Nuprl Lemma : coW-game-step-isom
∀[A:𝕌']
∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]). ∀t:coW-dom(a.B[a];w). ∀b:coW-dom(a.B[a];w').
sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))) ≅ coW-game(a.B[a];w;w')@<copath-cons(t;())
, copath-cons(b;())
>
Proof
Definitions occuring in Statement :
coW-game: coW-game(a.B[a];w;w'),
copath-cons: copath-cons(b;x),
copath-nil: (),
coW-item: coW-item(w;b),
coW-dom: coW-dom(a.B[a];w),
coW: coW(A;a.B[a]),
sg-normalize: sg-normalize(g),
sg-change-init: g@j,
isom-games: g1 ≅ g2,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
pair: <a, b>,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
sg-pos: Pos(g),
pi1: fst(t),
sg-normalize: sg-normalize(g),
sg-change-init: g@j,
spreadn: spread4,
coW-game: coW-game(a.B[a];w;w'),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
less_than: a < b,
ge: i ≥ j ,
guard: {T},
squash: ↓T,
cand: A c∧ B,
nat_plus: ℕ+,
nat: ℕ,
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),
implies: P ⇒ Q,
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,
and: P ∧ Q,
exists: ∃x:A. B[x],
sg-reachable: sg-reachable(g;x;y),
pi2: snd(t),
sg-init: InitialPos(g),
sq_stable: SqStable(P),
copath: copath(a.B[a];w),
sg-legal1: Legal1(x;y),
sg-legal2: Legal2(x;y),
coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),
copath-nil: (),
copath-length: copath-length(p),
copath-cons: copath-cons(b;x),
seq-comp: f o s,
seq-len: ||s||,
seq-item: s[i],
sequence: sequence(T),
respects-equality: respects-equality(S;T),
sq_type: SQType(T),
isom-games: g1 ≅ g2
Lemmas referenced :
copath-cons_wf,
sg-reachable_wf,
coW-game_wf,
copath-nil_wf,
coW-item_wf,
sg-pos_wf,
sg-normalize_wf,
coW-dom_wf,
coW_wf,
istype-universe,
omega-shadow,
minus-zero,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
iff_weakening_equal,
copath_wf,
subtype_rel_self,
true_wf,
equal_wf,
le-add-cancel2,
sg-legal2_wf,
nat_plus_wf,
mul-associates,
int_subtype_base,
le_wf,
set_subtype_base,
istype-sqequal,
sg-legal1_wf,
squash_wf,
istype-nat,
le-add-cancel-alt,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
istype-int,
minus-add,
istype-void,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
istype-le,
istype-false,
seq-item_wf,
seq-len_wf,
istype-less_than,
seq-comp_wf,
seq-comp-len,
seq-comp-item,
nat_properties,
nat_plus_properties,
sq_stable__le,
multiply_nat_wf,
add_nat_wf,
mul_bounds_1a,
copathAgree-cons,
copathAgree_wf,
copath-length_wf,
length-copath-cons,
subtract_nat_wf,
nat_plus_subtype_nat,
sq_stable__and,
less_than_wf,
copath-hd_wf,
sq_stable__less_than,
sq_stable__equal,
member-less_than,
coW-game-reachable,
sg-change-init_wf,
hd-copathAgree,
copath-hd-cons,
sg-pos-normalize,
sg-pos-change-init,
sg-init_wf,
subtype_rel-equal,
copath-tl_wf,
int_seg_wf,
seq-truncate_wf,
seq-len-truncate,
seq-truncate-item,
int_seg_properties,
coW-pos-agree_wf,
copath-tl-cons,
subtype-respects-equality,
le_antisymmetry_iff,
subtype_base_sq,
length-copath-tl,
copathAgree-tl,
multiply-is-int-iff,
mul_preserves_le,
sg-legal1-change-init,
sg-legal1-normalize,
sg-legal2-change-init,
sg-legal2-normalize,
copath-eta2,
subtype_rel_transitivity,
respects-equality-set-trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
sqequalRule,
setElimination,
thin,
rename,
productElimination,
Error :dependent_set_memberEquality_alt,
independent_pairEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
Error :universeIsType,
hypothesis,
Error :inhabitedIsType,
instantiate,
cumulativity,
Error :functionIsType,
universeEquality,
baseClosed,
imageMemberEquality,
productEquality,
spreadEquality,
imageElimination,
promote_hyp,
Error :equalityIsType1,
intEquality,
multiplyEquality,
minusEquality,
Error :isect_memberEquality_alt,
addEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
unionElimination,
dependent_functionElimination,
closedConclusion,
independent_pairFormation,
Error :equalityIstype,
natural_numberEquality,
Error :productIsType,
equalitySymmetry,
equalityTransitivity,
because_Cache,
Error :dependent_pairFormation_alt,
Error :unionIsType,
Error :inrFormation_alt,
sqequalBase,
Error :inlFormation_alt,
axiomEquality,
Error :functionIsTypeImplies,
applyLambdaEquality,
Error :dependent_pairEquality_alt,
hyp_replacement,
baseApply,
setEquality,
Error :setIsType
Latex:
\mforall{}[A:\mBbbU{}']
\mforall{}B:A {}\mrightarrow{} Type. \mforall{}w,w':coW(A;a.B[a]). \mforall{}t:coW-dom(a.B[a];w). \mforall{}b:coW-dom(a.B[a];w').
sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))) \mcong{}
coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>
Date html generated:
2019_06_20-PM-01_11_35
Last ObjectModification:
2019_01_02-PM-01_35_15
Theory : co-recursion-2
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