Nuprl Lemma : sg-normalize-win2
∀[g:SimpleGame]. win2(g) ≡ win2(sg-normalize(g))
Proof
Definitions occuring in Statement :
sg-normalize: sg-normalize(g),
win2: win2(g),
simple-game: SimpleGame,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
pi2: snd(t),
seq-item: s[i],
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
assert: ↑b,
bnot: ¬bb,
exists: ∃x:A. B[x],
sg-reachable: sg-reachable(g;x;y),
play-len: ||moves||,
less_than: a < b,
nat_plus: ℕ+,
cand: A c∧ B,
play-item: moves[i],
eq_int: (i =z j),
strat2play: strat2play(g;n;s),
it: ⋅,
unit: Unit,
bool: 𝔹,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
squash: ↓T,
sq_stable: SqStable(P),
win2strat: win2strat(g;n),
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
prop: ℙ,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
ext-eq: A ≡ B,
win2: win2(g),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
uall_wf,
dep-isect_wf,
subtype_rel_transitivity,
dep-isect-subtype,
mul-swap,
play-item-reachable,
sg-reachable_self,
strat2play-reachable,
subtype_rel_weakening,
ext-eq_inversion,
subtract-add-cancel,
iff_weakening_equal,
mod2-2n,
mod2-2n-plus-1,
nequal_wf,
true_wf,
modulus_wf,
nat_plus_properties,
seq-truncate-item,
seq-len-truncate,
seq-add-item,
seq-add-len,
assert-bnot,
bool_cases_sqequal,
assert_of_lt_int,
lt_int_wf,
mul_preserves_le,
nat_plus_subtype_nat,
nat_plus_wf,
squash_wf,
all_wf,
sequence_wf,
seq-truncate_wf,
seq-add_wf,
strat2play-invariant-1,
strat2play-invariant,
strat2play_subtype,
set_wf,
omega-shadow,
two-mul,
one-mul,
le_reflexive,
not-equal-implies-less,
mul-commutes,
mul-distributes,
mul-distributes-right,
play-item_wf,
sg-legal2_wf,
sg-legal2-normalize,
multiply_nat_wf,
add_nat_wf,
mul-associates,
add-is-int-iff,
mul_bounds_1a,
equal-wf-base,
le_weakening,
win2strat_subtype,
sg-legal1-normalize,
sg-init-normalize,
sg-pos-normalize,
sg-legal1_wf,
seq-item_wf,
seq-len_wf,
equal_functionality_wrt_subtype_rel2,
sg-init_wf,
sg-reachable_wf,
sg-pos_wf,
sequence_subtype,
subtype_rel-equal,
le_weakening2,
le-add-cancel2,
uiff_transitivity,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_cases,
play-len_wf,
equal_wf,
strat2play_wf,
minus-zero,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
bool_wf,
eq_int_wf,
nat_wf,
zero-mul,
add-mul-special,
sq_stable__le,
not-le-2,
le-add-cancel-alt,
not-equal-2,
not-lt-2,
decidable__lt,
sg-normalize_wf,
win2strat_wf,
le_wf,
int_seg_properties,
subtype_rel_self,
lelt_wf,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
decidable__int_equal,
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-ge-2,
false_wf,
subtract_wf,
decidable__le,
int_seg_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
simple-game_wf
Rules used in proof :
isectEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
addLevel,
promote_hyp,
sqequalIntensionalEquality,
functionEquality,
dependent_pairFormation,
applyLambdaEquality,
closedConclusion,
baseApply,
productEquality,
equalityElimination,
impliesFunctionality,
cumulativity,
setEquality,
functionExtensionality,
dependentIntersection_memberEquality,
dependentIntersectionElimination,
multiplyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
hypothesis_subsumption,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
instantiate,
because_Cache,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
applyEquality,
addEquality,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
lambdaEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesisEquality,
isectElimination,
lambdaFormation,
extract_by_obid,
hypothesis,
axiomEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[g:SimpleGame]. win2(g) \mequiv{} win2(sg-normalize(g))
Date html generated:
2018_07_25-PM-01_35_50
Last ObjectModification:
2018_06_20-PM-00_31_10
Theory : co-recursion
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