Nuprl Lemma : strat2play-invariant-1
∀g:SimpleGame. ∀n:ℕ. ∀s:win2strat(g;n). ∀moves:strat2play(g;n;s).
((moves[0] = InitialPos(g) ∈ Pos(g))
∧ (∀i:ℕn + 1
((↓Legal1(moves[2 * i];moves[(2 * i) + 1]))
∧ (i < n
⇒ ((↓Legal2(moves[(2 * i) + 1];moves[2 * (i + 1)]))
∧ (moves[2 * (i + 1)] = (s play-truncate(moves;2 * (i + 1))) ∈ Pos(g)))))))
Proof
Definitions occuring in Statement :
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-truncate: play-truncate(f;m),
play-item: moves[i],
sg-legal2: Legal2(x;y),
sg-legal1: Legal1(x;y),
sg-init: InitialPos(g),
sg-pos: Pos(g),
simple-game: SimpleGame,
seq-item: s[i],
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
multiply: n * m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
squash: ↓T,
int_seg: {i..j-},
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),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
true: True,
strat2play: strat2play(g;n;s),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
sq_type: SQType(T),
lelt: i ≤ j < k,
win2strat: win2strat(g;n),
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
play-len: ||moves||,
play-truncate: play-truncate(f;m),
play-item: moves[i]
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
int_seg_wf,
strat2play_wf,
win2strat_wf,
false_wf,
le_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,
nat_wf,
simple-game_wf,
decidable__int_equal,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
int_seg_subtype,
int_seg_cases,
eq_int_wf,
le_weakening,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
add-is-int-iff,
decidable__lt,
not-lt-2,
le-add-cancel-alt,
lelt_wf,
set_subtype_base,
not-equal-2,
minus-zero,
subtract-add-cancel,
mul-distributes,
mul-commutes,
not-le-2,
le-add-cancel2,
mul_preserves_le,
play-item_wf,
mul-associates,
mul-distributes-right,
zero-mul,
squash_wf,
true_wf,
sg-legal2_wf,
sg-pos_wf,
subtype_rel_self,
iff_weakening_equal,
play-truncate_wf,
equal-wf-T-base,
play-len_wf,
equal_wf,
set_wf,
seq-truncate-item,
seq-item_wf,
seq-len_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
imageElimination,
imageMemberEquality,
baseClosed,
addEquality,
because_Cache,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
dependentIntersectionElimination,
applyLambdaEquality,
promote_hyp,
impliesFunctionality,
baseApply,
closedConclusion,
multiplyEquality,
universeEquality,
functionEquality,
setEquality,
addLevel,
hyp_replacement,
levelHypothesis
Latex:
\mforall{}g:SimpleGame. \mforall{}n:\mBbbN{}. \mforall{}s:win2strat(g;n). \mforall{}moves:strat2play(g;n;s).
((moves[0] = InitialPos(g))
\mwedge{} (\mforall{}i:\mBbbN{}n + 1
((\mdownarrow{}Legal1(moves[2 * i];moves[(2 * i) + 1]))
\mwedge{} (i < n
{}\mRightarrow{} ((\mdownarrow{}Legal2(moves[(2 * i) + 1];moves[2 * (i + 1)]))
\mwedge{} (moves[2 * (i + 1)] = (s play-truncate(moves;2 * (i + 1)))))))))
Date html generated:
2018_07_25-PM-01_32_57
Last ObjectModification:
2018_06_12-PM-00_30_13
Theory : co-recursion
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