Nuprl Lemma : hdf-once-transformation1
∀[L,G,S,init:Base]. ∀[m:ℕ].
(hdf-once(fix((λmk-hdf,s. (inl (λa.cbva_seq(L[s;a]; λg.<mk-hdf S[g;s], G[g]>; m))))) init)
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(L[s;a]; λg.<case null(G[g]) of inl() => mk-hdf S[g;s] | inr() => inr Ax , G[g]>;
m)))))
init)
Proof
Definitions occuring in Statement :
hdf-once: hdf-once(X),
null: null(as),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
base: Base,
sqequal: s ~ t,
axiom: Ax,
cbva_seq: cbva_seq(L; F; m)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
hdf-once: hdf-once(X),
hdf-halted: hdf-halted(P),
hdf-halt: hdf-halt(),
bag-null: bag-null(bs),
ifthenelse: if b then t else f fi ,
hdf-ap: X(a),
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
it: ⋅,
hdf-run: hdf-run(P),
isr: isr(x),
btrue: tt,
bfalse: ff,
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
decidable: Dec(P),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
sq_type: SQType(T)
Latex:
\mforall{}[L,G,S,init:Base]. \mforall{}[m:\mBbbN{}].
(hdf-once(fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L[s;a]; \mlambda{}g.<mk-hdf S[g;s], G[g]> m))))) init)
\msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L[s;a]; \mlambda{}g.<case null(G[g])
of inl() =>
mk-hdf S[g;s]
| inr() =>
inr Ax
, G[g]
> m)))))
init)
Date html generated:
2016_05_16-AM-10_48_01
Last ObjectModification:
2016_01_17-AM-11_08_32
Theory : halting!dataflow
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