Nuprl Lemma : lg-all-append
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀g1,g2:LabeledGraph(T). (∀x∈lg-append(g1;g2).P[x] ⇐⇒ ∀x∈g1.P[x] ∧ ∀x∈g2.P[x])
Proof
Definitions occuring in Statement :
lg-all: ∀x∈G.P[x],
lg-append: lg-append(g1;g2),
labeled-graph: LabeledGraph(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
lg-all: ∀x∈G.P[x],
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
labeled-graph: LabeledGraph(T),
member: t ∈ T,
lg-size: lg-size(g),
lg-append: lg-append(g1;g2),
top: Top,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
le: A ≤ B,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
\mforall{}g1,g2:LabeledGraph(T). (\mforall{}x\mmember{}lg-append(g1;g2).P[x] \mLeftarrow{}{}\mRightarrow{} \mforall{}x\mmember{}g1.P[x] \mwedge{} \mforall{}x\mmember{}g2.P[x])
Date html generated:
2016_05_17-AM-10_18_20
Last ObjectModification:
2016_01_18-AM-00_21_35
Theory : process-model
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