author | wenzelm |
Sun, 02 Nov 2014 18:21:45 +0100 | |
changeset 58889 | 5b7a9633cfa8 |
parent 58622 | aa99568f56de |
child 59582 | 0fbed69ff081 |
permissions | -rw-r--r-- |
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more explicit HOL-Proofs sessions, including former ex/Hilbert_Classical.thy which works in parallel mode without the antiquotation option "margin" (which is still critical);
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1 |
(* Title: HOL/Proofs/Lambda/WeakNorm.thy |
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New proof of weak normalization with program extraction.
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2 |
Author: Stefan Berghofer |
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New proof of weak normalization with program extraction.
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3 |
Copyright 2003 TU Muenchen |
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New proof of weak normalization with program extraction.
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*) |
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New proof of weak normalization with program extraction.
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|
58889 | 6 |
section {* Weak normalization for simply-typed lambda calculus *} |
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New proof of weak normalization with program extraction.
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|
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theory WeakNorm |
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imports LambdaType NormalForm "~~/src/HOL/Library/Old_Datatype" |
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"~~/src/HOL/Library/Code_Target_Int" |
|
22512 | 11 |
begin |
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New proof of weak normalization with program extraction.
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|
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New proof of weak normalization with program extraction.
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text {* |
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New proof of weak normalization with program extraction.
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Formalization by Stefan Berghofer. Partly based on a paper proof by |
58622 | 15 |
Felix Joachimski and Ralph Matthes @{cite "Matthes-Joachimski-AML"}. |
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New proof of weak normalization with program extraction.
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16 |
*} |
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New proof of weak normalization with program extraction.
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17 |
|
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New proof of weak normalization with program extraction.
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|
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New proof of weak normalization with program extraction.
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subsection {* Main theorems *} |
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New proof of weak normalization with program extraction.
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20 |
|
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lemma norm_list: |
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22 |
assumes f_compat: "\<And>t t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> f t \<rightarrow>\<^sub>\<beta>\<^sup>* f t'" |
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and f_NF: "\<And>t. NF t \<Longrightarrow> NF (f t)" |
24 |
and uNF: "NF u" and uT: "e \<turnstile> u : T" |
|
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25 |
shows "\<And>Us. e\<langle>i:T\<rangle> \<tturnstile> as : Us \<Longrightarrow> |
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26 |
listall (\<lambda>t. \<forall>e T' u i. e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> |
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NF u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> (\<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')) as \<Longrightarrow> |
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28 |
\<exists>as'. \<forall>j. Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) as \<rightarrow>\<^sub>\<beta>\<^sup>* |
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Var j \<degree>\<degree> map f as' \<and> NF (Var j \<degree>\<degree> map f as')" |
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30 |
(is "\<And>Us. _ \<Longrightarrow> listall ?R as \<Longrightarrow> \<exists>as'. ?ex Us as as'") |
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31 |
proof (induct as rule: rev_induct) |
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32 |
case (Nil Us) |
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33 |
with Var_NF have "?ex Us [] []" by simp |
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thus ?case .. |
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35 |
next |
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36 |
case (snoc b bs Us) |
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have "e\<langle>i:T\<rangle> \<tturnstile> bs @ [b] : Us" by fact |
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38 |
then obtain Vs W where Us: "Us = Vs @ [W]" |
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and bs: "e\<langle>i:T\<rangle> \<tturnstile> bs : Vs" and bT: "e\<langle>i:T\<rangle> \<turnstile> b : W" |
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40 |
by (rule types_snocE) |
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41 |
from snoc have "listall ?R bs" by simp |
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42 |
with bs have "\<exists>bs'. ?ex Vs bs bs'" by (rule snoc) |
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43 |
then obtain bs' where |
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44 |
bsred: "\<And>j. Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) bs \<rightarrow>\<^sub>\<beta>\<^sup>* Var j \<degree>\<degree> map f bs'" |
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and bsNF: "\<And>j. NF (Var j \<degree>\<degree> map f bs')" by iprover |
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46 |
from snoc have "?R b" by simp |
22271 | 47 |
with bT and uNF and uT have "\<exists>b'. b[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* b' \<and> NF b'" |
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parents:
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48 |
by iprover |
22271 | 49 |
then obtain b' where bred: "b[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* b'" and bNF: "NF b'" |
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50 |
by iprover |
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from bsNF [of 0] have "listall NF (map f bs')" |
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52 |
by (rule App_NF_D) |
23464 | 53 |
moreover have "NF (f b')" using bNF by (rule f_NF) |
22271 | 54 |
ultimately have "listall NF (map f (bs' @ [b']))" |
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55 |
by simp |
22271 | 56 |
hence "\<And>j. NF (Var j \<degree>\<degree> map f (bs' @ [b']))" by (rule NF.App) |
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57 |
moreover from bred have "f (b[u/i]) \<rightarrow>\<^sub>\<beta>\<^sup>* f b'" |
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58 |
by (rule f_compat) |
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59 |
with bsred have |
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60 |
"\<And>j. (Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) bs) \<degree> f (b[u/i]) \<rightarrow>\<^sub>\<beta>\<^sup>* |
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61 |
(Var j \<degree>\<degree> map f bs') \<degree> f b'" by (rule rtrancl_beta_App) |
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62 |
ultimately have "?ex Us (bs @ [b]) (bs' @ [b'])" by simp |
eb3a7d3d874b
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63 |
thus ?case .. |
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parents:
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|
64 |
qed |
eb3a7d3d874b
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parents:
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|
65 |
|
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New proof of weak normalization with program extraction.
