Nuprl Lemma : transEquiv-trans-eq
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}].
(transEquivFun(p)
= (λI,a,J,f,u. ((snd((p(s(a)) I+new-name(I) 1 <new-name(I)>))) J new-name(J) f,new-name(I)=new-name(J) 0 ⋅
(compOp(A) J new-name(J) s(f(a)) 0 ⋅ u)))
∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Proof
Definitions occuring in Statement :
transEquiv-trans: transEquivFun(p),
universe-comp-op: compOp(t),
universe-decode: decode(t),
cubical-universe: c𝕌,
path-type: (Path_A a b),
cubical-fun: (A ⟶ B),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
face_lattice: face_lattice(I),
cube-set-restriction: f(s),
cubical_set: CubicalSet,
nc-e': g,i=j,
nc-s: s,
new-name: new-name(I),
add-name: I+i,
nh-id: 1,
dM_inc: <x>,
it: ⋅,
uall: ∀[x:A]. B[x],
pi2: snd(t),
apply: f a,
lambda: λx.A[x],
equal: s = t ∈ T,
lattice-0: 0
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
cubical-term: {X ⊢ _:A},
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
squash: ↓T,
lattice-join: a ∨ b,
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
fset-constrained-ac-lub: lub(P;ac1;ac2),
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
lattice-0: 0,
empty-fset: {},
nil: [],
it: ⋅,
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
implies: P ⇒ Q,
cubical-term-at: u(a),
true: True,
cube+: cube+(I;i),
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
names: names(I),
nat: ℕ,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nc-s: s,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
dM: dM(I),
dM-lift: dM-lift(I;J;f),
so_lambda: λ2x.t[x],
so_apply: x[s],
dM_inc: <x>,
dminc: <i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
DeMorgan-algebra: DeMorganAlgebra,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
universe-comp-op: compOp(t),
composition-op: Gamma ⊢ CompOp(A),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-point: Point(l),
face-presheaf: 𝔽,
cube-set-restriction: f(s),
pi2: snd(t),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
nc-e': g,i=j,
cubical-universe: c𝕌,
fibrant-type: FibrantType(X),
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
closed-cubical-universe: cc𝕌,
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
closed-type-to-type: closed-type-to-type(T),
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
universe-type: universe-type(t;I;a),
nc-0: (i0),
lattice-1: 1,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
nequal: a ≠ b ∈ T
Lemmas referenced :
cubical-term-at_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cubical-fun_wf,
universe-decode_wf,
transEquiv-trans_wf,
istype-cubical-term,
path-type_wf,
cubical-universe_wf,
istype-cubical-universe-term,
cubical_set_wf,
cubical_type_at_pair_lemma,
transEquiv-trans-sq,
cubical-type-at_wf,
cube-set-restriction_wf,
names-hom_wf,
istype-cubical-type-at,
cubical-type-ap-morph_wf,
nh-comp_wf,
subtype_rel-equal,
cube-set-restriction-comp,
I_cube_pair_redex_lemma,
eq_int_wf,
new-name_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-added-name,
equal_wf,
squash_wf,
true_wf,
istype-universe,
add-name_wf,
nc-s_wf,
f-subset-add-name,
nh-id-right,
subtype_rel_self,
iff_weakening_equal,
dM-lift-is-id2,
f-subset_weakening,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
f-subset_wf,
dM_inc_wf,
names-subtype,
names_wf,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM-point-subtype,
dM1_wf,
dma-neg_wf,
trivial-member-add-name1,
fset-member_wf,
nh-id_wf,
s-comp-if-lemma1,
dM-lift_wf2,
dM-lift-inc,
not-new-name,
universe-comp-op_wf,
lattice-0_wf,
face_lattice_wf,
face-presheaf_wf2,
empty-cubical-subset-term,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-term_wf,
cubical-subset_wf,
csm-ap-type_wf,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
cubical-path-0_wf,
cubical-path-1_wf,
nc-0_wf,
cube-set-restriction-when-id,
s-comp-nc-0-new,
cubical-path-condition-0,
cubical-path-condition_wf,
nc-e'_wf,
dM1-meet,
nh-id-left,
cube-set-restriction-id,
path-type-at,
pi2_wf,
cubical-type_wf,
composition-op_wf,
pi1_wf_top,
not_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
cube_set_map_wf,
cubical-term-eqcd,
istype-nat,
istype-void,
cube_set_restriction_pair_lemma,
nc-e'-lemma2,
universe-type-at,
nh-comp-assoc,
cubical_type_ap_morph_pair_lemma,
lifting-strict-spread,
strict4-spread,
csm-ap-type-at,
csm-ap-context-map,
dM0_wf,
dM0-sq-empty,
eq_int_eq_true,
btrue_wf,
not_assert_elim,
btrue_neq_bfalse,
intformeq_wf,
int_formula_prop_eq_lemma,
dM-lift-0-sq,
interval-type-ap-morph,
nc-1_wf,
s-comp-nc-1-new,
member-empty-cubical-subset,
nc-e'-lemma1,
universe-path-type-lemma-1,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
functionExtensionality,
sqequalRule,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_functionElimination,
universeIsType,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
Error :memTop,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination,
functionIsType,
natural_numberEquality,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation_alt,
promote_hyp,
cumulativity,
voidElimination,
universeEquality,
intEquality,
hyp_replacement,
productEquality,
isectEquality,
functionEquality,
independent_pairEquality,
setEquality,
approximateComputation,
int_eqEquality,
independent_pairFormation,
setIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[p:\{G \mvdash{} \_:(Path\_c\mBbbU{} A B)\}].
(transEquivFun(p)
= (\mlambda{}I,a,J,f,u. ((snd((p(s(a)) I+new-name(I) 1 <new-name(I)>))) J new-name(J)
f,new-name(I)=new-name(J)
0
\mcdot{}
(compOp(A) J new-name(J) s(f(a)) 0 \mcdot{} u))))
Date html generated:
2020_05_20-PM-07_37_09
Last ObjectModification:
2020_05_01-AM-10_11_18
Theory : cubical!type!theory
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