Nuprl Lemma : comp_term-subset
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}]. ∀[psi:{Gamma ⊢ _:𝔽}].
(comp cA [phi ⊢→ u] a0 = comp cA [phi ⊢→ u] a0 ∈ {Gamma, psi ⊢ _:(A)[1(𝕀)]})
Proof
Definitions occuring in Statement :
comp_term: comp cA [phi ⊢→ u] a0,
composition-structure: Gamma ⊢ Compositon(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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
guard: {T},
interval-1: 1(𝕀),
csm-ap-term: (t)s,
interval-type: 𝕀,
csm+: tau+,
csm-adjoin: (s;u),
csm-ap: (s)x,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-comp: G o F,
pi2: snd(t),
compose: f o g,
pi1: fst(t),
cubical-type: {X ⊢ _},
uimplies: b supposing a,
prop: ℙ,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
true: True,
all: ∀x:A. B[x],
implies: P ⇒ Q,
composition-structure: Gamma ⊢ Compositon(A),
comp_trm: comp_trm,
composition-function: composition-function{j:l,i:l}(Gamma;A),
csm-comp-structure: (cA)tau,
cube-context-adjoin: X.A,
cc-adjoin-cube: (v;u),
cubical-term-at: u(a),
and: P ∧ Q,
interval-0: 0(𝕀)
Lemmas referenced :
csm-comp_term,
context-subset_wf,
csm-id_wf,
csm-context-subset-subtype2,
constrained-cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-term_wf,
thin-context-subset-adjoin,
istype-cubical-term,
composition-structure_wf,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
thin-context-subset,
csm-id-adjoin_wf-interval-1,
cubical-term-eqcd,
equal_wf,
csm-face-type,
context-subset-term-subtype,
csm-id-adjoin_wf,
interval-1_wf,
cube_set_map_subtype3,
sub_cubical_set_self,
context-iterated-subset0,
subset-cubical-term,
csm-ap-id-term,
context-iterated-subset1,
sub_cubical_set_wf,
context-subset-is-subset,
squash_wf,
true_wf,
istype-universe,
comp_term_wf,
comp_trm_wf,
composition-function_wf,
csm-context-subset-subtype3,
cube_set_map_wf,
I_cube_wf,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
csm-ap_wf,
cc-adjoin-cube_wf,
subset-I_cube,
csm-equal,
csm-context-subset-subtype,
cubical-term-equal,
sub_cubical_set_functionality,
cubical-term-at_wf,
subset-cubical-type,
sub_cubical_set_transitivity,
context-subset-swap,
sub_cubical_set_functionality2,
context-iterated-subset2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
inhabitedIsType,
universeIsType,
instantiate,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
productElimination,
cumulativity,
setEquality,
independent_isectElimination,
Error :memTop,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality_alt,
universeEquality,
natural_numberEquality,
lambdaFormation_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
independent_pairFormation,
dependent_set_memberEquality_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} Compositon(A)].
\mforall{}[u:\{Gamma, phi.\mBbbI{} \mvdash{} \_:A\}]. \mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\}]. \mforall{}[psi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
(comp cA [phi \mvdash{}\mrightarrow{} u] a0 = comp cA [phi \mvdash{}\mrightarrow{} u] a0)
Date html generated:
2020_05_20-PM-04_40_06
Last ObjectModification:
2020_04_18-PM-01_33_22
Theory : cubical!type!theory
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