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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
altW: altW(A;a.B[a]),
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
squash: ↓T,
coW-wfdd: coW-wfdd(a.B[a];w),
sq_stable: SqStable(P),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
int_upper: {i...},
bdd-all: bdd-all(n;i.P[i]),
less_than: a < b,
copath-length: copath-length(p),
pi1: fst(t),
copath-nil: (),
band: p ∧b q,
copath-at: copath-at(w;p),
coPath-at: coPath-at(n;w;p),
altWind: altWind(h;w),
copath: copath(a.B[a];w),
coPath: coPath(a.B[a];w;n),
altW-item: altW-item(w;b),
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
altW_wf,
istype-universe,
coW-dom_wf,
altW-item_wf,
copath_wf,
less_than_wf,
equal_wf,
copath-length_wf,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
istype-void,
nat_wf,
minus-add,
istype-int,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
le_wf,
copathAgree_wf,
decidable__exists_int_seg,
not_wf,
int_seg_wf,
decidable__not,
decidable__int_equal,
exists_wf,
bdd_all_zero_lemma,
all_wf,
sq_stable__le,
add-subtract-cancel,
false_wf,
lelt_wf,
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,
uiff_transitivity,
eqtt_to_assert,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
int_subtype_base,
primrec1_lemma,
copath-nil_wf,
copath-at_wf,
int_upper_wf,
ge_wf,
coPath_wf,
istype-top,
subtract-1-ge-0,
equal-wf-base,
coW-item_wf,
copathAgree-extend,
copath-extend_wf,
set_subtype_base,
squash_wf,
true_wf,
eq_int_eq_false,
not-equal-2,
le-add-cancel2,
bfalse_wf,
subtype_rel_self,
iff_weakening_equal,
bool_cases,
copath-at-extend,
length-copath-extend,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaEquality_alt,
applyEquality,
instantiate,
Error :isect_memberEquality_alt,
because_Cache,
Error :functionIsType,
setElimination,
rename,
universeEquality,
strong_bar_Induction,
functionEquality,
natural_numberEquality,
intEquality,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
Error :lambdaFormation_alt,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
minusEquality,
Error :inhabitedIsType,
Error :productIsType,
Error :inlFormation_alt,
Error :inrFormation_alt,
baseClosed,
imageMemberEquality,
imageElimination,
functionExtensionality,
cumulativity,
voidEquality,
isect_memberEquality,
lambdaEquality,
lambdaFormation,
dependent_set_memberEquality,
multiplyEquality,
dependent_pairFormation,
promote_hyp,
allFunctionality,
equalityElimination,
impliesFunctionality,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
hypothesis_subsumption,
intWeakElimination,
baseApply,
closedConclusion,
Error :equalityIsType4,
int_eqReduceTrueSq,
Error :equalityIsType2,
int_eqReduceFalseSq
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:
2019_06_20-PM-01_12_35
Last ObjectModification:
2019_01_02-PM-01_35_50
Theory : co-recursion-2
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