From dea5a6fbe4e3abf1c354f678189236423d489a54 Mon Sep 17 00:00:00 2001
From: Francesco Mecca <me@francescomecca.eu>
Date: Tue, 30 Jun 2020 19:31:56 +0200
Subject: [PATCH] traces

---
 ...31ca755ec4bd798a690ed9350f68282bc11561.png | Bin 0 -> 1251 bytes
 ...cc44285208b6cf43ee96ac8e9f497e5f3191eb.png | Bin 0 -> 1384 bytes
 ...120fd1e8cd49ab0dab3ef97353929c15f4113a.png | Bin 0 -> 3467 bytes
 ...279e548f036c71f2640c10d3ac4556f571c8a3.png | Bin 0 -> 1364 bytes
 ...b96ab92fcbc42081a14294cafc9655d64124a1.png | Bin 0 -> 3593 bytes
 ...cddde4a95b736a274897266cb27c5b8a348b6f.png | Bin 0 -> 965 bytes
 ...6a52e070f6e7a9d353e2fb281d3431b02bdb3c.png | Bin 0 -> 1251 bytes
 ...f6cd40efd8589318c49e230274945c079c4ca8.png | Bin 0 -> 787 bytes
 ...2675337cc8345e6e71ecbfd98a2bd044cb53a2.png | Bin 0 -> 1404 bytes
 ...82b4110fb7414eb8af69a9211cfb2808fe4dd5.png | Bin 0 -> 1643 bytes
 ...705f10cbe564234adb81cafa072109c3faf2f5.png | Bin 0 -> 1080 bytes
 ...7c518e2def2cbaf55d84c945bd0b86d59e7037.png | Bin 0 -> 1823 bytes
 ...786361c751a1a74c01c332e65e4854d7be80e6.png | Bin 0 -> 633 bytes
 ...a68b35299cd5b8f2f8555b5b61d0f0803f6778.png | Bin 0 -> 550 bytes
 ...56f2a20f6be4475ef3496e1a2eb5e99cd0c727.png | Bin 0 -> 3452 bytes
 ...02030f255196914ab3a21c0b5eb83072e92d47.png | Bin 0 -> 1551 bytes
 ...429dea9befffee656ac9fc36b1df8509d3c7ff.png | Bin 0 -> 377 bytes
 ...116a10855bae8a6d0cb304a27220822ddaedb4.png | Bin 0 -> 329 bytes
 ...449751cf9a5061cafa5f18d30c08dbe8c20a11.png | Bin 0 -> 965 bytes
 ...43bcd039f21e4dbb99e46d60f3f9cf9f5c1311.png | Bin 0 -> 1929 bytes
 tesi/tesi_unicode.org                         | 127 ++++++++++--------
 tesi/traces/example0.trace                    |   4 +-
 tesi/traces/example3.lambda                   |  18 +++
 tesi/traces/example3.ml                       |  11 ++
 tesi/traces/example3.trace                    |  46 +++++++
 tesi/traces/example6.lambda                   |  70 ++++++++++
 tesi/traces/example6.ml                       |  11 ++
 tesi/traces/example9.lambda                   |   5 +
 tesi/traces/example9.ml                       |   9 ++
 tesi/traces/example9.trace                    |  18 +++
 tesi/traces/example9bis.lambda                |   9 ++
 tesi/traces/example9bis.ml                    |   6 +
 tesi/traces/example9bis.trace                 |  15 +++
 tesi/traces/guards2.lambda                    |   7 +
 tesi/traces/guards2.ml                        |   8 ++
 tesi/traces/guards2.trace                     |  24 ++++
 36 files changed, 332 insertions(+), 56 deletions(-)
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 create mode 100644 tesi/ltximg/org-ltximg_15cc44285208b6cf43ee96ac8e9f497e5f3191eb.png
 create mode 100644 tesi/ltximg/org-ltximg_20120fd1e8cd49ab0dab3ef97353929c15f4113a.png
 create mode 100644 tesi/ltximg/org-ltximg_25279e548f036c71f2640c10d3ac4556f571c8a3.png
 create mode 100644 tesi/ltximg/org-ltximg_38b96ab92fcbc42081a14294cafc9655d64124a1.png
 create mode 100644 tesi/ltximg/org-ltximg_41cddde4a95b736a274897266cb27c5b8a348b6f.png
 create mode 100644 tesi/ltximg/org-ltximg_5c6a52e070f6e7a9d353e2fb281d3431b02bdb3c.png
 create mode 100644 tesi/ltximg/org-ltximg_72f6cd40efd8589318c49e230274945c079c4ca8.png
 create mode 100644 tesi/ltximg/org-ltximg_8c2675337cc8345e6e71ecbfd98a2bd044cb53a2.png
 create mode 100644 tesi/ltximg/org-ltximg_8e82b4110fb7414eb8af69a9211cfb2808fe4dd5.png
 create mode 100644 tesi/ltximg/org-ltximg_96705f10cbe564234adb81cafa072109c3faf2f5.png
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 create mode 100644 tesi/ltximg/org-ltximg_bea68b35299cd5b8f2f8555b5b61d0f0803f6778.png
 create mode 100644 tesi/ltximg/org-ltximg_c356f2a20f6be4475ef3496e1a2eb5e99cd0c727.png
 create mode 100644 tesi/ltximg/org-ltximg_ca02030f255196914ab3a21c0b5eb83072e92d47.png
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 create mode 100644 tesi/ltximg/org-ltximg_f243bcd039f21e4dbb99e46d60f3f9cf9f5c1311.png
 create mode 100644 tesi/traces/example3.lambda
 create mode 100644 tesi/traces/example3.ml
 create mode 100644 tesi/traces/example3.trace
 create mode 100644 tesi/traces/example6.lambda
 create mode 100644 tesi/traces/example6.ml
 create mode 100644 tesi/traces/example9.lambda
 create mode 100644 tesi/traces/example9.ml
 create mode 100644 tesi/traces/example9.trace
 create mode 100644 tesi/traces/example9bis.lambda
 create mode 100644 tesi/traces/example9bis.ml
 create mode 100644 tesi/traces/example9bis.trace
 create mode 100644 tesi/traces/guards2.lambda
 create mode 100644 tesi/traces/guards2.ml
 create mode 100644 tesi/traces/guards2.trace

