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+{"url": "https://arxiv.org/abs/1412.7702", "title": "Convergence from below suffices", "abstract": "An elementary application of Fatou's lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in any introductory course on measure and integration.", "text": "\\section{The convergence from below theorem}\nThree famous convergence-related results appear in most introductory courses on measure and integration:\nthe monotone convergence theorem, Fatou's lemma and the dominated convergence theorem.\nIn teaching this material it is common to follow the approach taken in, for example, \\cite[Chapter 1]{Rudin}.\nThere Rudin begins by proving the monotone convergence theorem and then deduces Fatou's lemma. Finally, he deduces the dominated convergence theorem from Fatou's lemma. The result which we call the {\\em convergence from below theorem}\n(Theorem \\ref{CBT} below) is essentially distilled from this proof of the dominated convergence theorem (\\cite[pp. 26-27]{Rudin}).\nWe do not claim originality for this result, or for the related Theorem \\ref{infinite-integral}. They are presumably known, although we know of no explicit references for them. However, we wish to make a case that that they should be better known than they are. In particular, we suggest that Theorem \\ref{CBT} deserves a name and a place in the syllabus when this material is taught.\n\\vskip .2cm\nThroughout we discuss results concerning pointwise convergence. In the usual way, there are versions of all these results in terms of almost-everywhere convergence instead.\n\nFor convenience, we shall use the following terminology.\nLet $X$ be a set, let $(f_n)$ be a sequence of functions from $X$ to $[0,\\infty]$ and let\n$f$ be another function from $X$ to $[0,\\infty]$. We say that the functions $f_n$ converge to $f$\n{\\bf from below on $X$} if the functions $f_n$ tend to $f$ pointwise on $X$ and\n$f_n(x) \\leq f(x)$ $(n \\in {\\mathbb N},~x \\in X)$.\nWe say that the functions $f_n$ converge to $f$\n{\\bf monotonely from below on $X$} if the functions $f_n$ tend to $f$ pointwise on $X$ and, for all $x \\in X$, we have\n$f_1(x) \\leq f_2(x) \\leq f_3(x) \\leq \\cdots$.\n\n\\medskip\n\nWe begin by recalling the statement of the monotone convergence theorem.\n\\newpage\n\\begin{theorem}\n{\\bf (Monotone convergence theorem)}\nLet $(X,{\\cal F},\\mu )$\nbe a measure space, and let $f : X \\to [0,\\infty ]$ be a measurable function.\nLet $(f_n)$ be a sequence of measurable functions from $X$ to $[0,\\infty]$ which converge to $f$ monotonely from below on $X$.\nThen\n\\[\n\\int _X f\\,{\\rm d} \\mu = \\lim _{n\\to \\infty} \\int _X f_n\\,{\\rm d} \\mu\\,.\n\\]\n\\end{theorem}\n\nThe measurability assumption on\n$f$ is, of course, redundant here as it follows from the pointwise convergence of $f_n$ to $f$.\nWe now observe that an elementary application of Fatou's lemma shows that we may weaken the monotone convergence assumption. We have not found this result stated explicitly in the literature, and it does not appear to have a name. We propose to call it the {\\em convergence from below theorem}.\n\nThe concepts involved in the statements and applications of the monotone convergence theorem and the dominated convergence theorem are relatively simple. We suggest that convergence from below is a similarly simple concept, which should appeal to all levels of student. In particular, those students who find the concepts of $\\liminf$ and $\\limsup$ difficult may be happier applying the convergence from below theorem rather than Fatou's lemma (where possible).\n\\begin{theorem}\n{\\bf (Convergence from below theorem)}\n\\label{CBT}\nLet $(X,{\\cal F},\\mu )$\nbe a measure space, and let $f : X \\to [0,\\infty ]$ be a measurable function.\nLet $(f_n)$ be a sequence of measurable functions from $X$ to $[0,\\infty]$ which converge to $f$ from below on $X$.\nThen\n\\[\n\\int _X f\\,{\\rm d} \\mu = \\lim _{n\\to \\infty} \\int _X f_n\\,{\\rm d} \\mu\\,.\n\\]\n\\end{theorem}\n{\\bf Proof.}\nClearly\n\\[\n\\limsup_{n\\to \\infty}\\int_X f_n\\,{\\rm d} \\mu \\leq \\int_X f\\, {\\rm d} \\mu\\,.\n\\]\nHowever, by Fatou's lemma,\n\\[\n\\int_X f\\,{\\rm d} \\mu \\leq \\liminf_{n\\to\\infty}\\int_X f_n\\,{\\rm d} \\mu\\,.\n\\]\nThe result follows immediately.\n\\hfill$\\Box$\\par\n\\vskip .2cm\n\\noindent\n{\\bf Remarks.}\n\\begin{enumerate}\n \\item[(1)]\nThe monotone convergence theorem is now a special case of our stronger convergence from below theorem.\n\\item[(2)]\nIn the case where $\\int_X f\\,{\\rm d} \\mu < \\infty$, the convergence from below theorem is an immediate consequence of the dominated convergence theorem.\n\\item[(3)]\nIn the case where $\\int_X f\\,{\\rm d} \\mu = \\infty$, the result does not follow directly from either the monotone convergence theorem or the dominated convergence theorem. The following elementary result clarifies the situation in this case.