Nuprl Lemma : lg-all-map
∀[A,T:Type]. ∀f:A ⟶ T. ∀[P:T ⟶ ℙ]. ∀g:LabeledGraph(A). (∀x∈lg-map(f;g).P[x] ⇐⇒ ∀x∈g.P[f x])
Proof
Definitions occuring in Statement :
lg-all: ∀x∈G.P[x],
lg-map: lg-map(f;g),
labeled-graph: LabeledGraph(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
lg-all: ∀x∈G.P[x],
lg-size: lg-size(g),
lg-map: lg-map(f;g),
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
labeled-graph: LabeledGraph(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
lg-label: lg-label(g;x),
guard: {T},
nat: ℕ,
int_seg: {i..j-},
uimplies: b supposing a,
prop: ℙ,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
spreadn: spread3,
pi1: fst(t),
rev_implies: P ⇐ Q
Latex:
\mforall{}[A,T:Type]. \mforall{}f:A {}\mrightarrow{} T. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}g:LabeledGraph(A). (\mforall{}x\mmember{}lg-map(f;g).P[x] \mLeftarrow{}{}\mRightarrow{} \mforall{}x\mmember{}g.P[f x])
Date html generated:
2016_05_17-AM-10_18_31
Last ObjectModification:
2016_01_18-AM-00_21_06
Theory : process-model
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