author | haftmann |
Tue, 05 Jun 2007 19:23:09 +0200 | |
changeset 23263 | 0c227412b285 |
parent 23247 | b99dce43d252 |
child 23417 | 42c1a89b45c1 |
permissions | -rw-r--r-- |
15524 | 1 |
(* Title: HOL/Orderings.thy |
2 |
ID: $Id$ |
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson |
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*) |
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||
21329 | 6 |
header {* Syntactic and abstract orders *} |
15524 | 7 |
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theory Orderings |
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23247 | 9 |
imports HOL |
23263 | 10 |
uses |
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(*"~~/src/Provers/quasi.ML"*) |
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"~~/src/Provers/order.ML" |
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begin |
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||
21329 | 15 |
subsection {* Order syntax *} |
15524 | 16 |
|
22473 | 17 |
class ord = type + |
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) |
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parents:
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50) |
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begin |
21 |
||
22 |
notation |
|
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less_eq ("op \<^loc><=") and |
21620 | 24 |
less_eq ("(_/ \<^loc><= _)" [51, 51] 50) and |
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restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
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diff
changeset
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25 |
less ("op \<^loc><") and |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
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less ("(_/ \<^loc>< _)" [51, 51] 50) |
21620 | 27 |
|
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notation (xsymbols) |
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less_eq ("op \<^loc>\<le>") and |
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less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50) |
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|
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notation (HTML output) |
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parents:
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less_eq ("op \<^loc>\<le>") and |
21259
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modified less/less_eq syntax to avoid "x < y < z" etc.;
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parents:
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diff
changeset
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less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50) |
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abbreviation (input) |
|
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greater (infix "\<^loc>>" 50) where |
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"x \<^loc>> y \<equiv> y \<^loc>< x" |
39 |
||
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parents:
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40 |
abbreviation (input) |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
41 |
greater_eq (infix "\<^loc>>=" 50) where |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
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"x \<^loc>>= y \<equiv> y \<^loc><= x" |
21204 | 43 |
|
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43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
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notation (input) |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
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diff
changeset
|
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greater_eq (infix "\<^loc>\<ge>" 50) |
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text {* |
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syntactic min/max -- these definitions reach |
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their usual semantics in class linorder ahead. |
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*} |
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||
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definition |
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min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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"min a b = (if a \<^loc>\<le> b then a else b)" |
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definition |
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max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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22841 | 58 |
"max a b = (if a \<^loc>\<le> b then b else a)" |
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|
21204 | 60 |
end |
61 |
||
62 |
notation |
|
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restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
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changeset
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less_eq ("op <=") and |
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less_eq ("(_/ <= _)" [51, 51] 50) and |
21656
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
65 |
less ("op <") and |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
66 |
less ("(_/ < _)" [51, 51] 50) |
21204 | 67 |
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notation (xsymbols) |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21383
diff
changeset
|
69 |
less_eq ("op \<le>") and |
21259
63ab016c99ca
modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents:
21248
diff
changeset
|
70 |
less_eq ("(_/ \<le> _)" [51, 51] 50) |
15524 | 71 |
|
21204 | 72 |
notation (HTML output) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21383
diff
changeset
|
73 |
less_eq ("op \<le>") and |
21259
63ab016c99ca
modified less/less_eq syntax to avoid "x < y < z" etc.;
wenzelm
parents:
21248
diff
changeset
|
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
20714 | 75 |
|
19536 | 76 |
abbreviation (input) |
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restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
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greater (infix ">" 50) where |
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"x > y \<equiv> y < x" |
79 |
||
21656
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
80 |
abbreviation (input) |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
81 |
greater_eq (infix ">=" 50) where |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
82 |
"x >= y \<equiv> y <= x" |
21620 | 83 |
|
21656
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
84 |
notation (input) |
43d709faa9dc
restored notation for less/less_eq (observe proper order of mixfix annotations!);
wenzelm
parents:
21620
diff
changeset
|
85 |
greater_eq (infix "\<ge>" 50) |
15524 | 86 |
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23087 | 87 |
lemmas min_def [code func, code unfold, code inline del] = min_def [folded ord_class.min] |
