Nuprl Lemma : strat2play-invariant-type
∀g:SimpleGame. ∀n:ℕ. ∀s:win2strat(g;n). ∀moves:strat2play(g;n;s).
(∀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))))) ∈ ℙ)
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-pos: Pos(g),
simple-game: SimpleGame,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
prop: ℙ,
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
multiply: n * m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
pi1: fst(t),
seq-len: ||s||,
seq-truncate: seq-truncate(s;n),
play-len: ||moves||,
play-truncate: play-truncate(f;m),
sequence: sequence(T),
less_than: a < b,
so_apply: x[s],
nat_plus: ℕ+,
true: True,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
squash: ↓T,
sq_stable: SqStable(P),
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
seq-len_wf,
sequence_wf,
set_wf,
strat2play_subtype,
truncate-strat2play,
add-subtract-cancel,
nat_properties,
int_seg_properties,
minus-zero,
le-add-cancel-alt,
mul-swap,
omega-shadow,
two-mul,
add-mul-special,
one-mul,
le_reflexive,
simple-game_wf,
win2strat_wf,
mul-commutes,
mul-distributes,
zero-mul,
mul-distributes-right,
mul-associates,
le_weakening2,
play-len_wf,
minus-minus,
subtract_wf,
strat2play_wf,
le-add-cancel2,
less-iff-le,
win2strat_subtype,
not-lt-2,
decidable__lt,
win2strat-properties,
sg-pos_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
decidable__le,
sg-legal2_wf,
less_than_wf,
equal_wf,
sq_stable__le,
nat_wf,
multiply_nat_wf,
add_nat_wf,
lelt_wf,
int_seg_subtype_nat,
le_wf,
false_wf,
mul_bounds_1a,
play-item_wf,
sg-legal1_wf,
squash_wf,
int_seg_wf,
all_wf
Rules used in proof :
setEquality,
functionExtensionality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
voidElimination,
unionElimination,
functionEquality,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
independent_pairFormation,
multiplyEquality,
dependent_set_memberEquality,
hypothesisEquality,
productEquality,
lambdaEquality,
sqequalRule,
hypothesis,
because_Cache,
rename,
setElimination,
addEquality,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:SimpleGame. \mforall{}n:\mBbbN{}. \mforall{}s:win2strat(g;n). \mforall{}moves:strat2play(g;n;s).
(\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))))))) \mmember{} \mBbbP{})
Date html generated:
2018_07_25-PM-01_33_05
Last ObjectModification:
2018_06_20-PM-09_29_37
Theory : co-recursion
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