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parents:
diff
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|
66 |
lemma subst_type_NF: |
22271 | 67 |
"\<And>t e T u i. NF t \<Longrightarrow> e\<langle>i:U\<rangle> \<turnstile> t : T \<Longrightarrow> NF u \<Longrightarrow> e \<turnstile> u : U \<Longrightarrow> \<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" |
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New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
68 |
(is "PROP ?P U" is "\<And>t e T u i. _ \<Longrightarrow> PROP ?Q t e T u i U") |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
69 |
proof (induct U) |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
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70 |
fix T t |
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New proof of weak normalization with program extraction.
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parents:
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|
71 |
let ?R = "\<lambda>t. \<forall>e T' u i. |
22271 | 72 |
e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> NF u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> (\<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')" |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
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|
73 |
assume MI1: "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> PROP ?P T1" |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
74 |
assume MI2: "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> PROP ?P T2" |
22271 | 75 |
assume "NF t" |
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New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
76 |
thus "\<And>e T' u i. PROP ?Q t e T' u i T" |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
77 |
proof induct |
22271 | 78 |
fix e T' u i assume uNF: "NF u" and uT: "e \<turnstile> u : T" |
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New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
79 |
{ |
50241 | 80 |
case (App ts x e1 T'1 u1 i1) |
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eb3a7d3d874b
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parents:
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diff
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|
81 |
assume "e\<langle>i:T\<rangle> \<turnstile> Var x \<degree>\<degree> ts : T'" |
eb3a7d3d874b
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berghofe
parents:
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diff
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|
82 |
then obtain Us |
32960
69916a850301
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parents:
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diff
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83 |
where varT: "e\<langle>i:T\<rangle> \<turnstile> Var x : Us \<Rrightarrow> T'" |
69916a850301
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parents:
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diff
changeset
|
84 |
and argsT: "e\<langle>i:T\<rangle> \<tturnstile> ts : Us" |
69916a850301
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wenzelm
parents:
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diff
changeset
|
85 |
by (rule var_app_typesE) |
22271 | 86 |
from nat_eq_dec show "\<exists>t'. (Var x \<degree>\<degree> ts)[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" |
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New proof of weak normalization with program extraction.
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parents:
diff
changeset
|
87 |
proof |
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eliminated hard tabulators, guessing at each author's individual tab-width;
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parents:
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88 |
assume eq: "x = i" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
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diff
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|
89 |
show ?thesis |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
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diff
changeset
|
90 |
proof (cases ts) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
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parents:
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diff
changeset
|
91 |
case Nil |
69916a850301
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wenzelm
parents:
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diff
changeset
|
92 |
with eq have "(Var x \<degree>\<degree> [])[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* u" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
93 |
with Nil and uNF show ?thesis by simp iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
94 |
next |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
95 |
case (Cons a as) |
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eb3a7d3d874b
Factored out proof for normalization of applications (norm_list).
