Nuprl Lemma : lg-all-remove
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀g:LabeledGraph(T). ∀x:ℕlg-size(g). (∀x∈g.P[x] ⇐⇒ ∀x∈lg-remove(g;x).P[x] ∧ P[lg-label(g;x)])
Proof
Definitions occuring in Statement :
lg-all: ∀x∈G.P[x],
lg-label: lg-label(g;x),
lg-remove: lg-remove(g;n),
lg-size: lg-size(g),
labeled-graph: LabeledGraph(T),
int_seg: {i..j-},
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],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
lg-all: ∀x∈G.P[x],
lg-label2: lg-label2(g;x),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T},
int_seg: {i..j-},
squash: ↓T,
true: True,
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
ge: i ≥ j
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
\mforall{}g:LabeledGraph(T). \mforall{}x:\mBbbN{}lg-size(g). (\mforall{}x\mmember{}g.P[x] \mLeftarrow{}{}\mRightarrow{} \mforall{}x\mmember{}lg-remove(g;x).P[x] \mwedge{} P[lg-label(g;x)])
Date html generated:
2016_05_17-AM-10_18_28
Last ObjectModification:
2016_01_18-AM-00_21_54
Theory : process-model
Home
Index