Nuprl Lemma : composition-term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
(comp cA [phi ⊢→ u] a0 ∈ {Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ (u)[1(𝕀)]]})
Proof
Definitions occuring in Statement :
composition-term: comp cA [phi ⊢→ u] a0,
composition-op: Gamma ⊢ CompOp(A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
csm-id-adjoin: [u],
csm-id: 1(X),
guard: {T},
composition-term: comp cA [phi ⊢→ u] a0,
uimplies: b supposing a,
interval-presheaf: 𝕀,
names: names(I),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
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,
context-map: <rho>,
subset-iota: iota,
csm-comp: G o F,
compose: f o g,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
interval-0: 0(𝕀),
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
interval-type: 𝕀,
nc-0: (i0),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
empty-fset: {},
nil: [],
dM0: 0,
lattice-0: 0,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
DeMorgan-algebra: DeMorganAlgebra,
cubical-type-ap-morph: (u a f),
pi2: snd(t),
cube-set-restriction: f(s),
dM-lift: dM-lift(I;J;f),
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra-property,
free-dist-lattice-property,
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,
cc-adjoin-cube: (v;u),
csm-ap: (s)x,
csm-adjoin: (s;u),
context-subset: Gamma, phi,
bdd-distributive-lattice: BoundedDistributiveLattice,
name-morph-satisfies: (psi f) = 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
cube-context-adjoin: X.A,
composition-op: Gamma ⊢ CompOp(A),
cubical-term: {X ⊢ _:A},
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
composition-uniformity: composition-uniformity(Gamma;A;comp),
nc-1: (i1),
interval-1: 1(𝕀),
nc-e': g,i=j,
ge: i ≥ j ,
decidable: Dec(P),
dma-hom: dma-hom(dma1;dma2),
names-hom: I ⟶ J,
partial-term-1: u[1],
cubical-term-at: u(a),
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
csm-ap-term: (t)s,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s
Lemmas referenced :
composition-op_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical-type-cumulativity2,
cubical-type_wf,
istype-cubical-term,
face-type_wf,
cubical_set_wf,
context-subset-adjoin-subtype,
composition-type-lemma5,
constrained-cubical-term_wf,
csm-ap-type_wf,
csm-id-adjoin_wf-interval-0,
csm-ap-term_wf,
context-subset_wf,
thin-context-subset-adjoin,
cc-adjoin-cube_wf,
add-name_wf,
new-name_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
interval-type-at,
I_cube_pair_redex_lemma,
dM_inc_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
I_cube_wf,
fset_wf,
cubical-subset_wf,
face-presheaf_wf2,
cubical-term-at_wf,
subtype_rel_self,
context-map-lemma2,
csm-ap-type-at,
cubical-type-at_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-comp,
nc-0_wf,
iff_weakening_equal,
cube-set-restriction-when-id,
nh-comp_wf,
s-comp-nc-0-new,
csm-id-adjoin-ap,
cc-adjoin-cube-restriction,
trivial-equal,
dM0_wf,
interval-type-ap-inc,
interval-type-at-is-point,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
eq_int_eq_true,
btrue_wf,
not_assert_elim,
btrue_neq_bfalse,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-cubical-type-at,
cubical-subset-I_cube,
names-hom_wf,
cube-set-restriction-id,
s-comp-nc-0,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM-lift_wf2,
dM0-sq-empty,
dM-lift-0,
assert_elim,
bnot_wf,
bfalse_wf,
interval-type-ap-morph,
dM-lift-inc,
cubical-subset-I_cube-member,
cubical-type-ap-morph_wf,
subtype_rel-equal,
cubical-term-at-morph,
csm-cubical-type-ap-morph,
face_lattice_wf,
lattice-1_wf,
fl-morph_wf,
face-type-ap-morph,
thin-context-subset,
subset-I_cube,
context-subset-is-subset,
csm-ap-term-at,
arrow_pair_lemma,
s-comp-if-lemma1,
nh-comp-assoc,
nh-id-right,
nh-comp-sq,
cubical-path-condition_wf,
csm-id-adjoin_wf-interval-1,
composition-type-lemma2,
nc-1_wf,
nh-id_wf,
nh-id-left,
s-comp-nc-1,
dM1-sq-singleton-empty,
dM-lift-1,
s-comp-nc-1-new,
dM1_wf,
csm-ap_wf,
csm-ap-restriction,
nc-e'_wf,
nc-e'-lemma3,
face-term-at-restriction,
composition-type-lemma3,
cubical-path-0-ap-morph,
interval-presheaf_wf,
small_cubical_set_subtype,
nat_properties,
decidable__le,
intformand_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
istype-le,
dM-lift_wf,
eq_int_eq_true_intro,
names_wf,
composition-type-lemma4,
cubical-term-equal,
partial-term-1_wf,
subset-cubical-term,
csm-face-type,
cc-fst_wf_interval,
sub_cubical_set_transitivity,
sub_cubical_set_self,
context-adjoin-subset1,
name-morph-satisfies_wf,
name-morph-1-satisfies,
cubical-type-ap-morph-id,
csm-id-adjoin_wf,
interval-1_wf,
context-subset-term-subtype,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
lambdaFormation_alt,
cumulativity,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination,
independent_isectElimination,
Error :memTop,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
setElimination,
rename,
intEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeEquality,
productElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
voidElimination,
approximateComputation,
int_eqEquality,
productEquality,
isectEquality,
independent_pairFormation,
productIsType,
applyLambdaEquality,
dependent_pairEquality_alt,
functionExtensionality,
functionEquality,
functionIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)].
\mforall{}[u:\{Gamma, phi.\mBbbI{} \mvdash{} \_:A\}]. \mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\}].
(comp cA [phi \mvdash{}\mrightarrow{} u] a0 \mmember{} \{Gamma \mvdash{} \_:(A)[1(\mBbbI{})][phi |{}\mrightarrow{} (u)[1(\mBbbI{})]]\})
Date html generated:
2020_05_20-PM-04_11_53
Last ObjectModification:
2020_04_21-AM-00_49_38
Theory : cubical!type!theory
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