Nuprl Lemma : Wmul_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[zero,succ:A ⟶ 𝔹]. ∀[w2,w1:W(A;a.B[a])].
(w1 * w2) ∈ W(A;a.B[a]) supposing ∀a:A. ((↑(succ a)) ⇒ (Unit ⊆r B[a]))
Proof
Definitions occuring in Statement :
Wmul: (w1 * w2),
W: W(A;a.B[a]),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
Wmul: (w1 * w2),
Wsup: Wsup(a;b),
all: ∀x:A. B[x],
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
nat: ℕ,
pcw-pp-barred: Barred(pp),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: 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),
ext-eq: A ≡ B,
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+,
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,
sq_stable: SqStable(P)
Lemmas referenced :
W_wf,
bool_wf,
all_wf,
assert_wf,
subtype_rel_wf,
unit_wf2,
eqtt_to_assert,
Wadd_wf,
it_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
Wsup_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,
add-subtract-cancel,
W-ext,
param-co-W-ext,
param-co-W_wf,
ext-eq_inversion,
subtype_rel_weakening,
btrue_wf,
bfalse_wf,
pcw-steprel_wf,
subtype_rel_dep_function,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
extract_by_obid,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
strong_bar_Induction,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
addEquality,
voidEquality,
minusEquality,
intEquality,
lessCases,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
int_eqReduceTrueSq,
hypothesis_subsumption,
dependent_pairEquality,
productEquality,
inlEquality,
unionEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[zero,succ:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[w2,w1:W(A;a.B[a])].
(w1 * w2) \mmember{} W(A;a.B[a]) supposing \mforall{}a:A. ((\muparrow{}(succ a)) {}\mRightarrow{} (Unit \msubseteq{}r B[a]))
Date html generated:
2017_04_14-AM-07_44_51
Last ObjectModification:
2017_02_27-PM-03_16_13
Theory : co-recursion
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