Nuprl Lemma : coWtransInvariant_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w1,w2,w3:coW(A;a.B[a])]. ∀[k:ℕ]. ∀[X:win2strat(coW-game(a.B[a];w1;w2);k + 1)].
∀[Y:win2strat(coW-game(a.B[a];w2;w3);k + 1)]. ∀[a:strat2play(coW-game(a.B[a];w1;w2);k;X)].
∀[b:strat2play(coW-game(a.B[a];w2;w3);k;Y)]. ∀[m1:sequence(Pos(coW-game(a.B[a];w1;w3)))].
coWtransInvariant(a.B[a];w1;w2;w3;k;X;Y;a;b;m1) ∈ ℙ supposing ((2 * k) + 2) ≤ ||m1||
Proof
Definitions occuring in Statement :
coWtransInvariant: coWtransInvariant(x.B[x];w1;w2;w3;k;X;Y;a;b;moves),
coW-game: coW-game(a.B[a];w;w'),
coW: coW(A;a.B[a]),
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
sg-pos: Pos(g),
seq-len: ||s||,
sequence: sequence(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
member: t ∈ T,
function: x:A ⟶ B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
less_than: a < b,
ge: i ≥ j ,
exists: ∃x:A. B[x],
sq_type: SQType(T),
nat_plus: ℕ+,
pi2: snd(t),
pi1: fst(t),
sg-pos: Pos(g),
coW-game: coW-game(a.B[a];w;w'),
guard: {T},
lelt: i ≤ j < k,
int_seg: {i..j-},
true: True,
less_than': less_than'(a;b),
top: Top,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
prop: ℙ,
false: False,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
coWtransInvariant: coWtransInvariant(x.B[x];w1;w2;w3;k;X;Y;a;b;moves),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat_properties,
omega-shadow,
minus-zero,
two-mul,
one-mul,
le_reflexive,
le_weakening2,
sg-legal2_wf,
set_wf,
subtype_base_sq,
mul-commutes,
mul-distributes,
minus-minus,
subtract_wf,
less_than_wf,
win2strat-properties,
coW_wf,
win2strat_wf,
strat2play_wf,
copath-length_wf,
or_wf,
seq-len_wf,
sequence_wf,
subtype_rel_self,
seq-item_wf,
copath_wf,
play-len_wf,
sg-pos_wf,
lelt_wf,
le-add-cancel2,
mul-distributes-right,
mul-associates,
not-lt-2,
decidable__lt,
equal_wf,
nat_wf,
multiply_nat_wf,
add_nat_wf,
zero-mul,
add-mul-special,
multiply-is-int-iff,
set_subtype_base,
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,
int_subtype_base,
add-is-int-iff,
coW-game_wf,
win2strat_subtype,
play-item_wf
Rules used in proof :
promote_hyp,
sqequalIntensionalEquality,
dependent_pairFormation,
setEquality,
universeEquality,
functionEquality,
instantiate,
cumulativity,
axiomEquality,
independent_pairEquality,
functionExtensionality,
productEquality,
equalitySymmetry,
equalityTransitivity,
multiplyEquality,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
imageElimination,
imageMemberEquality,
independent_functionElimination,
voidElimination,
lambdaFormation,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
independent_isectElimination,
baseClosed,
closedConclusion,
baseApply,
productElimination,
natural_numberEquality,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
hypothesis,
lambdaEquality,
sqequalRule,
applyEquality,
hypothesisEquality,
because_Cache,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w1,w2,w3:coW(A;a.B[a])]. \mforall{}[k:\mBbbN{}]. \mforall{}[X:win2strat(coW-game(a.B[a];w1;w2);k
+ 1)].
\mforall{}[Y:win2strat(coW-game(a.B[a];w2;w3);k + 1)]. \mforall{}[a:strat2play(coW-game(a.B[a];w1;w2);k;X)].
\mforall{}[b:strat2play(coW-game(a.B[a];w2;w3);k;Y)]. \mforall{}[m1:sequence(Pos(coW-game(a.B[a];w1;w3)))].
coWtransInvariant(a.B[a];w1;w2;w3;k;X;Y;a;b;m1) \mmember{} \mBbbP{} supposing ((2 * k) + 2) \mleq{} ||m1||
Date html generated:
2018_07_25-PM-01_43_53
Last ObjectModification:
2018_07_09-PM-11_08_44
Theory : co-recursion
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