berghofe
parents:
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diff
changeset
|
96 |
with argsT obtain T'' Ts where Us: "Us = T'' # Ts" |
56073
29e308b56d23
enhanced simplifier solver for preconditions of rewrite rule, can now deal with conjunctions
nipkow
parents:
51143
diff
changeset
|
97 |
by (cases Us) (rule FalseE, simp) |
32960
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eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
98 |
from varT and Us have varT: "e\<langle>i:T\<rangle> \<turnstile> Var x : T'' \<Rightarrow> Ts \<Rrightarrow> T'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
99 |
by simp |
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New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
100 |
from varT eq have T: "T = T'' \<Rightarrow> Ts \<Rrightarrow> T'" by cases auto |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
101 |
with uT have uT': "e \<turnstile> u : T'' \<Rightarrow> Ts \<Rrightarrow> T'" by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
102 |
from argsT Us Cons have argsT': "e\<langle>i:T\<rangle> \<tturnstile> as : Ts" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
103 |
from argsT Us Cons have argT: "e\<langle>i:T\<rangle> \<turnstile> a : T''" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
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diff
changeset
|
104 |
from argT uT refl have aT: "e \<turnstile> a[u/i] : T''" by (rule subst_lemma) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
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diff
changeset
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105 |
from App and Cons have "listall ?R as" by simp (iprover dest: listall_conj2) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
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diff
changeset
|
106 |
with lift_preserves_beta' lift_NF uNF uT argsT' |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
107 |
have "\<exists>as'. \<forall>j. Var j \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as \<rightarrow>\<^sub>\<beta>\<^sup>* |
18331
eb3a7d3d874b
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parents:
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diff
changeset
|
108 |
Var j \<degree>\<degree> map (\<lambda>t. lift t 0) as' \<and> |
32960
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wenzelm
parents:
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diff
changeset
|
109 |
NF (Var j \<degree>\<degree> map (\<lambda>t. lift t 0) as')" by (rule norm_list) |
69916a850301
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wenzelm
parents:
32359
diff
changeset
|
110 |
then obtain as' where |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
111 |
asred: "Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as \<rightarrow>\<^sub>\<beta>\<^sup>* |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
112 |
Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
113 |
and asNF: "NF (Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')" by iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
114 |
from App and Cons have "?R a" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
115 |
with argT and uNF and uT have "\<exists>a'. a[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* a' \<and> NF a'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
116 |
by iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
117 |
then obtain a' where ared: "a[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* a'" and aNF: "NF a'" by iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
118 |
from uNF have "NF (lift u 0)" by (rule lift_NF) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
119 |
hence "\<exists>u'. lift u 0 \<degree> Var 0 \<rightarrow>\<^sub>\<beta>\<^sup>* u' \<and> NF u'" by (rule app_Var_NF) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
120 |
then obtain u' where ured: "lift u 0 \<degree> Var 0 \<rightarrow>\<^sub>\<beta>\<^sup>* u'" and u'NF: "NF u'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
121 |
by iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
122 |
from T and u'NF have "\<exists>ua. u'[a'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* ua \<and> NF ua" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
123 |
proof (rule MI1) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
124 |
have "e\<langle>0:T''\<rangle> \<turnstile> lift u 0 \<degree> Var 0 : Ts \<Rrightarrow> T'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
125 |
proof (rule typing.App) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
126 |
from uT' show "e\<langle>0:T''\<rangle> \<turnstile> lift u 0 : T'' \<Rightarrow> Ts \<Rrightarrow> T'" by (rule lift_type) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
127 |
show "e\<langle>0:T''\<rangle> \<turnstile> Var 0 : T''" by (rule typing.Var) simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
128 |
qed |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
129 |
with ured show "e\<langle>0:T''\<rangle> \<turnstile> u' : Ts \<Rrightarrow> T'" by (rule subject_reduction') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
130 |
from ared aT show "e \<turnstile> a' : T''" by (rule subject_reduction') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
131 |
show "NF a'" by fact |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
132 |
qed |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
133 |
then obtain ua where uared: "u'[a'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* ua" and uaNF: "NF ua" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
134 |
by iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
135 |
from ared have "(lift u 0 \<degree> Var 0)[a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* (lift u 0 \<degree> Var 0)[a'/0]" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
136 |
by (rule subst_preserves_beta2') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
