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
Lemmas :
or_wf,
subtype_base_sq,
subtype_rel_sets,
decidable_wf,
equal_wf,
set_wf,
not_wf,
sq_stable_from_decidable
\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:
2015_07_17-AM-09_01_39
Last ObjectModification:
2015_01_27-PM-00_55_55
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