Nuprl Lemma : truncate-strat2play
∀[g:SimpleGame]. ∀[n:ℕ]. ∀[s:win2strat(g;n)]. ∀[moves:strat2play(g;n;s)]. ∀[j:ℕ].
play-truncate(moves;2 * (j + 1)) ∈ strat2play(g;j;s) supposing j < n
Proof
Definitions occuring in Statement :
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-truncate: play-truncate(f;m),
simple-game: SimpleGame,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
play-len: ||moves||,
squash: ↓T,
true: True,
less_than': less_than'(a;b),
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
prop: ℙ,
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
guard: {T},
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
mul-commutes,
mul-distributes,
add-zero,
zero-mul,
mul-distributes-right,
mul-associates,
mul_preserves_le,
le-add-cancel2,
zero-add,
not-le-2,
decidable__le,
multiply-is-int-iff,
int_subtype_base,
le_wf,
set_subtype_base,
add-is-int-iff,
strat2play_subtype,
simple-game_wf,
win2strat_wf,
less_than_wf,
play-len_wf,
equal_wf,
strat2play_wf,
lelt_wf,
le-add-cancel,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
add-swap,
minus-one-mul,
nat_wf,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
false_wf,
decidable__lt,
strat2play_subtype_le,
le_weakening2,
win2strat_subtype,
play-truncate_wf
Rules used in proof :
closedConclusion,
baseApply,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
multiplyEquality,
setEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
independent_functionElimination,
voidElimination,
lambdaFormation,
unionElimination,
natural_numberEquality,
addEquality,
productElimination,
independent_pairFormation,
dependent_set_memberEquality,
sqequalRule,
rename,
setElimination,
dependent_functionElimination,
hypothesis,
independent_isectElimination,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[g:SimpleGame]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:win2strat(g;n)]. \mforall{}[moves:strat2play(g;n;s)]. \mforall{}[j:\mBbbN{}].
play-truncate(moves;2 * (j + 1)) \mmember{} strat2play(g;j;s) supposing j < n
Date html generated:
2018_07_25-PM-01_33_01
Last ObjectModification:
2018_06_17-PM-09_41_44
Theory : co-recursion
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