Nuprl Lemma : ctt-subterm-context_wf
∀[ctxt:CubicalContext]. ∀[f:CttOp]. ∀[n:ℕ]. ∀[vs:varname() List].
∀[m:Provisional''''(cttTerm(fst(ctxt)))
supposing ↑(((ctt-opr-is(f;"Glue") ∨bctt-opr-is(f;"case") ∨bctt-opr-is(f;"unglue")) ∧b ((n =z 2) ∨b(n =z 3)))
∨b(ctt-opr-is(f;"comp") ∧b (n =z 2))
∨b(ctt-opr-is(f;"glue") ∧b ((n =z 2) ∨b(n =z 3) ∨b(n =z 4)))) ⋂ Provisional''''(ctt-type-meaning1{i:l}(fst(ctxt)))
supposing ↑((ctt-opr-is(f;"pi")
∨bctt-opr-is(f;"sigma")
∨bctt-opr-is(f;"lambda")
∨bctt-opr-is(f;"apply")
∨bctt-opr-is(f;"pair")
∨bctt-opr-is(f;"fst")
∨bctt-opr-is(f;"snd"))
∧b (n =z 1))].
ctt-subterm-context{i:l}(ctxt;f;n;vs;m) ∈ ?CubicalContext
supposing (↑(((ctt-opr-is(f;"pi")
∨bctt-opr-is(f;"sigma")
∨bctt-opr-is(f;"lambda")
∨bctt-opr-is(f;"apply")
∨bctt-opr-is(f;"pair")
∨bctt-opr-is(f;"fst")
∨bctt-opr-is(f;"snd")
∨bctt-opr-is(f;"pathabs")
∨bctt-opr-is(f;"comp"))
∧b (n =z 1))
∨b(ctt-opr-is(f;"comp") ∧b (n =z 2))))
⇒ 0 < ||vs||
Proof
Definitions occuring in Statement :
ctt-subterm-context: ctt-subterm-context{i:l}(ctxt;f;n;vs;m),
ctt-opr-is: ctt-opr-is(f;s),
ctt-op: CttOp,
cubical-context: ?CubicalContext,
cubical_context: CubicalContext,
ctt-type-meaning1: ctt-type-meaning1{i:l}(X),
ctt-term-meaning: cttTerm(X),
varname: varname(),
length: ||as||,
list: T List,
isect2: T1 ⋂ T2,
bor: p ∨bq,
band: p ∧b q,
nat: ℕ,
assert: ↑b,
eq_int: (i =z j),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
implies: P ⇒ Q,
member: t ∈ T,
natural_number: $n,
token: "$token",
provisional-type: Provisional(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
cubical_context: CubicalContext,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi1: fst(t),
ctt-op: CttOp,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
prop: ℙ,
ctt-tokens: ctt-tokens(),
ctt-subterm-context: ctt-subterm-context{i:l}(ctxt;f;n;vs;m),
cubical-context: ?CubicalContext,
ctt-opr-is: ctt-opr-is(f;s),
band: p ∧b q,
bor: p ∨bq,
bfalse: ff,
uiff: uiff(P;Q),
nat: ℕ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
eq_int: (i =z j),
assert: ↑b,
cand: A c∧ B,
true: True,
exists: ∃x:A. B[x],
bnot: ¬bb,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
ctt-level-type: {X ⊢lvl _},
bool: 𝔹,
unit: Unit,
it: ⋅
Lemmas referenced :
cons_member,
cons_wf,
nil_wf,
subtype_base_sq,
atom_subtype_base,
provision_wf,
cubical_context_wf,
true_wf,
istype-true,
member_singleton,
istype-assert,
bor_wf,
ctt-opr-is_wf,
bool_cases,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
eq_int_wf,
bfalse_wf,
bool_wf,
istype-less_than,
length_wf,
varname_wf,
isect2_wf,
assert_wf,
provisional-type_wf,
ctt-term-meaning_wf,
pi1_wf_top,
cubical_set_wf,
ctt-type-meaning1_wf,
list_wf,
istype-nat,
ctt-op_wf,
equal-wf-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
isect2_decomp,
false_wf,
restrict-cubical-context_wf,
isect2_subtype_rel3,
uimplies_subtype,
subtype_rel_wf,
subtype_rel_self,
bnot_wf,
bool_cases_sqequal,
eqff_to_assert,
assert-bnot,
neg_assert_of_eq_int,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
