Nuprl Lemma : es-cut-induction
∀[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])))) ∨ (∀c:Cut(X;f). SqStable(P[c])))
⇒ P[{}]
⇒ (∀c:Cut(X;f). ∀e:E(X). (P[c] ⇒ P[c+e] supposing add-cut-conditions(c;e)))
⇒ {∀c:Cut(X;f). P[c]})
Proof
Definitions occuring in Statement :
add-cut-conditions: add-cut-conditions(c;e),
es-cut-add: c+e,
es-cut: Cut(X;f),
sys-antecedent: sys-antecedent(es;Sys),
es-E-interface: E(X),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
empty-fset: {},
linorder: Linorder(T;x,y.R[x; y]),
sq_stable: SqStable(P),
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
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,
add-cut-conditions: add-cut-conditions(c;e),
es-cut: Cut(X;f),
decidable: Dec(P),
or: P ∨ Q,
es-cut-add: c+e,
uiff: uiff(P;Q),
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A
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]))))
\mvee{} (\mforall{}c:Cut(X;f). SqStable(P[c])))
{}\mRightarrow{} P[\{\}]
{}\mRightarrow{} (\mforall{}c:Cut(X;f). \mforall{}e:E(X). (P[c] {}\mRightarrow{} P[c+e] supposing add-cut-conditions(c;e)))
{}\mRightarrow{} \{\mforall{}c:Cut(X;f). P[c]\})
Date html generated:
2016_05_17-AM-07_41_12
Last ObjectModification:
2016_01_17-PM-02_59_39
Theory : event-ordering
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