Nuprl Lemma : cw-pp-lemma
∀A:Type. ∀B:A ⟶ Type. ∀w:W(A;a.B[a]). ∀n:ℕ. ∀m:ℕn. ∀ss:ℕn ⟶ cw-step(A;a.B[a]).
((∀x:ℕn. (W-rel(A;a.B[a];w) x ss (ss x))) ⇒ (fst(snd((ss m))) ∈ W(A;a.B[a])))
Proof
Definitions occuring in Statement :
W-rel: W-rel(A;a.B[a];w),
W: W(A;a.B[a]),
cw-step: cw-step(A;a.B[a]),
int_seg: {i..j-},
nat: ℕ,
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
W: W(A;a.B[a]),
so_lambda: λ2x.t[x],
so_apply: x[s],
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],
subtype_rel: A ⊆r B,
cw-step: cw-step(A;a.B[a]),
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
guard: {T},
squash: ↓T,
W-rel: W-rel(A;a.B[a];w),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pi1: fst(t)
Lemmas referenced :
pcw-pp-lemma,
unit_wf2,
it_wf,
cw-step_wf,
all_wf,
int_seg_wf,
W-rel_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
int_seg_subtype,
sq_stable__le,
le_weakening2,
subtype_rel_self,
nat_wf,
W_wf,
subtype_rel-equal,
param-W_wf,
equal_wf,
equal-unit
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
isectElimination,
because_Cache,
independent_functionElimination,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type. \mforall{}w:W(A;a.B[a]). \mforall{}n:\mBbbN{}. \mforall{}m:\mBbbN{}n. \mforall{}ss:\mBbbN{}n {}\mrightarrow{} cw-step(A;a.B[a]).
((\mforall{}x:\mBbbN{}n. (W-rel(A;a.B[a];w) x ss (ss x))) {}\mRightarrow{} (fst(snd((ss m))) \mmember{} W(A;a.B[a])))
Date html generated:
2017_04_14-AM-07_43_30
Last ObjectModification:
2017_02_27-PM-03_14_11
Theory : co-recursion
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