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new file mode 100644
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new file mode 100644
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new file mode 100644
index 0000000000000000000000000000000000000000..aa26fbbe08b0df4bb45b1f28bdbe9ed78fa38f42
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diff --git a/tesi/ltximg/org-ltximg_5c6a52e070f6e7a9d353e2fb281d3431b02bdb3c.png b/tesi/ltximg/org-ltximg_5c6a52e070f6e7a9d353e2fb281d3431b02bdb3c.png
new file mode 100644
index 0000000000000000000000000000000000000000..a624488ea883d17c18e568204ba09424755ba92a
GIT binary patch
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new file mode 100644
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GIT binary patch
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diff --git a/tesi/ltximg/org-ltximg_8e82b4110fb7414eb8af69a9211cfb2808fe4dd5.png b/tesi/ltximg/org-ltximg_8e82b4110fb7414eb8af69a9211cfb2808fe4dd5.png
new file mode 100644
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GIT binary patch
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diff --git a/tesi/ltximg/org-ltximg_96705f10cbe564234adb81cafa072109c3faf2f5.png b/tesi/ltximg/org-ltximg_96705f10cbe564234adb81cafa072109c3faf2f5.png
new file mode 100644
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GIT binary patch
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new file mode 100644
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GIT binary patch
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diff --git a/tesi/tesi_unicode.org b/tesi/tesi_unicode.org
index c44dfc9..1e8ca0d 100644
--- a/tesi/tesi_unicode.org
+++ b/tesi/tesi_unicode.org
@@ -1879,7 +1879,7 @@ for content equality.
   {\EquivTEX S \Failure \Failure \emptyqueue}
 \\
   \infer
-  {\trel {t_S} {t_T}}
+  {\progrel {t_S} {t_T}}
   {\EquivTEX S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue}
 
 \end{mathpar}
@@ -1966,24 +1966,24 @@ is empty.
     {D_S} {\Guard {e_T} {D_0} {D_1}} {(e_S = b), G}}
 
