author | wenzelm |
Sun, 02 Nov 2014 17:16:01 +0100 | |
changeset 58880 | 0baae4311a9f |
parent 57492 | 74bf65a1910a |
child 62175 | 8ffc4d0e652d |
permissions | -rw-r--r-- |
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(* Title: HOL/HOLCF/Library/Stream.thy |
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Author: Franz Regensburger, David von Oheimb, Borislav Gajanovic |
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*) |
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||
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section {* General Stream domain *} |
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|
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theory Stream |
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Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
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imports "../HOLCF" "~~/src/HOL/Library/Extended_Nat" |
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begin |
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default_sort pcpo |
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domain (unsafe) 'a stream = scons (ft::'a) (lazy rt::"'a stream") (infixr "&&" 65) |
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|
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definition |
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smap :: "('a \<rightarrow> 'b) \<rightarrow> 'a stream \<rightarrow> 'b stream" where |
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"smap = fix\<cdot>(\<Lambda> h f s. case s of x && xs \<Rightarrow> f\<cdot>x && h\<cdot>f\<cdot>xs)" |
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definition |
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sfilter :: "('a \<rightarrow> tr) \<rightarrow> 'a stream \<rightarrow> 'a stream" where |
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"sfilter = fix\<cdot>(\<Lambda> h p s. case s of x && xs \<Rightarrow> |
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If p\<cdot>x then x && h\<cdot>p\<cdot>xs else h\<cdot>p\<cdot>xs)" |
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definition |
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slen :: "'a stream \<Rightarrow> enat" ("#_" [1000] 1000) where |
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"#s = (if stream_finite s then enat (LEAST n. stream_take n\<cdot>s = s) else \<infinity>)" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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28 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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(* concatenation *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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30 |
|
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definition |
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i_rt :: "nat => 'a stream => 'a stream" where (* chops the first i elements *) |
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"i_rt = (%i s. iterate i$rt$s)" |
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definition |
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i_th :: "nat => 'a stream => 'a" where (* the i-th element *) |
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"i_th = (%i s. ft$(i_rt i s))" |
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38 |
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definition |
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sconc :: "'a stream => 'a stream => 'a stream" (infixr "ooo" 65) where |
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"s1 ooo s2 = (case #s1 of |
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enat n \<Rightarrow> (SOME s. (stream_take n$s=s1) & (i_rt n s = s2)) |
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| \<infinity> \<Rightarrow> s1)" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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44 |
|
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primrec constr_sconc' :: "nat => 'a stream => 'a stream => 'a stream" |
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where |
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constr_sconc'_0: "constr_sconc' 0 s1 s2 = s2" |
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| constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft$s1 && |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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constr_sconc' n (rt$s1) s2" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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50 |
|
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definition |
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constr_sconc :: "'a stream => 'a stream => 'a stream" where (* constructive *) |
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"constr_sconc s1 s2 = (case #s1 of |
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enat n \<Rightarrow> constr_sconc' n s1 s2 |
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| \<infinity> \<Rightarrow> s1)" |
56 |
||
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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57 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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(* ----------------------------------------------------------------------- *) |
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(* theorems about scons *) |
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(* ----------------------------------------------------------------------- *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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61 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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62 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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63 |
section "scons" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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64 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma scons_eq_UU: "(a && s = UU) = (a = UU)" |
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66 |
by simp |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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67 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma scons_not_empty: "[| a && x = UU; a ~= UU |] ==> R" |
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69 |
by simp |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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70 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma stream_exhaust_eq: "(x ~= UU) = (EX a y. a ~= UU & x = a && y)" |
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72 |
by (cases x, auto) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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73 |
|
18109 | 74 |
lemma stream_neq_UU: "x~=UU ==> EX a a_s. x=a&&a_s & a~=UU" |
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by (simp add: stream_exhaust_eq,auto) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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76 |
|
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lemma stream_prefix: |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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"[| a && s << t; a ~= UU |] ==> EX b tt. t = b && tt & b ~= UU & s << tt" |
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79 |
by (cases t, auto) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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80 |
|
17291 | 81 |
lemma stream_prefix': |
82 |
"b ~= UU ==> x << b && z = |
|
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83 |
(x = UU | (EX a y. x = a && y & a ~= UU & a << b & y << z))" |
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84 |
by (cases x, auto) |
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85 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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|
86 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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|
87 |
(* |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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|
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lemma stream_prefix1: "[| x<<y; xs<<ys |] ==> x&&xs << y&&ys" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (insert stream_prefix' [of y "x&&xs" ys],force) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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90 |
*) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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|
91 |
|
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lemma stream_flat_prefix: |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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|
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"[| x && xs << y && ys; (x::'a::flat) ~= UU|] ==> x = y & xs << ys" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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94 |
apply (case_tac "y=UU",auto) |
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apply (drule ax_flat,simp) |
96 |
done |
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97 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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|
98 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
99 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
100 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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|
101 |
(* ----------------------------------------------------------------------- *) |
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102 |
(* theorems about stream_case *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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|
103 |
(* ----------------------------------------------------------------------- *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
104 |
|
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105 |
section "stream_case" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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|
106 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
107 |
|
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108 |
lemma stream_case_strictf: "stream_case$UU$s=UU" |
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109 |
by (cases s, auto) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
110 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
111 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
112 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
113 |
(* ----------------------------------------------------------------------- *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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|
114 |
(* theorems about ft and rt *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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changeset
|
115 |
(* ----------------------------------------------------------------------- *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
116 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
117 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
118 |
