Nuprl Lemma : comp-fun-to-comp-op_wf
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀[comp:Gamma ⊢ Compositon(A)]. (cfun-to-cop(Gamma;A;comp) ∈ Gamma ⊢ CompOp(A))
Proof
Definitions occuring in Statement :
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
composition-structure: Gamma ⊢ Compositon(A),
composition-op: Gamma ⊢ CompOp(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
composition-structure: Gamma ⊢ Compositon(A),
composition-op: Gamma ⊢ CompOp(A),
composition-uniformity: composition-uniformity(Gamma;A;comp),
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
comp-fun-to-comp-op1: comp-fun-to-comp-op1(Gamma;A;comp),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
subtype_rel: A ⊆r B,
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
formal-cube: formal-cube(I),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
unit: Unit,
trivial-cube-set: (),
face-presheaf: 𝔽,
lattice-point: Point(l),
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,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term-at: u(a),
ext-eq: A ≡ B,
lattice-hom: Hom(l1;l2),
bounded-lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
compose: f o g,
csm-ap: (s)x,
csm-comp: G o F,
subset-iota: iota,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
functor-arrow: arrow(F),
subset-trans: subset-trans(I;J;f;x),
context-map: <rho>,
interval-presheaf: 𝕀,
cube-context-adjoin: X.A,
cube+: cube+(I;i),
cc-fst: p,
pi2: snd(t),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
uiff: uiff(P;Q),
it: ⋅,
bool: 𝔹,
names: names(I),
DeMorgan-algebra: DeMorganAlgebra,
nc-s: s,
cat-comp: cat-comp(C),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
quotient: x,y:A//B[x; y],
fset: fset(T),
cube-cat: CubeCat,
spreadn: spread4,
op-cat: op-cat(C),
cat-ob: cat-ob(C),
nat-trans: nat-trans(C;D;F;G),
psc_map: A ⟶ B,
cube_set_map: A ⟶ B,
interval-type: 𝕀,
csm+: tau+,
csm-ap-term: (t)s,
cc-snd: q,
csm-adjoin: (s;u),
cubical-term: {X ⊢ _:A},
cubical-type-ap-morph: (u a 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,
context-subset: Gamma, phi,
nc-e': g,i=j,
nequal: a ≠ b ∈ T ,
name-morph-satisfies: (psi f) = 1,
top: Top,
dM-lift: dM-lift(I;J;f),
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra-property,
free-dist-lattice-property,
csm-id-adjoin: [u],
csm-id: 1(X),
interval-1: 1(𝕀),
composition-function: composition-function{j:l,i:l}(Gamma;A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
interval-0: 0(𝕀),
dM0: 0,
free-dist-lattice: free-dist-lattice(T; eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dM: dM(I),
lattice-0: 0,
nil: [],
empty-fset: {},
nc-0: (i0),
partial-term-0: u[0]
Lemmas referenced :
comp-fun-to-comp-op_wf1,
formal-cube_wf1,
context-map_wf,
subtype_rel_self,
I_cube_wf,
csm-comp_wf,
cube-context-adjoin_wf,
interval-type_wf,
add-name_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,
cube+_wf,
canonical-section_wf,
trivial-cube-set_wf,
face-type_wf,
it_wf,
cubical-type-at_wf_face-type,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
csm-face-type,
cubical-path-0_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
istype-cubical-term,
cubical-subset_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
subset-iota_wf,
names-hom_wf,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
fset_wf,
composition-uniformity_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
context-subset-is-cubical-subset,
iff_weakening_equal,
csm-ap-term-cube+,
canonical-section-cubical-path-0,
fl-morph-id,
face-type-ap-morph,
csm-comp-term,
csm-ap-term_wf,
cc-fst_wf_interval,
thin-context-subset,
context-adjoin-subset3,
nc-e'_wf,
thin-context-subset-adjoin,
context-adjoin-subset0,
context-subset_wf,
subset-cubical-term,
cubical-term-equal2,
subset-trans_wf,
fl-morph_wf,
nh-comp_wf,
nc-e'-lemma3,
lattice-join_wf,
lattice-meet_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
face_lattice_wf,
lattice-point_wf,
cube_set_restriction_pair_lemma,
fl-morph-comp2,
cubical-term_wf,
cube_set_map_wf,
csm-comp-type,
context-map_wf_cubical-subset,
csm-equal,
cubical-subset-I_cube,
cube-set-restriction-comp,
context-subset-map,
cubical-subset-is-context-subset-canonical,
csm-subset-domain,
csm-canonical-section-face-type,
cubical-term-equal,
csm-ap_wf,
csm-canonical-section-face,
interval-type-at,
I_cube_pair_redex_lemma,
face-type-at,
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,
eq_int_wf,
int_subtype_base,
assert_of_eq_int,
eqtt_to_assert,
names-subtype,
dM-lift-inc,
DeMorgan-algebra-axioms_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
DeMorgan-algebra-structure_wf,
dM_wf,
nh-comp-sq,
sub_cubical_set_wf,
csm-context-subset-subtype3,
csm+_wf_interval,
csm-subset-codomain,
csm-interval-type,
csm+_wf,
cubical-term-eqcd,
csm-context-subset-subtype2,
subtype_rel_wf,
subset-cubical-type,
context-map-cube+-csm+,
context-subset-is-subset,
arrow_pair_lemma,
trivial-member-add-name1,
int_formula_prop_eq_lemma,
intformeq_wf,
dM-lift-sq,
dM-lift_wf2,
cubical-type-cumulativity,
cubical-type-at_wf,
csm-ap-type-at,
cubical-term-at_wf,
uall_wf,
set_subtype_base,
not_wf,
name-morph-satisfies_wf,
constrained-cubical-term-eqcd,
csm-id-adjoin_wf-interval-1,
csm-id-adjoin_wf,
interval-1_wf,
csm-ap-comp-term,
nc-e'-lemma2,
cubical-type-ap-morph-comp-eq-general,
cubical-type-ap-morph_wf,
nc-0_wf,
csm-id-adjoin_wf-interval-0,
dM0_wf,
formal-cube-restriction,
cube-set-map-subtype,
subtype_rel-equal,
partial-term-0_wf,
member_wf,
interval-0_wf,
csm-comp-assoc,
context-subset-term-subtype,
csm-ap-comp-type,
nc-1_wf,
context-map-comp2,
cube+_interval-1,
cube-set-restriction-id,
subtype_rel_universe1,
nh-id_wf,
comp-fun-to-comp-op1_wf,
csm-ap-term-at,
nh-id-right,
csm-cubical-type-ap-morph,
istype-cubical-type-at,
nh-id-left,
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,
lambdaFormation_alt,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
hypothesis,
sqequalRule,
applyEquality,
instantiate,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
because_Cache,
setIsType,
functionIsType,
intEquality,
inhabitedIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
equalityIstype,
isectEquality,
cumulativity,
productEquality,
hyp_replacement,
functionExtensionality,
promote_hyp,
equalityElimination,
applyLambdaEquality,
isect_memberEquality_alt,
equalityIsType4,
baseApply,
closedConclusion,
equalityIsType1,
functionEquality
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}A:\{Gamma \mvdash{} \_\}.
\mforall{}[comp:Gamma \mvdash{} Compositon(A)]. (cfun-to-cop(Gamma;A;comp) \mmember{} Gamma \mvdash{} CompOp(A))
Date html generated:
2020_05_20-PM-04_32_16
Last ObjectModification:
2020_05_02-PM-07_25_25
Theory : cubical!type!theory
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