\n\\end{enumerate}\n\n\\begin{theorem}\n\\label{infinite-integral}\nLet $(X,{\\cal F},\\mu )$\nbe a measure space, and let $f : X \\to [0,\\infty ]$ be a measurable function with $\\int_X f\\,{\\rm d} \\mu = \\infty$.\nLet $(f_n)$ be a sequence of measurable functions from $X$ to $[0,\\infty]$ which converge to $f$ pointwise on $X$.\nThen\n\\[\n\\lim _{n\\to \\infty} \\int _X f_n\\,{\\rm d} \\mu\\, = \\infty.\n\\]\n\\end{theorem}\n{\\bf Proof.}\nBy Fatou's lemma,\n\\[\n\\infty = \\int_X f\\,{\\rm d} \\mu \\leq \\liminf_{n\\to\\infty}\\int_X f_n\\,{\\rm d} \\mu\\,.\n\\]\nIt follows immediately that\n$\\lim _{n\\to \\infty} \\int _X f_n\\,{\\rm d} \\mu\\, = \\infty$, as required.\n\\hfill$\\Box$\\par\n\\vskip .2cm\nWe suggest that the convergence from below theorem deserves a place between Fatou's lemma and the dominated convergence theorem: the dominated convergence theorem may be deduced from the convergence from below theorem as follows. This proof is based on the proof given in \\cite[pp. 26-27]{Rudin}, but applying the convergence from below theorem in the middle.\n\n\\begin{theorem} {\\bf (Dominated convergence theorem)}\nLet $(X,{\\cal F},\\mu )$\nbe a measure space, let $g : X \\to [0,\\infty]$ be a measurable function.\nwith $\\int_X f\\,{\\rm d} \\mu < \\infty$ and let $f$ be a measurable function from $X$ to ${\\mathbb C}$.\nLet $(f_n)$ be a sequence of measurable functions from $X$ to $C$ which converge to $f$ pointwise on $X$\nand such that $|f_n(x)| \\leq g(x)$ $(n \\in {\\mathbb N}, x \\in X)$.\nThen\n\\[\n\\lim_{n \\to \\infty}\\int_X|f_n-f|\\,{\\rm d} \\mu\\, = 0\n\\]\nand\n\\[\n\\int _X f\\,{\\rm d} \\mu = \\lim _{n\\to \\infty} \\int _X f_n\\,{\\rm d} \\mu\\,.\n\\]\n\\end{theorem}\n{\\bf Proof.}\nThe second equality follows quickly from the first.\nTo prove the first equality, observe that the non-negative, measurable functions $2g-|f_n-f|$ converge to the function $2g$ from below. Thus, by the convergence from below theorem,\n\\[\n\\lim_{n \\to \\infty}\\int_X\\left( 2g -|f_n-f|\\right)\\,{\\rm d} \\mu\\, = \\int_X 2g \\,{\\rm d} \\mu\\,.\n\\]\nThe result now follows by subtracting $\\int_X 2g \\,{\\rm d} \\mu\\,$ from both sides and rearranging.\n\\hfill$\\Box$\\par\n\\vskip .2cm\n\n\nAs discussed above, the convergence from below theorem is more than covered by a combination of the dominated convergence theorem and Theorem \\ref{infinite-integral}. Also, since the convergence from below theorem is such an elementary consequence of Fatou's lemma, any applications may also be deduced from that lemma.\nHowever, the monotone convergence theorem continues to be used in the literature, and any application of the monotone convergence theorem can be replaced directly by an application of the\nconvergence from below theorem.\nOf course, we then only need to check the weaker conditions of the latter theorem.\n\nAlso, the convergence from below theorem can be used to give elegant solutions to simple problems where neither the monotone convergence theorem nor the dominated convergence theorem apply directly. Here is such an application (an elementary undergraduate exercise).\n\\vskip .2cm\n\\noindent\n{\\bf Exercise.}\nLet $\\lambda$ denote Lebesgue measure on ${\\mathbb R}$.\nProve that, for every Lebesgue measurable subset $E$ of ${\\mathbb R}$, we have\n\\[\n\\int_E x^2\\, {\\rm d} \\lambda(x) = \\lim_{n \\to \\infty} \\int_E \\left( x^2 - \\frac{1}{n} | x \\sin n x|\\right) \\, {\\rm d} \\lambda(x)\\,.\n\\]\n\\vskip .2cm\n\\noindent\n{\\bf Solution.}\nSince $|x \\sin n x| \\leq  n x^2$ ($n \\in {\\mathbb N}$, $x \\in {\\mathbb R}$), the result is an immediate consequence of the convergence from\nbelow theorem.\n\n\nWe may, instead, apply Fatou's lemma directly. This does, of course, lead to a quick solution which essentially proves the convergence from below theorem again along the way.\n\nWe may also consider separately the cases where $\\int_E x^2\\, {\\rm d} \\lambda(x)<\\infty$ and\nwhere $\\int_E x^2\\, {\\rm d} \\lambda(x)=\\infty$.\nIn the first case we may apply the dominated convergence theorem, and in the second case we may use Theorem \\ref{infinite-integral}. However the use of the convergence from below theorem renders this splitting into two cases unnecessary.\n\n\n\n\\section{Proving the convergence from below theorem directly}\nAbove we suggested following the usual development of the theory, but inserting the convergence from below theorem between Fatou's lemma and the dominated convergence theorem. There are several alternatives, however. For example, we can prove Fatou's lemma directly first and then deduce the convergence from below theorem. The monotone convergence theorem and the dominated convergence theorem then follow easily.\n\nAnother approach is to modify the standard proof of the monotone convergence theorem (\\cite[1.26]{Rudin}) in order to give a direct proof of the convergence from below theorem. The monotone convergence theorem, dominated convergence theorem and Fatou's lemma are then corollaries of this.\nWe conclude with such a direct proof.