 \end{mathpar}
-Our equivalence-checking algorithm $\EquivTEX S {C_S} {C_T} G$ is
+Our equivalence-checking algorithm is
 a exactly decision procedure for the provability of the judgment
-$(\EquivTEX  S {C_S} {C_T} G)$, defined by the previous inference rules.
+$\EquivTEX  S {C_S} {C_T}$, defined by the previous inference rules.
 Running a program tₛ or its translation 〚tₛ〛 against an input vₛ
 produces as a result /r/ in the following way:
-| ( 〚tₛ〛ₛ(vₛ) ≡ Cₛ(vₛ) ) → r
+| ( 〚tₛ〛ₛ(vₛ) ≡ Dₛ(vₛ) ) → r
 | tₛ(vₛ) → r
 Likewise
-| ( 〚tₜ〛ₜ(vₜ) ≡ Cₜ(vₜ) ) → r
+| ( 〚tₜ〛ₜ(vₜ) ≡ Dₜ(vₜ) ) → r
 | tₜ(vₜ) → r
 | result r ::= guard list * (Match blackbox \vert{} NoMatch \vert{} Absurd)
 | guard ::= blackbox.
 Having defined equivalence between two inputs of which one is
 expressed in the source language and the other in the target language,
-vₛ ≃ vₜ, we can define the equivalence between a couple of programs or
+$\vrel {v_S} {v_T}$, we can define the equivalence between a couple of programs or
 a couple of decision trees 
-| tₛ ≃ tₜ := ∀vₛ≃vₜ, tₛ(vₛ) = tₜ(vₜ)
-| Cₛ ≃ Cₜ := ∀vₛ≃vₜ, Cₛ(vₛ) = Cₜ(vₜ)
+| $\progrel {t_S} {t_T} := \forall \vrel {v_S} {v_T}, t_S(v_S) = t_T(v_T)$
+| $D_S \approx D_T := \forall \vrel {v_S} {v_T}, D_S(v_S) = D_T(v_T)$
 The result of the proposed equivalence algorithm is /Yes/ or /No(vₛ,
 vₜ)/. In particular, in the negative case, vₛ and vₜ are a couple of
 possible counter examples for which the decision trees produce a
@@ -1991,19 +1991,19 @@ different result.
 \\
 In the presence of guards we can say that two results are
 equivalent modulo the guards queue, written /r₁ ≃gs r₂/, when:
-| (gs₁, r₁) ≃gs (gs₂, r₂)  ⇔  (gs₁, r₁) = (gs₂ ++ gs, r₂)
-We say that Cₜ covers the input space /S/, written
-/covers(Cₜ, S)/ when every value vₛ∈S is a valid input to the
-decision tree Cₜ. (TODO: rephrase)
+| (gs₁, r₁) ≃_{gs} (gs₂, r₂)  ⇔  (gs₁, r₁) = (gs₂ +++ gs, r₂)
+We say that Dₜ covers the input space /S/, written
+/covers(Dₜ, S)/ when every value vₛ∈S is a valid input to the
+decision tree Dₜ. (TODO: rephrase)
 Given an input space /S/ and a couple of decision trees, where
-the target decision tree Cₜ covers the input space /S/ we can define equivalence:
-| equiv(S, Cₛ, Cₜ, gs) = Yes ∧ covers(Cₜ, S) → ∀vₛ≃vₜ ∈ S, Cₛ(vₛ) ≃gs Cₜ(vₜ)
+the target decision tree Dₜ covers the input space /S/ we can define equivalence:
+| $equiv(S, C_S, C_T, gs) = Yes \wedge covers(C_T, S) \to \forall \vrel {v_S} {v_T} \in S, C_S(v_S) \simeq_{gs} C_T(v_T)$
 Similarly we say that a couple of decision trees in the presence of
 an input space /S/ are /not/ equivalent in the following way:
-| equiv(S, Cₛ, Cₜ, gs) = No(vₛ,vₜ) ∧ covers(Cₜ, S) → vₛ≃vₜ ∈ S ∧ Cₛ(vₛ) ≠gs Cₜ(vₜ)
+| $equiv(S, C_S, C_T, gs) = No(v_S, v_T) \wedge covers(C_T, S) \to \vrel {v_S} {v_T} \in S \wedge C_S(v_S) \ne_{gs} C_T(v_T)$
 Corollary: For a full input space /S/, that is the universe of the
 target program:
-| equiv(S, 〚tₛ〛ₛ, 〚tₜ〛ₜ, ∅) = Yes  ⇔  tₛ ≃ tₜ
+| $equiv(S, \llbracket t_S \rrbracket{_S}, \llbracket t_T \rrbracket{_T}, \emptyset) = Yes \Leftrightarrow  \progrel {t_S} {t_T}$
 
 
 \begin{comment}
@@ -2012,35 +2012,35 @@ TODO: put ^i∈I where needed
 \subsubsection{The trimming lemma}
 The trimming lemma allows to reduce the size of a decision tree given
 an accessor /a/ → π relation (TODO: expand)
-| ∀vₜ ∈ (a→π), Cₜ(vₜ) = C_{t/a→π}(vₜ)
+| ∀vₜ ∈ (a→π), Dₜ(vₜ) = D_{t/a→π}(vₜ)
 We prove this by induction on Cₜ:
 