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lemmas max_def [code func, code unfold, code inline del] = max_def [folded ord_class.max] |
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subsection {* Partial orders *} |
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class order = ord + |
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assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y" |
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and order_refl [iff]: "x \<sqsubseteq> x" |
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and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z" |
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assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y" |
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begin |
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||
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text {* Reflexivity. *} |
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||
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lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y" |
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-- {* This form is useful with the classical reasoner. *} |
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by (erule ssubst) (rule order_refl) |
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|
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lemma less_irrefl [iff]: "\<not> x \<^loc>< x" |
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by (simp add: less_le) |
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lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y" |
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-- {* NOT suitable for iff, since it can cause PROOF FAILED. *} |
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by (simp add: less_le) blast |
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|
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lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y" |
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unfolding less_le by blast |
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lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y" |
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unfolding less_le by blast |
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lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y" |
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by (erule contrapos_pn, erule subst, rule less_irrefl) |
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text {* Useful for simplification, but too risky to include by default. *} |
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lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
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by auto |
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lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False" |
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by auto |
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text {* Transitivity rules for calculational reasoning *} |
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||
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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b" |
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by (simp add: less_le) |
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|
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lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b" |
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by (simp add: less_le) |
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text {* Asymmetry. *} |
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||
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lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)" |
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by (simp add: less_le antisym) |
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|
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lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P" |
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by (drule less_not_sym, erule contrapos_np) simp |
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|
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lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x" |
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by (blast intro: antisym) |
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|
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lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y" |
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by (blast intro: antisym) |
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|
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lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y" |
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by (erule contrapos_pn, erule subst, rule less_irrefl) |
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text {* Transitivity. *} |
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||
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lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z" |
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by (simp add: less_le) (blast intro: order_trans antisym) |
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|
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lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z" |
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by (simp add: less_le) (blast intro: order_trans antisym) |
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|
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lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z" |
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by (simp add: less_le) (blast intro: order_trans antisym) |
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text {* Useful for simplification, but too risky to include by default. *} |
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||
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lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
15524 | 176 |
|
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lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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|
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text {* Transitivity rules for calculational reasoning *} |
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|
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lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P" |
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by (rule less_asym) |
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|
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text {* Reverse order *} |
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||
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lemma order_reverse: |
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"order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)" |
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by unfold_locales |
