Nuprl Lemma : win2strat-strat2play-wf
∀[g:SimpleGame]. ∀[n:ℕ].
((win2strat(g;n) ∈ Type)
∧ (∀[s:win2strat(g;n)]. (strat2play(g;n;s) ∈ Type))
∧ (∀[s:win2strat(g;n)]. ∀[f:strat2play(g;n;s)]. (||f|| ∈ ℤ))
∧ (∀[s:win2strat(g;n)]. ∀[f:strat2play(g;n;s)]. ∀[k:{(2 * n) + 2..||f|| + 1-}].
(play-truncate(f;k) ∈ strat2play(g;n;s))))
Proof
Definitions occuring in Statement :
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-truncate: play-truncate(f;m),
play-len: ||moves||,
simple-game: SimpleGame,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
play-truncate: play-truncate(f;m),
play-len: ||moves||,
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,
all: ∀x:A. B[x],
true: True,
top: Top,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
play-item: moves[i],
strat2play: strat2play(g;n;s),
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
win2strat: win2strat(g;n),
cand: A c∧ B,
squash: ↓T,
sq_stable: SqStable(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
pi2: snd(t),
pi1: fst(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
less_than: a < b,
seq-item: s[i],
seq-truncate: seq-truncate(s;n)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
subtract-1-ge-0,
nat_wf,
simple-game_wf,
le-add-cancel2,
sg-legal1_wf,
sg-init_wf,
lelt_wf,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
decidable__lt,
false_wf,
seq-item_wf,
equal_wf,
seq-len_wf,
le_wf,
sg-pos_wf,
sequence_wf,
top_wf,
int_seg_wf,
add-associates,
less-iff-le,
sq_stable__le,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
decidable__le,
seq-truncate_wf,
set_wf,
seq-len-truncate,
seq-truncate-item,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
int_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
dep-isect-wf,
equal-wf-T-base,
subtract_wf,
sg-legal2_wf,
istype-false,
not-equal-2,
le_antisymmetry_iff,
mul-associates,
istype-void,
minus-minus,
le-add-cancel-alt,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
subtract_nat_wf,
set_subtype_base,
add-is-int-iff,
mul-distributes,
mul-commutes,
mul-distributes-right,
zero-mul,
add-zero,
not-equal-implies-less,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
omega-shadow,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
le_weakening,
minus-zero,
mul_bounds_1a,
le_weakening2,
add_nat_wf,
multiply_nat_wf,
uiff_transitivity,
int_seg_subtype_nat,
seq-truncate-truncate
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
unionElimination,
productElimination,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
applyEquality,
because_Cache,
productEquality,
setEquality,
isect_memberFormation,
multiplyEquality,
independent_pairEquality,
imageElimination,
baseClosed,
imageMemberEquality,
minusEquality,
addEquality,
applyLambdaEquality,
Error :inhabitedIsType,
equalityElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType2,
baseApply,
closedConclusion,
promote_hyp,
instantiate,
cumulativity,
functionEquality,
Error :equalityIsType1,
Error :dependent_set_memberEquality_alt,
Error :isect_memberEquality_alt,
Error :productIsType,
Error :equalityIsType4,
dependentIntersectionElimination,
sqequalIntensionalEquality,
dependentIntersection_memberEquality
Latex:
\mforall{}[g:SimpleGame]. \mforall{}[n:\mBbbN{}].
((win2strat(g;n) \mmember{} Type)
\mwedge{} (\mforall{}[s:win2strat(g;n)]. (strat2play(g;n;s) \mmember{} Type))
\mwedge{} (\mforall{}[s:win2strat(g;n)]. \mforall{}[f:strat2play(g;n;s)]. (||f|| \mmember{} \mBbbZ{}))
\mwedge{} (\mforall{}[s:win2strat(g;n)]. \mforall{}[f:strat2play(g;n;s)]. \mforall{}[k:\{(2 * n) + 2..||f|| + 1\msupminus{}\}].
(play-truncate(f;k) \mmember{} strat2play(g;n;s))))
Date html generated:
2019_06_20-PM-00_52_21
Last ObjectModification:
2019_01_02-PM-01_31_55
Theory : co-recursion-2
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