Nuprl Lemma : conditional-apply
∀[T,V:Type]. ∀[A,B:T ⟶ ℙ]. ∀[dcd_A:t:T ⟶ Dec(A t)]. ∀[f:{t:T| A t} ⟶ V]. ∀[g:{t:T| B t} ⟶ V]. ∀[t:{t:T|
(A t) ∨ (B t)} ].
(([A? f : g] t) = (f t) ∈ V supposing A t ∧ ([A? f : g] t) = (g t) ∈ V supposing ¬(A t))
Proof
Definitions occuring in Statement :
conditional: [P? f : g],
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
conditional: [P? f : g],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
branch: if p:P then A[p] else B fi ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Latex:
\mforall{}[T,V:Type]. \mforall{}[A,B:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[dcd$_{A}$:t:T {}\mrightarrow{} Dec(A t)]. \mforall{}[f:\{t:T| A t\} {}\mrightarrow{} V].\000C \mforall{}[g:\{t:T| B t\} {}\mrightarrow{} V].
\mforall{}[t:\{t:T| (A t) \mvee{} (B t)\} ].
(([A? f : g] t) = (f t) supposing A t \mwedge{} ([A? f : g] t) = (g t) supposing \mneg{}(A t))
Date html generated:
2016_05_16-AM-10_15_42
Last ObjectModification:
2015_12_28-PM-09_24_54
Theory : new!event-ordering
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