Nuprl Lemma : es-cut-induction-ordered
∀[Info:Type]
∀es:EO+(Info). ∀X:EClass(Top). ∀f:sys-antecedent(es;X).
∀[P:Cut(X;f) ⟶ ℙ]
((∃R:E(X) ⟶ E(X) ⟶ ℙ. (Linorder(E(X);x,y.R[x;y]) ∧ (∀x,y:E(X). Dec(R[x;y]))))
⇒ P[{}]
⇒ (∀c:Cut(X;f). ∀e:E(X).
(P[c]
⇒ (P[c+e]) supposing (prior(X)(e) ∈ c supposing ↑e ∈b prior(X) and f e ∈ c supposing ¬((f e) = e ∈ E(X)))))
⇒ (∀c:Cut(X;f). P[c]))
Proof
Definitions occuring in Statement :
es-cut-add: c+e,
es-cut: Cut(X;f),
es-prior-interface: prior(X),
sys-antecedent: sys-antecedent(es;Sys),
es-E-interface: E(X),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-eq: es-eq(es),
empty-fset: {},
fset-member: a ∈ s,
linorder: Linorder(T;x,y.R[x; y]),
assert: ↑b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
sys-antecedent: sys-antecedent(es;Sys),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
top: Top,
es-E-interface: E(X),
and: P ∧ Q,
guard: {T},
int_seg: {i..j-},
exists: ∃x:A. B[x],
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
es-cut: Cut(X;f),
ge: i ≥ j ,
es-cut-add: c+e,
fset-add: fset-add(eq;x;s),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
sq_type: SQType(T),
uiff: uiff(P;Q),
true: True,
rev_uimplies: rev_uimplies(P;Q),
fset-closed: (s closed under fs),
es-interface-pred: X-pred,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
fset-member: a ∈ s,
assert: ↑b
Latex:
\mforall{}[Info:Type]
\mforall{}es:EO+(Info). \mforall{}X:EClass(Top). \mforall{}f:sys-antecedent(es;X).
\mforall{}[P:Cut(X;f) {}\mrightarrow{} \mBbbP{}]
((\mexists{}R:E(X) {}\mrightarrow{} E(X) {}\mrightarrow{} \mBbbP{}. (Linorder(E(X);x,y.R[x;y]) \mwedge{} (\mforall{}x,y:E(X). Dec(R[x;y]))))
{}\mRightarrow{} P[\{\}]
{}\mRightarrow{} (\mforall{}c:Cut(X;f). \mforall{}e:E(X).
(P[c]
{}\mRightarrow{} (P[c+e]) supposing
(prior(X)(e) \mmember{} c supposing \muparrow{}e \mmember{}\msubb{} prior(X) and
f e \mmember{} c supposing \mneg{}((f e) = e))))
{}\mRightarrow{} (\mforall{}c:Cut(X;f). P[c]))
Date html generated:
2016_05_17-AM-07_39_54
Last ObjectModification:
2016_01_17-PM-03_00_26
Theory : event-ordering
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