Nuprl Lemma : csm-composition-id
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)]. ((comp)1(Gamma) = comp ∈ Gamma ⊢ CompOp(A))
Proof
Definitions occuring in Statement :
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
cubical-type: {X ⊢ _},
csm-id: 1(X),
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
composition-op: Gamma ⊢ CompOp(A),
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
csm-composition: (comp)sigma,
csm-ap: (s)x,
csm-id: 1(X)
Lemmas referenced :
composition-uniformity_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
composition-op_wf,
cubical-type_wf,
cubical_set_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,
csm-ap-type_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionExtensionality,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setEquality,
intEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:Gamma \mvdash{} CompOp(A)]. ((comp)1(Gamma) = comp)
Date html generated:
2020_05_20-PM-03_51_49
Last ObjectModification:
2020_04_09-PM-01_11_49
Theory : cubical!type!theory
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