Nuprl Lemma : composition-term-uniformity
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[phi:{H ⊢ _:𝔽}]. ∀[A:{H.𝕀 ⊢ _}]. ∀[u:{H, phi.𝕀 ⊢ _:A}].
∀[a0:{H ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}]. ∀[cA:H.𝕀 ⊢ CompOp(A)].
((comp cA [phi ⊢→ u] a0)tau = comp (cA)tau+ [(phi)tau ⊢→ (u)tau+] (a0)tau ∈ {K ⊢ _:((A)[1(𝕀)])tau})
Proof
Definitions occuring in Statement :
composition-term: comp cA [phi ⊢→ u] a0,
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
csm+: tau+,
csm-comp: G o F,
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
uimplies: b supposing a,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-ap-term: (t)s,
csm-ap-type: (AF)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
guard: {T},
interval-type: 𝕀,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
pi2: snd(t),
compose: f o g,
pi1: fst(t),
same-cubical-type: Gamma ⊢ A = B,
dM0: 0,
lattice-0: 0,
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],
ifthenelse: if b then t else f fi ,
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),
btrue: tt,
empty-fset: {},
nil: [],
it: ⋅,
interval-1: 1(𝕀),
cubical-term-at: u(a),
composition-term: comp cA [phi ⊢→ u] a0,
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
cc-adjoin-cube: (v;u),
squash: ↓T,
prop: ℙ,
true: True,
interval-presheaf: 𝕀,
names: names(I),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-type-at: A(a),
face-type: 𝔽,
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
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),
subset-iota: iota,
formal-cube: formal-cube(I),
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q,
cube-context-adjoin: X.A,
context-map: <rho>,
names-hom: I ⟶ J,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
DeMorgan-algebra: DeMorganAlgebra,
nc-0: (i0),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
bnot: ¬bb,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
assert: ↑b,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cubical-type-ap-morph: (u a f),
cube-set-restriction: f(s),
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,
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,
name-morph-satisfies: (psi f) = 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
sub_cubical_set: Y ⊆ X,
functor-arrow: arrow(F),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u)
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
interval-type_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
csm+_wf_interval,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-ap-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
context-subset_wf,
thin-context-subset-adjoin,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term-eqcd,
istype-cubical-term,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
cubical-term-equal2,
interval-1_wf,
composition-term_wf,
csm-composition_wf,
context-subset-map,
csm-constrained-cubical-term,
subset-cubical-term2,
sub_cubical_set_self,
thin-context-subset,
subset-cubical-type,
context-subset-is-subset,
equal_functionality_wrt_subtype_rel2,
csm-ap-term-wf-subset,
face-term-implies-same,
cubical-term-eqcd,
csm-ap-term-at,
I_cube_wf,
fset_wf,
nat_wf,
new-name_wf,
cc-adjoin-cube_wf,
squash_wf,
true_wf,
istype-cubical-type-at,
add-name_wf,
csm-ap-restriction,
nc-s_wf,
f-subset-add-name,
interval-type-at,
I_cube_pair_redex_lemma,
dM_inc_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
cubical-term-at_wf,
subtype_rel_self,
face-presheaf_wf2,
csm-comp_wf,
formal-cube_wf1,
context-map_wf,
cube-set-restriction_wf,
cubical-term_wf,
cubical-subset-is-context-subset,
csm-ap_wf,
cubical-term-equal,
face-type-at,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
arrow_pair_lemma,
cubical_type_ap_morph_pair_lemma,
cubical-type-at_wf_face-type,
names-hom_wf,
istype-universe,
subtype_rel-equal,
iff_weakening_equal,
face-type-comp-at-lemma,
csm-comp-context-map,
context-subset-subtype-simple,
context-subset-map-equal,
cubical-type-at_wf,
composition-type-lemma1,
interval-type-at-is-point,
csm-ap-type-at,
cubical-subset_wf,
cc-adjoin-cube-restriction,
cube-set-restriction-comp,
nc-0_wf,
cubical-subset-I_cube-member,
nh-comp_wf,
s-comp-if-lemma1,
nh-comp-assoc,
s-comp-nc-0-new,
nh-id-right,
cubical-subset-I_cube,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
dM-lift_wf2,
interval-type-ap-morph,
dM-lift-inc,
dM-lift-0,
dM0-sq-empty,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
assert_elim,
bnot_wf,
bool_wf,
eq_int_eq_true,
bfalse_wf,
btrue_neq_bfalse,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
btrue_wf,
not_assert_elim,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
dM0_wf,
cubical-type-ap-morph_wf,
cube-set-restriction-when-id,
cubical-term-at-morph,
csm-cubical-type-ap-morph,
lattice-1_wf,
fl-morph_wf,
face-type-ap-morph,
subset-I_cube,
nh-comp-sq,
context-adjoin-subset1,
cube_set_restriction_pair_lemma,
fl-morph-comp2,
cubical-path-condition_wf,
csm-ap-interval-1-adjoin-lemma,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
sqequalRule,
Error :memTop,
inhabitedIsType,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
setElimination,
rename,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
intEquality,
functionExtensionality,
productEquality,
cumulativity,
isectEquality,
universeEquality,
dependent_pairEquality_alt,
unionElimination,
equalityElimination,
independent_pairFormation,
productIsType,
applyLambdaEquality,
voidElimination,
dependent_pairFormation_alt,
promote_hyp,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[H,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} H]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{H.\mBbbI{} \mvdash{} \_\}]. \mforall{}[u:\{H, phi.\mBbbI{} \mvdash{} \_:A\}].
\mforall{}[a0:\{H \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\}]. \mforall{}[cA:H.\mBbbI{} \mvdash{} CompOp(A)].
((comp cA [phi \mvdash{}\mrightarrow{} u] a0)tau = comp (cA)tau+ [(phi)tau \mvdash{}\mrightarrow{} (u)tau+] (a0)tau)
Date html generated:
2020_05_20-PM-04_25_42
Last ObjectModification:
2020_05_02-AM-10_10_30
Theory : cubical!type!theory
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