-- Cₜ = Leaf_{bb}: when the decision tree is a leaf terminal, the result of trimming on a Leaf is the Leaf itself
+- Dₜ = Leaf_{bb}: when the decision tree is a leaf terminal, the result of trimming on a Leaf is the Leaf itself
     | Leaf_{bb/a→π}(v) = Leaf_{bb}(v)
 - The same applies to Failure terminal
     | Failure_{/a→π}(v) = Failure(v)
-- When Cₜ = Switch(b, (π→Cᵢ)ⁱ)_{/a→π} then we look at the accessor
-    /a/ of the subtree Cᵢ and we define πᵢ' = πᵢ if a≠b else πᵢ∩π Trimming
+- When Dₜ = Switch(b, (π→Dᵢ)ⁱ)_{/a→π} then we look at the accessor
+    /a/ of the subtree Dᵢ and we define πᵢ' = πᵢ if a≠b else πᵢ∩π Trimming
     a switch node yields the following result:
-    | Switch(b, (π→Cᵢ)^{i∈I})_{/a→π} :=  Switch(b, (π'ᵢ→C_{i/a→π})^{i∈I})
+    | Switch(b, (π→Dᵢ)^{i∈I})_{/a→π} :=  Switch(b, (π'ᵢ→D_{i/a→π})^{i∈I})
 \begin{comment}
 TODO: understand how to properly align lists
 check that every list is aligned
 \end{comment}
 For the trimming lemma we have to prove that running the value vₜ against
-the decision tree Cₜ is the same as running vₜ against the tree
-C_{trim} that is the result of the trimming operation on Cₜ
-| Cₜ(vₜ) = C_{trim}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ)
+the decision tree Dₜ is the same as running vₜ against the tree
+D_{trim} that is the result of the trimming operation on Dₜ
+| Dₜ(vₜ) = D_{trim}(vₜ) = Switch(b, (πᵢ'→D_{i/a→π})^{i∈I})(vₜ)
 We can reason by first noting that when vₜ∉(b→πᵢ)ⁱ the node must be a Failure node.
 In the case where ∃k \vert{} vₜ∈(b→πₖ) then we can prove that
-| C_{k/a→π}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ)
+| D_{k/a→π}(vₜ) = Switch(b, (πᵢ'→D_{i/a→π})^{i∈I})(vₜ)
 because when a ≠ b then πₖ'= πₖ and this means that vₜ∈πₖ'
 while when a = b  then πₖ'=(πₖ∩π) and vₜ∈πₖ' because:
 - by the hypothesis, vₜ∈π
 - we are in the case where vₜ∈πₖ
 So vₜ ∈ πₖ' and by induction
-| Cₖ(vₜ) = C_{k/a→π}(vₜ)
-We also know that ∀vₜ∈(b→πₖ) → Cₜ(vₜ) = Cₖ(vₜ) 
+| Dₖ(vₜ) = D_{k/a→π}(vₜ)
+We also know that ∀vₜ∈(b→πₖ) → Dₜ(vₜ) = Dₖ(vₜ) 
 By putting together the last two steps, we have proven the trimming
 lemma.
 
@@ -2056,21 +2056,21 @@ Covering lemma:
 
