Nuprl Lemma : equal-composition-op
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[c1:Gamma ⊢ CompOp(A)]. ∀[c2:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u)].
c1 = c2 ∈ Gamma ⊢ CompOp(A)
supposing c1
= c2
∈ (I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ cubical-path-1(Gamma;A;I;i;rho;phi;u))
Proof
Definitions occuring in Statement :
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-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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
composition-op: Gamma ⊢ CompOp(A),
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q
Lemmas referenced :
composition-uniformity_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
fset_wf,
nat_wf,
istype-nat,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
istype-void,
I_cube_wf,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
face-presheaf_wf2,
cubical-term_wf,
cubical-subset_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
cubical-type-cumulativity,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
cubical-path-0_wf,
cubical-path-1_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
equalityIstype,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsType,
setIsType,
intEquality,
independent_isectElimination,
lambdaEquality_alt,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[c1:Gamma \mvdash{} CompOp(A)].
\mforall{}[c2:I:fset(\mBbbN{})
{}\mrightarrow{} i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\}
{}\mrightarrow{} rho:Gamma(I+i)
{}\mrightarrow{} phi:\mBbbF{}(I)
{}\mrightarrow{} u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}
{}\mrightarrow{} cubical-path-0(Gamma;A;I;i;rho;phi;u)
{}\mrightarrow{} cubical-path-1(Gamma;A;I;i;rho;phi;u)].
c1 = c2 supposing c1 = c2
Date html generated:
2020_05_20-PM-03_49_43
Last ObjectModification:
2020_04_09-PM-01_10_48
Theory : cubical!type!theory
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