Nuprl Lemma : same-cubical-type-by-list-cases
∀[Gamma:j⊢]. ∀L:{Gamma ⊢ _:𝔽} List. ∀A,B:{Gamma, \/(L) ⊢ _}. Gamma, \/(L) ⊢ A = B supposing (∀phi∈L.Gamma, phi ⊢ A = B)
Proof
Definitions occuring in Statement :
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-or-list: \/(L),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
l_all: (∀x∈L.P[x]),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
bdd-distributive-lattice: BoundedDistributiveLattice,
exists: ∃x:A. B[x],
cand: A c∧ B,
face-or-list: \/(L),
l_all: (∀x∈L.P[x]),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
same-cubical-type: Gamma ⊢ A = B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
subtract: n - m,
ge: i ≥ j ,
less_than': less_than'(a;b),
nat_plus: ℕ+,
guard: {T},
cons: [a / b],
less_than: a < b,
squash: ↓T,
l_exists: (∃x∈L. P[x]),
true: True,
respects-equality: respects-equality(S;T)
Lemmas referenced :
list_induction,
list_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
cubical-type_wf,
context-subset_wf,
face-or-list_wf,
l_all_wf2,
same-cubical-type_wf,
context-subset-subtype,
face-or-list-eq-1,
l_exists_iff,
equal_wf,
lattice-point_wf,
face_lattice_wf,
cubical-term-at_wf,
subtype_rel_self,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-1_wf,
l_member_wf,
I_cube_wf,
fset_wf,
nat_wf,
reduce_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
same-cubical-type-0,
int_seg_wf,
int_seg_properties,
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,
face-0_wf,
reduce_cons_lemma,
face-or_wf,
same-cubical-type-by-cases,
context-subset-subtype-or2,
length_wf,
add-member-int_seg2,
cons_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
length_of_cons_lemma,
non_neg_length,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
istype-le,
istype-less_than,
select_cons_tl_sq2,
int_seg_subtype_nat,
istype-false,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
select_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
iff_weakening_equal,
face-type-at,
respects-equality_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
independent_functionElimination,
universeIsType,
hypothesisEquality,
hypothesis,
cumulativity,
lambdaEquality_alt,
functionEquality,
isectEquality,
setElimination,
rename,
applyEquality,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
productElimination,
productEquality,
setIsType,
inhabitedIsType,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :memTop,
baseClosed,
axiomEquality,
functionIsType,
natural_numberEquality,
approximateComputation,
int_eqEquality,
voidElimination,
closedConclusion,
dependent_set_memberEquality_alt,
unionElimination,
addEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
isectIsType,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]
\mforall{}L:\{Gamma \mvdash{} \_:\mBbbF{}\} List. \mforall{}A,B:\{Gamma, \mbackslash{}/(L) \mvdash{} \_\}.
Gamma, \mbackslash{}/(L) \mvdash{} A = B supposing (\mforall{}phi\mmember{}L.Gamma, phi \mvdash{} A = B)
Date html generated:
2020_05_20-PM-03_02_16
Last ObjectModification:
2020_04_06-PM-08_27_03
Theory : cubical!type!theory
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