137 |
also from ured have "(lift u 0 \<degree> Var 0)[a'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* u'[a'/0]" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
138 |
by (rule subst_preserves_beta') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
139 |
also note uared |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
140 |
finally have "(lift u 0 \<degree> Var 0)[a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* ua" . |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
141 |
hence uared': "u \<degree> a[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* ua" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
142 |
from T asNF _ uaNF have "\<exists>r. (Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[ua/0] \<rightarrow>\<^sub>\<beta>\<^sup>* r \<and> NF r" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
143 |
proof (rule MI2) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
144 |
have "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as : T'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
145 |
proof (rule list_app_typeI) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
146 |
show "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 : Ts \<Rrightarrow> T'" by (rule typing.Var) simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
147 |
from uT argsT' have "e \<tturnstile> map (\<lambda>t. t[u/i]) as : Ts" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
148 |
by (rule substs_lemma) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
149 |
hence "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<tturnstile> map (\<lambda>t. lift t 0) (map (\<lambda>t. t[u/i]) as) : Ts" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
150 |
by (rule lift_types) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
151 |
thus "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<tturnstile> map (\<lambda>t. lift (t[u/i]) 0) as : Ts" |
33640 | 152 |
by (simp_all add: o_def) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
153 |
qed |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
154 |
with asred show "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as' : T'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
155 |
by (rule subject_reduction') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
156 |
from argT uT refl have "e \<turnstile> a[u/i] : T''" by (rule subst_lemma) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
157 |
with uT' have "e \<turnstile> u \<degree> a[u/i] : Ts \<Rrightarrow> T'" by (rule typing.App) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
158 |
with uared' show "e \<turnstile> ua : Ts \<Rrightarrow> T'" by (rule subject_reduction') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
159 |
qed |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
160 |
then obtain r where rred: "(Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[ua/0] \<rightarrow>\<^sub>\<beta>\<^sup>* r" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
161 |
and rnf: "NF r" by iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
162 |
from asred have |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
163 |
"(Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as)[u \<degree> a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
164 |
(Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[u \<degree> a[u/i]/0]" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
165 |
by (rule subst_preserves_beta') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
166 |
also from uared' have "(Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[u \<degree> a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
167 |
(Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[ua/0]" by (rule subst_preserves_beta2') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
168 |
also note rred |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
169 |
finally have "(Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as)[u \<degree> a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* r" . |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
170 |
with rnf Cons eq show ?thesis |
33640 | 171 |
by (simp add: o_def) iprover |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
172 |
qed |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
173 |
next |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
174 |
assume neq: "x \<noteq> i" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
175 |
from App have "listall ?R ts" by (iprover dest: listall_conj2) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
176 |
with TrueI TrueI uNF uT argsT |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
177 |
have "\<exists>ts'. \<forall>j. Var j \<degree>\<degree> map (\<lambda>t. t[u/i]) ts \<rightarrow>\<^sub>\<beta>\<^sup>* Var j \<degree>\<degree> ts' \<and> |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
178 |
NF (Var j \<degree>\<degree> ts')" (is "\<exists>ts'. ?ex ts'") |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
179 |
by (rule norm_list [of "\<lambda>t. t", simplified]) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
180 |
then obtain ts' where NF: "?ex ts'" .. |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
181 |
from nat_le_dec show ?thesis |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
182 |
proof |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
183 |
assume "i < x" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
184 |
with NF show ?thesis by simp iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
185 |
next |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
186 |
assume "\<not> (i < x)" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
187 |
with NF neq show ?thesis by (simp add: subst_Var) iprover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
188 |
qed |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
189 |
