Nuprl Lemma : equiv_comp-apply-sq
∀[H,A,E,cA,cE,I,a,b,c,d:Top].
(equiv_comp(H;A;E;cA;cE) formal-cube(I) (λx,y. <a[x;y], b[x;y]>) (λK,f. c[K;f]) (λI@0,a. ⋅) (λK,f. d[K;f]) I 1
~ equiv_comp_apply(H;A;E;cA;cE;I;a;b;c;d))
Proof
Definitions occuring in Statement :
equiv_comp_apply: equiv_comp_apply(H;A;E;cA;cE;I;a;b;c;d),
equiv_comp: equiv_comp(H;A;E;cA;cE),
formal-cube: formal-cube(I),
nh-id: 1,
it: ⋅,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
apply: f a,
lambda: λx.A[x],
pair: <a, b>,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
formal-cube: formal-cube(I),
cubical-lambda: (λb),
let: let,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
equiv_comp_apply: equiv_comp_apply(H;A;E;cA;cE;I;a;b;c;d),
cubical-sigma: Σ A B,
cubical-fiber: Fiber(w;a),
top: Top,
csm-ap: (s)x,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda4,
csm-ap-type: (AF)s,
bfalse: ff,
ifthenelse: if b then t else f fi
Lemmas referenced :
equiv_comp-sq,
sigma_comp-sq,
pi_comp-sq,
cube_set_restriction_pair_lemma,
istype-top,
contractible_comp-sq,
strict4-spread,
lifting-strict-spread,
fl-eq-0-1-false
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
Error :memTop,
dependent_functionElimination,
axiomSqEquality,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
voidEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
baseClosed
Latex:
\mforall{}[H,A,E,cA,cE,I,a,b,c,d:Top].
(equiv\_comp(H;A;E;cA;cE) formal-cube(I) (\mlambda{}x,y. <a[x;y], b[x;y]>) (\mlambda{}K,f. c[K;f]) (\mlambda{}I@0,a. \mcdot{}) (\mlambda{}K,f.\000C d[K;f]) I 1
\msim{} equiv\_comp\_apply(H;A;E;cA;cE;I;a;b;c;d))
Date html generated:
2020_05_20-PM-07_19_24
Last ObjectModification:
2020_04_27-PM-01_57_05
Theory : cubical!type!theory
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