Nuprl Lemma : sigmacomp_wf1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(sigmacomp(Gamma;A;B;cA;cB) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ mu:{I+i,s(phi) ⊢ _:(Σ A B)<rho> o iota}
⟶ lambda:cubical-path-0(Gamma;Σ A B;I;i;rho;phi;mu)
⟶ cubical-path-1(Gamma;Σ A B;I;i;rho;phi;mu))
Proof
Definitions occuring in Statement :
sigmacomp: sigmacomp(Gamma;A;B;cA;cB),
composition-op: Gamma ⊢ CompOp(A),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {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-s: s,
add-name: I+i,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
sigmacomp: sigmacomp(Gamma;A;B;cA;cB),
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
composition-op: Gamma ⊢ CompOp(A),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
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,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-type: {X ⊢ _},
cc-snd: q,
subset-iota: iota,
csm-comp: G o F,
csm-ap-type: (AF)s,
cc-fst: p,
csm-ap: (s)x,
compose: f o g,
squash: ↓T,
guard: {T},
true: True,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-sigma: Σ A B,
pi1: fst(t),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
pi2: snd(t),
cubical-type-ap-morph: (u a f),
iff: P ⇐⇒ Q,
filling-op: filling-op(Gamma;A),
let: let,
rev_implies: P ⇐ Q,
cube-context-adjoin: X.A,
context-map: <rho>,
csm-adjoin: (s;u),
functor-arrow: arrow(F),
cc-adjoin-cube: (v;u),
section-iota: section-iota(Gamma;A;I;rho;a),
canonical-section: canonical-section(Gamma;A;I;rho;a),
csm-ap-term: (t)s,
cube-set-restriction: f(s),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cubical-term-at: u(a)
Lemmas referenced :
fill_from_comp_wf,
cubical-path-0_wf,
cubical-sigma_wf,
cubical-type-cumulativity2,
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,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity,
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,
composition-op_wf,
cube-context-adjoin_wf,
cubical-type_wf,
cubical_set_wf,
csm-cubical-sigma,
cubical-fst_wf,
csm-adjoin_wf,
cc-fst_wf,
cc-snd_wf,
squash_wf,
true_wf,
equal_functionality_wrt_subtype_rel2,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
cubical-path-condition_wf,
pi1_wf_top,
cubical-type-at_wf,
cubical-subset-I_cube-member,
nc-0_wf,
cubical-subset-I_cube,
equal_wf,
istype-universe,
cubical-fst-at,
subtype_rel_self,
iff_weakening_equal,
cubical-snd_wf,
csm-id-adjoin-ap-type,
cc-adjoin-cube_wf,
cube_set_map_wf,
csm-equal2,
istype-cubical-term,
I_cube_pair_redex_lemma,
arrow_pair_lemma,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
cc-adjoin-cube-restriction,
cubical-snd-at,
subtype_rel-equal,
nc-1_wf,
nh-comp_wf,
name-morph-satisfies_wf,
lattice-point_wf,
face_lattice_wf,
nh-id_wf,
nh-id-right,
uiff_transitivity2,
name-morph-satisfies-comp,
names-hom_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
nh-comp-assoc,
s-comp-nc-1,
csm-ap-type-at,
cube-set-restriction-comp,
cubical-term-at_wf,
cubical-type-ap-morph-comp-general,
pair-eta,
cubical-path-condition'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionExtensionality,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
rename,
setElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
instantiate,
applyEquality,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
setEquality,
intEquality,
productElimination,
imageElimination,
cumulativity,
universeEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
independent_pairEquality,
dependent_pairEquality_alt,
productIsType,
productEquality,
isectEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[B:\{Gamma.A \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)].
\mforall{}[cB:Gamma.A \mvdash{} CompOp(B)].
(sigmacomp(Gamma;A;B;cA;cB) \mmember{} I:fset(\mBbbN{})
{}\mrightarrow{} i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\}
{}\mrightarrow{} rho:Gamma(I+i)
{}\mrightarrow{} phi:\mBbbF{}(I)
{}\mrightarrow{} mu:\{I+i,s(phi) \mvdash{} \_:(\mSigma{} A B)<rho> o iota\}
{}\mrightarrow{} lambda:cubical-path-0(Gamma;\mSigma{} A B;I;i;rho;phi;mu)
{}\mrightarrow{} cubical-path-1(Gamma;\mSigma{} A B;I;i;rho;phi;mu))
Date html generated:
2020_05_20-PM-04_05_10
Last ObjectModification:
2020_04_17-AM-08_49_46
Theory : cubical!type!theory
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