qed |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
190 |
next |
50241 | 191 |
case (Abs r e1 T'1 u1 i1) |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
192 |
assume absT: "e\<langle>i:T\<rangle> \<turnstile> Abs r : T'" |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
193 |
then obtain R S where "e\<langle>0:R\<rangle>\<langle>Suc i:T\<rangle> \<turnstile> r : S" by (rule abs_typeE) simp |
23464 | 194 |
moreover have "NF (lift u 0)" using `NF u` by (rule lift_NF) |
195 |
moreover have "e\<langle>0:R\<rangle> \<turnstile> lift u 0 : T" using uT by (rule lift_type) |
|
22271 | 196 |
ultimately have "\<exists>t'. r[lift u 0/Suc i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" by (rule Abs) |
197 |
thus "\<exists>t'. Abs r[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
198 |
by simp (iprover intro: rtrancl_beta_Abs NF.Abs) |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
199 |
} |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
200 |
qed |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
201 |
qed |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
202 |
|
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
203 |
|
22271 | 204 |
-- {* A computationally relevant copy of @{term "e \<turnstile> t : T"} *} |
23750 | 205 |
inductive rtyping :: "(nat \<Rightarrow> type) \<Rightarrow> dB \<Rightarrow> type \<Rightarrow> bool" ("_ \<turnstile>\<^sub>R _ : _" [50, 50, 50] 50) |
22271 | 206 |
where |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
207 |
Var: "e x = T \<Longrightarrow> e \<turnstile>\<^sub>R Var x : T" |
22271 | 208 |
| Abs: "e\<langle>0:T\<rangle> \<turnstile>\<^sub>R t : U \<Longrightarrow> e \<turnstile>\<^sub>R Abs t : (T \<Rightarrow> U)" |
209 |
| App: "e \<turnstile>\<^sub>R s : T \<Rightarrow> U \<Longrightarrow> e \<turnstile>\<^sub>R t : T \<Longrightarrow> e \<turnstile>\<^sub>R (s \<degree> t) : U" |
|
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
210 |
|
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
211 |
lemma rtyping_imp_typing: "e \<turnstile>\<^sub>R t : T \<Longrightarrow> e \<turnstile> t : T" |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
212 |
apply (induct set: rtyping) |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
213 |
apply (erule typing.Var) |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
214 |
apply (erule typing.Abs) |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
215 |
apply (erule typing.App) |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
216 |
apply assumption |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
217 |
done |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
218 |
|
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
219 |
|
18513 | 220 |
theorem type_NF: |
221 |
assumes "e \<turnstile>\<^sub>R t : T" |
|
23464 | 222 |
shows "\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" using assms |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
223 |
proof induct |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
224 |
case Var |
17589 | 225 |
show ?case by (iprover intro: Var_NF) |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
226 |
next |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
227 |
case Abs |
17589 | 228 |
thus ?case by (iprover intro: rtrancl_beta_Abs NF.Abs) |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
229 |
next |
22271 | 230 |
case (App e s T U t) |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
231 |
from App obtain s' t' where |
23464 | 232 |
sred: "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'" and "NF s'" |
22271 | 233 |
and tred: "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" and tNF: "NF t'" by iprover |
234 |
have "\<exists>u. (Var 0 \<degree> lift t' 0)[s'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* u \<and> NF u" |
|
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
235 |
proof (rule subst_type_NF) |
23464 | 236 |
have "NF (lift t' 0)" using tNF by (rule lift_NF) |
22271 | 237 |
hence "listall NF [lift t' 0]" by (rule listall_cons) (rule listall_nil) |
238 |
hence "NF (Var 0 \<degree>\<degree> [lift t' 0])" by (rule NF.App) |
|
239 |
thus "NF (Var 0 \<degree> lift t' 0)" by simp |
|
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
240 |
show "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 \<degree> lift t' 0 : U" |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
241 |
proof (rule typing.App) |
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
242 |
show "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 : T \<Rightarrow> U" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
243 |
by (rule typing.Var) simp |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
244 |
from tred have "e \<turnstile> t' : T" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
245 |
by (rule subject_reduction') (rule rtyping_imp_typing, rule App.hyps) |
14063
e61a310cde02
New proof of weak normalization with program extraction.
berghofe
parents:
diff
changeset
|
246 |
thus "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> lift t' 0 : T" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32359
diff
changeset
|
247 |
by (rule lift_type) |
14063
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qed |
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from sred show "e \<turnstile> s' : T \<Rightarrow> U" |
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by (rule subject_reduction') (rule rtyping_imp_typing, rule App.hyps) |
251 |
show "NF s'" by fact |
|
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qed |
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then obtain u where ured: "s' \<degree> t' \<rightarrow>\<^sub>\<beta>\<^sup>* u" and unf: "NF u" by simp iprover |
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from sred tred have "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<degree> t'" by (rule rtrancl_beta_App) |
