Nuprl Lemma : comp-fun-to-comp-op-inverse
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)].
(cop-to-cfun(cfun-to-cop(Gamma;A;cA)) = cA ∈ Gamma ⊢ Compositon(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-structure: Gamma ⊢ Compositon(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-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
prop: ℙ,
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
comp-op-to-comp-fun: cop-to-cfun(cA),
comp-fun-to-comp-op1: comp-fun-to-comp-op1(Gamma;A;comp),
csm-composition: (comp)sigma,
composition-term: comp cA [phi ⊢→ u] a0,
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
names-hom: I ⟶ J,
formal-cube: formal-cube(I),
pi1: fst(t),
functor-ob: ob(F),
I_cube: A(I),
true: True,
squash: ↓T,
implies: P ⇒ Q,
cubical-term-at: u(a),
csm+: tau+,
csm-comp: G o F,
cube-context-adjoin: X.A,
cube-set-restriction: f(s),
pi2: snd(t),
constant-cubical-type: (X),
cc-fst: p,
cc-snd: q,
compose: f o g,
interval-type: 𝕀,
cc-adjoin-cube: (v;u),
functor-arrow: arrow(F),
cube+: cube+(I;i),
context-map: <rho>,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
false: False,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
so_apply: x[s],
guard: {T},
and: P ∧ Q,
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
free-dist-lattice: free-dist-lattice(T; eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
bfalse: ff,
eq_atom: x =a y,
record-update: r[x := v],
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dM: dM(I),
record-select: r.x,
lattice-point: Point(l),
interval-presheaf: 𝕀,
cubical-type-at: A(a),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nat: ℕ,
names: names(I),
nc-s: s,
dM-lift: dM-lift(I;J;f),
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
nh-comp: g ⋅ f,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
not: ¬A,
nequal: a ≠ b ∈ T ,
csm-ap-term: (t)s,
lattice-hom: Hom(l1;l2),
bounded-lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
face-lattice: face-lattice(T;eq),
face_lattice: face_lattice(I),
face-presheaf: 𝔽,
face-type: 𝔽,
subset-iota: iota,
context-subset: Gamma, phi,
interval-0: 0(𝕀),
canonical-section: canonical-section(Gamma;A;I;rho;a),
nc-0: (i0),
nat-trans: nat-trans(C;D;F;G),
psc_map: A ⟶ B,
cube_set_map: A ⟶ B,
type-cat: TypeCat,
cat-arrow: cat-arrow(C)
Lemmas referenced :
istype-cubical-term,
context-subset_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-1,
csm-context-subset-subtype2,
csm-ap-term_wf,
csm-context-subset-subtype3,
subset-cubical-term2,
sub_cubical_set_self,
thin-context-subset-adjoin,
csm-id-adjoin_wf,
interval-1_wf,
subset-cubical-term,
context-subset-is-subset,
interval-0_wf,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term-eqcd,
cubical-term-eqcd,
face-type_wf,
cube_set_map_wf,
cubical_set_wf,
uniform-comp-function_wf,
composition-structure_wf,
cubical-type_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
formal-cube_wf1,
context-map_wf,
nh-id_wf,
cubical-term-at_wf,
cubical_set_cumulativity-i-j,
csm-ap_wf,
cubical-type-at_wf,
csm-ap-type-at,
cube-set-restriction-id,
csm-ap-context-map,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-ap-term-at,
subtype_rel-equal,
cubical-type-cumulativity2,
csm-comp_wf,
csm+_wf_interval,
csm-equal,
I_cube_pair_redex_lemma,
iff_weakening_equal,
names_wf,
not-added-name,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
dM_wf,
lattice-point_wf,
subtype_rel_self,
eq_int_wf,
new-name_wf,
add-name_wf,
csm-ap-restriction,
interval-type-at,
cube-set-restriction_wf,
strong-subtype-self,
istype-int,
le_wf,
strong-subtype-set3,
strong-subtype-deq-subtype,
int-deq_wf,
fset-member_wf,
trivial-member-add-name1,
dM_inc_wf,
f-subset-add-name,
nc-s_wf,
istype-cubical-type-at,
cube_set_restriction_pair_lemma,
cube-set-restriction-comp,
assert_of_eq_int,
eqtt_to_assert,
names-subtype,
dM-lift-inc,
not_wf,
set_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
deq_wf,
interval-type-ap-morph,
not_assert_elim,
btrue_wf,
btrue_neq_bfalse,
bfalse_wf,
eq_int_eq_true,
bnot_wf,
assert_elim,
csm-face-type,
canonical-section-at,
face-type-ap-morph,
face-type-at,
fl-morph_wf,
face_lattice_wf,
cubical-term-at-morph,
context-subset-map,
csm-ap-comp-type-sq,
arrow_pair_lemma,
names-hom_wf,
int_subtype_base,
new-name-property,
csm-ap-csm-comp,
cubical_type_at_pair_lemma,
cubical-term_wf,
cubical-type-cumulativity,
s-comp-nc-0-new,
nh-comp_wf,
cube-set-restriction-when-id,
nc-0_wf,
dM0_wf,
dM0-sq-empty,
cc-adjoin-cube_wf,
cubical-type-ap-morph_wf,
csm-cubical-type-ap-morph,
context-subset-term-subtype,
constrained-cubical-term_wf,
dM1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
dependent_set_memberEquality_alt,
functionExtensionality,
equalityIstype,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
productElimination,
equalityTransitivity,
universeIsType,
lambdaEquality_alt,
inhabitedIsType,
dependent_functionElimination,
applyLambdaEquality,
hyp_replacement,
natural_numberEquality,
baseClosed,
imageMemberEquality,
imageElimination,
Error :memTop,
universeEquality,
independent_functionElimination,
lambdaFormation_alt,
voidElimination,
promote_hyp,
dependent_pairFormation_alt,
isectEquality,
cumulativity,
productEquality,
equalityElimination,
unionElimination,
dependent_pairEquality_alt,
intEquality,
lambdaEquality,
lambdaFormation,
levelHypothesis,
independent_pairFormation,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
dependent_set_memberEquality,
addLevel,
productIsType,
functionEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} Compositon(A)].
(cop-to-cfun(cfun-to-cop(Gamma;A;cA)) = cA)
Date html generated:
2020_05_20-PM-04_35_07
Last ObjectModification:
2020_05_01-PM-09_20_18
Theory : cubical!type!theory
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