section "ft & rt" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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changeset
|
119 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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diff
changeset
|
120 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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changeset
|
121 |
lemma ft_defin: "s~=UU ==> ft$s~=UU" |
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122 |
by simp |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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changeset
|
123 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
124 |
lemma rt_strict_rev: "rt$s~=UU ==> s~=UU" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
125 |
by auto |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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diff
changeset
|
126 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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changeset
|
127 |
lemma surjectiv_scons: "(ft$s)&&(rt$s)=s" |
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128 |
by (cases s, auto) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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diff
changeset
|
129 |
|
18075 | 130 |
lemma monofun_rt_mult: "x << s ==> iterate i$rt$x << iterate i$rt$s" |
131 |
by (rule monofun_cfun_arg) |
|
15188
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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changeset
|
132 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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changeset
|
133 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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14981
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changeset
|
134 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
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diff
changeset
|
135 |
(* ----------------------------------------------------------------------- *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
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diff
changeset
|
136 |
(* theorems about stream_take *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
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diff
changeset
|
137 |
(* ----------------------------------------------------------------------- *) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
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diff
changeset
|
138 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
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diff
changeset
|
139 |
|
17291 | 140 |
section "stream_take" |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
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diff
changeset
|
141 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
142 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
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diff
changeset
|
143 |
lemma stream_reach2: "(LUB i. stream_take i$s) = s" |
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diff
changeset
|
144 |
by (rule stream.reach) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
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diff
changeset
|
145 |
|
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
146 |
lemma chain_stream_take: "chain (%i. stream_take i$s)" |
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changeset
|
147 |
by simp |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
148 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
149 |
lemma stream_take_prefix [simp]: "stream_take n$s << s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
150 |
apply (insert stream_reach2 [of s]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
151 |
apply (erule subst) back |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
152 |
apply (rule is_ub_thelub) |
49521 | 153 |
apply (simp only: chain_stream_take) |
154 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
155 |
|
17291 | 156 |
lemma stream_take_more [rule_format]: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
157 |
"ALL x. stream_take n$x = x --> stream_take (Suc n)$x = x" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
158 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
159 |
apply (case_tac "x=UU",auto) |
49521 | 160 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
161 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
162 |
|
17291 | 163 |
lemma stream_take_lemma3 [rule_format]: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
164 |
"ALL x xs. x~=UU --> stream_take n$(x && xs) = x && xs --> stream_take n$xs=xs" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
165 |
apply (induct_tac n,clarsimp) |
16745 | 166 |
(*apply (drule sym, erule scons_not_empty, simp)*) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
167 |
apply (clarify, rule stream_take_more) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
168 |
apply (erule_tac x="x" in allE) |
49521 | 169 |
apply (erule_tac x="xs" in allE,simp) |
170 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
171 |
|
17291 | 172 |
lemma stream_take_lemma4: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
173 |
"ALL x xs. stream_take n$xs=xs --> stream_take (Suc n)$(x && xs) = x && xs" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
174 |
by auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
175 |
|
17291 | 176 |
lemma stream_take_idempotent [rule_format, simp]: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
177 |
"ALL s. stream_take n$(stream_take n$s) = stream_take n$s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
178 |
apply (induct_tac n, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
179 |
apply (case_tac "s=UU", auto) |
49521 | 180 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
181 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
182 |
|
17291 | 183 |
lemma stream_take_take_Suc [rule_format, simp]: |
184 |
"ALL s. stream_take n$(stream_take (Suc n)$s) = |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
185 |
stream_take n$s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
186 |
apply (induct_tac n, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
187 |
apply (case_tac "s=UU", auto) |
49521 | 188 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
189 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
190 |
|
17291 | 191 |
lemma mono_stream_take_pred: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
192 |
"stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==> |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
193 |
stream_take n$s1 << stream_take n$s2" |
17291 | 194 |
by (insert monofun_cfun_arg [of "stream_take (Suc n)$s1" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
195 |
"stream_take (Suc n)$s2" "stream_take n"], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
196 |
(* |
17291 | 197 |
lemma mono_stream_take_pred: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
198 |
"stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==> |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
199 |
stream_take n$s1 << stream_take n$s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
200 |
by (drule mono_stream_take [of _ _ n],simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
201 |
*) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
202 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
203 |
lemma stream_take_lemma10 [rule_format]: |
17291 | 204 |
"ALL k<=n. stream_take n$s1 << stream_take n$s2 |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
205 |
--> stream_take k$s1 << stream_take k$s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
206 |
apply (induct_tac n,simp,clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
207 |
apply (case_tac "k=Suc n",blast) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
208 |
apply (erule_tac x="k" in allE) |
49521 | 209 |
apply (drule mono_stream_take_pred,simp) |
210 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
211 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
212 |
lemma stream_take_le_mono : "k<=n ==> stream_take k$s1 << stream_take n$s1" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
213 |
apply (insert chain_stream_take [of s1]) |
49521 | 214 |
apply (drule chain_mono,auto) |
215 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
216 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
217 |
lemma mono_stream_take: "s1 << s2 ==> stream_take n$s1 << stream_take n$s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
218 |
by (simp add: monofun_cfun_arg) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
219 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
220 |
(* |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
221 |
lemma stream_take_prefix [simp]: "stream_take n$s << s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
222 |
apply (subgoal_tac "s=(LUB n. stream_take n$s)") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
223 |
apply (erule ssubst, rule is_ub_thelub) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
224 |
apply (simp only: chain_stream_take) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
225 |
by (simp only: stream_reach2) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
226 |
*) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
227 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
228 |
lemma stream_take_take_less:"stream_take k$(stream_take n$s) << stream_take k$s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
229 |
by (rule monofun_cfun_arg,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
230 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
231 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
232 |
(* ------------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
233 |
(* special induction rules *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
234 |
(* ------------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
235 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
236 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
237 |
section "induction" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
238 |
|
17291 | 239 |
lemma stream_finite_ind: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
240 |
"[| stream_finite x; P UU; !!a s. [| a ~= UU; P s |] ==> P (a && s) |] ==> P x" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
241 |
apply (simp add: stream.finite_def,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
242 |
apply (erule subst) |
49521 | 243 |
apply (drule stream.finite_induct [of P _ x], auto) |
244 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
245 |
|
17291 | 246 |
lemma stream_finite_ind2: |
247 |
"[| P UU; !! x. x ~= UU ==> P (x && UU); !! y z s. [| y ~= UU; z ~= UU; P s |] ==> P (y && z && s )|] ==> |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
248 |
!s. P (stream_take n$s)" |
29855
e0ab6cf95539
Repaired a proof that did, after all, refer to the theorem nat_induct2.
paulson
parents:
29530
diff
changeset
|
249 |
apply (rule nat_less_induct [of _ n],auto) |
e0ab6cf95539
Repaired a proof that did, after all, refer to the theorem nat_induct2.
paulson
parents:
29530
diff
changeset
|
250 |
apply (case_tac n, auto) |
e0ab6cf95539
Repaired a proof that did, after all, refer to the theorem nat_induct2.
