Nuprl Lemma : pres-c1_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cA:composition-function{j:l,i:l}(G.𝕀;A)].
(pres-c1(G;phi;f;t;t0;cA) ∈ {G ⊢ _:(A)[1(𝕀)][phi |⟶ app(f; t)[1]]})
Proof
Definitions occuring in Statement :
pres-c1: pres-c1(G;phi;f;t;t0;cA),
composition-function: composition-function{j:l,i:l}(Gamma;A),
partial-term-1: u[1],
partial-term-0: u[0],
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,
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
csm-id-adjoin: [u],
cc-fst: p,
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,
pres-c1: pres-c1(G;phi;f;t;t0;cA),
subtype_rel: A ⊆r B,
partial-term-1: u[1],
partial-term-0: u[0],
guard: {T},
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
csm-id-adjoin: [u],
csm-id: 1(X),
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
context-subset-term-subtype,
cube-context-adjoin_wf,
interval-type_wf,
cubical-fun_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
cubical-app_wf_fun,
context-subset_wf,
thin-context-subset,
cubical-fun-subset,
comp_term_wf,
composition-function_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
partial-term-0_wf,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf,
subset-cubical-term,
context-adjoin-subset4,
csm-cubical-fun,
csm-id-adjoin_wf,
interval-0_wf,
cubical-term-eqcd,
csm-cubical-app,
context-subset-is-subset,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
hypothesisEquality,
applyEquality,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
instantiate,
hypothesis,
sqequalRule,
Error :memTop,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeIsType,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
lambdaEquality_alt,
cumulativity,
universeEquality,
hyp_replacement,
dependent_set_memberEquality_alt,
imageElimination,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityIstype
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A,T:\{G.\mBbbI{} \mvdash{} \_\}]. \mforall{}[f:\{G.\mBbbI{} \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[t:\{G.\mBbbI{}, (phi)p \mvdash{} \_:T\}].
\mforall{}[t0:\{G \mvdash{} \_:(T)[0(\mBbbI{})][phi |{}\mrightarrow{} t[0]]\}]. \mforall{}[cA:composition-function\{j:l,i:l\}(G.\mBbbI{};A)].
(pres-c1(G;phi;f;t;t0;cA) \mmember{} \{G \mvdash{} \_:(A)[1(\mBbbI{})][phi |{}\mrightarrow{} app(f; t)[1]]\})
Date html generated:
2020_05_20-PM-05_25_34
Last ObjectModification:
2020_04_18-PM-10_56_42
Theory : cubical!type!theory
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