Nuprl Lemma : implies-sg-win2
∀g:SimpleGame
((∃Good:Pos(g) ⟶ ℙ
∃F:p:Pos(g) ⟶ q:Pos(g) ⟶ Pos(g)
(Good[InitialPos(g)] ∧ (∀p:{p:Pos(g)| Good[p]} . ∀q:{q:Pos(g)| Legal1(p;q)} . (Good[F[p;q]] ∧ Legal2(q;F[p;q]))))\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[s1;s2],
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 ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
win2strat: win2strat(g;n),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
squash: ↓T,
play-item: moves[i],
int_seg: {i..j-},
lelt: i ≤ j < k,
play-truncate: play-truncate(f;m),
play-len: ||moves||,
cand: A c∧ B,
nat_plus: ℕ+,
less_than: a < b,
sq_stable: SqStable(P),
seq-item: s[i],
pi2: snd(t)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
nat_wf,
exists_wf,
sg-pos_wf,
sg-init_wf,
subtype_rel_self,
all_wf,
sg-legal1_wf,
sg-legal2_wf,
simple-game_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
bnot_wf,
not_wf,
int_subtype_base,
subtype_base_sq,
strat2play_wf,
not-le-2,
le_wf,
equal-wf-T-base,
play-len_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
strat2play-invariant-1,
le-add-cancel-alt,
decidable__lt,
not-lt-2,
lelt_wf,
le-add-cancel2,
subtract-add-cancel,
strat2play_subtype,
le_weakening2,
seq-truncate-item,
seq-len-truncate,
squash_wf,
true_wf,
iff_weakening_equal,
add-is-int-iff,
mul-distributes,
mul-commutes,
seq-item_wf,
mul_bounds_1a,
seq-len_wf,
set_subtype_base,
multiply-is-int-iff,
mul-associates,
mul-distributes-right,
zero-mul,
not-equal-implies-less,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
omega-shadow,
add_nat_wf,
multiply_nat_wf,
sq_stable__le,
play-item_wf,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_pairFormation,
addEquality,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
because_Cache,
instantiate,
functionEquality,
cumulativity,
universeEquality,
productEquality,
setEquality,
baseClosed,
baseApply,
closedConclusion,
dependentIntersection_memberEquality,
dependent_set_memberEquality,
equalityElimination,
impliesFunctionality,
imageElimination,
imageMemberEquality,
addLevel,
hyp_replacement,
levelHypothesis,
applyLambdaEquality,
multiplyEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
promote_hyp,
functionExtensionality
Latex:
\mforall{}g:SimpleGame
((\mexists{}Good:Pos(g) {}\mrightarrow{} \mBbbP{}
\mexists{}F:p:Pos(g) {}\mrightarrow{} q:Pos(g) {}\mrightarrow{} Pos(g)
(Good[InitialPos(g)]
\mwedge{} (\mforall{}p:\{p:Pos(g)| Good[p]\} . \mforall{}q:\{q:Pos(g)| Legal1(p;q)\} . (Good[F[p;q]] \mwedge{} Legal2(q;F[p;q])))))
{}\mRightarrow{} win2(g))
Date html generated:
2018_07_25-PM-01_34_55
Last ObjectModification:
2018_07_11-AM-11_51_42
Theory : co-recursion
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