paulson
parents:
29530
diff
changeset
|
251 |
apply (case_tac nat, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
252 |
apply (case_tac "s=UU",clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
253 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
254 |
apply (case_tac "s=UU",clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
255 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
256 |
apply (case_tac "y=UU",clarsimp) |
49521 | 257 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
258 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
259 |
|
17291 | 260 |
lemma stream_ind2: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
261 |
"[| adm P; P UU; !!a. a ~= UU ==> P (a && UU); !!a b s. [| a ~= UU; b ~= UU; P s |] ==> P (a && b && s) |] ==> P x" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
262 |
apply (insert stream.reach [of x],erule subst) |
35494
45c9a8278faf
domain package no longer generates copy functions; all proofs use take functions instead
huffman
parents:
35444
diff
changeset
|
263 |
apply (erule admD, rule chain_stream_take) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
264 |
apply (insert stream_finite_ind2 [of P]) |
35494
45c9a8278faf
domain package no longer generates copy functions; all proofs use take functions instead
huffman
parents:
35444
diff
changeset
|
265 |
by simp |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
266 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
267 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
268 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
269 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
270 |
(* simplify use of coinduction *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
271 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
272 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
273 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
274 |
section "coinduction" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
275 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
276 |
lemma stream_coind_lemma2: "!s1 s2. R s1 s2 --> ft$s1 = ft$s2 & R (rt$s1) (rt$s2) ==> stream_bisim R" |
30807 | 277 |
apply (simp add: stream.bisim_def,clarsimp) |
35497 | 278 |
apply (drule spec, drule spec, drule (1) mp) |
279 |
apply (case_tac "x", simp) |
|
40025 | 280 |
apply (case_tac "y", simp) |
49521 | 281 |
apply auto |
282 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
283 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
284 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
285 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
286 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
287 |
(* theorems about stream_finite *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
288 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
289 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
290 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
291 |
section "stream_finite" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
292 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
293 |
lemma stream_finite_UU [simp]: "stream_finite UU" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
294 |
by (simp add: stream.finite_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
295 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
296 |
lemma stream_finite_UU_rev: "~ stream_finite s ==> s ~= UU" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
297 |
by (auto simp add: stream.finite_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
298 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
299 |
lemma stream_finite_lemma1: "stream_finite xs ==> stream_finite (x && xs)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
300 |
apply (simp add: stream.finite_def,auto) |
35557
5da670d57118
uniformly use variable names m and n in take-related lemmas; use export_without_context where appropriate
huffman
parents:
35524
diff
changeset
|
301 |
apply (rule_tac x="Suc n" in exI) |
49521 | 302 |
apply (simp add: stream_take_lemma4) |
303 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
304 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
305 |
lemma stream_finite_lemma2: "[| x ~= UU; stream_finite (x && xs) |] ==> stream_finite xs" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
306 |
apply (simp add: stream.finite_def, auto) |
35557
5da670d57118
uniformly use variable names m and n in take-related lemmas; use export_without_context where appropriate
huffman
parents:
35524
diff
changeset
|
307 |
apply (rule_tac x="n" in exI) |
49521 | 308 |
apply (erule stream_take_lemma3,simp) |
309 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
310 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
311 |
lemma stream_finite_rt_eq: "stream_finite (rt$s) = stream_finite s" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
312 |
apply (cases s, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
313 |
apply (rule stream_finite_lemma1, simp) |
49521 | 314 |
apply (rule stream_finite_lemma2,simp) |
315 |
apply assumption |
|
316 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
317 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
318 |
lemma stream_finite_less: "stream_finite s ==> !t. t<<s --> stream_finite t" |
19440 | 319 |
apply (erule stream_finite_ind [of s], auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
320 |
apply (case_tac "t=UU", auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
321 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
322 |
apply (erule_tac x="y" in allE, simp) |
49521 | 323 |
apply (rule stream_finite_lemma1, simp) |
324 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
325 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
326 |
lemma stream_take_finite [simp]: "stream_finite (stream_take n$s)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
327 |
apply (simp add: stream.finite_def) |
49521 | 328 |
apply (rule_tac x="n" in exI,simp) |
329 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
330 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
331 |
lemma adm_not_stream_finite: "adm (%x. ~ stream_finite x)" |
25833 | 332 |
apply (rule adm_upward) |
333 |
apply (erule contrapos_nn) |
|
334 |
apply (erule (1) stream_finite_less [rule_format]) |
|
335 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
336 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
337 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
338 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
339 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
340 |
(* theorems about stream length *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
341 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
342 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
343 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
344 |
section "slen" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
345 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
346 |
lemma slen_empty [simp]: "#\<bottom> = 0" |
43919 | 347 |
by (simp add: slen_def stream.finite_def zero_enat_def Least_equality) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
348 |
|
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
349 |
lemma slen_scons [simp]: "x ~= \<bottom> ==> #(x&&xs) = eSuc (#xs)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
350 |
apply (case_tac "stream_finite (x && xs)") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
351 |
apply (simp add: slen_def, auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
352 |
apply (simp add: stream.finite_def, auto simp add: eSuc_enat) |
27111 | 353 |
apply (rule Least_Suc2, auto) |
16745 | 354 |
(*apply (drule sym)*) |
355 |
(*apply (drule sym scons_eq_UU [THEN iffD1],simp)*) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
356 |
apply (erule stream_finite_lemma2, simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
357 |
apply (simp add: slen_def, auto) |
49521 | 358 |
apply (drule stream_finite_lemma1,auto) |
359 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
360 |
|
43924 | 361 |
lemma slen_less_1_eq: "(#x < enat (Suc 0)) = (x = \<bottom>)" |
49521 | 362 |
by (cases x) (auto simp add: enat_0 eSuc_enat[THEN sym]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
363 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
364 |
lemma slen_empty_eq: "(#x = 0) = (x = \<bottom>)" |
49521 | 365 |
by (cases x) auto |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
366 |
|
43924 | 367 |
lemma slen_scons_eq: "(enat (Suc n) < #x) = (? a y. x = a && y & a ~= \<bottom> & enat n < #y)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
368 |
apply (auto, case_tac "x=UU",auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
369 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
27111 | 370 |
apply (case_tac "#y") apply simp_all |
371 |
apply (case_tac "#y") apply simp_all |
|
372 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
373 |
|
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
374 |
lemma slen_eSuc: "#x = eSuc n --> (? a y. x = a&&y & a ~= \<bottom> & #y = n)" |
49521 | 375 |
by (cases x) auto |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
376 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
377 |
lemma slen_stream_take_finite [simp]: "#(stream_take n$s) ~= \<infinity>" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
378 |