\n\nIn this proof we\navoid explicit reference to $\\liminf$ and $\\limsup$ in order to make the proof more accessible to students who have\ndifficulty with these concepts. However, only minor changes are needed to give a direct proof of Fatou's lemma instead.\n\\vskip .2cm\n\\noindent{\\bf Direct proof of Theorem \\ref{CBT}.}\nFirst note that we have $\\int_X f_n\\,{\\rm d} \\mu \\leq \\int_X f\\,{\\rm d} \\mu$ $(n \\in {\\mathbb N})$. Thus it is sufficient to prove that,\nfor all $\\alpha < \\int_X f\\,{\\rm d} \\mu$, $\\int_X f_n\\,{\\rm d} \\mu$ is eventually greater than $\\alpha$, i.e., there is an $N \\in {\\mathbb N}$ such that, for all $n \\geq N$, we have $\\int_X f_n\\,{\\rm d} \\mu > \\alpha$.\nGiven such an $\\alpha$, the definition of the integral tells us that there is a nonnegative, simple measurable function $s$ with $s(x)\\leq f(x)$ $(x \\in X)$ and such that\n$\\int_X s \\,{\\rm d} \\mu > \\alpha$.\nChoose $c \\in (0,1)$ large enough that\n$\\int_X c s \\,{\\rm d} \\mu > \\alpha$.\nSet\n$A_n = \\{x \\in X: cs(x) \\leq f_n(x)\\}$\nand, for each $k \\in {\\mathbb N}$, set\n\\[\nB_k = \\bigcap_{n \\geq k} A_n = \\{x \\in X: cs(x) \\leq f_n(x) \\jtext{for all} n \\geq k\\}.\n\\]\nClearly, $B_1\\subseteq B_2\\subseteq \\cdots$. We claim that $\\bigcup_{k=1}^\\infty B_k = X$.\nLet $x \\in X$. If $s(x)>0$, then $cs(x)<f(x)$, and so\n$x \\in B_k$ provided that $k$ is large enough. On the other hand, if $s(x)=0$, then $x \\in B_k$ for all $k \\in {\\mathbb N}$.\nThis proves our claim.\nBy standard continuity properties of measures, we have\n\\[\n\\int_X{cs\\,{\\rm d} \\mu} = \\lim_{k\\to \\infty} \\int_{B_k} cs\\,{\\rm d}\\mu\\,.\n\\]\nChoose $N \\in {\\mathbb N}$ such that $\\int_{B_N} cs\\,{\\rm d}\\mu > \\alpha$.\nFor all $n \\geq N$ and $x \\in B_N$ we have $cs(x) \\leq f_n(x)$.\nThus, for $n \\geq N$, we have\n\\[\n\\int_X f_n \\,{\\rm d}\\mu\\, \\geq \\int_{B_N} f_n \\,{\\rm d}\\mu\\, \\geq  \\int_{B_N} cs \\,{\\rm d}\\mu\\, > \\alpha,\n\\]\nas required.\n\\hfill$\\Box$\\par\n\n", "meta": {"timestamp": "2014-12-25T02:09:06", "yymm": "1412", "arxiv_id": "1412.7702", "language": "en", "url": "https://arxiv.org/abs/1412.7702", "abstract": "An elementary application of Fatou's lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in any introductory course on measure and integration.", "subjects": "Functional Analysis (math.FA)", "title": "Convergence from below suffices", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9920620077761345, "lm_q2_score": 0.8807970764133561, "lm_q1q2_score": 0.8738053160699835}}

data/code/agda/0.10-0.15.jsonl

@@ -0,0 +1 @@
+{"text": "open import Nat\nopen import Prelude\nopen import List\n\nopen import contexts\nopen import core\n\n-- This file is a glorious testament to all of agda's greatest strengths -\n-- a beautiful work that will shine brightly throughout the ages and instill\n-- hope for the future in many generations to come. May I never use any other\n-- proof assistant.\nmodule decidability where\n  mutual\n    -- We define context decidability in contexts.agda,\n    -- because it's an important metatheorem. However, it accepts a\n    -- decidability theorem for its value type, and for environments that\n    -- value type is result, so the termination checker complains. To fix\n    -- this, we redefine type-specific versions of this theorem here.\n    -- Note that we really should not be destructing contexts via `_::_` -\n    -- here we only do that because we copied this code from the contexts file.\n\n    ctx<result>-==-dec : (Γ1 Γ2 : result ctx) →\n                          Γ1 == Γ2 ∨ Γ1 ≠ Γ2\n    ctx<result>-==-dec [] []       = Inl refl\n    ctx<result>-==-dec [] (_ :: _) = Inr (λ ())\n    ctx<result>-==-dec (_ :: _) [] = Inr (λ ())\n    ctx<result>-==-dec ((hx1 , hr1) :: t1) ((hx2 , hr2) :: t2)\n      with natEQ hx1 hx2 | result-==-dec hr1 hr2 | ctx<result>-==-dec t1 t2\n    ... | Inl refl | Inl refl | Inl refl = Inl refl\n    ... | Inl refl | Inl refl | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl | Inr ne   | _        = Inr λ where refl → ne refl\n    ... | Inr ne   | _        | _        = Inr λ where refl → ne refl\n\n    ctx<rule>-==-dec : (rules1 rules2 : rule ctx) →\n                        rules1 == rules2 ∨ rules1 ≠ rules2\n    ctx<rule>-==-dec [] [] = Inl refl\n    ctx<rule>-==-dec [] (_ :: _) = Inr (λ ())\n    ctx<rule>-==-dec (_ :: _) [] = Inr (λ ())\n    ctx<rule>-==-dec ((hx1 , hrule1) :: t1) ((hx2 , hrule2) :: t2)\n      with natEQ hx1 hx2 | rule-==-dec hrule1 hrule2 | ctx<rule>-==-dec t1 t2\n    ... | Inl refl | Inl refl | Inl refl = Inl refl\n    ... | Inl refl | Inl refl | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl | Inr ne   | _        = Inr λ where refl → ne refl\n    ... | Inr ne   | _        | _        = Inr λ where refl → ne refl\n\n    rule-==-dec : (rule1 rule2 : rule) →\n                   rule1 == rule2 ∨ rule1 ≠ rule2\n    rule-==-dec (|C parm1 => branch1) (|C parm2 => branch2)\n      with natEQ parm1 parm2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with