 \subsubsection{Equivalence checking}
 The equivalence checking algorithm takes as parameters an input space
-/S/, a source decision tree /Cₛ/ and a target decision tree /Cₜ/:
-| equiv(S, Cₛ, Cₜ) → Yes \vert{} No(vₛ, vₜ)
+/S/, a source decision tree /Dₛ/ and a target decision tree /Dₜ/:
+| equiv(S, Dₛ, Dₜ) → Yes \vert{} No(vₛ, vₜ)
 \\
-When the algorithm returns Yes and the input space is covered by /Cₛ/
+When the algorithm returns Yes and the input space is covered by /Dₛ/
 we can say that the couple of decision trees are the same for
 every couple of source value /vₛ/ and target value /vₜ/ that are equivalent.
 \begin{comment}
 Define "covered"
 Is it correct to say the same? How to correctly distinguish in words ≃ and = ?
 \end{comment}
-| equiv(S, Cₛ, Cₜ) = Yes and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) = Cₜ(vₜ)
+| equiv(S, Dₛ, Dₜ) = Yes and cover(Dₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Dₛ(vₛ) = Dₜ(vₜ)
 In the case where the algorithm returns No we have at least a couple
 of counter example values vₛ and vₜ for which the two decision trees
 outputs a different result.
-| equiv(S, Cₛ, Cₜ) = No(vₛ,vₜ) and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) ≠ Cₜ(vₜ)
+| equiv(S, Dₛ, Cₜ) = No(vₛ,vₜ) and cover(Dₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Dₛ(vₛ) ≠ Dₜ(vₜ)
 We define  the following
 | Forall(Yes) = Yes
 | Forall(Yes::l) = Forall(l)
@@ -2092,9 +2092,9 @@ I start numbering from zero to leave the numbers as they were on the blackboard,
 I think the unreachable case should go at the end.
 \end{comment}
 0. in case of unreachable: 
-| Cₛ(vₛ) = Absurd(Unreachable) ≠ Cₜ(vₜ) ∀vₛ,vₜ
+| Dₛ(vₛ) = Absurd(Unreachable) ≠ Dₜ(vₜ) ∀vₛ,vₜ
 1. In the case of an empty input space
-   | equiv(∅, Cₛ, Cₜ) := Yes
+   | equiv(∅, Dₛ, Dₜ) := Yes
    and that is trivial to prove because there is no pair of values (vₛ, vₜ) that could be
    tested against the decision trees.
    In the other subcases S is always non-empty.
@@ -2102,48 +2102,46 @@ I think the unreachable case should go at the end.
    |equiv(S, Failure, Failure) := Yes
    Given that ∀v, Failure(v) = Failure, the statement holds.
 3. When we have a Leaf or a Failure at the left side:
-   | equiv(S, Failure as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I})
-   | equiv(S, Leaf bbₛ as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I})
+   | equiv(S, Failure as Dₛ, Switch(a, (πᵢ → Dₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Dₛ, Dₜᵢ)^{i∈I})
+   | equiv(S, Leaf bbₛ as Dₛ, Switch(a, (πᵢ → Dₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Dₛ, Dₜᵢ)^{i∈I})
    Our algorithm either returns Yes for every sub-input space Sᵢ := S∩(a→π(kᵢ)) and
    subtree Cₜᵢ
-   | equiv(Sᵢ, Cₛ, Cₜᵢ) = Yes ∀i
-   or we have a counter example vₛ, vₜ for which
-   | vₛ≃vₜ∈Sₖ ∧ cₛ(vₛ) ≠ Cₜₖ(vₜ)
-   then because 
-   | vₜ∈(a→πₖ) → Cₜ(vₜ) = Cₜₖ(vₜ) ,
-   | vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ) 
-   we can say that
+   | equiv(Sᵢ, Dₛ, Dₜᵢ) = Yes ∀i
+   or we have a counter example v_S, v_T for which
+   | $\vrel {v_S} {v_T \in S_k} \wedge D_S(v_S) \ne D_{Tk}(v_T)$
+   then because $v_T \in (a \to \pi_k) \to D_T(v_T) = D_{Tk}(v_T)$ and $\vrel v_S {v_T \in S} \wedge D_S(v_S) \ne D_T(v_T)$ we can say that
    | equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I
 4. When we have a Switch on the right we define π_{fallback} as the domain of
    values not covered but the union of the constructors kᵢ
    | $\pi_{fallback} = \neg\bigcup\limits_{i\in I}\pi(k_i)$
    Our algorithm proceeds by trimming
-   | equiv(S, Switch(a, (kᵢ → Cₛᵢ)^{i∈I}, C_{sf}), Cₜ) := 
-   | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} +++   equiv(S∩(a→πₙ), Cₛ, C_{a→π_{fallback}}))
+   | equiv(S, Switch(a, (kᵢ → Dₛᵢ)^{i∈I}, D_{sf}), Dₜ) := 
+   | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Dₛᵢ, D_{t/a→π(kᵢ)})^{i∈I} +++   equiv(S∩(a→πₙ), Dₛ, D_{a→π_{fallback}}))
     The statement still holds and we show this by first analyzing the
    /Yes/ case:
-    | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} = Yes
+    | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Dₛᵢ, D_{t/a→π(kᵢ)})^{i∈I} = Yes
     The constructor k is either included in the set of constructors kᵢ:
-    | k \vert k∈(kᵢ)ⁱ ∧ Cₛ(vₛ) = Cₛᵢ(vₛ)
+    | k \vert k∈(kᵢ)ⁱ ∧ Dₛ(vₛ) = Dₛᵢ(vₛ)
     We also know that 
-    | (1) Cₛᵢ(vₛ) = C_{t/a→πᵢ}(vₜ)      
-    | (2) C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)      
+    | (1) Dₛᵢ(vₛ) = D_{t/a→πᵢ}(vₜ)      
+    | (2) D_{T/a→πᵢ}(vₜ) = Dₜ(vₜ)      
     (1) is true by induction and (2) is a consequence of the trimming lemma.
     Putting everything together:
-    | Cₛ(vₛ) = Cₛᵢ(vₛ) = C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)
+    | Dₛ(vₛ) = Dₛᵢ(vₛ) = D_{T/a→πᵢ}(vₜ) = Dₜ(vₜ)
             
     When the k∉(kᵢ)ⁱ [TODO]
 
     The auxiliary Forall function returns /No(vₛ, vₜ)/ when, for a minimum  k,
-    | equiv(Sₖ, Cₛₖ, C_{T/a→πₖ} = No(vₛ, vₜ)
+    | equiv(Sₖ, Dₛₖ, D_{T/a→πₖ} = No(vₛ, vₜ)
     Then we can say that 
-    | Cₛₖ(vₛ) ≠ C_{t/a→πₖ}(vₜ)
+    | Dₛₖ(vₛ) ≠ D_{t/a→πₖ}(vₜ)
     that is enough for proving that 
-    | Cₛₖ(vₛ) ≠ (C_{t/a→πₖ}(vₜ) = Cₜ(vₜ))
+    | Dₛₖ(vₛ) ≠ (D_{t/a→πₖ}(vₜ) = Dₜ(vₜ))
 
 * Examples
 In this section we discuss some examples given as input and output of
-the prototype tool.
+the prototype tool. Source and target files are taken from the
+internship git repository\cite{internship}.
 
