Nuprl Lemma : coW-game-reachable
∀[A:𝕌']
∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]). ∀p,q:Pos(coW-game(a.B[a];w;w')).
(sg-reachable(coW-game(a.B[a];w;w');p;q) ⇒ coW-pos-agree(a.B[a];w;w';p;q))
Proof
Definitions occuring in Statement :
coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),
coW-game: coW-game(a.B[a];w;w'),
coW: coW(A;a.B[a]),
sg-reachable: sg-reachable(g;x;y),
sg-pos: Pos(g),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
sg-legal2: Legal2(x;y),
pi2: snd(t),
sg-legal1: Legal1(x;y),
coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),
pi1: fst(t),
sg-pos: Pos(g),
coW-game: coW-game(a.B[a];w;w'),
ge: i ≥ j ,
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
less_than: a < b,
sq_stable: SqStable(P),
less_than': less_than'(a;b),
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
nat_plus: ℕ+,
false: False,
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
sg-reachable: sg-reachable(g;x;y),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
copathAgree_refl,
copathAgree_wf,
copath_wf,
and_wf,
copath-length_wf,
sq_stable__coW-pos-agree,
add-is-int-iff,
not-equal-implies-less,
add_functionality_wrt_eq,
mul-commutes,
equal-wf-T-base,
or_wf,
div_bounds_1,
rem_bounds_1,
nequal_wf,
equal-wf-base,
div_rem_sum,
le_weakening,
coW-pos-agree_transitivity,
coW-pos-agree_refl,
subtype_base_sq,
seq-truncate-item,
seq-len-truncate,
le_antisymmetry_iff,
seq-truncate_wf,
not-equal-2,
decidable__int_equal,
nat_properties,
nat_plus_properties,
omega-shadow,
minus-zero,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
primrec-wf2,
set_wf,
le-add-cancel-alt,
zero-mul,
add-mul-special,
add-zero,
minus-minus,
zero-add,
not-le-2,
decidable__le,
int_subtype_base,
set_subtype_base,
sq_stable__le,
multiply_nat_wf,
add_nat_wf,
le-add-cancel,
mul-associates,
minus-add,
nat_plus_subtype_nat,
mul_bounds_1a,
lelt_wf,
le-add-cancel2,
add-commutes,
add_functionality_wrt_le,
less-iff-le,
minus-one-mul-top,
add-swap,
minus-one-mul,
add-associates,
condition-implies-le,
not-lt-2,
false_wf,
decidable__lt,
equal_wf,
le_weakening2,
subtract_wf,
sequence_wf,
le_wf,
sg-legal1_wf,
nat_wf,
seq-item_wf,
sg-legal2_wf,
seq-len_wf,
less_than_wf,
nat_plus_wf,
all_wf,
less_than_irreflexivity,
less_than_transitivity1,
sg-reachable_wf,
iff_weakening_equal,
subtype_rel_self,
coW_wf,
coW-game_wf,
sg-pos_wf,
true_wf,
squash_wf,
coW-pos-agree_wf
Rules used in proof :
applyLambdaEquality,
hyp_replacement,
levelHypothesis,
inrFormation,
inlFormation,
remainderEquality,
closedConclusion,
baseApply,
productEquality,
divideEquality,
addLevel,
intEquality,
voidEquality,
isect_memberEquality,
minusEquality,
unionElimination,
promote_hyp,
sqequalIntensionalEquality,
dependent_pairFormation,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
addEquality,
functionExtensionality,
rename,
setElimination,
multiplyEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
because_Cache,
universeEquality,
functionEquality,
sqequalRule,
cumulativity,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
imageElimination,
lambdaEquality,
instantiate,
applyEquality,
cut,
thin,
productElimination,
sqequalHypSubstitution,
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{}p,q:Pos(coW-game(a.B[a];w;w')).
(sg-reachable(coW-game(a.B[a];w;w');p;q) {}\mRightarrow{} coW-pos-agree(a.B[a];w;w';p;q))
Date html generated:
2018_07_25-PM-01_48_21
Last ObjectModification:
2018_06_20-PM-03_17_30
Theory : co-recursion
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