exp-==-dec branch1 branch2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n\n    ex-==-dec : (ex1 ex2 : ex) →\n                 ex1 == ex2 ∨ ex1 ≠ ex2\n    ex-==-dec ⟨⟩ ⟨⟩ = Inl refl\n    ex-==-dec ⟨⟩ ⟨ ex2 , ex3 ⟩ = Inr (λ ())\n    ex-==-dec ⟨⟩ (C[ x ] ex2) = Inr (λ ())\n    ex-==-dec ⟨⟩ ¿¿ = Inr (λ ())\n    ex-==-dec ⟨⟩ (x ↦ ex2) = Inr (λ ())\n    ex-==-dec ⟨ ex1a , ex1b ⟩ ⟨ ex2a , ex2b ⟩\n      with ex-==-dec ex1a ex2a\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with ex-==-dec ex1b ex2b\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    ex-==-dec ⟨ ex1 , ex2 ⟩ ⟨⟩ = Inr (λ ())\n    ex-==-dec ⟨ _ , _ ⟩ (C[ _ ] _) = Inr (λ ())\n    ex-==-dec ⟨ ex1 , ex2 ⟩ ¿¿ = Inr (λ ())\n    ex-==-dec ⟨ ex1 , ex2 ⟩ (x ↦ ex3) = Inr (λ ())\n    ex-==-dec (C[ c1 ] ex1) (C[ c2 ] ex2)\n      with natEQ c1 c2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with ex-==-dec ex1 ex2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    ex-==-dec (C[ x ] ex1) ⟨⟩ = Inr (λ ())\n    ex-==-dec (C[ _ ] _) ⟨ _ , _ ⟩ = Inr (λ ())\n    ex-==-dec (C[ x ] ex1) ¿¿ = Inr (λ ())\n    ex-==-dec (C[ x ] ex1) (x₁ ↦ ex2) = Inr (λ ())\n    ex-==-dec ¿¿ ¿¿ = Inl refl\n    ex-==-dec ¿¿ ⟨⟩ = Inr (λ ())\n    ex-==-dec ¿¿ ⟨ ex2 , ex3 ⟩ = Inr (λ ())\n    ex-==-dec ¿¿ (C[ x ] ex2) = Inr (λ ())\n    ex-==-dec ¿¿ (x ↦ ex2) = Inr (λ ())\n    ex-==-dec (v1 ↦ ex1) (v2 ↦ ex2)\n      with result-==-dec v1 v2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with ex-==-dec ex1 ex2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    ex-==-dec (x ↦ ex1) ⟨⟩ = Inr (λ ())\n    ex-==-dec (x ↦ ex1) ⟨ ex2 , ex3 ⟩ = Inr (λ ())\n    ex-==-dec (x ↦ ex1) (C[ x₁ ] ex2) = Inr (λ ())\n    ex-==-dec (x ↦ ex1) ¿¿ = Inr (λ ())\n\n    exp-==-dec : (e1 e2 : exp) → e1 == e2 ∨ e1 ≠ e2\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ fix f2 ⦇·λ x2 => e2 ·⦈\n      with natEQ f1 f2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with natEQ x1 x2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with exp-==-dec e1 e2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ X[ x ] = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ ⟨⟩ = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ ⟨ _ , _ ⟩ = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ (fst _) = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ (snd _) = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ (C[ x ] e2) = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ ??[ x ] = Inr (λ ())\n    exp-==-dec fix f1 ⦇·λ x1 => e1 ·⦈ (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec X[ x1 ] X[ x2 ]\n      with natEQ x1 x2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec X[ x1 ] fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec X[ x1 ] (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec X[ x1 ] ⟨⟩ = Inr (λ ())\n    exp-==-dec X[ x1 ] ⟨ _ , _ ⟩ = Inr (λ ())\n    exp-==-dec X[ x1 ] (fst _) = Inr (λ ())\n    exp-==-dec X[ x1 ] (snd _) = Inr (λ ())\n    exp-==-dec X[ x1 ] (C[ x ] e2) = Inr (λ ())\n    exp-==-dec X[ x1 ] case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec X[ x1 ] ??[ x ] = Inr (λ ())\n    exp-==-dec X[ x1 ] (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) (ef2 ∘ earg2)\n      with exp-==-dec ef1 ef2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with exp-==-dec earg1 earg2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec (ef1 ∘ earg1) fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) X[ x ] = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) ⟨⟩ = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) ⟨ _ , _ ⟩ = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) (fst _) = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) (snd _) = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) (C[ x ] e2) = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) ??[ x ] = Inr (λ ())\n    exp-==-dec (ef1 ∘ earg1) (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec ⟨⟩ ⟨⟩ = Inl refl\n    exp-==-dec ⟨⟩ fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec ⟨⟩ X[ x ] = Inr (λ ())\n    exp-==-dec ⟨⟩ (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec ⟨⟩ ⟨ e2 , e3 ⟩ = Inr (λ ())\n    exp-==-dec ⟨⟩ (fst e2) = Inr (λ ())\n    exp-==-dec ⟨⟩ (snd e2) = Inr (λ ())\n    exp-==-dec ⟨⟩ (C[ x ] e2) = Inr (λ ())\n    exp-==-dec ⟨⟩ case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec ⟨⟩ ??[ x ] = Inr (λ ())\n    exp-==-dec ⟨⟩ (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec ⟨ es1a , es1b ⟩ ⟨ es2a , es2b ⟩\n      with exp-==-dec es1a es2a\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with exp-==-dec es1b es2b\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec ⟨ _ , _ ⟩ fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ X[ x ] = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ ⟨⟩ = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ (fst _) = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ (snd _) = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ (C[ x ] e2) = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ ??