Nuprl Lemma : isom-preserves-win2
∀g1,g2:SimpleGame. (g1 ≅ g2 ⇒ win2(g1) ⇒ win2(g2))
Proof
Definitions occuring in Statement :
isom-games: g1 ≅ g2,
win2: win2(g),
simple-game: SimpleGame,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
win2: win2(g),
uall: ∀[x:A]. B[x],
member: t ∈ T,
isom-games: g1 ≅ g2,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
win2strat: win2strat(g;n),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
strat2play: strat2play(g;n;s),
squash: ↓T,
play-item: moves[i],
int_seg: {i..j-},
lelt: i ≤ j < k,
seq-item: s[i],
pi2: snd(t),
nat_plus: ℕ+,
less_than: a < b,
sq_type: SQType(T),
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
play-len: ||moves||,
so_lambda: λ2x.t[x],
so_apply: x[s],
play-truncate: play-truncate(f;m)
Lemmas referenced :
nat_wf,
win2_wf,
isom-games_wf,
simple-game_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
strat2play_wf,
win2strat_wf,
false_wf,
le_wf,
top_wf,
decidable__le,
subtract_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,
le_weakening2,
seq-comp_wf,
sg-pos_wf,
seq-comp-len,
seq-comp-item,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
seq-len_wf,
seq-item_wf,
decidable__lt,
not-lt-2,
lelt_wf,
sg-legal1_wf,
le-add-cancel2,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
not-le-2,
omega-shadow,
eq_int_wf,
le_weakening,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
equal-wf-T-base,
play-len_wf,
uiff_transitivity,
strat2play_subtype,
sequence_wf,
subtype_rel_transitivity,
set_wf,
sg-legal2_wf,
play-item_wf,
add-is-int-iff,
mul-associates,
mul-distributes,
mul-commutes,
le-add-cancel-alt,
seq-comp-truncate,
play-truncate_wf,
seq-truncate-item
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
isect_memberFormation,
cut,
hypothesis,
isectElimination,
thin,
hypothesisEquality,
rename,
productElimination,
introduction,
extract_by_obid,
equalityTransitivity,
equalitySymmetry,
setElimination,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
because_Cache,
dependent_set_memberEquality,
unionElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
baseClosed,
instantiate,
productEquality,
promote_hyp,
multiplyEquality,
dependentIntersectionElimination,
cumulativity,
impliesFunctionality,
baseApply,
closedConclusion,
dependentIntersection_memberEquality,
setEquality,
equalityElimination,
functionExtensionality,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}g1,g2:SimpleGame. (g1 \mcong{} g2 {}\mRightarrow{} win2(g1) {}\mRightarrow{} win2(g2))
Date html generated:
2018_07_25-PM-01_34_37
Last ObjectModification:
2018_07_11-PM-00_03_55
Theory : co-recursion
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