 ;; include_file example0.ml
 We can see from this first source file the usage of the /observe/
@@ -2154,6 +2152,23 @@ The prototype tool states that the compilation was successful and the
 two programs are equivalent.
 ;; include_file example0.trace
 
+The following example shows how the prototype handles ADT.
+;; include_file example3.ml
+;; include_file example3.lambda
+;; include_file example3.trace
+This trivial pattern matching code shows how guards are handled.
+;; include_file guards2.ml
+;; include_file guards2.lambda
+;; include_file guards2.trace
+The following source code shows the usage of the OCaml refutation
+clause.
+;; include_file example9.ml
+;; include_file example9.lambda
+;; include_file example9.trace
+In this example the tool correctly handles /Failure/ nodes on both decision trees.
+;; include_file example9bis.ml
+;; include_file example9bis.lambda
+;; include_file example9bis.trace
 
 \begin{thebibliography}{9}
 \bibitem{cpp_pat} 
@@ -2240,4 +2255,8 @@ https://github.com/janestreet/ocaml-compiler-libs
 \bibitem{menhir}
 Régis-Gianas, François Pottier Yann.
 \textit{Menhir Reference Manual}.
+
+\bibitem{internship}
+https://github.com/FraMecca/inria-internship
+
 \end{thebibliography}
diff --git a/tesi/traces/example0.trace b/tesi/traces/example0.trace
index 1f0d6da..cd860ae 100644
--- a/tesi/traces/example0.trace
+++ b/tesi/traces/example0.trace
@@ -1,4 +1,4 @@
-Target program constraint tree
+Target program decision tree
 Switch ({ var=AcAdd(-2 AcRoot=param/82); dom=Int [-inf; -1] [3; +inf] v Tag _; }) =
 	Leaf=VConstant:5
 Switch ({ var=AcAdd(-2 AcRoot=param/82); dom=Int [0; 2]; }) =
@@ -12,7 +12,7 @@ Switch ({ var=AcAdd(-2 AcRoot=param/82); dom=Int [0; 2]; }) =
 Fallback=None
 
