Nuprl Lemma : strat2play-reachable
∀g:SimpleGame. ∀n:ℕ. ∀s:win2strat(g;n). ∀moves:strat2play(g;n;s).
((||moves|| ≤ ((2 * n) + 2)) ⇒ (moves ∈ sequence({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-len: ||moves||,
sg-init: InitialPos(g),
sg-pos: Pos(g),
simple-game: SimpleGame,
sequence: sequence(T),
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
guard: {T},
uiff: uiff(P;Q),
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
pi2: snd(t),
pi1: fst(t),
seq-len: ||s||,
seq-item: s[i],
play-len: ||moves||,
play-item: moves[i],
sequence: sequence(T),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
squash: ↓T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
squash_wf,
all_wf,
sg-init_wf,
sg-reachable_wf,
lelt_wf,
less_than_transitivity1,
multiply-is-int-iff,
int_subtype_base,
set_subtype_base,
add-is-int-iff,
simple-game_wf,
win2strat_wf,
strat2play_wf,
play-len_wf,
equal_wf,
nat_wf,
seq-len_wf,
le_wf,
sg-pos_wf,
sequence_wf,
set_wf,
int_seg_wf,
strat2play_subtype,
play-item-reachable
Rules used in proof :
setEquality,
functionEquality,
independent_isectElimination,
intEquality,
closedConclusion,
baseApply,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
dependent_pairEquality,
productElimination,
independent_functionElimination,
lambdaEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
multiplyEquality,
addEquality,
natural_numberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
rename,
setElimination,
applyLambdaEquality,
sqequalRule,
applyEquality,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}g:SimpleGame. \mforall{}n:\mBbbN{}. \mforall{}s:win2strat(g;n). \mforall{}moves:strat2play(g;n;s).
((||moves|| \mleq{} ((2 * n) + 2)) {}\mRightarrow{} (moves \mmember{} sequence(\{p:Pos(g)| sg-reachable(g;InitialPos(g);p)\} )))
Date html generated:
2018_07_25-PM-01_33_58
Last ObjectModification:
2018_06_20-AM-10_53_45
Theory : co-recursion
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