Nuprl Lemma : csm-composition-comp
∀[X,Y,Z:j⊢]. ∀[s1:Z j⟶ Y]. ∀[s2:Y j⟶ X]. ∀[A:{X ⊢ _}]. ∀[comp:X ⊢ CompOp(A)].
(((comp)s2)s1 = (comp)s2 o s1 ∈ Z ⊢ CompOp((A)s2 o s1))
Proof
Definitions occuring in Statement :
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
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,
subtype_rel: A ⊆r B,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
uimplies: b supposing a,
squash: ↓T,
cubical-type: {X ⊢ _},
csm-comp: G o F,
csm-ap-type: (AF)s,
csm-ap: (s)x,
true: True,
composition-op: Gamma ⊢ CompOp(A),
csm-composition: (comp)sigma,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
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],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal-composition-op,
csm-ap-type_wf,
csm-comp_wf,
csm-composition_wf,
cubical_set_cumulativity-i-j,
subtype_rel_self,
cube_set_map_wf,
cubical-type-cumulativity2,
subtype_rel-equal,
composition-op_wf,
cubical-path-0_wf,
cubical-term_wf,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
cubical-type-cumulativity,
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,
cubical-type_wf,
cubical_set_wf,
csm-cubical-path-0-subtype,
csm-ap-csm-comp,
csm-ap_wf,
squash_wf,
true_wf,
equal_wf,
csm-ap-comp-type,
iff_weakening_equal,
csm-comp-context-map,
istype-universe,
csm-cubical-path-1-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
setElimination,
rename,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
dependent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
setEquality,
intEquality,
lambdaFormation_alt,
equalityIstype,
hyp_replacement,
universeEquality
Latex:
\mforall{}[X,Y,Z:j\mvdash{}]. \mforall{}[s1:Z j{}\mrightarrow{} Y]. \mforall{}[s2:Y j{}\mrightarrow{} X]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[comp:X \mvdash{} CompOp(A)].
(((comp)s2)s1 = (comp)s2 o s1)
Date html generated:
2020_05_20-PM-03_52_05
Last ObjectModification:
2020_04_09-PM-01_31_49
Theory : cubical!type!theory
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