Nuprl Lemma : altWind_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[P:altW(A;a.B[a]) ⟶ ℙ]. ∀[h:∀w:altW(A;a.B[a])
((∀b:coW-dom(a.B[a];w). P[altW-item(w;b)]) ⇒ P[w])].
∀[w:altW(A;a.B[a])].
(altWind(h;w) ∈ P[w])
Proof
Definitions occuring in Statement :
altWind: altWind(h;w),
altW-item: altW-item(w;b),
altW: altW(A;a.B[a]),
coW-dom: coW-dom(a.B[a];w),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
altW-item: altW-item(w;b),
coPath: coPath(a.B[a];w;n),
copath: copath(a.B[a];w),
altWind: altWind(h;w),
coPath-at: coPath-at(n;w;p),
copath-at: copath-at(w;p),
band: p ∧b q,
copath-nil: (),
pi1: fst(t),
copath-length: copath-length(p),
less_than: a < b,
bdd-all: bdd-all(n;i.P[i]),
int_upper: {i...},
ge: i ≥ j ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
coW-wfdd: coW-wfdd(a.B[a];w),
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
guard: {T},
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
altW: altW(A;a.B[a]),
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
add_functionality_wrt_eq,
length-copath-extend,
set_subtype_base,
copath-at-extend,
bool_cases,
iff_weakening_equal,
subtype_rel_self,
or_wf,
le-add-cancel2,
not-equal-2,
eq_int_eq_false,
true_wf,
squash_wf,
copath-extend_wf,
copathAgree-extend,
coW-item_wf,
equal-wf-base,
not-ge-2,
top_wf,
coPath_wf,
ge_wf,
int_upper_wf,
copath-at_wf,
copath-nil_wf,
primrec1_lemma,
int_subtype_base,
nequal-le-implies,
nat_properties,
upper_subtype_nat,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
eqtt_to_assert,
uiff_transitivity,
it_wf,
assert_of_eq_int,
assert-bdd-all,
bdd-all_wf,
bnot_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
assert_wf,
equal-wf-T-base,
bool_wf,
eq_int_wf,
le_wf,
sq_stable__le,
add-subtract-cancel,
bdd_all_zero_lemma,
exists_wf,
decidable__int_equal,
decidable__not,
not_wf,
decidable__exists_int_seg,
copathAgree_wf,
lelt_wf,
le-add-cancel-alt,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
nat_wf,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
false_wf,
decidable__le,
subtract_wf,
int_seg_wf,
copath-length_wf,
equal_wf,
less_than_wf,
copath_wf,
altW-item_wf,
coW-dom_wf,
all_wf,
altW_wf
Rules used in proof :
int_eqReduceFalseSq,
int_eqReduceTrueSq,
int_eqEquality,
orFunctionality,
addLevel,
productEquality,
closedConclusion,
baseApply,
intWeakElimination,
hypothesis_subsumption,
impliesFunctionality,
equalityElimination,
promote_hyp,
allFunctionality,
multiplyEquality,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
imageElimination,
inrFormation,
inlFormation,
minusEquality,
voidEquality,
addEquality,
independent_isectElimination,
independent_functionElimination,
productElimination,
voidElimination,
lambdaFormation,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
intEquality,
natural_numberEquality,
strong_bar_Induction,
universeEquality,
rename,
setElimination,
functionEquality,
cumulativity,
instantiate,
because_Cache,
isect_memberEquality,
applyEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:altW(A;a.B[a]) {}\mrightarrow{} \mBbbP{}]. \mforall{}[h:\mforall{}w:altW(A;a.B[a])
((\mforall{}b:coW-dom(a.B[a];w). P[altW-item(w;b)])
{}\mRightarrow{} P[w])]. \mforall{}[w:altW(A;a.B[a])].
(altWind(h;w) \mmember{} P[w])
Date html generated:
2018_07_29-AM-09_22_26
Last ObjectModification:
2018_07_26-PM-09_33_46
Theory : co-recursion
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