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hence "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* u" using ured by (rule rtranclp_trans) |
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with unf show ?case by iprover |
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qed |
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|
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|
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subsection {* Extracting the program *} |
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|
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declare NF.induct [ind_realizer] |
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declare rtranclp.induct [ind_realizer irrelevant] |
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declare rtyping.induct [ind_realizer] |
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lemmas [extraction_expand] = conj_assoc listall_cons_eq |
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|
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extract type_NF |
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|
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lemma rtranclR_rtrancl_eq: "rtranclpR r a b = r\<^sup>*\<^sup>* a b" |
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apply (rule iffI) |
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apply (erule rtranclpR.induct) |
272 |
apply (rule rtranclp.rtrancl_refl) |
|
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apply (erule rtranclp.rtrancl_into_rtrancl) |
|
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apply assumption |
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apply (erule rtranclp.induct) |
276 |
apply (rule rtranclpR.rtrancl_refl) |
|
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apply (erule rtranclpR.rtrancl_into_rtrancl) |
|
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apply assumption |
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done |
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|
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lemma NFR_imp_NF: "NFR nf t \<Longrightarrow> NF t" |
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apply (erule NFR.induct) |
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apply (rule NF.intros) |
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apply (simp add: listall_def) |
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apply (erule NF.intros) |
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done |
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|
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text_raw {* |
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\begin{figure} |
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\renewcommand{\isastyle}{\scriptsize\it}% |
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@{thm [display,eta_contract=false,margin=100] subst_type_NF_def} |
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\renewcommand{\isastyle}{\small\it}% |
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\caption{Program extracted from @{text subst_type_NF}} |
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\label{fig:extr-subst-type-nf} |
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\end{figure} |
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|
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\begin{figure} |
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\renewcommand{\isastyle}{\scriptsize\it}% |
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@{thm [display,margin=100] subst_Var_NF_def} |
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@{thm [display,margin=100] app_Var_NF_def} |
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@{thm [display,margin=100] lift_NF_def} |
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@{thm [display,eta_contract=false,margin=100] type_NF_def} |
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\renewcommand{\isastyle}{\small\it}% |
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\caption{Program extracted from lemmas and main theorem} |
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\label{fig:extr-type-nf} |
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\end{figure} |
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*} |
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|
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text {* |
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The program corresponding to the proof of the central lemma, which |
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performs substitution and normalization, is shown in Figure |
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\ref{fig:extr-subst-type-nf}. The correctness |
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theorem corresponding to the program @{text "subst_type_NF"} is |
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@{thm [display,margin=100] subst_type_NF_correctness |
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[simplified rtranclR_rtrancl_eq Collect_mem_eq, no_vars]} |
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where @{text NFR} is the realizability predicate corresponding to |
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the datatype @{text NFT}, which is inductively defined by the rules |
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\pagebreak |
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@{thm [display,margin=90] NFR.App [of ts nfs x] NFR.Abs [of nf t]} |
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320 |
|
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The programs corresponding to the main theorem @{text "type_NF"}, as |
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well as to some lemmas, are shown in Figure \ref{fig:extr-type-nf}. |
