Nuprl Lemma : pcw-pp-lemma
∀P:Type. ∀A:P ⟶ Type. ∀B:p:P ⟶ A[p] ⟶ Type. ∀C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P. ∀par:P. ∀w:pW par. ∀n:ℕ. ∀m:ℕn.
∀ss:ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]).
((∀x:ℕn. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) x ss (ss x)))
⇒ (fst(snd((ss m))) ∈ pW (fst((ss m)))))
Proof
Definitions occuring in Statement :
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
param-W: pW,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
sq_stable: SqStable(P),
spreadn: spread3,
pcw-step-agree: StepAgree(s;p1;w),
pcw-steprel: StepRel(s1;s2),
squash: ↓T,
less_than: a < b,
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
true: True,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
int_seg: {i..j-},
pi2: snd(t),
pi1: fst(t),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
subtype_rel: A ⊆r B,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
le: A ≤ B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
ext-eq: A ≡ B,
ext-family: F ≡ G,
cand: A c∧ B,
param-W: pW,
pcw-path: Path,
exists: ∃x:A. B[x]
Lemmas referenced :
nat_wf,
subtype_rel_self,
le_weakening2,
sq_stable__le,
top_wf,
decidable__lt,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
subtract_wf,
decidable__le,
int_seg_subtype,
subtype_rel_dep_function,
false_wf,
int_seg_subtype_nat,
equal_wf,
param-W_wf,
subtype_rel-equal,
pcw-step_wf,
param-W-rel_wf,
int_seg_wf,
all_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
le-add-cancel2,
zero-mul,
add-mul-special,
lelt_wf,
le-add-cancel-alt,
not-lt-2,
not-le-2,
param-W-ext,
iff_weakening_equal,
squash_wf,
true_wf,
param-co-W_wf,
param-co-W-ext,
equal_functionality_wrt_subtype_rel2,
equal-implies-member-param-W,
subtype_rel_wf,
pcw-step-agree_wf,
istype-void,
istype-le,
pcw-path_wf,
pcw-pp-barred_wf,
pcw-partial_wf,
istype-universe,
exists_wf,
le_wf,
le_reflexive
Rules used in proof :
universeEquality,
imageElimination,
baseClosed,
imageMemberEquality,
axiomSqEquality,
isect_memberFormation,
lessCases,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
addEquality,
unionElimination,
independent_pairFormation,
functionEquality,
because_Cache,
functionExtensionality,
cumulativity,
applyEquality,
productElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
multiplyEquality,
dependent_set_memberEquality,
instantiate,
productEquality,
hypothesis_subsumption,
comment,
applyLambdaEquality,
hyp_replacement,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
Error :universeIsType,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
Error :functionIsType,
levelHypothesis,
addLevel
Latex:
\mforall{}P:Type. \mforall{}A:P {}\mrightarrow{} Type. \mforall{}B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type. \mforall{}C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P. \mforall{}par:P. \mforall{}w:pW par.
\mforall{}n:\mBbbN{}. \mforall{}m:\mBbbN{}n. \mforall{}ss:\mBbbN{}n {}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]).
((\mforall{}x:\mBbbN{}n. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) x ss (ss x)))
{}\mRightarrow{} (fst(snd((ss m))) \mmember{} pW (fst((ss m)))))
Date html generated:
2019_06_20-PM-00_36_03
Last ObjectModification:
2019_04_15-PM-10_32_59
Theory : co-recursion
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