Nuprl Lemma : strat2play-longer
∀[g:SimpleGame]. ∀[n:ℕ]. ∀[s:win2strat(g;n)]. ∀[moves:strat2play(g;n;s)]. ∀[x:sequence(Pos(g))].
((x ∈ strat2play(g;n;s)) ∧ (seq-truncate(x;||moves||) = moves ∈ strat2play(g;n;s))) supposing
((seq-truncate(x;||moves||) = moves ∈ sequence(Pos(g))) and
(||moves|| ≤ ||x||))
Proof
Definitions occuring in Statement :
strat2play: strat2play(g;n;s)
,
win2strat: win2strat(g;n)
,
sg-pos: Pos(g)
,
simple-game: SimpleGame
,
seq-truncate: seq-truncate(s;n)
,
seq-len: ||s||
,
sequence: sequence(T)
,
nat: ℕ
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
le: A ≤ B
,
and: P ∧ Q
,
member: t ∈ T
,
equal: s = t ∈ T
Definitions unfolded in proof :
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
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
not: ¬A
,
subtype_rel: A ⊆r B
,
all: ∀x:A. B[x]
,
cand: A c∧ B
,
squash: ↓T
,
int_seg: {i..j-}
,
lelt: i ≤ j < k
,
true: True
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
top: Top
,
uiff: uiff(P;Q)
,
or: P ∨ Q
,
decidable: Dec(P)
,
play-item: moves[i]
,
btrue: tt
,
ifthenelse: if b then t else f fi
,
subtract: n - m
,
eq_int: (i =z j)
,
strat2play: strat2play(g;n;s)
,
sq_stable: SqStable(P)
,
sq_type: SQType(T)
,
bfalse: ff
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
play-truncate: play-truncate(f;m)
,
play-len: ||moves||
,
win2strat: win2strat(g;n)
,
less_than: a < b
,
nat_plus: ℕ+
,
so_apply: x[s]
,
so_lambda: λ2x.t[x]
,
exists: ∃x:A. B[x]
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
strat2play_subtype,
istype-false,
istype-le,
seq-item_wf,
less_than_wf,
squash_wf,
true_wf,
istype-int,
seq-len_wf,
sg-pos_wf,
subtype_rel_self,
iff_weakening_equal,
seq-truncate-item,
istype-void,
int_seg_wf,
seq-truncate_wf,
sequence_wf,
strat2play_wf,
win2strat_wf,
subtract-1-ge-0,
le_weakening2,
istype-nat,
simple-game_wf,
le_wf,
le-add-cancel2,
sg-legal1_wf,
nat_wf,
lelt_wf,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
decidable__lt,
false_wf,
equal_wf,
le_transitivity,
istype-universe,
sq_stable__le,
multiply_nat_wf,
add_nat_wf,
mul-associates,
mul_bounds_1a,
add-is-int-iff,
zero-mul,
add-mul-special,
add-zero,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
win2strat_subtype,
bool_wf,
int_subtype_base,
equal-wf-base,
not_wf,
bnot_wf,
assert_wf,
le_weakening,
eq_int_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
uiff_transitivity,
le_reflexive,
seq-len-truncate,
play-len_wf,
equal-wf-T-base,
play-truncate_wf,
seq-truncate-truncate,
omega-shadow,
minus-zero,
two-mul,
one-mul,
not-equal-implies-less,
le-add-cancel-alt,
mul-commutes,
mul-distributes,
mul-distributes-right,
set_subtype_base,
play-item_wf,
sg-legal2_wf,
istype-assert,
istype-sqequal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
because_Cache,
applyEquality,
Error :productIsType,
Error :equalityIstype,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
independent_pairEquality,
productEquality,
intEquality,
voidEquality,
isect_memberEquality,
unionElimination,
lambdaFormation,
lambdaEquality,
dependent_set_memberEquality,
hyp_replacement,
multiplyEquality,
minusEquality,
addEquality,
dependentIntersection_memberEquality,
closedConclusion,
baseApply,
dependentIntersectionElimination,
promote_hyp,
cumulativity,
impliesFunctionality,
equalityElimination,
setEquality,
sqequalIntensionalEquality,
dependent_pairFormation,
sqequalBase,
Error :functionIsType,
Error :dependent_pairFormation_alt
Latex:
\mforall{}[g:SimpleGame]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:win2strat(g;n)]. \mforall{}[moves:strat2play(g;n;s)]. \mforall{}[x:sequence(Pos(g))].
((x \mmember{} strat2play(g;n;s)) \mwedge{} (seq-truncate(x;||moves||) = moves)) supposing
((seq-truncate(x;||moves||) = moves) and
(||moves|| \mleq{} ||x||))
Date html generated:
2019_06_20-PM-00_52_45
Last ObjectModification:
2019_01_02-PM-01_32_08
Theory : co-recursion-2
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