Nuprl Lemma : comp-op-to-comp-fun-inverse
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. (cfun-to-cop(Gamma;A;cop-to-cfun(cA)) = cA ∈ Gamma ⊢ CompOp(A))
Proof
Definitions occuring in Statement :
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
comp-op-to-comp-fun: cop-to-cfun(cA),
composition-op: Gamma ⊢ CompOp(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
composition-op: Gamma ⊢ CompOp(A),
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
interval-presheaf: 𝕀,
names: names(I),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
comp-op-to-comp-fun: cop-to-cfun(cA),
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
comp-fun-to-comp-op1: comp-fun-to-comp-op1(Gamma;A;comp),
csm-composition: (comp)sigma,
composition-term: comp cA [phi ⊢→ u] a0,
cubical-term-at: u(a),
canonical-section: canonical-section(Gamma;A;I;rho;a),
composition-uniformity: composition-uniformity(Gamma;A;comp),
names-hom: I ⟶ J,
nc-e': g,i=j,
cube+: cube+(I;i),
cc-adjoin-cube: (v;u),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
guard: {T},
cubical-type-at: A(a),
pi1: fst(t),
interval-type: 𝕀,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
DeMorgan-algebra: DeMorganAlgebra,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
pi2: snd(t),
nh-id: 1,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nc-s: s,
dM_inc: <x>,
dminc: <i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
context-map: <rho>,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g,
functor-arrow: arrow(F),
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
subset-iota: iota,
csm-ap-term: (t)s,
subset-trans: subset-trans(I;J;f;x),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
dM-lift: dM-lift(I;J;f),
cube-context-adjoin: X.A,
face-presheaf: 𝔽,
bdd-distributive-lattice: BoundedDistributiveLattice,
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),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
cubical-type-ap-morph: (u a f),
face-type: 𝔽,
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),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
nc-0: (i0),
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra-property,
free-dist-lattice-property,
empty-fset: {},
nil: [],
lattice-0: 0,
dM0: 0,
nequal: a ≠ b ∈ T ,
dma-hom: dma-hom(dma1;dma2),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
rev_uimplies: rev_uimplies(P;Q),
nc-1: (i1)
Lemmas referenced :
cubical-path-0_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-term_wf,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
sq_stable__composition-uniformity,
composition-uniformity_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
interval-type-at,
I_cube_pair_redex_lemma,
dM_inc_wf,
new-name_wf,
trivial-member-add-name1,
nh-id_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_rel_self,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-added-name,
names_wf,
istype-top,
squash_wf,
true_wf,
istype-universe,
nh-id-right,
names-subtype,
iff_weakening_equal,
names-hom_wf,
nc-e'_wf,
cubical-type-at_wf,
cube-set-restriction-id,
nc-1_wf,
cubical-type-ap-morph-id,
cubical-term-equal,
csm-ap-type-at,
cubical-subset-I_cube,
cube-set-restriction-comp,
csm-ap-context-map,
cubical-term-at_wf,
context-map_wf_cubical-subset,
interval-type-ap-morph,
cube_set_restriction_pair_lemma,
dM-lift_wf2,
dM-lift-inc,
nh-comp_wf,
name-morph-satisfies_wf,
face_lattice_wf,
fl-morph-comp2,
fl-morph_wf,
int_subtype_base,
name-morph-satisfies-comp,
nc-e'-lemma3,
csm-ap-restriction,
csm-ap_wf,
cubical-type-equal,
subtype_rel_product,
subtype_rel_universe1,
subtype_rel_dep_function,
equal_functionality_wrt_subtype_rel2,
csm-ap-term_wf,
subset-trans_wf,
cube_set_map_wf,
face-type-ap-morph,
nc-0_wf,
cube-context-adjoin_wf,
interval-type_wf,
cube+_wf,
cc-adjoin-cube_wf,
cat-ob_wf,
op-cat_wf,
cube-cat_wf,
cubical-type-ap-morph_wf,
cc-adjoin-cube-restriction,
formal-cube-restriction,
dM0_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
interval-type-ap-inc,
eq_int_eq_true,
btrue_wf,
not_assert_elim,
btrue_neq_bfalse,
dM-lift_wf,
set_subtype_base,
sq_stable__fset-member,
sq_stable__not,
subtype_rel-equal,
apply-fl-morph-id,
nh-comp-assoc,
nc-e'-1,
nc-e-comp-nc-0,
istype-void,
istype-cubical-type-at,
cube-set-restriction-when-id,
s-comp-nc-0,
csm-ap-term-at,
istype-cubical-term,
nc-e'-lemma2,
cubical-path-condition_wf,
cubical-path-1_wf,
dM1-sq-singleton-empty,
dM1_wf,
sq_stable__cubical-path-condition',
cubical-path-condition'_wf,
face-lattice-property,
free-dist-lattice-with-constraints-property,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
functionExtensionality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
setEquality,
intEquality,
imageMemberEquality,
baseClosed,
imageElimination,
inhabitedIsType,
equalityTransitivity,
lambdaFormation_alt,
equalityElimination,
productElimination,
productEquality,
cumulativity,
isectEquality,
equalityIstype,
promote_hyp,
universeEquality,
functionEquality,
hyp_replacement,
dependent_pairEquality_alt,
functionIsType,
closedConclusion,
sqequalBase,
applyLambdaEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)]. (cfun-to-cop(Gamma;A;cop-to-cfun(cA)) = cA)
Date html generated:
2020_05_20-PM-04_33_53
Last ObjectModification:
2020_04_21-AM-00_59_29
Theory : cubical!type!theory
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