 
-Source program constraint tree
+Source program decision tree
 Switch AcRoot:{
 	Int 3 -> 
 		Leaf='Int 3 '
diff --git a/tesi/traces/example3.lambda b/tesi/traces/example3.lambda
new file mode 100644
index 0000000..fee7a34
--- /dev/null
+++ b/tesi/traces/example3.lambda
@@ -0,0 +1,18 @@
+(setglobal Example3!
+  (let
+    (a/85 =
+       (function t/86
+         (switch* t/86
+          case int 0: (observe 3)
+          case tag 0:
+           (let (*match*/90 =a (field 0 t/86))
+             (catch
+               (if (!= *match*/90 1)
+                 (if (!= *match*/90 2) (exit 1) (apply (observe 1) 2))
+                 (apply (observe 1) 1))
+              with (1) (apply (observe 1) 0a)))
+          case tag 1:
+           (let (*match*/91 =a (field 0 t/86))
+             (if (!= *match*/91 0) (apply (observe 2) 1a)
+               (apply (observe 2) 0a))))))
+    (makeblock 0 a/85)))
diff --git a/tesi/traces/example3.ml b/tesi/traces/example3.ml
new file mode 100644
index 0000000..63dbca1
--- /dev/null
+++ b/tesi/traces/example3.ml
@@ -0,0 +1,11 @@
+external observe : 'a -> 'b = "observe"
+
+type t = K1 of int | K2 of bool | K3
+
+let a = fun t -> match t with
+  | K1 1 -> observe 1 1
+  | K1 2 -> observe 1 2
+  | K1 _ -> observe 1 ()
+  | K2 true -> observe 2 true
+  | K2 false -> observe 2 false
+  | K3 -> observe 3
diff --git a/tesi/traces/example3.trace b/tesi/traces/example3.trace
new file mode 100644
index 0000000..3a177bc
--- /dev/null
+++ b/tesi/traces/example3.trace
@@ -0,0 +1,46 @@
+Target program decision tree
+Switch ({ var=AcRoot=t/86; dom=Int 0; }) =
+	Leaf=VConstant:3
+Switch ({ var=AcRoot=t/86; dom=Tag 0; }) =
+	Switch ({ var=AcField(0 AcRoot=t/86); dom=Int [-inf; 0] [2; +inf] v Tag _; }) =
+		Switch ({ var=AcField(0 AcRoot=t/86); dom=Int [-inf; 1] [3; +inf] v Tag _; }) =
+			Leaf=VConstant:1, VConstant:0
+		Switch ({ var=AcField(0 AcRoot=t/86); dom=Int 2; }) =
+			Leaf=VConstant:1, VConstant:2
+		Fallback=None
+	Switch ({ var=AcField(0 AcRoot=t/86); dom=Int 1; }) =
+		Leaf=VConstant:1, VConstant:1
+	Fallback=None
+Switch ({ var=AcRoot=t/86; dom=Tag 1; }) =
+	Switch ({ var=AcField(0 AcRoot=t/86); dom=Int [-inf; -1] [1; +inf] v Tag _; }) =
+		Leaf=VConstant:2, VConstant:1
+	Switch ({ var=AcField(0 AcRoot=t/86); dom=Int 0; }) =
+		Leaf=VConstant:2, VConstant:0
+	Fallback=None
+Fallback=None
+
+Source program decision tree
+Switch AcRoot:{
+	Variant K2 -> 
+		Switch AcRoot.0:{
+		Bool false -> 
+			Leaf='Int 2 Bool false '
+		
+		Bool true -> 
+			Leaf='Int 2 Bool true ' 
+	} Fallback: Unreachable
+	
+	Variant K1 -> 
+		Switch AcRoot.0:{
+		Int 2 -> 
+			Leaf='Int 1 Int 2 '
+		
+		Int 1 -> 
+			Leaf='Int 1 Int 1 ' 
+	} Fallback: Leaf='Int 1 Unit '
+	
+	Variant K3 -> 
+		Leaf='Int 3 ' 
+} Fallback: Unreachable
+
+The two programs are equivalent.
\ No newline at end of file
diff --git a/tesi/traces/example6.lambda b/tesi/traces/example6.lambda
new file mode 100644
index 0000000..ce901f5
--- /dev/null
+++ b/tesi/traces/example6.lambda
@@ -0,0 +1,70 @@
+(setglobal Example6!
+  (let
+    (mm/81 =
+       (function x/89
+         (catch
+           (if x/89
+             (let (*match*/92 =a (field 0 x/89))
+               (if *match*/92
+                 (let
+                   (x/83 =a (field 0 *match*/92)
+                    x/86 =a (field 0 x/83)
+                    *match*/93 =a (field 0 x/86))
+                   (catch
+                     (if (!= *match*/93 1) (exit 2)
+                       (let (*match*/94 =a (field 1 x/86))
+                         (if (!= *match*/94 2) (exit 2)
+                           (let (x/82 =a (field 1 x/83))
+                             (catch
+                               (if (!= x/82 2) (exit 3)
+                                 (let (*match*/95 =a (field 1 *match*/92))
+                                   (if *match*/95 (exit 1)
+                                     (let (*match*/96 =a (field 1 x/89))
+                                       (catch
+                                         (if *match*/96 (exit 4)
+                                           (observe
+                                             [0: [0: [0: [0: 1 2] 2] 0a] 0a]))
+                                        with (4)
+                                         (observe [0: [0: [0: 1 2] 2] 0a]))))))
+                              with (3)
+                               (let (*match*/97 =a (field 1 *match*/92))
+                                 (if *match*/97 (exit 1)
+                                   (let (*match*/98 =a (field 1 x/89))
+                                     (observe
+                                       (makeblock 0
+                                         (makeblock 0 (*,int) [0: 1 2] x/82)
+                                         0a))))))))))
+                    with (2)
+                     (let (*match*/100 =a (field 1 *match*/92))
+                       (if *match*/100 (exit 1)
+                         (let
+                           (z/88 =a (field 1 x/89)
+                            y/87 =a (field 1 x/83)
+                            *match*/99 =a (field 1 x/86))
+                           (catch
+                             (if z/88
+                               (let (*match*/101 =a (field 0 z/88))
+                                 (if *match*/101
+                                   (let
+                                     (*match*/102 =a (field 1 *match*/101))
+                                     (if *match*/102 (exit 5)
+                                       (let (*match*/103 =a (field 1 z/88))