by (simp add: slen_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
379 |
|
43924 | 380 |
lemma slen_scons_eq_rev: "(#x < enat (Suc (Suc n))) = (!a y. x ~= a && y | a = \<bottom> | #y < enat (Suc n))" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
381 |
apply (cases x, auto) |
43919 | 382 |
apply (simp add: zero_enat_def) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
383 |
apply (case_tac "#stream") apply (simp_all add: eSuc_enat) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
384 |
apply (case_tac "#stream") apply (simp_all add: eSuc_enat) |
27111 | 385 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
386 |
|
17291 | 387 |
lemma slen_take_lemma4 [rule_format]: |
43924 | 388 |
"!s. stream_take n$s ~= s --> #(stream_take n$s) = enat n" |
389 |
apply (induct n, auto simp add: enat_0) |
|
27111 | 390 |
apply (case_tac "s=UU", simp) |
49521 | 391 |
apply (drule stream_exhaust_eq [THEN iffD1], auto simp add: eSuc_enat) |
392 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
393 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
394 |
(* |
17291 | 395 |
lemma stream_take_idempotent [simp]: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
396 |
"stream_take n$(stream_take n$s) = stream_take n$s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
397 |
apply (case_tac "stream_take n$s = s") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
398 |
apply (auto,insert slen_take_lemma4 [of n s]); |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
399 |
by (auto,insert slen_take_lemma1 [of "stream_take n$s" n],simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
400 |
|
17291 | 401 |
lemma stream_take_take_Suc [simp]: "stream_take n$(stream_take (Suc n)$s) = |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
402 |
stream_take n$s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
403 |
apply (simp add: po_eq_conv,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
404 |
apply (simp add: stream_take_take_less) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
405 |
apply (subgoal_tac "stream_take n$s = stream_take n$(stream_take n$s)") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
406 |
apply (erule ssubst) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
407 |
apply (rule_tac monofun_cfun_arg) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
408 |
apply (insert chain_stream_take [of s]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
409 |
by (simp add: chain_def,simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
410 |
*) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
411 |
|
43924 | 412 |
lemma slen_take_eq: "ALL x. (enat n < #x) = (stream_take n\<cdot>x ~= x)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
413 |
apply (induct_tac n, auto) |
43924 | 414 |
apply (simp add: enat_0, clarsimp) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
415 |
apply (drule not_sym) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
416 |
apply (drule slen_empty_eq [THEN iffD1], simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
417 |
apply (case_tac "x=UU", simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
418 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
419 |
apply (erule_tac x="y" in allE, auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
420 |
apply (simp_all add: not_less eSuc_enat) |
27111 | 421 |
apply (case_tac "#y") apply simp_all |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
422 |
apply (case_tac "x=UU", simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
423 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
424 |
apply (erule_tac x="y" in allE, simp) |
49521 | 425 |
apply (case_tac "#y") |
426 |
apply simp_all |
|
427 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
428 |
|
43924 | 429 |
lemma slen_take_eq_rev: "(#x <= enat n) = (stream_take n\<cdot>x = x)" |
26102 | 430 |
by (simp add: linorder_not_less [symmetric] slen_take_eq) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
431 |
|
43924 | 432 |
lemma slen_take_lemma1: "#x = enat n ==> stream_take n\<cdot>x = x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
433 |
by (rule slen_take_eq_rev [THEN iffD1], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
434 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
435 |
lemma slen_rt_mono: "#s2 <= #s1 ==> #(rt$s2) <= #(rt$s1)" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
436 |
apply (cases s1) |
49521 | 437 |
apply (cases s2, simp+)+ |
438 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
439 |
|
43924 | 440 |
lemma slen_take_lemma5: "#(stream_take n$s) <= enat n" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
441 |
apply (case_tac "stream_take n$s = s") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
442 |
apply (simp add: slen_take_eq_rev) |
49521 | 443 |
apply (simp add: slen_take_lemma4) |
444 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
445 |
|
43924 | 446 |
lemma slen_take_lemma2: "!x. ~stream_finite x --> #(stream_take i\<cdot>x) = enat i" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
447 |
apply (simp add: stream.finite_def, auto) |
49521 | 448 |
apply (simp add: slen_take_lemma4) |
449 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
450 |
|
43921 | 451 |
lemma slen_infinite: "stream_finite x = (#x ~= \<infinity>)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
452 |
by (simp add: slen_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
453 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
454 |
lemma slen_mono_lemma: "stream_finite s ==> ALL t. s << t --> #s <= #t" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
455 |
apply (erule stream_finite_ind [of s], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
456 |
apply (case_tac "t=UU", auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
457 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
30807 | 458 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
459 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
460 |
lemma slen_mono: "s << t ==> #s <= #t" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
461 |
apply (case_tac "stream_finite t") |
17291 | 462 |
apply (frule stream_finite_less) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
463 |
apply (erule_tac x="s" in allE, simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
464 |
apply (drule slen_mono_lemma, auto) |
49521 | 465 |
apply (simp add: slen_def) |
466 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
467 |
|
18075 | 468 |
lemma iterate_lemma: "F$(iterate n$F$x) = iterate n$F$(F$x)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
469 |
by (insert iterate_Suc2 [of n F x], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
470 |
|
43924 | 471 |
lemma slen_rt_mult [rule_format]: "!x. enat (i + j) <= #x --> enat j <= #(iterate i$rt$x)" |
27111 | 472 |
apply (induct i, auto) |
43919 | 473 |
apply (case_tac "x=UU", auto simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
474 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
475 |
apply (erule_tac x="y" in allE, auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
476 |
apply (simp add: not_le) apply (case_tac "#y") apply (simp_all add: eSuc_enat) |
49521 | 477 |
apply (simp add: iterate_lemma) |
478 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
479 |
|
17291 | 480 |
lemma slen_take_lemma3 [rule_format]: |
43924 | 481 |
"!(x::'a::flat stream) y. enat n <= #x --> x << y --> stream_take n\<cdot>x = stream_take n\<cdot>y" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
482 |
apply (induct_tac n, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
483 |
apply (case_tac "x=UU", auto) |
43919 | 484 |
apply (simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
485 |
apply (simp add: Suc_ile_eq) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
486 |
apply (case_tac "y=UU", clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
487 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)+ |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
488 |
apply (erule_tac x="ya" in allE, simp) |
25920 | 489 |
by (drule ax_flat, simp) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
490 |
|
17291 | 491 |
lemma slen_strict_mono_lemma: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
492 |
"stream_finite t ==> !s. #(s::'a::flat stream) = #t & s << t --> s = t" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
493 |
apply (erule stream_finite_ind, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
494 |
apply (case_tac "sa=UU", auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
495 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp) |
49521 | 496 |
apply (drule ax_flat, simp) |
497 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
498 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
499 |
lemma slen_strict_mono: "[|stream_finite t; s ~= t; s << (t::'a::flat stream) |] ==> #s < #t" |
27111 | 500 |
by (auto simp add: slen_mono less_le dest: slen_strict_mono_lemma) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
501 |
|
17291 | 502 |
lemma stream_take_Suc_neq: "stream_take (Suc n)$s ~=s ==> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
503 |
stream_take n$s ~= stream_take (Suc n)$s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
504 |
apply auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
505 |
apply (subgoal_tac "stream_take n$s ~=s") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
506 |
