-coPath(n;p),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
copath-nil: (),
guard: {T},
pcw-pp-barred: Barred(pp),
pcw-partial: pcw-partial(path;n),
let: let,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
spreadn: spread3,
isr: isr(x),
assert: ↑b,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
copath: copath(a.B[a];w),
copath-extend: copath-extend(q;t),
cand: A c∧ B
Lemmas referenced :
coW_wf,
coW-wfdd_wf,
pcw-partial_wf,
pcw-pp-barred_wf,
nat_wf,
exists_wf,
squash_wf,
all_wf,
pcw-path_wf,
le_wf,
false_wf,
it_wf,
unit_wf2,
pcw-step-agree_wf,
copathAgree_wf,
copath-length_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,
decidable__le,
equal_wf,
pcw-path-copathAgree,
pcw-path-coPath_wf,
not_wf,
set_wf,
less_than_wf,
primrec-wf2,
decidable__int_equal,
subtract_wf,
less-iff-le,
minus-minus,
le_weakening2,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
bnot_wf,
pcw-step_wf,
copath_length_nil_lemma,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal-wf-T-base,
copath_wf,
not-equal-2,
le_antisymmetry_iff,
member-less_than
Rules used in proof :
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
functionEquality,
independent_pairFormation,
natural_numberEquality,
rename,
setElimination,
because_Cache,
functionExtensionality,
universeEquality,
cumulativity,
lambdaEquality,
applyEquality,
isectElimination,
extract_by_obid,
instantiate,
baseClosed,
thin,
imageMemberEquality,
imageElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
hypothesisEquality,
dependent_set_memberEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
minusEquality,
voidEquality,
voidElimination,
unionElimination,
addEquality,
intEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
promote_hyp,
dependent_pairFormation,
baseApply,
closedConclusion,
equalityElimination,
impliesFunctionality,
independent_pairEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. w \mmember{} W(A;a.B[a]) supposing coW-wfdd(a.B[a];w)
Date html generated:
2019_06_20-PM-00_57_40
Last ObjectModification:
2019_01_02-PM-01_34_20
Theory : co-recursion-2
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