[ x ] = Inr (λ ())\n    exp-==-dec ⟨ _ , _ ⟩ (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec (fst e1) (fst e2)\n      with exp-==-dec e1 e2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec (fst e1) fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec (fst e1) X[ x ] = Inr (λ ())\n    exp-==-dec (fst e1) (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec (fst e1) ⟨⟩ = Inr (λ ())\n    exp-==-dec (fst e1) ⟨ e2 , e3 ⟩ = Inr (λ ())\n    exp-==-dec (fst e1) (snd e2) = Inr (λ ())\n    exp-==-dec (fst e1) (C[ x ] e2) = Inr (λ ())\n    exp-==-dec (fst e1) case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec (fst e1) ??[ x ] = Inr (λ ())\n    exp-==-dec (fst e1) (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec (snd e1) (snd e2)\n      with exp-==-dec e1 e2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec (snd e1) fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec (snd e1) X[ x ] = Inr (λ ())\n    exp-==-dec (snd e1) (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec (snd e1) ⟨⟩ = Inr (λ ())\n    exp-==-dec (snd e1) ⟨ e2 , e3 ⟩ = Inr (λ ())\n    exp-==-dec (snd e1) (fst e2) = Inr (λ ())\n    exp-==-dec (snd e1) (C[ x ] e2) = Inr (λ ())\n    exp-==-dec (snd e1) case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec (snd e1) ??[ x ] = Inr (λ ())\n    exp-==-dec (snd e1) (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) (C[ c2 ] e2)\n      with natEQ c1 c2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with exp-==-dec e1 e2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec (C[ c1 ] e1) fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) X[ x ] = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) ⟨⟩ = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) ⟨ _ , _ ⟩ = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) (fst _) = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) (snd _) = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) ??[ x ] = Inr (λ ())\n    exp-==-dec (C[ c1 ] e1) (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ case e2 of⦃· rules2 ·⦄\n      with exp-==-dec e1 e2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with ctx<rule>-==-dec rules1 rules2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec case e1 of⦃· rules1 ·⦄ fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ X[ x ] = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ ⟨⟩ = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ ⟨ _ , _ ⟩ = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ (fst _) = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ (snd _) = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ (C[ x ] e2) = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ ??[ x ] = Inr (λ ())\n    exp-==-dec case e1 of⦃· rules1 ·⦄ (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec ??[ u1 ] ??[ u2 ]\n      with natEQ u1 u2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec ??[ u1 ] fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec ??[ u1 ] X[ x ] = Inr (λ ())\n    exp-==-dec ??[ u1 ] (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec ??[ u1 ] ⟨⟩ = Inr (λ ())\n    exp-==-dec ??[ u1 ] ⟨ _ , _ ⟩ = Inr (λ ())\n    exp-==-dec ??[ u1 ] (fst _) = Inr (λ ())\n    exp-==-dec ??[ u1 ] (snd _) = Inr (λ ())\n    exp-==-dec ??[ u1 ] (C[ x ] e2) = Inr (λ ())\n    exp-==-dec ??[ u1 ] case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec ??[ u1 ] (PBE:assert e2 e3) = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) (PBE:assert e2a e2b)\n      with exp-==-dec e1a e2a\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl\n      with exp-==-dec e1b e2b\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    exp-==-dec (PBE:assert e1a e1b) fix x ⦇·λ x₁ => e2 ·⦈ = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) X[ x ] = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) (e2 ∘ e3) = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) ⟨⟩ = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) ⟨ _ , _ ⟩ = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) (fst _) = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) (snd _) = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) (C[ x ] e2) = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) case e2 of⦃· x ·⦄ = Inr (λ ())\n    exp-==-dec (PBE:assert e1a e1b) ??[ x ] = Inr (λ ())\n\n    result-==-dec : (r1 r2 : result) → r1 == r2 ∨ r1 ≠ r2\n    result-==-dec [ E1 ]fix f1 ⦇·λ x1 => e1 ·⦈ [ E2 ]fix f2 ⦇·λ x2 => e2 ·⦈\n      with ctx<result>-==-dec E1 E2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with natEQ f1 f2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with natEQ x1 x2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with exp-==-dec e1 e2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ ⟨⟩ = Inr (λ ())\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ ⟨ _ , _ ⟩ = Inr (λ ())\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ (C[ _ ] _) = Inr (λ ())\n    result-==-dec [ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈ (C⁻[ x₄ ] r2) = Inr (λ ())\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ [ _ ]??