apply (insert slen_take_lemma4 [of n s],auto) |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
507 |
apply (cases s, simp) |
49521 | 508 |
apply (simp add: slen_take_lemma4 eSuc_enat) |
509 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
510 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
511 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
512 |
(* theorems about smap *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
513 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
514 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
515 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
516 |
section "smap" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
517 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
518 |
lemma smap_unfold: "smap = (\<Lambda> f t. case t of x&&xs \<Rightarrow> f$x && smap$f$xs)" |
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
27361
diff
changeset
|
519 |
by (insert smap_def [where 'a='a and 'b='b, THEN eq_reflection, THEN fix_eq2], auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
520 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
521 |
lemma smap_empty [simp]: "smap\<cdot>f\<cdot>\<bottom> = \<bottom>" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
522 |
by (subst smap_unfold, simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
523 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
524 |
lemma smap_scons [simp]: "x~=\<bottom> ==> smap\<cdot>f\<cdot>(x&&xs) = (f\<cdot>x)&&(smap\<cdot>f\<cdot>xs)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
525 |
by (subst smap_unfold, force) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
526 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
527 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
528 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
529 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
530 |
(* theorems about sfilter *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
531 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
532 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
533 |
section "sfilter" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
534 |
|
17291 | 535 |
lemma sfilter_unfold: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
536 |
"sfilter = (\<Lambda> p s. case s of x && xs \<Rightarrow> |
40322
707eb30e8a53
make syntax of continuous if-then-else consistent with HOL if-then-else
huffman
parents:
40213
diff
changeset
|
537 |
If p\<cdot>x then x && sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs)" |
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
27361
diff
changeset
|
538 |
by (insert sfilter_def [where 'a='a, THEN eq_reflection, THEN fix_eq2], auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
539 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
540 |
lemma strict_sfilter: "sfilter\<cdot>\<bottom> = \<bottom>" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
37110
diff
changeset
|
541 |
apply (rule cfun_eqI) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
542 |
apply (subst sfilter_unfold, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
543 |
apply (case_tac "x=UU", auto) |
49521 | 544 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
545 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
546 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
547 |
lemma sfilter_empty [simp]: "sfilter\<cdot>f\<cdot>\<bottom> = \<bottom>" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
548 |
by (subst sfilter_unfold, force) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
549 |
|
17291 | 550 |
lemma sfilter_scons [simp]: |
551 |
"x ~= \<bottom> ==> sfilter\<cdot>f\<cdot>(x && xs) = |
|
40322
707eb30e8a53
make syntax of continuous if-then-else consistent with HOL if-then-else
huffman
parents:
40213
diff
changeset
|
552 |
If f\<cdot>x then x && sfilter\<cdot>f\<cdot>xs else sfilter\<cdot>f\<cdot>xs" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
553 |
by (subst sfilter_unfold, force) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
554 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
555 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
556 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
557 |
section "i_rt" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
558 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
559 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
560 |
lemma i_rt_UU [simp]: "i_rt n UU = UU" |
34941 | 561 |
by (induct n) (simp_all add: i_rt_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
562 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
563 |
lemma i_rt_0 [simp]: "i_rt 0 s = s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
564 |
by (simp add: i_rt_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
565 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
566 |
lemma i_rt_Suc [simp]: "a ~= UU ==> i_rt (Suc n) (a&&s) = i_rt n s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
567 |
by (simp add: i_rt_def iterate_Suc2 del: iterate_Suc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
568 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
569 |
lemma i_rt_Suc_forw: "i_rt (Suc n) s = i_rt n (rt$s)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
570 |
by (simp only: i_rt_def iterate_Suc2) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
571 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
572 |
lemma i_rt_Suc_back:"i_rt (Suc n) s = rt$(i_rt n s)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
573 |
by (simp only: i_rt_def,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
574 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
575 |
lemma i_rt_mono: "x << s ==> i_rt n x << i_rt n s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
576 |
by (simp add: i_rt_def monofun_rt_mult) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
577 |
|
43924 | 578 |
lemma i_rt_ij_lemma: "enat (i + j) <= #x ==> enat j <= #(i_rt i x)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
579 |
by (simp add: i_rt_def slen_rt_mult) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
580 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
581 |
lemma slen_i_rt_mono: "#s2 <= #s1 ==> #(i_rt n s2) <= #(i_rt n s1)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
582 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
583 |
apply (simp add: i_rt_Suc_back) |
49521 | 584 |
apply (drule slen_rt_mono,simp) |
585 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
586 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
587 |
lemma i_rt_take_lemma1 [rule_format]: "ALL s. i_rt n (stream_take n$s) = UU" |
17291 | 588 |
apply (induct_tac n) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
589 |
apply (simp add: i_rt_Suc_back,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
590 |
apply (case_tac "s=UU",auto) |
49521 | 591 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
592 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
593 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
594 |
lemma i_rt_slen: "(i_rt n s = UU) = (stream_take n$s = s)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
595 |
apply auto |
17291 | 596 |
apply (insert i_rt_ij_lemma [of n "Suc 0" s]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
597 |
apply (subgoal_tac "#(i_rt n s)=0") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
598 |
apply (case_tac "stream_take n$s = s",simp+) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
599 |
apply (insert slen_take_eq [rule_format,of n s],simp) |
43919 | 600 |
apply (cases "#s") apply (simp_all add: zero_enat_def) |
27111 | 601 |
apply (simp add: slen_take_eq) |
602 |
apply (cases "#s") |
|
603 |
using i_rt_take_lemma1 [of n s] |
|
43919 | 604 |
apply (simp_all add: zero_enat_def) |
27111 | 605 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
606 |
|
43924 | 607 |
lemma i_rt_lemma_slen: "#s=enat n ==> i_rt n s = UU" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
608 |
by (simp add: i_rt_slen slen_take_lemma1) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
609 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
610 |
lemma stream_finite_i_rt [simp]: "stream_finite (i_rt n s) = stream_finite s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
611 |
apply (induct_tac n, auto) |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
612 |
apply (cases s, auto simp del: i_rt_Suc) |
49521 | 613 |
apply (simp add: i_rt_Suc_back stream_finite_rt_eq)+ |
614 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
615 |
|
43924 | 616 |
lemma take_i_rt_len_lemma: "ALL sl x j t. enat sl = #x & n <= sl & |
617 |
#(stream_take n$x) = enat t & #(i_rt n x)= enat j |
|
618 |
--> enat (j + t) = #x" |
|
27111 | 619 |
apply (induct n, auto) |
43919 | 620 |
apply (simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
621 |
apply (case_tac "x=UU",auto) |
43919 | 622 |
apply (simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
623 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
43924 | 624 |
apply (subgoal_tac "EX k. enat k = #y",clarify) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
625 |
apply (erule_tac x="k" in allE) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
626 |
apply (erule_tac x="y" in allE,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
627 |
apply (erule_tac x="THE p. Suc p = t" in allE,auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
