Nuprl Lemma : strat2play-invariant
∀g:SimpleGame. ∀n:ℕ. ∀s:win2strat(g;n). ∀moves:strat2play(g;n;s). ∀i:ℕ(2 * n) + 1.
((((i mod 2) = 0 ∈ ℤ) ⇒ (↓Legal1(moves[i];moves[i + 1])))
∧ (((i mod 2) = 1 ∈ ℤ) ⇒ (↓Legal2(moves[i];moves[i + 1]))))
Proof
Definitions occuring in Statement :
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-item: moves[i],
sg-legal2: Legal2(x;y),
sg-legal1: Legal1(x;y),
simple-game: SimpleGame,
modulus: a mod n,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_seg: {i..j-},
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
lelt: i ≤ j < k,
exists: ∃x:A. B[x],
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
strat2play-invariant-1,
int_seg_wf,
strat2play_wf,
win2strat_wf,
nat_wf,
simple-game_wf,
div_rem_sum,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rem_bounds_1,
int_seg_subtype_nat,
false_wf,
less_than_wf,
div_bounds_1,
equal_wf,
lelt_wf,
mul-commutes,
add-commutes,
subtract_wf,
add-associates,
minus-add,
minus-one-mul,
mul-associates,
add-swap,
add-mul-special,
mul-distributes-right,
two-mul,
zero-mul,
zero-add,
add-zero,
one-mul,
add-is-int-iff,
add_functionality_wrt_le,
le_reflexive,
less-iff-le,
not-lt-2,
omega-shadow,
mul-distributes,
int_seg_properties,
nat_properties,
decidable__lt,
modulus-is-rem,
equal-wf-T-base,
set_subtype_base,
sq_stable__le,
squash_wf,
add_functionality_wrt_eq,
subtype_rel_self,
iff_weakening_equal,
le-add-cancel
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
natural_numberEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
baseClosed,
applyEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
because_Cache,
divideEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
promote_hyp,
isect_memberEquality,
voidEquality,
baseApply,
closedConclusion,
minusEquality,
unionElimination,
remainderEquality,
lambdaEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}g:SimpleGame. \mforall{}n:\mBbbN{}. \mforall{}s:win2strat(g;n). \mforall{}moves:strat2play(g;n;s). \mforall{}i:\mBbbN{}(2 * n) + 1.
((((i mod 2) = 0) {}\mRightarrow{} (\mdownarrow{}Legal1(moves[i];moves[i + 1])))
\mwedge{} (((i mod 2) = 1) {}\mRightarrow{} (\mdownarrow{}Legal2(moves[i];moves[i + 1]))))
Date html generated:
2018_07_25-PM-01_33_11
Last ObjectModification:
2018_06_12-PM-00_38_49
Theory : co-recursion
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