Nuprl Lemma : play-item-reachable
∀g:SimpleGame. ∀n:ℕ. ∀s:win2strat(g;n). ∀moves:strat2play(g;n;s). ∀i:ℕ(2 * n) + 2.
(moves[i] ∈ {p:Pos(g)| sg-reachable(g;InitialPos(g);p)} )
Proof
Definitions occuring in Statement :
sg-reachable: sg-reachable(g;x;y),
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-item: moves[i],
sg-init: InitialPos(g),
sg-pos: Pos(g),
simple-game: SimpleGame,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
sq_type: SQType(T),
less_than: a < b,
play-item: moves[i],
cand: A c∧ B,
nat_plus: ℕ+,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
less_than': less_than'(a;b),
top: Top,
subtract: n - m,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
prop: ℙ,
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
uimplies: b supposing a,
uiff: uiff(P;Q),
le: A ≤ B,
int_seg: {i..j-},
nat: ℕ,
exists: ∃x:A. B[x],
sg-reachable: sg-reachable(g;x;y),
squash: ↓T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
subtype_base_sq,
nat_plus_properties,
nat_properties,
int_seg_properties,
seq-truncate-item,
seq-len-truncate,
minus-zero,
mul-commutes,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
add-subtract-cancel,
simple-game_wf,
win2strat_wf,
strat2play_wf,
int_seg_wf,
sg-init_wf,
sg-reachable_wf,
nat_plus_subtype_nat,
le_weakening2,
sg-legal2_wf,
nat_plus_wf,
multiply_nat_wf,
add_nat_wf,
mul_bounds_1a,
sg-legal1_wf,
squash_wf,
nat_wf,
all_wf,
le-add-cancel-alt,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
minus-minus,
multiply-is-int-iff,
set_subtype_base,
subtract_wf,
lelt_wf,
seq-item_wf,
equal_wf,
less_than_wf,
le-add-cancel2,
seq-len_wf,
mul-associates,
less-iff-le,
le_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,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
int_subtype_base,
add-is-int-iff,
sg-pos_wf,
seq-truncate_wf,
play-item_wf,
strat2play_subtype,
strat2play-invariant-1
Rules used in proof :
cumulativity,
instantiate,
levelHypothesis,
addLevel,
promote_hyp,
sqequalIntensionalEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
productEquality,
multiplyEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
independent_functionElimination,
voidElimination,
independent_pairFormation,
unionElimination,
independent_isectElimination,
closedConclusion,
baseApply,
natural_numberEquality,
addEquality,
dependent_pairFormation,
dependent_set_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
rename,
setElimination,
applyLambdaEquality,
sqequalRule,
hypothesis,
applyEquality,
productElimination,
because_Cache,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
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{}(2 * n) + 2.
(moves[i] \mmember{} \{p:Pos(g)| sg-reachable(g;InitialPos(g);p)\} )
Date html generated:
2018_07_25-PM-01_33_53
Last ObjectModification:
2018_06_20-AM-10_51_25
Theory : co-recursion
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