628 |
apply (simp add: eSuc_def split: enat.splits) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
629 |
apply (simp add: eSuc_def split: enat.splits) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
630 |
apply (simp only: the_equality) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
631 |
apply (simp add: eSuc_def split: enat.splits) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
632 |
apply force |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
633 |
apply (simp add: eSuc_def split: enat.splits) |
27111 | 634 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
635 |
|
17291 | 636 |
lemma take_i_rt_len: |
43924 | 637 |
"[| enat sl = #x; n <= sl; #(stream_take n$x) = enat t; #(i_rt n x) = enat j |] ==> |
638 |
enat (j + t) = #x" |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
639 |
by (blast intro: take_i_rt_len_lemma [rule_format]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
640 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
641 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
642 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
643 |
section "i_th" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
644 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
645 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
646 |
lemma i_th_i_rt_step: |
17291 | 647 |
"[| i_th n s1 << i_th n s2; i_rt (Suc n) s1 << i_rt (Suc n) s2 |] ==> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
648 |
i_rt n s1 << i_rt n s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
649 |
apply (simp add: i_th_def i_rt_Suc_back) |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
650 |
apply (cases "i_rt n s1", simp) |
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
651 |
apply (cases "i_rt n s2", auto) |
30807 | 652 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
653 |
|
17291 | 654 |
lemma i_th_stream_take_Suc [rule_format]: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
655 |
"ALL s. i_th n (stream_take (Suc n)$s) = i_th n s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
656 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
657 |
apply (simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
658 |
apply (case_tac "s=UU",auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
659 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
660 |
apply (case_tac "s=UU",simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
661 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
49521 | 662 |
apply (simp add: i_th_def i_rt_Suc_forw) |
663 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
664 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
665 |
lemma i_th_last: "i_th n s && UU = i_rt n (stream_take (Suc n)$s)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
666 |
apply (insert surjectiv_scons [of "i_rt n (stream_take (Suc n)$s)"]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
667 |
apply (rule i_th_stream_take_Suc [THEN subst]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
668 |
apply (simp add: i_th_def i_rt_Suc_back [symmetric]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
669 |
by (simp add: i_rt_take_lemma1) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
670 |
|
17291 | 671 |
lemma i_th_last_eq: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
672 |
"i_th n s1 = i_th n s2 ==> i_rt n (stream_take (Suc n)$s1) = i_rt n (stream_take (Suc n)$s2)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
673 |
apply (insert i_th_last [of n s1]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
674 |
apply (insert i_th_last [of n s2]) |
49521 | 675 |
apply auto |
676 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
677 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
678 |
lemma i_th_prefix_lemma: |
17291 | 679 |
"[| k <= n; stream_take (Suc n)$s1 << stream_take (Suc n)$s2 |] ==> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
680 |
i_th k s1 << i_th k s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
681 |
apply (insert i_th_stream_take_Suc [of k s1, THEN sym]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
682 |
apply (insert i_th_stream_take_Suc [of k s2, THEN sym],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
683 |
apply (simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
684 |
apply (rule monofun_cfun, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
685 |
apply (rule i_rt_mono) |
49521 | 686 |
apply (blast intro: stream_take_lemma10) |
687 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
688 |
|
17291 | 689 |
lemma take_i_rt_prefix_lemma1: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
690 |
"stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==> |
17291 | 691 |
i_rt (Suc n) s1 << i_rt (Suc n) s2 ==> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
692 |
i_rt n s1 << i_rt n s2 & stream_take n$s1 << stream_take n$s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
693 |
apply auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
694 |
apply (insert i_th_prefix_lemma [of n n s1 s2]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
695 |
apply (rule i_th_i_rt_step,auto) |
49521 | 696 |
apply (drule mono_stream_take_pred,simp) |
697 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
698 |
|
17291 | 699 |
lemma take_i_rt_prefix_lemma: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
700 |
"[| stream_take n$s1 << stream_take n$s2; i_rt n s1 << i_rt n s2 |] ==> s1 << s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
701 |
apply (case_tac "n=0",simp) |
25161 | 702 |
apply (auto) |
17291 | 703 |
apply (subgoal_tac "stream_take 0$s1 << stream_take 0$s2 & |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
704 |
i_rt 0 s1 << i_rt 0 s2") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
705 |
defer 1 |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
706 |
apply (rule zero_induct,blast) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
707 |
apply (blast dest: take_i_rt_prefix_lemma1) |
49521 | 708 |
apply simp |
709 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
710 |
|
17291 | 711 |
lemma streams_prefix_lemma: "(s1 << s2) = |
712 |
(stream_take n$s1 << stream_take n$s2 & i_rt n s1 << i_rt n s2)" |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
713 |
apply auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
714 |
apply (simp add: monofun_cfun_arg) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
715 |
apply (simp add: i_rt_mono) |
49521 | 716 |
apply (erule take_i_rt_prefix_lemma,simp) |
717 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
718 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
719 |
lemma streams_prefix_lemma1: |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
720 |
"[| stream_take n$s1 = stream_take n$s2; i_rt n s1 = i_rt n s2 |] ==> s1 = s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
721 |
apply (simp add: po_eq_conv,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
722 |
apply (insert streams_prefix_lemma) |
49521 | 723 |
apply blast+ |
724 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
725 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
726 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
727 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
728 |
section "sconc" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
729 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
730 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
731 |
lemma UU_sconc [simp]: " UU ooo s = s " |
43919 | 732 |
by (simp add: sconc_def zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
733 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
734 |
lemma scons_neq_UU: "a~=UU ==> a && s ~=UU" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
735 |
by auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
736 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
737 |
lemma singleton_sconc [rule_format, simp]: "x~=UU --> (x && UU) ooo y = x && y" |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
738 |
apply (simp add: sconc_def zero_enat_def eSuc_def split: enat.splits, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
739 |
apply (rule someI2_ex,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
740 |
apply (rule_tac x="x && y" in exI,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
741 |
apply (simp add: i_rt_Suc_forw) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
742 |
apply (case_tac "xa=UU",simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
743 |
by (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
744 |
|
17291 | 745 |
lemma ex_sconc [rule_format]: |
43924 | 746 |
"ALL k y. #x = enat k --> (EX w. stream_take k$w = x & i_rt k w = y)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
747 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
748 |
apply (rule stream_finite_ind [of x],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
749 |
apply (simp add: stream.finite_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
750 |
apply (drule slen_take_lemma1,blast) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
751 |
apply (simp_all add: zero_enat_def eSuc_def split: enat.splits) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
752 |
apply (erule_tac x="y" in allE,auto) |
49521 | 753 |
apply (rule_tac x="a && w" in exI,auto) |
754 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
755 |
|
