Nuprl Lemma : W_subtype
∀[A1,A2:Type]. ∀[B1:A1 ⟶ Type]. ∀[B2:A2 ⟶ Type].
(W(A1;a.B1[a]) ⊆r W(A2;a.B2[a])) supposing ((∀a:A1. (B2[a] ⊆r B1[a])) and (A1 ⊆r A2))
Proof
Definitions occuring in Statement :
W: W(A;a.B[a]),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
and: P ∧ Q,
nat: ℕ,
implies: P ⇒ Q,
pcw-pp-barred: Barred(pp),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
iff: 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,
cw-step: cw-step(A;a.B[a]),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
spreadn: spread3,
less_than: a < b,
squash: ↓T,
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
ext-eq: A ≡ B,
unit: Unit,
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
pi1: fst(t),
nat_plus: ℕ+,
guard: {T},
W-rel: W-rel(A;a.B[a];w),
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
pcw-steprel: StepRel(s1;s2),
pi2: snd(t),
isl: isl(x),
pcw-step-agree: StepAgree(s;p1;w),
cand: A c∧ B,
Wsup: Wsup(a;b),
sq_type: SQType(T),
sq_stable: SqStable(P)
Lemmas referenced :
W_wf,
all_wf,
subtype_rel_wf,
W-elimination-facts,
int_seg_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
nat_wf,
minus-add,
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,
lelt_wf,
top_wf,
less_than_wf,
true_wf,
equal_wf,
add-subtract-cancel,
W-ext,
param-co-W-ext,
unit_wf2,
it_wf,
param-co-W_wf,
ext-eq_inversion,
subtype_rel_weakening,
assert_wf,
btrue_wf,
bfalse_wf,
pcw-steprel_wf,
subtype_rel_dep_function,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__int_equal,
not-equal-2,
minus-zero,
le-add-cancel2,
int_seg_subtype,
sq_stable__le,
Wsup_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
applyEquality,
functionExtensionality,
hypothesis,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_functionElimination,
productElimination,
strong_bar_Induction,
natural_numberEquality,
setElimination,
rename,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
lambdaFormation,
voidElimination,
independent_isectElimination,
addEquality,
voidEquality,
minusEquality,
intEquality,
lessCases,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
int_eqReduceTrueSq,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
dependent_pairEquality,
productEquality,
inlEquality,
unionEquality,
hyp_replacement,
applyLambdaEquality,
instantiate
Latex:
\mforall{}[A1,A2:Type]. \mforall{}[B1:A1 {}\mrightarrow{} Type]. \mforall{}[B2:A2 {}\mrightarrow{} Type].
(W(A1;a.B1[a]) \msubseteq{}r W(A2;a.B2[a])) supposing ((\mforall{}a:A1. (B2[a] \msubseteq{}r B1[a])) and (A1 \msubseteq{}r A2))
Date html generated:
2017_04_14-AM-07_43_50
Last ObjectModification:
2017_02_27-PM-03_14_54
Theory : co-recursion
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