Nuprl Lemma : coW-trans_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w1,w2,w3:coW(A;a.B[a])]. ∀[n:ℕ]. ∀[X:win2strat(coW-game(a.B[a];w1;w2);n)].
∀[Y:win2strat(coW-game(a.B[a];w2;w3);n)].
(coW-trans(X; Y) ∈ win2strat(coW-game(a.B[a];w1;w3);n))
Proof
Definitions occuring in Statement :
coW-trans: coW-trans(X; Y),
coW-game: coW-game(a.B[a];w;w'),
coW: coW(A;a.B[a]),
win2strat: win2strat(g;n),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
coW-pos-lens: coW-pos-lens(p;i;j),
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
coW-trans: coW-trans(X; Y),
play-truncate: play-truncate(f;m),
copathAgree: copathAgree(a.B[a];w;x;y),
copath: copath(a.B[a];w),
copath-length: copath-length(p),
label: ...$L... t,
copath-nil: (),
coWtransInvariant: coWtransInvariant(x.B[x];w1;w2;w3;k;X;Y;a;b;moves),
sg-legal2: Legal2(x;y),
cand: A c∧ B,
seq-item: s[i],
assert: ↑b,
bnot: ¬bb,
exists: ∃x:A. B[x],
seq-nil: seq-nil(),
seq-add: seq-add(s;x),
let: let,
lt_int: i <z j,
pi2: snd(t),
sg-legal1: Legal1(x;y),
sg-init: InitialPos(g),
sg-pos: Pos(g),
coW-game: coW-game(a.B[a];w;w'),
squash: ↓T,
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
play-item: moves[i],
sequence: sequence(T),
strat2play: strat2play(g;n;s),
seq-truncate: seq-truncate(s;n),
pi1: fst(t),
seq-len: ||s||,
play-len: ||moves||,
transMoves: transMoves(X;Y;moves),
sq_type: SQType(T),
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
true: True,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
not: ¬A,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
top: Top,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
win2strat: win2strat(g;n),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
neg_assert_of_eq_int,
le_weakening,
strat2play-longer,
subtype_rel_set,
seq-add_wf,
seq-add-len,
copathAgree_refl,
add-subtract-cancel,
strat2play-add,
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
le_int_wf,
mod2-2n,
mod2-2n-plus-1,
strat2play-invariant,
coW-play-invariant,
seq-truncate-item,
mul-distributes-right,
win2strat-properties,
bfalse_wf,
not-equal-2,
eq_int_eq_false,
subtract-add-cancel,
seq-truncate_wf,
add-is-int-iff,
seq-len-truncate,
zero-mul,
add-mul-special,
strat2play_subtype_le,
play-truncate_wf,
mul-associates,
mul_preserves_le,
le-add-cancel-alt,
sequence_wf,
strat2play_subtype,
or_wf,
subtype_rel_wf,
member_wf,
pi2_wf,
pi1_wf,
and_wf,
coWtransInvariant_wf,
le_antisymmetry_iff,
copathAgree-nil,
sg-legal2_wf,
minus-zero,
le-add-cancel2,
sg-init_wf,
not-lt-2,
decidable__lt,
seq-item_wf,
seq-len_wf,
copathAgree_wf,
copath_wf,
equal-wf-base-T,
int_seg_wf,
copath-nil_wf,
assert-bnot,
bool_cases_sqequal,
assert_of_lt_int,
lt_int_wf,
copath-length_wf,
set_subtype_base,
iff_weakening_equal,
subtype_rel_self,
simple-game_wf,
true_wf,
squash_wf,
sg-legal1_wf,
strat2play-invariant-1,
sg-pos_wf,
lelt_wf,
play-item_wf,
copath_length_nil_lemma,
le_transitivity,
set_wf,
mul-commutes,
mul-distributes,
le_reflexive,
bool_subtype_base,
subtype_base_sq,
bool_cases,
equal_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
play-len_wf,
equal-wf-T-base,
not-le-2,
strat2play_wf,
win2strat_subtype,
not_wf,
bnot_wf,
assert_wf,
int_subtype_base,
equal-wf-base,
bool_wf,
eq_int_wf,
coW_wf,
nat_wf,
le_weakening2,
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-ge-2,
subtract_wf,
decidable__le,
le_wf,
false_wf,
coW-game_wf,
win2strat_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties
Rules used in proof :
spreadEquality,
orFunctionality,
addLevel,
inrFormation,
levelHypothesis,
equalityUniverse,
dependentIntersectionEqElimination,
productEquality,
inlFormation,
promote_hyp,
dependent_pairFormation,
dependent_pairEquality,
independent_pairEquality,
applyLambdaEquality,
hyp_replacement,
imageElimination,
imageMemberEquality,
multiplyEquality,
impliesFunctionality,
equalityElimination,
setEquality,
dependentIntersection_memberEquality,
dependentIntersectionElimination,
baseClosed,
closedConclusion,
baseApply,
universeEquality,
functionEquality,
instantiate,
minusEquality,
intEquality,
addEquality,
productElimination,
unionElimination,
independent_pairFormation,
dependent_set_memberEquality,
voidEquality,
because_Cache,
functionExtensionality,
applyEquality,
cumulativity,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
isect_memberEquality,
dependent_functionElimination,
lambdaEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
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{}[n:\mBbbN{}].
\mforall{}[X:win2strat(coW-game(a.B[a];w1;w2);n)]. \mforall{}[Y:win2strat(coW-game(a.B[a];w2;w3);n)].
(coW-trans(X; Y) \mmember{} win2strat(coW-game(a.B[a];w1;w3);n))
Date html generated:
2018_07_25-PM-01_47_52
Last ObjectModification:
2018_07_11-AM-10_48_10
Theory : co-recursion
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