+                                         (if *match*/103 (exit 5)
+                                           (let
+                                             (x/84 =a x/83
+                                              y/85 =a (field 0 *match*/101))
+                                             (observe
+                                               (makeblock 0
+                                                 (makeblock 0 x/84 y/85) 0a)))))))
+                                   (exit 5)))
+                               (observe (makeblock 0 x/83 0a)))
+                            with (5)
+                             (observe
+                               (makeblock 0
+                                 (makeblock 0 (makeblock 0 (*,int) x/86 y/87)
+                                   z/88)
+                                 0a))))))))
+                 (exit 1)))
+             (exit 1))
+          with (1) (observe x/89))))
+    (makeblock 0 mm/81)))
diff --git a/tesi/traces/example6.ml b/tesi/traces/example6.ml
new file mode 100644
index 0000000..6044120
--- /dev/null
+++ b/tesi/traces/example6.ml
@@ -0,0 +1,11 @@
+[@@@warning "-20"]
+external observe : 'a -> 'b = "observe"
+
+let mm = function
+  | [(1, 2), 2]::[] -> observe ([(1, 2), 2]::[])
+  | [(1, 2), 2]::_ -> observe [(1, 2), 2]
+  | [(1, 2), x]::_ -> observe [(1, 2), x]
+  | [x]::[] -> observe [x]
+  | [x]::[y]::[] -> observe [x, y]
+  | [x, y]::z -> observe [(x, y), z]
+  | x -> observe x
diff --git a/tesi/traces/example9.lambda b/tesi/traces/example9.lambda
new file mode 100644
index 0000000..96c593c
--- /dev/null
+++ b/tesi/traces/example9.lambda
@@ -0,0 +1,5 @@
+(setglobal Example9!
+  (let
+    (test/81 =
+       (function param/82 (if (!= param/82 0) (observe 0) (observe 1))))
+    (makeblock 0 test/81)))
diff --git a/tesi/traces/example9.ml b/tesi/traces/example9.ml
new file mode 100644
index 0000000..7b163ea
--- /dev/null
+++ b/tesi/traces/example9.ml
@@ -0,0 +1,9 @@
+[@@@warning "-20"]
+external observe : 'a -> 'b = "observe"
+
+let test = function
+  | true -> observe 0
+  | false -> observe 1
+  | _ -> .
+    (* Unreachable; if this annotation was incorrect,
+       the OCaml compiler would error at pattern-checking-time *)
diff --git a/tesi/traces/example9.trace b/tesi/traces/example9.trace
new file mode 100644
index 0000000..78619ed
--- /dev/null
+++ b/tesi/traces/example9.trace
@@ -0,0 +1,18 @@
+Target program decision tree
+Switch ({ var=AcRoot=param/82; dom=Int [-inf; -1] [1; +inf] v Tag _; }) =
+	Leaf=VConstant:0
+Switch ({ var=AcRoot=param/82; dom=Int 0; }) =
+	Leaf=VConstant:1
+Fallback=None
+
+
+Source program decision tree
+Switch AcRoot:{
+	Bool false -> 
+		Leaf='Int 1 '
+	
+	Bool true -> 
+		Leaf='Int 0 ' 
+} Fallback: Unreachable
+
+The two programs are equivalent.
\ No newline at end of file
diff --git a/tesi/traces/example9bis.lambda b/tesi/traces/example9bis.lambda
new file mode 100644
index 0000000..5203342
--- /dev/null
+++ b/tesi/traces/example9bis.lambda
@@ -0,0 +1,9 @@
+(setglobal Example9bis!
+  (let
+    (test/81 =
+       (function param/82
+         (catch (if (!= param/82 0) (observe 0) (exit 1)) with (1)
+           (raise
+             (makeblock 0 (global Match_failure/18!)
+               [0: "example9bis.ml" 4 11])))))
+    (makeblock 0 test/81)))
diff --git a/tesi/traces/example9bis.ml b/tesi/traces/example9bis.ml
new file mode 100644
index 0000000..adf0901
--- /dev/null
+++ b/tesi/traces/example9bis.ml
@@ -0,0 +1,6 @@
+[@@@warning "-20"]
+external observe : 'a -> 'b = "observe"
+
+let test = function
+  | true -> observe 0
+  (* we expect a Match_failure node for 'false' in the lambda representation *)
diff --git a/tesi/traces/example9bis.trace b/tesi/traces/example9bis.trace
new file mode 100644
index 0000000..7afb94d
--- /dev/null
+++ b/tesi/traces/example9bis.trace
@@ -0,0 +1,15 @@
+Target program decision tree
+Switch ({ var=AcRoot=param/82; dom=Int [-inf; -1] [1; +inf] v Tag _; }) =
+	Leaf=VConstant:0
+Switch ({ var=AcRoot=param/82; dom=Int 0; }) =
+	Failure
+    Fallback=None
+
+
+Source program decision tree
+Switch AcRoot:{
+	Bool true -> 
+		Leaf='Int 0 ' 
+} Fallback: Failure
+
+The two programs are equivalent.
diff --git a/tesi/traces/guards2.lambda b/tesi/traces/guards2.lambda
new file mode 100644
index 0000000..2436972
--- /dev/null
+++ b/tesi/traces/guards2.lambda
@@ -0,0 +1,7 @@
+(setglobal Guards2!
+  (let
+      (ff/82 =
+             (function x/83
+                      (if (guard 0a) (observe 1)
+                                 (if (guard 1) (observe 2) (observe 3)))))
+                                     (makeblock 0 ff/82)))
diff --git a/tesi/traces/guards2.ml b/tesi/traces/guards2.ml
new file mode 100644
index 0000000..452e39e
--- /dev/null
+++ b/tesi/traces/guards2.ml
@@ -0,0 +1,8 @@
+[@@@warning "-20"]
+external observe : 'a -> 'b = "observe"
+external guard : 'a -> 'b = "guard"
+
+let ff = function
+  | x when guard () -> observe 1
+  | _ when guard 1 -> observe 2
+  | _ -> observe 3
diff --git a/tesi/traces/guards2.trace b/tesi/traces/guards2.trace
new file mode 100644
index 0000000..6b774fa
--- /dev/null
+++ b/tesi/traces/guards2.trace
@@ -0,0 +1,24 @@
+Target program decision tree
+Guard (VConstant:0):
+guard(true) =
+	Leaf=VConstant:1
+guard(false) =
+	Guard (VConstant:1):
+	guard(true) =
+		Leaf=VConstant:2
+	guard(false) =
+		Leaf=VConstant:3
+
+Source program decision tree
+Switch AcRoot:{ 
+} Fallback: Guard (Unit ) =
+	guard(true) =
+		Leaf='Int 1 '
+	guard(false) =
+		Guard (Int 1 ) =
+		guard(true) =
+			Leaf='Int 2 '
+		guard(false) =
+			Leaf='Int 3 '
+
+The two programs are equivalent.