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The correctness statement for the main function @{text "type_NF"} is |
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@{thm [display,margin=100] type_NF_correctness |
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[simplified rtranclR_rtrancl_eq Collect_mem_eq, no_vars]} |
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where the realizability predicate @{text "rtypingR"} corresponding to the |
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327 |
computationally relevant version of the typing judgement is inductively |
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328 |
defined by the rules |
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@{thm [display,margin=100] rtypingR.Var [no_vars] |
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rtypingR.Abs [of ty, no_vars] rtypingR.App [of ty e s T U ty' t]} |
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*} |
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|
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subsection {* Generating executable code *} |
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|
27982 | 335 |
instantiation NFT :: default |
336 |
begin |
|
337 |
||
338 |
definition "default = Dummy ()" |
|
339 |
||
340 |
instance .. |
|
341 |
||
342 |
end |
|
343 |
||
344 |
instantiation dB :: default |
|
345 |
begin |
|
346 |
||
347 |
definition "default = dB.Var 0" |
|
348 |
||
349 |
instance .. |
|
350 |
||
351 |
end |
|
352 |
||
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instantiation prod :: (default, default) default |
27982 | 354 |
begin |
355 |
||
356 |
definition "default = (default, default)" |
|
357 |
||
358 |
instance .. |
|
359 |
||
360 |
end |
|
361 |
||
362 |
instantiation list :: (type) default |
|
363 |
begin |
|
364 |
||
365 |
definition "default = []" |
|
366 |
||
367 |
instance .. |
|
368 |
||
369 |
end |
|
370 |
||
371 |
instantiation "fun" :: (type, default) default |
|
372 |
begin |
|
373 |
||
374 |
definition "default = (\<lambda>x. default)" |
|
375 |
||
376 |
instance .. |
|
377 |
||
378 |
end |
|
379 |
||
380 |
definition int_of_nat :: "nat \<Rightarrow> int" where |
|
381 |
"int_of_nat = of_nat" |
|
382 |
||
383 |
text {* |
|
384 |
The following functions convert between Isabelle's built-in {\tt term} |
|
385 |
datatype and the generated {\tt dB} datatype. This allows to |
|
386 |
generate example terms using Isabelle's parser and inspect |
|
387 |
normalized terms using Isabelle's pretty printer. |
|
388 |
*} |
|
389 |
||
390 |
ML {* |
|
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val nat_of_integer = @{code nat} o @{code int_of_integer}; |
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392 |
|
27982 | 393 |
fun dBtype_of_typ (Type ("fun", [T, U])) = |
394 |
@{code Fun} (dBtype_of_typ T, dBtype_of_typ U) |
|
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|
395 |
| dBtype_of_typ (TFree (s, _)) = (case raw_explode s of |
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396 |
["'", a] => @{code Atom} (nat_of_integer (ord a - 97)) |
27982 | 397 |
| _ => error "dBtype_of_typ: variable name") |
398 |
| dBtype_of_typ _ = error "dBtype_of_typ: bad type"; |
|
399 |
||
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400 |
fun dB_of_term (Bound i) = @{code dB.Var} (nat_of_integer i) |
27982 | 401 |
| dB_of_term (t $ u) = @{code dB.App} (dB_of_term t, dB_of_term u) |
402 |
| dB_of_term (Abs (_, _, t)) = @{code dB.Abs} (dB_of_term t) |
|
403 |
| dB_of_term _ = error "dB_of_term: bad term"; |
|
404 |
||
405 |
fun term_of_dB Ts (Type ("fun", [T, U])) (@{code dB.Abs} dBt) = |
|
406 |
Abs ("x", T, term_of_dB (T :: Ts) U dBt) |
|
407 |
| term_of_dB Ts _ dBt = term_of_dB' Ts dBt |
|
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408 |
and term_of_dB' Ts (@{code dB.Var} n) = Bound (@{code integer_of_nat} n) |
27982 | 409 |
| term_of_dB' Ts (@{code dB.App} (dBt, dBu)) = |
410 |
let val t = term_of_dB' Ts dBt |
|
411 |
in case fastype_of1 (Ts, t) of |
|
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412 |
Type ("fun", [T, _]) => t $ term_of_dB Ts T dBu |
27982 | 413 |
| _ => error "term_of_dB: function type expected" |
414 |
end |
|
415 |
| term_of_dB' _ _ = error "term_of_dB: term not in normal form"; |
|
416 |
||
417 |
fun typing_of_term Ts e (Bound i) = |
|
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418 |
@{code Var} (e, nat_of_integer i, dBtype_of_typ (nth Ts i)) |
27982 | 419 |
| typing_of_term Ts e (t $ u) = (case fastype_of1 (Ts, t) of |
420 |
Type ("fun", [T, U]) => @{code App} (e, dB_of_term t, |
|
421 |
dBtype_of_typ T, dBtype_of_typ U, dB_of_term u, |
|
422 |
typing_of_term Ts e t, typing_of_term Ts e u) |
|
423 |
| _ => error "typing_of_term: function type expected") |
|
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424 |
| typing_of_term Ts e (Abs (_, T, t)) = |
27982 | 425 |
let val dBT = dBtype_of_typ T |
426 |
in @{code Abs} (e, dBT, dB_of_term t, |
|
427 |
dBtype_of_typ (fastype_of1 (T :: Ts, t)), |
|
428 |
typing_of_term (T :: Ts) (@{code shift} e @{code "0::nat"} dBT) t) |
|
429 |
end |
|
430 |
| typing_of_term _ _ _ = error "typing_of_term: bad term"; |
|
431 |
||
432 |
fun dummyf _ = error "dummy"; |
|
433 |
||
434 |
val ct1 = @{cterm "%f. ((%f x. f (f (f x))) ((%f x. f (f (f (f x)))) f))"}; |
|
435 |
val (dB1, _) = @{code type_NF} (typing_of_term [] dummyf (term_of ct1)); |
|
32010 | 436 |
val ct1' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct1)) dB1); |
27982 | 437 |
|
438 |
val ct2 = @{cterm "%f x. (%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) x)))))"}; |
|
439 |
val (dB2, _) = @{code type_NF} (typing_of_term [] dummyf (term_of ct2)); |
|
32010 | 440 |
val ct2' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct2)) dB2); |
27982 | 441 |
*} |
442 |
||
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443 |
end |