Nuprl Lemma : ctt-level-comp-cumulativity
∀[X:⊢''']. ∀[a:ℕ4]. ∀[T:{X ⊢a _}]. ∀[b:ℕ4]. Comp(X;a;T) ⊆r Comp(X;b;T) supposing a ≤ b
Proof
Definitions occuring in Statement :
ctt-level-comp: Comp(X;lvl;T),
ctt-level-type: {X ⊢lvl _},
cubical_set: CubicalSet,
int_seg: {i..j-},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
ctt-level-type: {X ⊢lvl _},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
ctt-level-comp: Comp(X;lvl;T),
bfalse: ff,
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
false: False,
less_than': less_than'(a;b),
true: True,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
squash: ↓T,
nat: ℕ,
less_than: a < b
Lemmas referenced :
sq_stable__subtype_rel,
ctt-level-comp_wf,
ctt-level-type-cumulativity2,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
composition-structure-subset,
sub_cubical_set_self,
istype-le,
int_seg_subtype_special,
int_seg_cases,
subtype_rel_self,
composition-structure_wf,
cubical-type-cumulativity,
cubical-type-cumulativity2,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
ctt-level-type_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
int_seg_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
setElimination,
rename,
productElimination,
unionElimination,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
lambdaFormation_alt,
hypothesis_subsumption,
voidElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt
Latex:
\mforall{}[X:\mvdash{}''']. \mforall{}[a:\mBbbN{}4]. \mforall{}[T:\{X \mvdash{}a \_\}]. \mforall{}[b:\mBbbN{}4]. Comp(X;a;T) \msubseteq{}r Comp(X;b;T) supposing a \mleq{} b
Date html generated:
2020_05_20-PM-07_47_15
Last ObjectModification:
2020_05_11-PM-01_27_01
Theory : cubical!type!theory
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