istype-void,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
update-cubical-context2_wf,
hd_wf,
decidable__le,
intformle_wf,
intformless_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
update-cubical-context_wf,
int_seg_wf,
ctt-level-type_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_universe1,
decidable__lt,
istype-le,
interval-type_wf,
int_seg_properties,
update-provisional-context-I_wf,
assert_of_bor,
bnot_thru_bor,
assert_of_band,
uiff_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
hypothesisEquality,
inhabitedIsType,
hypothesis,
lambdaFormation_alt,
sqequalRule,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
introduction,
extract_by_obid,
isectElimination,
atomEquality,
tokenEquality,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
isect_memberEquality_alt,
functionIsType,
closedConclusion,
because_Cache,
natural_numberEquality,
universeIsType,
isectEquality,
Error :memTop,
baseApply,
baseClosed,
applyEquality,
intEquality,
lambdaEquality_alt,
unionEquality,
sqequalBase,
unionIsType,
independent_pairFormation,
inlFormation_alt,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
productEquality,
productIsType,
approximateComputation,
int_eqEquality,
imageElimination,
universeEquality,
dependent_pairEquality_alt,
dependent_set_memberEquality_alt,
equalityElimination,
inrFormation_alt
Latex:
\mforall{}[ctxt:CubicalContext]. \mforall{}[f:CttOp]. \mforall{}[n:\mBbbN{}]. \mforall{}[vs:varname() List].
\mforall{}[m:Provisional''''(cttTerm(fst(ctxt)))
supposing \muparrow{}(((ctt-opr-is(f;"Glue") \mvee{}\msubb{}ctt-opr-is(f;"case") \mvee{}\msubb{}ctt-opr-is(f;"unglue"))
\mwedge{}\msubb{} ((n =\msubz{} 2) \mvee{}\msubb{}(n =\msubz{} 3)))
\mvee{}\msubb{}(ctt-opr-is(f;"comp") \mwedge{}\msubb{} (n =\msubz{} 2))
\mvee{}\msubb{}(ctt-opr-is(f;"glue")
\mwedge{}\msubb{} ((n =\msubz{} 2) \mvee{}\msubb{}(n =\msubz{} 3) \mvee{}\msubb{}(n =\msubz{} 4)))) \mcap{} Provisional''''(ctt-type-meaning1\{i:l\}(fst(ctxt)))
supposing \muparrow{}((ctt-opr-is(f;"pi")
\mvee{}\msubb{}ctt-opr-is(f;"sigma")
\mvee{}\msubb{}ctt-opr-is(f;"lambda")
\mvee{}\msubb{}ctt-opr-is(f;"apply")
\mvee{}\msubb{}ctt-opr-is(f;"pair")
\mvee{}\msubb{}ctt-opr-is(f;"fst")
\mvee{}\msubb{}ctt-opr-is(f;"snd"))
\mwedge{}\msubb{} (n =\msubz{} 1))].
ctt-subterm-context\{i:l\}(ctxt;f;n;vs;m) \mmember{} ?CubicalContext
supposing (\muparrow{}(((ctt-opr-is(f;"pi")
\mvee{}\msubb{}ctt-opr-is(f;"sigma")
\mvee{}\msubb{}ctt-opr-is(f;"lambda")
\mvee{}\msubb{}ctt-opr-is(f;"apply")
\mvee{}\msubb{}ctt-opr-is(f;"pair")
\mvee{}\msubb{}ctt-opr-is(f;"fst")
\mvee{}\msubb{}ctt-opr-is(f;"snd")
\mvee{}\msubb{}ctt-opr-is(f;"pathabs")
\mvee{}\msubb{}ctt-opr-is(f;"comp"))
\mwedge{}\msubb{} (n =\msubz{} 1))
\mvee{}\msubb{}(ctt-opr-is(f;"comp") \mwedge{}\msubb{} (n =\msubz{} 2))))
{}\mRightarrow{} 0 < ||vs||
Date html generated:
2020_05_21-AM-10_37_08
Last ObjectModification:
2020_05_18-AM-11_26_02
Theory : cubical!type!theory
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