Nuprl Lemma : case-type-comp-false-true
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
(∀[B:{Gamma ⊢ _}]. ∀[cB:Gamma ⊢ Compositon(B)]. ∀[A:{Gamma, phi ⊢ _}]. ∀[cA:Gamma, phi ⊢ Compositon(A)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB) = cB ∈ Gamma ⊢ Compositon(B))) supposing
(Gamma ⊢ (1(𝔽) ⇒ psi) and
(phi = 0(𝔽) ∈ {Gamma ⊢ _:𝔽}))
Proof
Definitions occuring in Statement :
case-type-comp: case-type-comp(G; phi; psi; A; B; cA; cB),
composition-structure: Gamma ⊢ Compositon(A),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
context-subset: Gamma, phi,
face-1: 1(𝔽),
face-0: 0(𝔽),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
same-cubical-type: Gamma ⊢ A = B,
cubical-type: {X ⊢ _},
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
case-type-comp: case-type-comp(G; phi; psi; A; B; cA; cB),
case-term: (u ∨ v),
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
csm-id-adjoin: [u],
csm-id: 1(X),
bdd-distributive-lattice: BoundedDistributiveLattice,
cubical-term-at: u(a),
face-forall: (∀ phi),
fl_all: (∀i.phi),
fl-all-hom: fl-all-hom(I;i),
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
csm-ap-term: (t)s,
face-0: 0(𝔽),
lattice-0: 0,
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,
empty-fset: {},
nil: [],
it: ⋅,
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),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
sq_type: SQType(T),
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
context-subset: Gamma, phi,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
interval-presheaf: 𝕀,
names: names(I),
nat: ℕ,
face-1: 1(𝔽),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b]
Lemmas referenced :
composition-structure_wf,
context-subset_wf,
cubical-type_wf,
face-term-implies_wf,
face-1_wf,
face-0_wf,
istype-cubical-term,
face-type_wf,
cubical_set_wf,
subset-cubical-type,
context-subset-is-subset,
composition-structure-subset,
empty-context-subset-lemma6,
subtype_rel_product,
fset_wf,
nat_wf,
I_cube_wf,
names-hom_wf,
cube-set-restriction_wf,
istype-universe,
top_wf,
istype-top,
face-and_wf,
face-term-implies-subset,
face-term-and-implies1,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
case-type-comp_wf,
compatible-composition-disjoint,
thin-context-subset,
subtype_rel_wf,
face-or_wf,
case-type_wf,
same-cubical-type_wf,
face-term-and-implies2,
sub_cubical_set_self,
face-1-implies-subset,
face-0-or,
case-type-same2,
subtype-context-subset-0,
composition-structure-equal,
cubical-term-equal,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-1,
constrained-cubical-term_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-term_wf,
thin-context-subset-adjoin,
csm-context-subset-subtype3,
cube_set_map_wf,
lattice-point_wf,
face_lattice_wf,
lattice-0_wf,
equal_wf,
cubical-term-at_wf,
face-forall_wf,
subset-cubical-term2,
csm-face-type,
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,
fl-eq_wf,
lattice-1_wf,
eqtt_to_assert,
assert-fl-eq,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
iff_weakening_uiff,
assert_wf,
face-lattice-0-not-1,
composition-in-subset,
csm-id-adjoin_wf,
interval-1_wf,
csm-context-subset-subtype2,
I_cube_pair_redex_lemma,
csm-face-term-implies,
csm-face-1,
add-name_wf,
new-name_wf,
cc-adjoin-cube_wf,
nc-s_wf,
f-subset-add-name,
interval-type-at,
dM_inc_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
fl_all-1,
fl_all_wf,
istype-nat,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
equalityIstype,
because_Cache,
applyEquality,
sqequalRule,
independent_isectElimination,
setElimination,
rename,
functionEquality,
cumulativity,
universeEquality,
lambdaEquality_alt,
functionIsType,
Error :memTop,
lambdaFormation_alt,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
hyp_replacement,
functionExtensionality,
productEquality,
isectEquality,
unionElimination,
equalityElimination,
dependent_functionElimination,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt,
intEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
(\mforall{}[B:\{Gamma \mvdash{} \_\}]. \mforall{}[cB:Gamma \mvdash{} Compositon(B)]. \mforall{}[A:\{Gamma, phi \mvdash{} \_\}].
\mforall{}[cA:Gamma, phi \mvdash{} Compositon(A)].
(case-type-comp(Gamma; phi; psi; A; B; cA; cB) = cB)) supposing
(Gamma \mvdash{} (1(\mBbbF{}) {}\mRightarrow{} psi) and
(phi = 0(\mBbbF{})))
Date html generated:
2020_05_20-PM-05_19_00
Last ObjectModification:
2020_04_18-PM-07_59_12
Theory : cubical!type!theory
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