Nuprl Lemma : composition-type-lemma4
∀Gamma:j⊢. ∀phi:{Gamma ⊢ _:𝔽}. ∀A:{Gamma.𝕀 ⊢ _}. ∀u:{Gamma, phi.𝕀 ⊢ _:A}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀rho:Gamma(I).
∀K:fset(ℕ). ∀g:J,phi(f(rho))(K).
((u)<(s(f(rho));<new-name(J)>)> o iota((new-name(J)0) ⋅ g)
= ((u)<(s(rho);<new-name(I)>)> o iota)subset-trans(I+new-name(I);J+new-name(J);f,new-name(I)=new-name(J);
s(phi(rho)))((new-name(J)0) ⋅ g)
∈ A(g((new-name(J)0)((s(f(rho));<new-name(J)>)))))
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
interval-type: 𝕀,
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
subset-trans: subset-trans(I;J;f;x),
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
csm-comp: G o F,
context-map: <rho>,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-e': g,i=j,
nc-0: (i0),
nc-s: s,
new-name: new-name(I),
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
dM_inc: <x>,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
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,
guard: {T},
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
subset-iota: iota,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g,
context-map: <rho>,
functor-arrow: arrow(F),
cube-context-adjoin: X.A,
cc-adjoin-cube: (v;u),
pi2: snd(t),
implies: P ⇒ Q,
interval-presheaf: 𝕀,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
names: names(I),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nc-0: (i0),
sq_type: SQType(T),
name-morph-satisfies: (psi f) = 1,
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
composition-type-lemma3,
context-subset-adjoin-subtype,
interval-type_wf,
I_cube_wf,
cubical-subset_wf,
cubical-term-at_wf,
face-type_wf,
cube-set-restriction_wf,
subtype_rel_self,
face-presheaf_wf2,
names-hom_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cube-context-adjoin_wf,
context-subset_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
subtype_rel_transitivity,
cubical-type_wf,
cubical_set_wf,
csm-ap-type-at,
cubical-type-at_wf,
squash_wf,
true_wf,
cc-adjoin-cube-restriction,
interval-type-ap-morph,
cc-adjoin-cube_wf,
istype-cubical-type-at,
new-name_wf,
cubical-subset-I_cube,
cube-set-restriction-comp,
add-name_wf,
nc-0_wf,
nc-s_wf,
f-subset-add-name,
interval-type-at,
I_cube_pair_redex_lemma,
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,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_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,
nh-comp_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
dM-lift_wf2,
istype-universe,
dM-lift-inc,
iff_weakening_equal,
nh-comp-sq,
dM0-sq-empty,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eq_int_eq_true_intro,
btrue_wf,
dM0_wf,
dM-lift-0,
csm-ap-type_wf,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
dM_inc_wf,
name-morph-satisfies_wf,
face-type-at,
face_lattice_wf,
fl-morph_wf,
lattice-1_wf,
cube_set_restriction_pair_lemma,
nh-id-right,
fl-morph-comp2,
nh-comp-assoc,
s-comp-nc-0
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyLambdaEquality,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
inhabitedIsType,
cumulativity,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
Error :memTop,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
rename,
equalityIstype,
independent_functionElimination,
productEquality,
isectEquality,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
intEquality,
universeEquality,
productElimination
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}phi:\{Gamma \mvdash{} \_:\mBbbF{}\}. \mforall{}A:\{Gamma.\mBbbI{} \mvdash{} \_\}. \mforall{}u:\{Gamma, phi.\mBbbI{} \mvdash{} \_:A\}. \mforall{}I,J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I.
\mforall{}rho:Gamma(I). \mforall{}K:fset(\mBbbN{}). \mforall{}g:J,phi(f(rho))(K).
((u)<(s(f(rho));<new-name(J)>)> o iota((new-name(J)0) \mcdot{} g)
= ((u)<(s(rho);<new-name(I)>)> o iota)subset-trans(I+new-name(I);J+new-name(J);
f,new-name(I)=new-name(J);
s(phi(rho)))((new-name(J)0) \mcdot{} g))
Date html generated:
2020_05_20-PM-04_09_29
Last ObjectModification:
2020_04_11-PM-06_37_25
Theory : cubical!type!theory
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