43924 | 756 |
lemma rt_sconc1: "enat n = #x ==> i_rt n (x ooo y) = y" |
43919 | 757 |
apply (simp add: sconc_def split: enat.splits, arith?,auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
758 |
apply (rule someI2_ex,auto) |
49521 | 759 |
apply (drule ex_sconc,simp) |
760 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
761 |
|
43924 | 762 |
lemma sconc_inj2: "\<lbrakk>enat n = #x; x ooo y = x ooo z\<rbrakk> \<Longrightarrow> y = z" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
763 |
apply (frule_tac y=y in rt_sconc1) |
49521 | 764 |
apply (auto elim: rt_sconc1) |
765 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
766 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
767 |
lemma sconc_UU [simp]:"s ooo UU = s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
768 |
apply (case_tac "#s") |
27111 | 769 |
apply (simp add: sconc_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
770 |
apply (rule someI2_ex) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
771 |
apply (rule_tac x="s" in exI) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
772 |
apply auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
773 |
apply (drule slen_take_lemma1,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
774 |
apply (simp add: i_rt_lemma_slen) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
775 |
apply (drule slen_take_lemma1,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
776 |
apply (simp add: i_rt_slen) |
49521 | 777 |
apply (simp add: sconc_def) |
778 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
779 |
|
43924 | 780 |
lemma stream_take_sconc [simp]: "enat n = #x ==> stream_take n$(x ooo y) = x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
781 |
apply (simp add: sconc_def) |
27111 | 782 |
apply (cases "#x") |
783 |
apply auto |
|
784 |
apply (rule someI2_ex, auto) |
|
49521 | 785 |
apply (drule ex_sconc,simp) |
786 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
787 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
788 |
lemma scons_sconc [rule_format,simp]: "a~=UU --> (a && x) ooo y = a && x ooo y" |
27111 | 789 |
apply (cases "#x",auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
790 |
apply (simp add: sconc_def eSuc_enat) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
791 |
apply (rule someI2_ex) |
27111 | 792 |
apply (drule ex_sconc, simp) |
793 |
apply (rule someI2_ex, auto) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
794 |
apply (simp add: i_rt_Suc_forw) |
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
49521
diff
changeset
|
795 |
apply (rule_tac x="a && xa" in exI, auto) |
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
49521
diff
changeset
|
796 |
apply (case_tac "xaa=UU",auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
797 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
798 |
apply (drule streams_prefix_lemma1,simp+) |
49521 | 799 |
apply (simp add: sconc_def) |
800 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
801 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
802 |
lemma ft_sconc: "x ~= UU ==> ft$(x ooo y) = ft$x" |
49521 | 803 |
by (cases x) auto |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
804 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
805 |
lemma sconc_assoc: "(x ooo y) ooo z = x ooo y ooo z" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
806 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
807 |
apply (rule stream_finite_ind [of x],auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
808 |
apply (simp add: stream.finite_def del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
809 |
apply (drule slen_take_lemma1,auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
810 |
apply (case_tac "a = UU", auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
811 |
by (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
812 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
813 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
814 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
815 |
|
25833 | 816 |
lemma cont_sconc_lemma1: "stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)" |
817 |
by (erule stream_finite_ind, simp_all) |
|
818 |
||
819 |
lemma cont_sconc_lemma2: "\<not> stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)" |
|
820 |
by (simp add: sconc_def slen_def) |
|
821 |
||
822 |
lemma cont_sconc: "cont (\<lambda>y. x ooo y)" |
|
823 |
apply (cases "stream_finite x") |
|
824 |
apply (erule cont_sconc_lemma1) |
|
825 |
apply (erule cont_sconc_lemma2) |
|
826 |
done |
|
827 |
||
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
828 |
lemma sconc_mono: "y << y' ==> x ooo y << x ooo y'" |
25833 | 829 |
by (rule cont_sconc [THEN cont2mono, THEN monofunE]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
830 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
831 |
lemma sconc_mono1 [simp]: "x << x ooo y" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
832 |
by (rule sconc_mono [of UU, simplified]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
833 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
834 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
835 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
836 |
lemma empty_sconc [simp]: "(x ooo y = UU) = (x = UU & y = UU)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
837 |
apply (case_tac "#x",auto) |
17291 | 838 |
apply (insert sconc_mono1 [of x y]) |
49521 | 839 |
apply auto |
840 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
841 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
842 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
843 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
844 |
lemma rt_sconc [rule_format, simp]: "s~=UU --> rt$(s ooo x) = rt$s ooo x" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
845 |
by (cases s, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
846 |
|
17291 | 847 |
lemma i_th_sconc_lemma [rule_format]: |
43924 | 848 |
"ALL x y. enat n < #x --> i_th n (x ooo y) = i_th n x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
849 |
apply (induct_tac n, auto) |
43924 | 850 |
apply (simp add: enat_0 i_th_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
851 |
apply (simp add: slen_empty_eq ft_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
852 |
apply (simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
853 |
apply (case_tac "x=UU",auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
854 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
855 |
apply (erule_tac x="ya" in allE) |
49521 | 856 |
apply (case_tac "#ya") |
857 |
apply simp_all |
|
858 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
859 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
860 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
861 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
862 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
863 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
864 |
lemma sconc_lemma [rule_format, simp]: "ALL s. stream_take n$s ooo i_rt n s = s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
865 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
866 |
apply (case_tac "s=UU",auto) |
49521 | 867 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
868 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
869 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
870 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
871 |
subsection "pointwise equality" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
872 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
873 |
|
17291 | 874 |
lemma ex_last_stream_take_scons: "stream_take (Suc n)$s = |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
875 |
stream_take n$s ooo i_rt n (stream_take (Suc n)$s)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
876 |
by (insert sconc_lemma [of n "stream_take (Suc n)$s"],simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
877 |
|
17291 | 878 |
lemma i_th_stream_take_eq: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
879 |
"!!n. ALL n. i_th n s1 = i_th n s2 ==> stream_take n$s1 = stream_take n$s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
880 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
881 |
apply (subgoal_tac "stream_take (Suc na)$s1 = |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
882 |
stream_take na$s1 ooo i_rt na (stream_take (Suc na)$s1)") |
17291 | 883 |
apply (subgoal_tac "i_rt na (stream_take (Suc na)$s1) = |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
884 |
i_rt na (stream_take (Suc na)$s2)") |
17291 | 885 |
apply (subgoal_tac "stream_take (Suc na)$s2 = |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
886 |
stream_take na$s2 ooo i_rt na (stream_take (Suc na)$s2)") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
887 |
apply (insert ex_last_stream_take_scons,simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
888 |
apply blast |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
889 |
apply (erule_tac x="na" in allE) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
890 |
apply (insert i_th_last_eq [of _ s1 s2]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
891 |
by blast+ |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
892 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
893 |
lemma pointwise_eq_lemma[rule_format]: "ALL n. i_th n s1 = i_th n s2 ==> s1 = s2" |
