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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
win2: win2(g),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
win2strat: win2strat(g;n),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
subtract: n - m,
less_than': less_than'(a;b),
le: A ≤ B,
cand: A c∧ B,
play-item: moves[i],
eq_int: (i =z j),
strat2play: strat2play(g;n;s),
sq_stable: SqStable(P),
play-len: ||moves||,
sg-reachable: sg-reachable(g;x;y),
less_than: a < b,
nat_plus: ℕ+,
bnot: ¬bb,
assert: ↑b,
pi2: snd(t),
seq-item: s[i]
Lemmas referenced :
simple-game_wf,
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,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
win2strat_wf,
sg-normalize_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
eq_int_wf,
equal-wf-base,
bool_wf,
le_wf,
assert_wf,
bnot_wf,
not_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_true,
btrue_wf,
iff_weakening_equal,
btrue_neq_bfalse,
istype-assert,
subtype_rel-equal,
nat_wf,
base_wf,
eqtt_to_assert,
assert_of_eq_int,
strat2play_wf,
play-len_wf,
bool_cases,
bool_subtype_base,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
uiff_transitivity,
le_weakening2,
le-add-cancel2,
not-lt-2,
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,
less_than_wf,
less_than_irreflexivity,
less_than_transitivity1,
sg-legal1_wf,
lelt_wf,
seq-item_wf,
seq-len_wf,
equal_functionality_wrt_subtype_rel2,
sg-init_wf,
sg-reachable_wf,
sg-pos_wf,
sequence_subtype,
sg-pos-normalize,
sg-init-normalize,
sg-legal1-normalize,
zero-mul,
add-mul-special,
not-le-2,
win2strat_subtype,
le_weakening,
sq_stable__le,
multiply_nat_wf,
add_nat_wf,
mul-associates,
add-is-int-iff,
mul_bounds_1a,
le-add-cancel-alt,
minus-zero,
not-equal-2,
sg-legal2-normalize,
sg-legal2_wf,
play-item_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
strat2play_subtype,
strat2play-invariant,
strat2play-invariant-1,
seq-add_wf,
seq-truncate_wf,
subtract-add-cancel,
nat_plus_wf,
nat_plus_properties,
mul_preserves_le,
lt_int_wf,
assert_of_lt_int,
bool_cases_sqequal,
assert-bnot,
seq-add-len,
seq-add-item,
seq-len-truncate,
seq-truncate-item,
mod2-2n-plus-1,
subtype_rel_weakening,
ext-eq_inversion,
mul-commutes,
mul-distributes,
mul-distributes-right,
strat2play-reachable,
le_reflexive,
omega-shadow,
two-mul,
one-mul,
not-equal-implies-less,
sg-reachable_self,
mul-swap,
play-item-reachable,
sequence_wf,
dep-isect_wf,
subtype_rel_transitivity,
dep-isect-subtype,
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
Error :universeIsType,
extract_by_obid,
Error :lambdaFormation_alt,
isectElimination,
hypothesisEquality,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
addEquality,
baseApply,
closedConclusion,
baseClosed,
intEquality,
imageElimination,
universeEquality,
imageMemberEquality,
Error :equalityIstype,
Error :functionIsType,
sqequalBase,
dependentIntersectionElimination,
dependentIntersection_memberEquality,
functionExtensionality,
setEquality,
cumulativity,
equalityElimination,
minusEquality,
voidEquality,
dependent_set_memberEquality,
isect_memberEquality,
lambdaEquality,
lambdaFormation,
productEquality,
multiplyEquality,
impliesFunctionality,
promote_hyp,
sqequalIntensionalEquality,
dependent_pairFormation,
functionEquality,
isectEquality
Latex:
\mforall{}[g:SimpleGame]. win2(g) \mequiv{} win2(sg-normalize(g))
Date html generated:
2019_06_20-PM-00_54_31
Last ObjectModification:
2019_01_02-PM-03_39_29
Theory : co-recursion-2
Home
Index