Nuprl Lemma : good-sg-win2
∀g:SimpleGame
  ((∃Good:Pos(g) ⟶ ℙ'
     (Good[InitialPos(g)] ∧ (∀p:Pos(g). ∀q:{q:Pos(g)| Legal1(p;q)} . ∀gd:Good[p].  ∃r:{r:Pos(g)| Legal2(q;r)} . Good[r])\000C))
  
⇒ win2(g))
Proof
Definitions occuring in Statement : 
win2: win2(g)
, 
sg-legal2: Legal2(x;y)
, 
sg-legal1: Legal1(x;y)
, 
sg-init: InitialPos(g)
, 
sg-pos: Pos(g)
, 
simple-game: SimpleGame
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
win2: win2(g)
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
nat: ℕ
, 
false: False
, 
ge: i ≥ j 
, 
uimplies: b supposing a
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
prop: ℙ
, 
win2strat: win2strat(g;n)
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
sq_type: SQType(T)
, 
guard: {T}
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
pi1: fst(t)
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
squash: ↓T
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
subtract: n - m
, 
goodAux: goodAux(g0;G;moves)
, 
eq_int: (i =z j)
, 
play-len: ||moves||
, 
play-truncate: play-truncate(f;m)
, 
true: True
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
less_than: a < b
, 
nat_plus: ℕ+
, 
play-item: moves[i]
, 
pi2: snd(t)
, 
seq-item: s[i]
, 
nequal: a ≠ b ∈ T 
, 
let: let, 
sq_stable: SqStable(P)
, 
seq-len: ||s||
, 
seq-truncate: seq-truncate(s;n)
, 
sequence: sequence(T)
Lemmas referenced : 
nat_properties, 
full-omega-unsat, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
ge_wf, 
istype-less_than, 
subtract-1-ge-0, 
istype-nat, 
sg-pos_wf, 
sg-init_wf, 
subtype_rel_self, 
sg-legal1_wf, 
sg-legal2_wf, 
simple-game_wf, 
eq_int_wf, 
equal-wf-base, 
bool_wf, 
assert_wf, 
bnot_wf, 
not_wf, 
istype-assert, 
intformnot_wf, 
intformeq_wf, 
int_formula_prop_not_lemma, 
int_formula_prop_eq_lemma, 
int_subtype_base, 
subtype_base_sq, 
strat2play_wf, 
subtract_wf, 
decidable__le, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
istype-le, 
play-len_wf, 
subtype_rel_function, 
uiff_transitivity, 
eqtt_to_assert, 
assert_of_eq_int, 
iff_transitivity, 
iff_weakening_uiff, 
eqff_to_assert, 
assert_of_bnot, 
play-item_wf, 
itermMultiply_wf, 
int_term_value_mul_lemma, 
decidable__lt, 
itermAdd_wf, 
int_term_value_add_lemma, 
strat2play-invariant-1, 
seq-len-truncate, 
le-add-cancel, 
add-zero, 
add_functionality_wrt_le, 
add-commutes, 
add-swap, 
add-associates, 
minus-minus, 
minus-add, 
minus-one-mul-top, 
zero-add, 
minus-one-mul, 
condition-implies-le, 
less-iff-le, 
not-le-2, 
istype-false, 
strat2play_subtype, 
mul_bounds_1b, 
mul-commutes, 
mul-distributes, 
subtract-add-cancel, 
bool_subtype_base, 
equal_wf, 
squash_wf, 
true_wf, 
istype-universe, 
eq_int_eq_false, 
bfalse_wf, 
iff_weakening_equal, 
seq-truncate-truncate, 
seq-truncate-item, 
mul-swap, 
mul-associates, 
not-lt-2, 
omega-shadow, 
minus-zero, 
zero-mul, 
mul-distributes-right, 
two-mul, 
add-mul-special, 
one-mul, 
le_reflexive, 
not-equal-implies-less, 
istype-sqequal, 
sq_stable__le, 
subtract_nat_wf, 
subtype_rel-equal, 
le-add-cancel-alt
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :lambdaFormation_alt, 
sqequalHypSubstitution, 
productElimination, 
thin, 
rename, 
cut, 
hypothesis, 
promote_hyp, 
Error :isect_memberFormation_alt, 
introduction, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
setElimination, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
Error :dependent_pairFormation_alt, 
Error :lambdaEquality_alt, 
int_eqEquality, 
dependent_functionElimination, 
Error :isect_memberEquality_alt, 
voidElimination, 
sqequalRule, 
independent_pairFormation, 
Error :universeIsType, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
Error :functionIsTypeImplies, 
Error :inhabitedIsType, 
Error :productIsType, 
Error :functionIsType, 
universeEquality, 
applyEquality, 
instantiate, 
because_Cache, 
Error :setIsType, 
baseClosed, 
intEquality, 
Error :equalityIstype, 
sqequalBase, 
baseApply, 
closedConclusion, 
dependentIntersection_memberEquality, 
cumulativity, 
Error :dependent_set_memberEquality_alt, 
unionElimination, 
functionExtensionality, 
functionEquality, 
setEquality, 
productEquality, 
equalityElimination, 
imageElimination, 
multiplyEquality, 
addEquality, 
applyLambdaEquality, 
minusEquality, 
hyp_replacement, 
imageMemberEquality
Latex:
\mforall{}g:SimpleGame
    ((\mexists{}Good:Pos(g)  {}\mrightarrow{}  \mBbbP{}'
          (Good[InitialPos(g)]
          \mwedge{}  (\mforall{}p:Pos(g).  \mforall{}q:\{q:Pos(g)|  Legal1(p;q)\}  .  \mforall{}gd:Good[p].    \mexists{}r:\{r:Pos(g)|  Legal2(q;r)\}  .  Good[r]))\000C)
    {}\mRightarrow{}  win2(g))
Date html generated:
2019_06_20-PM-00_53_46
Last ObjectModification:
2019_01_02-PM-03_35_18
Theory : co-recursion-2
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