35642
f478d5a9d238
generate separate qualified theorem name for each type's reach and take_lemma
huffman
parents:
35557
diff
changeset
|
894 |
by (insert i_th_stream_take_eq [THEN stream.take_lemma],blast) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
895 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
896 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
897 |
subsection "finiteness" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
898 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
899 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
900 |
lemma slen_sconc_finite1: |
43924 | 901 |
"[| #(x ooo y) = \<infinity>; enat n = #x |] ==> #y = \<infinity>" |
43921 | 902 |
apply (case_tac "#y ~= \<infinity>",auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
903 |
apply (drule_tac y=y in rt_sconc1) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
904 |
apply (insert stream_finite_i_rt [of n "x ooo y"]) |
49521 | 905 |
apply (simp add: slen_infinite) |
906 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
907 |
|
43921 | 908 |
lemma slen_sconc_infinite1: "#x=\<infinity> ==> #(x ooo y) = \<infinity>" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
909 |
by (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
910 |
|
43921 | 911 |
lemma slen_sconc_infinite2: "#y=\<infinity> ==> #(x ooo y) = \<infinity>" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
912 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
913 |
apply (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
914 |
apply (rule someI2_ex) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
915 |
apply (drule ex_sconc,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
916 |
apply (erule contrapos_pp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
917 |
apply (insert stream_finite_i_rt) |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44019
diff
changeset
|
918 |
apply (fastforce simp add: slen_infinite,auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
919 |
by (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
920 |
|
43921 | 921 |
lemma sconc_finite: "(#x~=\<infinity> & #y~=\<infinity>) = (#(x ooo y)~=\<infinity>)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
922 |
apply auto |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
923 |
apply (metis not_infinity_eq slen_sconc_finite1) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
924 |
apply (metis not_infinity_eq slen_sconc_infinite1) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
925 |
apply (metis not_infinity_eq slen_sconc_infinite2) |
31084 | 926 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
927 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
928 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
929 |
|
43924 | 930 |
lemma slen_sconc_mono3: "[| enat n = #x; enat k = #(x ooo y) |] ==> n <= k" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
931 |
apply (insert slen_mono [of "x" "x ooo y"]) |
27111 | 932 |
apply (cases "#x") apply simp_all |
933 |
apply (cases "#(x ooo y)") apply simp_all |
|
934 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
935 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
936 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
937 |
subsection "finite slen" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
938 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
939 |
|
43924 | 940 |
lemma slen_sconc: "[| enat n = #x; enat m = #y |] ==> #(x ooo y) = enat (n + m)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
941 |
apply (case_tac "#(x ooo y)") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
942 |
apply (frule_tac y=y in rt_sconc1) |
43924 | 943 |
apply (insert take_i_rt_len [of "THE j. enat j = #(x ooo y)" "x ooo y" n n m],simp) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
944 |
apply (insert slen_sconc_mono3 [of n x _ y],simp) |
49521 | 945 |
apply (insert sconc_finite [of x y],auto) |
946 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
947 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
948 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
949 |
subsection "flat prefix" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
950 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
951 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
952 |
lemma sconc_prefix: "(s1::'a::flat stream) << s2 ==> EX t. s1 ooo t = s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
953 |
apply (case_tac "#s1") |
17291 | 954 |
apply (subgoal_tac "stream_take nat$s1 = stream_take nat$s2") |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
955 |
apply (rule_tac x="i_rt nat s2" in exI) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
956 |
apply (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
957 |
apply (rule someI2_ex) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
958 |
apply (drule ex_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
959 |
apply (simp,clarsimp,drule streams_prefix_lemma1) |
17291 | 960 |
apply (simp+,rule slen_take_lemma3 [of _ s1 s2]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
961 |
apply (simp+,rule_tac x="UU" in exI) |
17291 | 962 |
apply (insert slen_take_lemma3 [of _ s1 s2]) |
49521 | 963 |
apply (rule stream.take_lemma,simp) |
964 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
965 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
966 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
967 |
subsection "continuity" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
968 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
969 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
970 |
lemma chain_sconc: "chain S ==> chain (%i. (x ooo S i))" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
971 |
by (simp add: chain_def,auto simp add: sconc_mono) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
972 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
973 |
lemma chain_scons: "chain S ==> chain (%i. a && S i)" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
974 |
apply (simp add: chain_def,auto) |
49521 | 975 |
apply (rule monofun_cfun_arg,simp) |
976 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
977 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
978 |
lemma contlub_scons_lemma: "chain S ==> (LUB i. a && S i) = a && (LUB i. S i)" |
40327 | 979 |
by (rule cont2contlubE [OF cont_Rep_cfun2, symmetric]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
980 |
|
17291 | 981 |
lemma finite_lub_sconc: "chain Y ==> (stream_finite x) ==> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
982 |
(LUB i. x ooo Y i) = (x ooo (LUB i. Y i))" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
983 |
apply (rule stream_finite_ind [of x]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
984 |
apply (auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
985 |
apply (subgoal_tac "(LUB i. a && (s ooo Y i)) = a && (LUB i. s ooo Y i)") |
49521 | 986 |
apply (force,blast dest: contlub_scons_lemma chain_sconc) |
987 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
988 |
|
17291 | 989 |
lemma contlub_sconc_lemma: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
990 |
"chain Y ==> (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))" |
43921 | 991 |
apply (case_tac "#x=\<infinity>") |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
992 |
apply (simp add: sconc_def) |
18075 | 993 |
apply (drule finite_lub_sconc,auto simp add: slen_infinite) |
994 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
995 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
996 |
lemma monofun_sconc: "monofun (%y. x ooo y)" |
16218 | 997 |
by (simp add: monofun_def sconc_mono) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
998 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
999 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1000 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1001 |
section "constr_sconc" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1002 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1003 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1004 |
lemma constr_sconc_UUs [simp]: "constr_sconc UU s = s" |
43919 | 1005 |
by (simp add: constr_sconc_def zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1006 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1007 |
lemma "x ooo y = constr_sconc x y" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1008 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1009 |
apply (rule stream_finite_ind [of x],auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1010 |
defer 1 |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1011 |
apply (simp add: constr_sconc_def del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1012 |
apply (case_tac "#s") |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
1013 |
apply (simp add: eSuc_enat) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1014 |
apply (case_tac "a=UU",auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1015 |
apply (simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1016 |
apply (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1017 |
apply (simp add: constr_sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1018 |
apply (simp add: stream.finite_def) |
49521 | 1019 |
apply (drule slen_take_lemma1,auto) |
1020 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1021 |
|
2570 | 1022 |
end |