[ _ ] = Inr (λ ())\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ (_ ∘ _) = Inr (λ ())\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ (fst _) = Inr (λ ())\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ (snd _) = Inr (λ ())\n    result-==-dec [ E ]fix f ⦇·λ x => e ·⦈ [ _ ]case _ of⦃· _ ·⦄ = Inr (λ ())\n    result-==-dec ⟨⟩ ⟨⟩ = Inl refl\n    result-==-dec ⟨⟩ [ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈ = Inr (λ ())\n    result-==-dec ⟨⟩ ⟨ r2 , r3 ⟩ = Inr (λ ())\n    result-==-dec ⟨⟩ (C[ x ] r2) = Inr (λ ())\n    result-==-dec ⟨⟩ (C⁻[ x ] r2) = Inr (λ ())\n    result-==-dec ⟨⟩ [ x ]??[ x₁ ] = Inr (λ ())\n    result-==-dec ⟨⟩ (r2 ∘ r3) = Inr (λ ())\n    result-==-dec ⟨⟩ (fst r2) = Inr (λ ())\n    result-==-dec ⟨⟩ (snd r2) = Inr (λ ())\n    result-==-dec ⟨⟩ [ x ]case r2 of⦃· x₁ ·⦄ = Inr (λ ())\n    result-==-dec ⟨ rs1a , rs1b ⟩ ⟨ rs2a , rs2b ⟩\n      with result-==-dec rs1a rs2a\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with result-==-dec rs1b rs2b\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n    result-==-dec ⟨ _ , _ ⟩ [ _ ]fix _ ⦇·λ _ => _ ·⦈ = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ ⟨⟩ = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ (C[ _ ] _) = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ (C⁻[ x ] r3) = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ [ _ ]??[ _ ] = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ (_ ∘ _) = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ (snd _) = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ (fst _) = Inr (λ ())\n    result-==-dec ⟨ _ , _ ⟩ [ _ ]case _ of⦃· _ ·⦄ = Inr (λ ())\n    result-==-dec (C[ c1 ] r1) (C[ c2 ] r2)\n      with natEQ c1 c2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with result-==-dec r1 r2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n    result-==-dec (C[ c ] r) [ _ ]fix _ ⦇·λ _ => _ ·⦈ = Inr (λ ())\n    result-==-dec (C[ c ] r) ⟨⟩ = Inr (λ ())\n    result-==-dec (C[ c ] r) ⟨ _ , _ ⟩ = Inr (λ ())\n    result-==-dec (C[ x ] r1) (C⁻[ x₁ ] r2) = Inr (λ ())\n    result-==-dec (C[ c ] r) [ _ ]??[ _ ] = Inr (λ ())\n    result-==-dec (C[ c ] r) (_ ∘ _) = Inr (λ ())\n    result-==-dec (C[ c ] r) (fst _) = Inr (λ ())\n    result-==-dec (C[ c ] r) (snd _) = Inr (λ ())\n    result-==-dec (C[ c ] r) [ _ ]case _ of⦃· _ ·⦄ = Inr (λ ())\n    result-==-dec (C⁻[ c1 ] r1) (C⁻[ c2 ] r2)\n      with natEQ c1 c2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with result-==-dec r1 r2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n    result-==-dec (C⁻[ x ] r1) [ x₁ ]fix x₂ ⦇·λ x₃ => x₄ ·⦈ = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) ⟨⟩ = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) ⟨ r2 , r3 ⟩ = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) (C[ x₁ ] r2) = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) [ x₁ ]??[ x₂ ] = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) (r2 ∘ r3) = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) (fst r2) = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) (snd r2) = Inr (λ ())\n    result-==-dec (C⁻[ x ] r1) [ x₁ ]case r2 of⦃· x₂ ·⦄ = Inr (λ ())\n    result-==-dec [ E1 ]??[ u1 ] [ E2 ]??[ u2 ]\n      with ctx<result>-==-dec E1 E2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with natEQ u1 u2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n    result-==-dec [ E ]??[ u1 ] [ _ ]fix _ ⦇·λ _ => _ ·⦈ = Inr (λ ())\n    result-==-dec [ E ]??[ u1 ] ⟨⟩ = Inr (λ ())\n    result-==-dec [ E ]??[ u1 ] ⟨ _ , _ ⟩ = Inr (λ ())\n    result-==-dec [ E ]??[ u1 ] (C[ _ ] _) = Inr (λ ())\n    result-==-dec [ x ]??[ x₁ ] (C⁻[ x₂ ] r2) = Inr (λ ())\n    result-==-dec [ E ]??[ u1 ] (_ ∘ _) = Inr (λ ())\n    result-==-dec [ E ]??[ u1 ] (fst _) = Inr (λ ())\n    result-==-dec [ E ]??[ u1 ] (snd _) = Inr (λ ())\n    result-==-dec [ E ]??[ u1 ] [ _ ]case _ of⦃· _ ·⦄ = Inr (λ ())\n    result-==-dec (rf1 ∘ rarg1) (rf2 ∘ rarg2)\n      with result-==-dec rf1 rf2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with result-==-dec rarg1 rarg2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n    result-==-dec (rf ∘ rarg) [ _ ]fix _ ⦇·λ _ => _ ·⦈ = Inr (λ ())\n    result-==-dec (rf ∘ rarg) ⟨⟩ = Inr (λ ())\n    result-==-dec (rf ∘ rarg) ⟨ _ , _ ⟩ = Inr (λ ())\n    result-==-dec (rf ∘ rarg) (C[ _ ] _) = Inr (λ ())\n    result-==-dec (r1 ∘ r2) (C⁻[ x ] r3) = Inr (λ ())\n    result-==-dec (rf ∘ rarg) [ _ ]??[ _ ] = Inr (λ ())\n    result-==-dec (rf ∘ rarg) (fst _) = Inr (λ ())\n    result-==-dec (rf ∘ rarg) (snd _) = Inr (λ ())\n    result-==-dec (rf ∘ rarg) [ _ ]case _ of⦃· _ ·⦄ = Inr (λ ())\n    result-==-dec (fst r1) (fst r2)\n      with result-==-dec r1 r2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    result-==-dec (fst r1) [ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈ = Inr (λ ())\n    result-==-dec (fst r1) ⟨⟩ = Inr (λ ())\n    result-==-dec (fst r1) ⟨ r2 , r3 ⟩ = Inr (λ ())\n    result-==-dec (fst r1) (C[ x ] r2) = Inr (λ ())\n    result-==-dec (fst r1) (C⁻[ x ] r2) = Inr (λ ())\n    result-==-dec (fst r1) [ x ]??