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(simp add: less_le, auto intro: antisym order_trans) |
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|
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end |
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subsection {* Linear (total) orders *} |
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||
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class linorder = order + |
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200 |
assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
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begin |
202 |
||
22841 | 203 |
lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x" |
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unfolding less_le using less_le linear by blast |
21248 | 205 |
|
22841 | 206 |
lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x" |
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by (simp add: le_less less_linear) |
21248 | 208 |
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lemma le_cases [case_names le ge]: |
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22841 | 210 |
"(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P" |
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using linear by blast |
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lemma linorder_cases [case_names less equal greater]: |
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"(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P" |
215 |
using less_linear by blast |
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21248 | 216 |
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22841 | 217 |
lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x" |
23212 | 218 |
apply (simp add: less_le) |
219 |
using linear apply (blast intro: antisym) |
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220 |
done |
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221 |
||
222 |
lemma not_less_iff_gr_or_eq: |
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223 |
"\<not>(x \<^loc>< y) \<longleftrightarrow> (x \<^loc>> y | x = y)" |
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apply(simp add:not_less le_less) |
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225 |
apply blast |
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226 |
done |
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|
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lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x" |
23212 | 229 |
apply (simp add: less_le) |
230 |
using linear apply (blast intro: antisym) |
|
231 |
done |
|
15524 | 232 |
|
22841 | 233 |
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x" |
23212 | 234 |
by (cut_tac x = x and y = y in less_linear, auto) |
15524 | 235 |
|
22841 | 236 |
lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R" |
23212 | 237 |
by (simp add: neq_iff) blast |
15524 | 238 |
|
22841 | 239 |
lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y" |
23212 | 240 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 241 |
|
22841 | 242 |
lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y" |
23212 | 243 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 244 |
|
22841 | 245 |
lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y" |
23212 | 246 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 247 |
|
16796 | 248 |
text{*Replacing the old Nat.leI*} |
22841 | 249 |
lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x" |
23212 | 250 |
unfolding not_less . |
16796 | 251 |
|
22841 | 252 |
lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y" |
23212 | 253 |
unfolding not_less . |
16796 | 254 |
|
255 |
(*FIXME inappropriate name (or delete altogether)*) |
|
22841 | 256 |
lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y" |
23212 | 257 |
unfolding not_le . |
21248 | 258 |
|
22916 | 259 |
|
260 |
text {* Reverse order *} |
|
261 |
||
262 |
lemma linorder_reverse: |
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263 |
"linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)" |
23212 | 264 |
by unfold_locales |
265 |
(simp add: less_le, auto intro: antisym order_trans simp add: linear) |
|
22916 | 266 |
|
267 |
||
22738 | 268 |
text {* min/max properties *} |
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269 |
|
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270 |
lemma min_le_iff_disj: |
22841 | 271 |
"min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z" |
23212 | 272 |
unfolding min_def using linear by (auto intro: order_trans) |
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273 |
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274 |
lemma le_max_iff_disj: |
22841 | 275 |
"z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y" |
23212 | 276 |
unfolding max_def using linear by (auto intro: order_trans) |
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277 |
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278 |
lemma min_less_iff_disj: |
22841 | 279 |
"min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z" |
23212 | 280 |
unfolding min_def le_less using less_linear by (auto intro: less_trans) |
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281 |
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282 |
lemma less_max_iff_disj: |
22841 | 283 |
"z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y" |
23212 | 284 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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285 |
|
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|
286 |
lemma min_less_iff_conj [simp]: |
22841 | 287 |
"z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y" |
23212 | 288 |
unfolding min_def le_less using less_linear by (auto intro: less_trans) |
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|
289 |
|
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|
290 |
lemma max_less_iff_conj [simp]: |
22841 | 291 |
"max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z" |
23212 | 292 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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|
293 |
|
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|
294 |
lemma split_min: |
22841 | 295 |
"P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)" |
23212 | 296 |
by (simp add: min_def) |
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|
297 |
|
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|
298 |
lemma split_max: |
22841 | 299 |
"P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)" |
23212 | 300 |
by (simp add: max_def) |
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changeset
|
301 |
|
21248 | 302 |
end |
303 |
||
22916 | 304 |
subsection {* Name duplicates -- including min/max interpretation *} |
21248 | 305 |
|
22384
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|
306 |
lemmas order_less_le = less_le |
22841 | 307 |
lemmas order_eq_refl = order_class.eq_refl |
308 |
lemmas order_less_irrefl = order_class.less_irrefl |
|
309 |
lemmas order_le_less = order_class.le_less |
|
310 |
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq |
|
311 |
lemmas order_less_imp_le = order_class.less_imp_le |
|
312 |
