Nuprl Lemma : Game-induction
∀[P:Game ⟶ ℙ']
  ((∀g:Game. ((∀m:Game. ((left-option{i:l}(g;m) ∨ right-option{i:l}(g;m)) 
⇒ P[m])) 
⇒ P[g])) 
⇒ {∀g:Game. P[g]})
Proof
Definitions occuring in Statement : 
right-option: right-option{i:l}(g;m)
, 
left-option: left-option{i:l}(g;m)
, 
Game: Game
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
guard: {T}
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
function: x:A ⟶ B[x]
Definitions unfolded in proof : 
guard: {T}
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
Game: Game
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
or: P ∨ Q
, 
Wsup: Wsup(a;b)
, 
left-option: left-option{i:l}(g;m)
, 
left-move: left-move(g;x)
, 
left-indices: left-indices(g)
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
exists: ∃x:A. B[x]
, 
right-option: right-option{i:l}(g;m)
, 
right-move: right-move(g;x)
, 
right-indices: right-indices(g)
, 
GameB: GameB(p)
, 
GameA: GameA{i:l}()
, 
uimplies: b supposing a
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
W-induction, 
GameA_wf, 
GameB_wf, 
Game_wf, 
all_wf, 
or_wf, 
left-option_wf, 
right-option_wf, 
Wsup_wf, 
subtype_rel_self, 
W_wf, 
subtype_rel_union, 
iff_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
lambdaFormation, 
cut, 
thin, 
instantiate, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesis, 
lambdaEquality, 
cumulativity, 
hypothesisEquality, 
independent_functionElimination, 
dependent_functionElimination, 
functionEquality, 
applyEquality, 
universeEquality, 
because_Cache, 
functionExtensionality, 
equalityTransitivity, 
equalitySymmetry, 
unionElimination, 
productElimination, 
inlEquality, 
voidEquality, 
independent_isectElimination, 
voidElimination, 
inrEquality
Latex:
\mforall{}[P:Game  {}\mrightarrow{}  \mBbbP{}']
    ((\mforall{}g:Game.  ((\mforall{}m:Game.  ((left-option\{i:l\}(g;m)  \mvee{}  right-option\{i:l\}(g;m))  {}\mRightarrow{}  P[m]))  {}\mRightarrow{}  P[g]))
    {}\mRightarrow{}  \{\mforall{}g:Game.  P[g]\})
Date html generated:
2018_05_22-PM-09_52_41
Last ObjectModification:
2018_05_20-PM-10_37_08
Theory : Numbers!and!Games
Home
Index