[ x₁ ] = Inr (λ ())\n    result-==-dec (fst r1) (r2 ∘ r3) = Inr (λ ())\n    result-==-dec (fst r1) (snd r2) = Inr (λ ())\n    result-==-dec (fst r1) [ x ]case r2 of⦃· x₁ ·⦄ = Inr (λ ())\n    result-==-dec (snd r1) (snd r2)\n      with result-==-dec r1 r2\n    ... | Inr ne   = Inr (λ where refl → ne refl)\n    ... | Inl refl = Inl refl\n    result-==-dec (snd r1) [ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈ = Inr (λ ())\n    result-==-dec (snd r1) ⟨⟩ = Inr (λ ())\n    result-==-dec (snd r1) ⟨ r2 , r3 ⟩ = Inr (λ ())\n    result-==-dec (snd r1) (C[ x ] r2) = Inr (λ ())\n    result-==-dec (snd r1) (C⁻[ x ] r2) = Inr (λ ())\n    result-==-dec (snd r1) [ x ]??[ x₁ ] = Inr (λ ())\n    result-==-dec (snd r1) (r2 ∘ r3) = Inr (λ ())\n    result-==-dec (snd r1) (fst r2) = Inr (λ ())\n    result-==-dec (snd r1) [ x ]case r2 of⦃· x₁ ·⦄ = Inr (λ ())\n    result-==-dec [ E1 ]case r1 of⦃· rules1 ·⦄ [ E2 ]case r2 of⦃· rules2 ·⦄\n      with ctx<result>-==-dec E1 E2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with result-==-dec r1 r2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl\n      with ctx<rule>-==-dec rules1 rules2\n    ... | Inr ne   = Inr λ where refl → ne refl\n    ... | Inl refl = Inl refl\n    result-==-dec [ E ]case r of⦃· rules ·⦄ [ _ ]fix _ ⦇·λ _ => _ ·⦈ = Inr (λ ())\n    result-==-dec [ E ]case r of⦃· rules ·⦄ ⟨⟩ = Inr (λ ())\n    result-==-dec [ E ]case r of⦃· rules ·⦄ ⟨ _ , _ ⟩ = Inr (λ ())\n    result-==-dec [ E ]case r of⦃· rules ·⦄ (C[ _ ] _) = Inr (λ ())\n    result-==-dec [ E ]case r of⦃· rules ·⦄ [ _ ]??[ _ ] = Inr (λ ())\n    result-==-dec [ E ]case r of⦃· rules ·⦄ (_ ∘ _) = Inr (λ ())\n    result-==-dec [ E ]case r of⦃· rules ·⦄ (fst _) = Inr (λ ())\n    result-==-dec [ E ]case r of⦃· rules ·⦄ (snd _) = Inr (λ ())\n    result-==-dec [ x ]case r1 of⦃· x₁ ·⦄ (C⁻[ x₂ ] r2) = Inr (λ ())\n\n  typ-==-dec : (τ1 τ2 : typ) → τ1 == τ2 ∨ τ1 ≠ τ2\n  typ-==-dec (τ1i ==> τ1o) (τ2i ==> τ2o)\n    with typ-==-dec τ1i τ2i\n  ... | Inr ne   = Inr λ where refl → ne refl\n  ... | Inl refl\n    with typ-==-dec τ1o τ2o\n  ... | Inr ne   = Inr λ where refl → ne refl\n  ... | Inl refl = Inl refl\n  typ-==-dec (τ1i ==> τ1o) ⟨⟩ = Inr (λ ())\n  typ-==-dec (τ1i ==> τ1o) ⟨ _ × _ ⟩ = Inr (λ ())\n  typ-==-dec (τ1i ==> τ1o) D[ _ ] = Inr (λ ())\n  typ-==-dec ⟨⟩ ⟨⟩ = Inl refl\n  typ-==-dec ⟨⟩ ⟨ τ2 × τ3 ⟩ = Inr (λ ())\n  typ-==-dec ⟨⟩ (_ ==> _) = Inr (λ ())\n  typ-==-dec ⟨⟩ D[ _ ] = Inr (λ ())\n  typ-==-dec ⟨ τ1a × τ1b ⟩ ⟨ τ2a × τ2b ⟩\n    with typ-==-dec τ1a τ2a\n  ... | Inr ne   = Inr λ where refl → ne refl\n  ... | Inl refl\n    with typ-==-dec τ1b τ2b\n  ... | Inr ne   = Inr λ where refl → ne refl\n  ... | Inl refl = Inl refl\n  typ-==-dec ⟨ τ1 × τ2 ⟩ ⟨⟩ = Inr (λ ())\n  typ-==-dec ⟨ _ × _ ⟩ (_ ==> _) = Inr (λ ())\n  typ-==-dec ⟨ _ × _ ⟩ D[ _ ] = Inr (λ ())\n  typ-==-dec D[ d1 ] D[ d2 ]\n    with natEQ d1 d2\n  ... | Inr ne   = Inr λ where refl → ne refl\n  ... | Inl refl = Inl refl\n  typ-==-dec D[ _ ] (_ ==> _) = Inr (λ ())\n  typ-==-dec D[ _ ] ⟨⟩ = Inr (λ ())\n  typ-==-dec D[ _ ] ⟨ _ × _ ⟩ = Inr (λ ())\n\n  final-dec : ∀{r} → r final ∨ (r final → ⊥)\n\n  env-final-dec : ∀{E} → E env-final ∨ (E env-final → ⊥)\n\n  final-dec {[ E ]fix x₁ ⦇·λ x₂ => e ·⦈}\n    with env-final-dec {E}\n  ... | Inr nf  = Inr (λ {(FFix f) → nf f})\n  ... | Inl E-f = Inl (FFix E-f)\n  final-dec {⟨⟩} = Inl FUnit\n  final-dec {⟨ r1 , r2 ⟩}\n    with final-dec {r1}\n  ... | Inr nf   = Inr λ {(FPair h1 h2) → nf h1}\n  ... | Inl r1-f\n    with final-dec {r2}\n  ... | Inr nf   = Inr λ {(FPair h1 h2) → nf h2}\n  ... | Inl r2-f = Inl (FPair r1-f r2-f)\n  final-dec {C[ x ] r}\n    with final-dec {r}\n  ... | Inr nf  = Inr λ {(FCon h1) → nf h1}\n  ... | Inl r-f = Inl (FCon r-f)\n  final-dec {C⁻[ x ] r} = Inr (λ ())\n  final-dec {[ E ]??[ x₁ ]}\n    with env-final-dec {E}\n  ... | Inr nf  = Inr λ {(FHole h1) → nf h1}\n  ... | Inl E-f = Inl (FHole E-f)\n  final-dec {r1 ∘ r2}\n    with final-dec {r1}\n  ... | Inr nf   = Inr λ {(FAp h _ _) → nf h}\n  ... | Inl r1-f\n    with final-dec {r2}\n  ... | Inr nf   = Inr λ {(FAp _ h _) → nf h}\n  final-dec {[ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈ ∘ r2} | Inl r1-f | Inl r2-f\n    = Inr λ {(FAp _ _ h) → abort (h refl)}\n  final-dec {⟨⟩ ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f (λ {E} {f} {x} {e} ()))\n  final-dec {⟨ r1 , r3 ⟩ ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f (λ {E} {f} {x} {e} ()))\n  final-dec {(C[ x ] r1) ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f (λ {E} {f} {x} {e} ()))\n  final-dec {[ x ]??