Nuprl Lemma : strat2play-add
∀[g:SimpleGame]. ∀[n:ℕ]. ∀[s:win2strat(g;n + 1)]. ∀[moves:strat2play(g;n;s)].
∀[x,y:Pos(g)].
(seq-add(seq-add(moves;x);y) ∈ strat2play(g;n + 1;s)) supposing ((x = (s moves) ∈ Pos(g)) and Legal1(x;y))
supposing ||moves|| = ((2 * n) + 2) ∈ ℤ
Proof
Definitions occuring in Statement :
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-len: ||moves||,
sg-legal1: Legal1(x;y),
sg-pos: Pos(g),
simple-game: SimpleGame,
seq-add: seq-add(s;x),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
cand: A c∧ B,
play-truncate: play-truncate(f;m),
assert: ↑b,
bnot: ¬bb,
pi1: fst(t),
seq-truncate: seq-truncate(s;n),
seq-add: seq-add(s;x),
seq-len: ||s||,
sequence: sequence(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
play-item: moves[i],
play-len: ||moves||,
it: ⋅,
unit: Unit,
bool: 𝔹,
strat2play: strat2play(g;n;s),
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
less_than: a < b,
nat_plus: ℕ+,
ge: i ≥ j ,
exists: ∃x:A. B[x],
lelt: i ≤ j < k,
int_seg: {i..j-},
sq_type: SQType(T),
guard: {T},
top: Top,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
prop: ℙ,
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
win2strat: win2strat(g;n),
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
seq-add-item,
multiply_nat_wf,
add_nat_wf,
mul_bounds_1a,
seq-item_wf,
int_seg_wf,
assert-bnot,
bool_cases_sqequal,
assert_of_lt_int,
lt_int_wf,
sequence_wf,
seq-add-len,
seq-len_wf,
seq-add_wf,
strat2play-longer,
multiply-is-int-iff,
set_subtype_base,
add-is-int-iff,
strat2play_subtype,
strat2play-add1,
uiff_transitivity,
sg-legal1_wf,
nat_properties,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
bool_cases,
less-iff-le,
less_than_wf,
omega-shadow,
minus-zero,
two-mul,
one-mul,
le_reflexive,
subtype_rel_self,
not-equal-implies-less,
lelt_wf,
le-add-cancel2,
mul-distributes-right,
mul-associates,
not-lt-2,
decidable__lt,
le_weakening2,
subtract_wf,
play-item_wf,
sg-legal2_wf,
sg-pos_wf,
mul-commutes,
mul-distributes,
int_subtype_base,
subtype_base_sq,
add-subtract-cancel,
equal-wf-T-base,
not_wf,
bnot_wf,
assert_wf,
le_antisymmetry_iff,
eq_int_wf,
simple-game_wf,
win2strat_wf,
nat_wf,
strat2play_wf,
zero-mul,
add-mul-special,
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,
win2strat_subtype,
play-len_wf,
equal_wf
Rules used in proof :
hyp_replacement,
productEquality,
functionEquality,
functionExtensionality,
dependent_pairEquality,
closedConclusion,
baseApply,
applyLambdaEquality,
equalityElimination,
dependentIntersection_memberEquality,
axiomEquality,
impliesFunctionality,
promote_hyp,
sqequalIntensionalEquality,
dependent_pairFormation,
setEquality,
cumulativity,
instantiate,
dependentIntersectionElimination,
equalitySymmetry,
equalityTransitivity,
voidEquality,
isect_memberEquality,
lambdaEquality,
multiplyEquality,
minusEquality,
because_Cache,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_isectElimination,
independent_functionElimination,
productElimination,
voidElimination,
lambdaFormation,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
applyEquality,
hypothesisEquality,
intEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
sqequalHypSubstitution,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[g:SimpleGame]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:win2strat(g;n + 1)]. \mforall{}[moves:strat2play(g;n;s)].
\mforall{}[x,y:Pos(g)].
(seq-add(seq-add(moves;x);y) \mmember{} strat2play(g;n + 1;s)) supposing
((x = (s moves)) and
Legal1(x;y))
supposing ||moves|| = ((2 * n) + 2)
Date html generated:
2018_07_25-PM-01_33_33
Last ObjectModification:
2018_06_25-AM-11_35_04
Theory : co-recursion
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