Nuprl Lemma : altW-item_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:altW(A;a.B[a])]. ∀[b:coW-dom(a.B[a];w)]. (altW-item(w;b) ∈ altW(A;a.B[a]))
Proof
Definitions occuring in Statement :
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],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
altW: altW(A;a.B[a]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
altW-item: altW-item(w;b),
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtract: n - m,
sq_stable: SqStable(P),
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,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
squash: ↓T,
all: ∀x:A. B[x],
coW-wfdd: coW-wfdd(a.B[a];w),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
ge: i ≥ j ,
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 ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
copath-nil: (),
pi1: fst(t),
copath-length: copath-length(p),
copath: copath(a.B[a];w)
Lemmas referenced :
altW_wf,
coW-dom_wf,
coW-wfdd_wf,
coW-item_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,
not-equal-2,
minus-minus,
subtract_wf,
copath-cons_wf,
nequal-le-implies,
nat_properties,
upper_subtype_nat,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
copath-nil_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
uiff_transitivity,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
copathAgree-nil,
le_antisymmetry_iff,
le-add-cancel2,
minus-zero,
copathAgree-cons,
general_arith_equation1,
bool_cases,
length-copath-cons,
istype-false,
istype-void,
istype-int,
istype-le,
int_subtype_base,
istype-nat,
set_subtype_base,
decidable__int_equal,
top_wf,
coPath_wf,
subtype_rel_product
Rules used in proof :
universeEquality,
functionEquality,
because_Cache,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
extract_by_obid,
dependent_set_memberEquality,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
minusEquality,
voidEquality,
independent_isectElimination,
independent_functionElimination,
productElimination,
voidElimination,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
addEquality,
intEquality,
functionExtensionality,
cumulativity,
baseClosed,
imageMemberEquality,
imageElimination,
lambdaFormation,
hypothesis_subsumption,
instantiate,
promote_hyp,
dependent_pairFormation,
equalityElimination,
impliesFunctionality,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :inhabitedIsType,
Error :universeIsType,
Error :equalityIstype,
baseApply,
closedConclusion,
sqequalBase
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:altW(A;a.B[a])]. \mforall{}[b:coW-dom(a.B[a];w)].
(altW-item(w;b) \mmember{} altW(A;a.B[a]))
Date html generated:
2019_06_20-PM-01_12_24
Last ObjectModification:
2019_01_02-PM-01_35_47
Theory : co-recursion-2
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