Nuprl Lemma : coW-play-invariant
∀[A:𝕌']
∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]). ∀n:ℕ. ∀s:win2strat(coW-game(a.B[a];w;w');n).
∀moves:strat2play(coW-game(a.B[a];w;w');n;s). ∀i:ℕ.
(((i ≤ n)
⇒ (coW-pos-lens(moves[2 * i];i;i)
∧ (coW-pos-lens(moves[(2 * i) + 1];i;i + 1) ∨ coW-pos-lens(moves[(2 * i) + 1];i + 1;i))))
∧ ((i ≤ ((2 * n) + 1)) ⇒ (∀j:ℕi + 1. coW-pos-agree(a.B[a];w;w';moves[j];moves[i]))))
Proof
Definitions occuring in Statement :
coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),
coW-pos-lens: coW-pos-lens(p;i;j),
coW-game: coW-game(a.B[a];w;w'),
coW: coW(A;a.B[a]),
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-item: moves[i],
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
exists: ∃x:A. B[x],
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sg-legal2: Legal2(x;y),
coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),
sq_type: SQType(T),
sg-legal1: Legal1(x;y),
sg-pos: Pos(g),
copath-length: copath-length(p),
copath-nil: (),
pi2: snd(t),
pi1: fst(t),
coW-pos-lens: coW-pos-lens(p;i;j),
sg-init: InitialPos(g),
coW-game: coW-game(a.B[a];w;w'),
play-item: moves[i],
cand: A c∧ B,
less_than: a < b,
nat_plus: ℕ+,
squash: ↓T,
sq_stable: SqStable(P),
guard: {T},
true: True,
top: Top,
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW-pos-agree_transitivity,
not-equal-implies-less,
not-equal-2,
sq_stable__copathAgree,
and_wf,
rem_bounds_1,
div_bounds_1,
nequal_wf,
equal-wf-base,
div_rem_sum,
le_weakening2,
subtract-add-cancel,
less_than_irreflexivity,
less_than_transitivity1,
int_seg_cases,
int_seg_subtype,
copathAgree_refl,
minus-zero,
int_seg_properties,
equal-wf-base-T,
copath-nil_wf,
copathAgree_wf,
copath_wf,
equal-wf-T-base,
le_antisymmetry_iff,
copath-length_wf,
decidable__int_equal,
subtype_base_sq,
simple-game_wf,
sg-legal1_wf,
iff_weakening_equal,
subtype_rel_self,
sg-pos_wf,
true_wf,
squash_wf,
nat_properties,
le-add-cancel-alt,
mul-swap,
omega-shadow,
zero-mul,
mul-distributes-right,
two-mul,
add-mul-special,
one-mul,
le_reflexive,
mul-commutes,
mul-distributes,
coW_wf,
win2strat_wf,
strat2play_wf,
primrec-wf2,
less_than_wf,
set_wf,
le-add-cancel2,
mul-associates,
not-lt-2,
decidable__lt,
multiply-is-int-iff,
int_subtype_base,
set_subtype_base,
add-is-int-iff,
coW-pos-agree_wf,
int_seg_wf,
all_wf,
equal_wf,
sq_stable__le,
nat_wf,
multiply_nat_wf,
add_nat_wf,
or_wf,
lelt_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
decidable__le,
false_wf,
mul_bounds_1a,
play-item_wf,
coW-pos-lens_wf,
subtract_wf,
le_wf,
coW-game_wf,
strat2play-invariant-1
Rules used in proof :
applyLambdaEquality,
hyp_replacement,
remainderEquality,
sqequalIntensionalEquality,
dependent_pairFormation,
divideEquality,
andLevelFunctionality,
orFunctionality,
equalityUniverse,
promote_hyp,
axiomEquality,
independent_pairEquality,
hypothesis_subsumption,
inrFormation,
inlFormation,
levelHypothesis,
addLevel,
closedConclusion,
baseApply,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
addEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
unionElimination,
independent_pairFormation,
multiplyEquality,
dependent_set_memberEquality,
functionExtensionality,
instantiate,
universeEquality,
cumulativity,
because_Cache,
natural_numberEquality,
functionEquality,
productEquality,
setElimination,
rename,
productElimination,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']
\mforall{}B:A {}\mrightarrow{} Type. \mforall{}w,w':coW(A;a.B[a]). \mforall{}n:\mBbbN{}. \mforall{}s:win2strat(coW-game(a.B[a];w;w');n).
\mforall{}moves:strat2play(coW-game(a.B[a];w;w');n;s). \mforall{}i:\mBbbN{}.
(((i \mleq{} n)
{}\mRightarrow{} (coW-pos-lens(moves[2 * i];i;i)
\mwedge{} (coW-pos-lens(moves[(2 * i) + 1];i;i + 1) \mvee{} coW-pos-lens(moves[(2 * i) + 1];i + 1;i))))
\mwedge{} ((i \mleq{} ((2 * n) + 1)) {}\mRightarrow{} (\mforall{}j:\mBbbN{}i + 1. coW-pos-agree(a.B[a];w;w';moves[j];moves[i]))))
Date html generated:
2018_07_25-PM-01_43_44
Last ObjectModification:
2018_06_16-PM-00_20_17
Theory : co-recursion
Home
Index