[ x₁ ] ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f  (λ {E} {f} {x} {e} ()))\n  final-dec {(r1 ∘ r3) ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f (λ {E} {f} {x} {e} ()))\n  final-dec {fst r1 ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f (λ {E} {f} {x} {e} ()))\n  final-dec {snd r1 ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f (λ {E} {f} {x} {e} ()))\n  final-dec {[ x ]case r1 of⦃· x₁ ·⦄ ∘ r2} | Inl r1-f | Inl r2-f = Inl (FAp r1-f r2-f (λ {E} {f} {x} {e} ()))\n  final-dec {fst r}\n    with final-dec {r}\n  ... | Inr nf  = Inr λ {(FFst h _) → nf h}\n  final-dec {fst ⟨ r , r₁ ⟩} | Inl r-f\n    = Inr λ {(FFst _ h) → abort (h refl)}\n  final-dec {fst [ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {fst ⟨⟩} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {fst (C[ x ] r)} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {fst [ x ]??[ x₁ ]} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {fst (r ∘ r₁)} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {fst (fst r)} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {fst (snd r)} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {fst [ x ]case r of⦃· x₁ ·⦄} | Inl r-f = Inl (FFst r-f (λ {r1} {r2} ()))\n  final-dec {snd r}\n    with final-dec {r}\n  ... | Inr nf  = Inr λ {(FSnd h _) → nf h}\n  final-dec {snd ⟨ r , r₁ ⟩} | Inl r-f\n    = Inr λ {(FSnd _ h) → abort (h refl)}\n  final-dec {snd [ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {snd ⟨⟩} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {snd (C[ x ] r)} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {snd [ x ]??[ x₁ ]} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {snd (r ∘ r₁)} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {snd (fst r)} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {snd (snd r)} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {snd [ x ]case r of⦃· x₁ ·⦄} | Inl r-f = Inl (FSnd r-f (λ {r1} {r2} ()))\n  final-dec {[ E ]case r of⦃· x₁ ·⦄}\n    with final-dec {r}\n  ... | Inr nf  = Inr λ {(FCase h _ _) → nf h}\n  ... | Inl r-f\n    with env-final-dec {E}\n  ... | Inr nf  = Inr λ {(FCase _ _ h) → nf h}\n  final-dec {[ E ]case C[ x ] r of⦃· x₁ ·⦄} | Inl r-f | Inl E-f\n    = Inr λ {(FCase _ h _) → abort (h refl)}\n  final-dec {[ E ]case [ x ]fix x₂ ⦇·λ x₃ => x₄ ·⦈ of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n  final-dec {[ E ]case ⟨⟩ of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n  final-dec {[ E ]case ⟨ r , r₁ ⟩ of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n  final-dec {[ E ]case [ x ]??[ x₂ ] of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n  final-dec {[ E ]case r ∘ r₁ of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n  final-dec {[ E ]case fst r of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n  final-dec {[ E ]case snd r of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n  final-dec {[ E ]case [ x ]case r of⦃· x₂ ·⦄ of⦃· x₁ ·⦄} | Inl r-f | Inl E-f = Inl (FCase r-f (λ {c} {r'} ()) E-f)\n\n  env-final-dec {[]} = Inl (EF (λ ()))\n  env-final-dec {(_ , r) :: E}\n    with final-dec {r}\n  ... | Inr nf       = Inr (λ {(EF h) → nf (h InH)})\n  ... | Inl r-f\n    with env-final-dec {E}\n  ... | Inr nf       = Inr (λ {(EF h) → nf (EF (λ z → h (InT z)))})\n  ... | Inl (EF E-f) = Inl (EF λ {InH → r-f ; (InT h) → E-f h})\n\n  Coerce-dec : ∀{r} → Σ[ ex ∈ ex ] (Coerce r := ex) ∨ (Σ[ ex ∈ ex ] (Coerce r := ex) → ⊥)\n  Coerce-dec {[ x ]fix x₁ ⦇·λ x₂ => x₃ ·⦈} = Inr (λ where (_ , ()))\n  Coerce-dec {⟨⟩} = Inl (⟨⟩ , CoerceUnit)\n  Coerce-dec {⟨ r1 , r2 ⟩}\n    with Coerce-dec {r1}\n  ...  | Inr nc          = Inr λ {(_ , CoercePair c-r1 _) → nc (_ , c-r1)}\n  ...  | Inl (_ , c-r1)\n    with Coerce-dec {r2}\n  ...  | Inr nc          = Inr λ {(_ , CoercePair _ c-r2) → nc (_ , c-r2)}\n  ...  | Inl (_ , c-r2)  = Inl (_ , CoercePair c-r1 c-r2)\n  Coerce-dec {C[ c ] r}\n    with Coerce-dec {r}\n  ...  | Inr nc         = Inr λ {(_ , CoerceCtor c-r) → nc (_ , c-r)}\n  ...  | Inl (_ , c-r)  = Inl (_ , CoerceCtor c-r)\n  Coerce-dec {C⁻[ x ] r} = Inr (λ where (_ , ()))\n  Coerce-dec {[ x ]??[ x₁ ]} = Inr (λ where (_ , ()))\n  Coerce-dec {r ∘ r₁} = Inr (λ where (_ , ()))\n  Coerce-dec {fst r} = Inr (λ where (_ , ()))\n  Coerce-dec {snd r} = Inr (λ where (_ , ()))\n  Coerce-dec {[ x ]case r of⦃· x₁ ·⦄} = Inr (λ where (_ , ()))\n\n  value-dec : ∀{r} → r value ∨ (r value → ⊥)\n  value-dec {[ E ]fix x₁ ⦇·λ x₂ => x₃ ·⦈}\n    with env-final-dec {E}\n  ... | Inr nf  = Inr (λ {(VFix E-f) → nf E-f})\n  ... | Inl E-f = Inl (VFix E-f)\n  value-dec {⟨⟩} = Inl VUnit\n  value-dec {⟨ r1 , r2 ⟩}\n    with value-dec {r1}\n  ... | Inr nv  = Inr λ {(VPair h _) → nv h}\n  ... | Inl r1v\n    with value-dec {r2}\n  ... | Inr nv  = Inr λ {(VPair _ h) → nv h}\n  ... | Inl r2v = Inl (VPair r1v r2v)\n  value-dec {C[ x ] r}\n    with value-dec {r}\n  ... | Inl rv = Inl (VCon rv)\n  ... | Inr nv = Inr (λ {(VCon rv) → nv rv})\n  value-dec {C⁻[ x ] r} = Inr (λ ())\n  value-dec {[ x ]??[ x₁ ]} = Inr (λ ())\n  value-dec {r ∘ r₁} = Inr (λ ())\n  value-dec {fst r} = Inr (λ ())\n  value-dec {snd r} = Inr (λ ())\n  value-dec {[ x ]case r of⦃· x₁ ·⦄} = Inr (�� ())\n\n  ex-¿¿-dec : ∀{ex} → ex == ¿¿ ∨ ex ≠ ¿¿\n  ex-¿¿-dec {⟨⟩} = Inr (λ ())\n  ex-¿¿-dec {⟨ ex₁ , ex₂ ⟩} = Inr (λ ())\n  ex-¿¿-dec {C[ x ] ex₁} = Inr (λ ())\n  ex-¿¿-dec {¿¿} = Inl refl\n  ex-¿¿-dec {x ↦ ex₁} = Inr 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