Nuprl Lemma : hdf-compose1-transformation1
∀[f,L:Base]. ∀[m:ℕ+].
(f o fix((λmk-hdf.(inl (λa.simple-cbva-seq(L[a];λout.<mk-hdf, out>;m)))))
~ fix((λmk-hdf.(inl (λa.simple-cbva-seq(λn.if (n =z m)
then mk_lambdas(λout.bag-map(f;out);m - 1)
else L[a] n
fi ;λout.<mk-hdf, out>;m + 1))))))
Proof
Definitions occuring in Statement :
hdf-compose1: f o X,
nat_plus: ℕ+,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
subtract: n - m,
add: n + m,
natural_number: $n,
base: Base,
sqequal: s ~ t,
bag-map: bag-map(f;bs),
simple-cbva-seq: simple-cbva-seq(L;F;m),
mk_lambdas: mk_lambdas(F;m)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
hdf-compose1: f o X,
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
all: ∀x:A. B[x],
implies: P ⇒ Q,
ifthenelse: if b then t else f fi ,
hdf-halted: hdf-halted(P),
isr: isr(x),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
bfalse: ff,
hdf-run: hdf-run(P),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
hdf-ap: X(a),
bag-map: bag-map(f;bs),
empty-bag: {},
nil: [],
it: ⋅,
btrue: tt,
hdf-halt: hdf-halt(),
eq_int: (i =z j),
subtract: n - m,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
map: map(f;as),
list_ind: list_ind,
nat: ℕ,
decidable: Dec(P),
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
sq_type: SQType(T),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k
Latex:
\mforall{}[f,L:Base]. \mforall{}[m:\mBbbN{}\msupplus{}].
(f o fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.simple-cbva-seq(L[a];\mlambda{}out.<mk-hdf, out>m)))))
\msim{} fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.simple-cbva-seq(\mlambda{}n.if (n =\msubz{} m)
then mk\_lambdas(\mlambda{}out.bag-map(f;out);m - 1)
else L[a] n
fi ;\mlambda{}out.<mk-hdf, out>m + 1))))))
Date html generated:
2016_05_16-AM-10_45_52
Last ObjectModification:
2016_01_17-AM-11_11_21
Theory : halting!dataflow
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