Nuprl Lemma : ctt-term-meaning-subtype
∀[X,Y:⊢''']. cttTerm(X) ⊆r cttTerm(Y) supposing sub_cubical_set{i''':l}(Y; X)
Proof
Definitions occuring in Statement :
ctt-term-meaning: cttTerm(X),
sub_cubical_set: Y ⊆ X,
cubical_set: CubicalSet,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
ctt-term-meaning: cttTerm(X),
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),
implies: P ⇒ Q,
guard: {T},
ctt-level-type: {X ⊢lvl _},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ
Lemmas referenced :
ctt-term-meaning_wf,
sub_cubical_set_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
subset-cubical-type,
subset-cubical-term,
istype-cubical-term,
int_seg_subtype_special,
int_seg_cases,
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,
cubical-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairEquality_alt,
hypothesisEquality,
universeIsType,
extract_by_obid,
isectElimination,
hypothesis,
sqequalRule,
axiomEquality,
instantiate,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
unionElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
hypothesis_subsumption,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
productIsType
Latex:
\mforall{}[X,Y:\mvdash{}''']. cttTerm(X) \msubseteq{}r cttTerm(Y) supposing sub\_cubical\_set\{i''':l\}(Y; X)
Date html generated:
2020_05_20-PM-07_53_46
Last ObjectModification:
2020_05_04-PM-05_42_46
Theory : cubical!type!theory
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