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq |
|
313 |
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 |
|
314 |
lemmas order_neq_le_trans = order_class.neq_le_trans |
|
315 |
lemmas order_le_neq_trans = order_class.le_neq_trans |
|
22316 | 316 |
|
22384
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|
317 |
lemmas order_antisym = antisym |
22316 | 318 |
lemmas order_less_not_sym = order_class.less_not_sym |
319 |
lemmas order_less_asym = order_class.less_asym |
|
320 |
lemmas order_eq_iff = order_class.eq_iff |
|
321 |
lemmas order_antisym_conv = order_class.antisym_conv |
|
322 |
lemmas order_less_trans = order_class.less_trans |
|
323 |
lemmas order_le_less_trans = order_class.le_less_trans |
|
324 |
lemmas order_less_le_trans = order_class.less_le_trans |
|
325 |
lemmas order_less_imp_not_less = order_class.less_imp_not_less |
|
326 |
lemmas order_less_imp_triv = order_class.less_imp_triv |
|
327 |
lemmas order_less_asym' = order_class.less_asym' |
|
328 |
||
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diff
changeset
|
329 |
lemmas linorder_linear = linear |
22316 | 330 |
lemmas linorder_less_linear = linorder_class.less_linear |
331 |
lemmas linorder_le_less_linear = linorder_class.le_less_linear |
|
332 |
lemmas linorder_le_cases = linorder_class.le_cases |
|
333 |
lemmas linorder_not_less = linorder_class.not_less |
|
334 |
lemmas linorder_not_le = linorder_class.not_le |
|
335 |
lemmas linorder_neq_iff = linorder_class.neq_iff |
|
336 |
lemmas linorder_neqE = linorder_class.neqE |
|
337 |
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 |
|
338 |
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 |
|
339 |
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 |
|
16796 | 340 |
|
23087 | 341 |
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [folded ord_class.min] |
342 |
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [folded ord_class.max] |
|
343 |
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [folded ord_class.min] |
|
344 |
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [folded ord_class.max] |
|
345 |
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [folded ord_class.min] |
|
346 |
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [folded ord_class.max] |
|
347 |
lemmas split_min = linorder_class.split_min [folded ord_class.min] |
|
348 |
lemmas split_max = linorder_class.split_max [folded ord_class.max] |
|
22916 | 349 |
|
21083 | 350 |
|
351 |
subsection {* Reasoning tools setup *} |
|
352 |
||
21091 | 353 |
ML {* |
354 |
local |
|
355 |
||
356 |
fun decomp_gen sort thy (Trueprop $ t) = |
|
21248 | 357 |
let |
358 |
fun of_sort t = |
|
359 |
let |
|
360 |
val T = type_of t |
|
361 |
in |
|
21091 | 362 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
21248 | 363 |
T <> HOLogic.natT andalso T <> HOLogic.intT |
364 |
andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort) |
|
365 |
end; |
|
22916 | 366 |
fun dec (Const (@{const_name Not}, _) $ t) = (case dec t |
21248 | 367 |
of NONE => NONE |
368 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
|
22916 | 369 |
| dec (Const (@{const_name "op ="}, _) $ t1 $ t2) = |
21248 | 370 |
if of_sort t1 |
371 |
then SOME (t1, "=", t2) |
|
372 |
else NONE |
|
22997 | 373 |
| dec (Const (@{const_name Orderings.less_eq}, _) $ t1 $ t2) = |
21248 | 374 |
if of_sort t1 |
375 |
then SOME (t1, "<=", t2) |
|
376 |
else NONE |
|
22997 | 377 |
| dec (Const (@{const_name Orderings.less}, _) $ t1 $ t2) = |
21248 | 378 |
if of_sort t1 |
379 |
then SOME (t1, "<", t2) |
|
380 |
else NONE |
|
381 |
| dec _ = NONE; |
|
21091 | 382 |
in dec t end; |
383 |
||
384 |
in |
|
385 |
||
22841 | 386 |
(* sorry - there is no preorder class |
21248 | 387 |
structure Quasi_Tac = Quasi_Tac_Fun ( |
388 |
struct |
|
389 |
val le_trans = thm "order_trans"; |
|
390 |
val le_refl = thm "order_refl"; |
|
391 |
val eqD1 = thm "order_eq_refl"; |
|
392 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
393 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
394 |
val less_imp_le = thm "order_less_imp_le"; |
|
395 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
396 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
397 |
val less_imp_neq = thm "less_imp_neq"; |
|
22738 | 398 |
val decomp_trans = decomp_gen ["Orderings.preorder"]; |
399 |
val decomp_quasi = decomp_gen ["Orderings.preorder"]; |
|
22841 | 400 |
end);*) |
21091 | 401 |
|
402 |
structure Order_Tac = Order_Tac_Fun ( |
|
21248 | 403 |
struct |
404 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
405 |
val le_refl = thm "order_refl"; |
|
406 |
val less_imp_le = thm "order_less_imp_le"; |
|
407 |
val not_lessI = thm "linorder_not_less" RS thm "iffD2"; |
|
408 |
val not_leI = thm "linorder_not_le" RS thm "iffD2"; |
|
409 |
val not_lessD = thm "linorder_not_less" RS thm "iffD1"; |
|
410 |
val not_leD = thm "linorder_not_le" RS thm "iffD1"; |
|
411 |
val eqI = thm "order_antisym"; |
|
412 |
val eqD1 = thm "order_eq_refl"; |
|
413 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
414 |
val less_trans = thm "order_less_trans"; |
|
415 |
val less_le_trans = thm "order_less_le_trans"; |
|
416 |
val le_less_trans = thm "order_le_less_trans"; |
|
417 |
val le_trans = thm "order_trans"; |
|
418 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
419 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
420 |
val less_imp_neq = thm "less_imp_neq"; |
|
421 |
val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq"; |
|
422 |
val not_sym = thm "not_sym"; |
|
423 |
val decomp_part = decomp_gen ["Orderings.order"]; |
|
424 |
val decomp_lin = decomp_gen ["Orderings.linorder"]; |
|
425 |
end); |
|
21091 | 426 |
|
427 |
end; |
|
428 |
*} |
|
429 |
||
21083 | 430 |
setup {* |
431 |
let |
|
432 |
||
433 |
fun prp t thm = (#prop (rep_thm thm) = t); |
|
15524 | 434 |
|
21083 | 435 |
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = |
436 |
let val prems = prems_of_ss ss; |
|
22916 | 437 |
val less = Const (@{const_name less}, T); |
21083 | 438 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
439 |
in case find_first (prp t) prems of |
|
440 |
NONE => |
|
441 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) |
|
442 |
in case find_first (prp t) prems of |
|
443 |
NONE => NONE |
|
22738 | 444 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1})) |
21083 | 445 |
end |
22738 | 446 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv})) |
21083 | 447 |
end |
448 |
handle THM _ => NONE; |
|
15524 | 449 |
|
21083 | 450 |
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = |
451 |
let val prems = prems_of_ss ss; |
|
22916 | 452 |
val le = Const (@{const_name less_eq}, T); |
21083 | 453 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
454 |
in case find_first (prp t) prems of |
|
455 |
NONE => |
|
456 |
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) |
|
457 |
in case find_first (prp t) prems of |
|
458 |
NONE => NONE |
|
22738 | 459 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3})) |
21083 | 460 |
end |
22738 | 461 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2})) |
21083 | 462 |
end |
463 |
handle THM _ => NONE; |
|
15524 | 464 |
|
21248 | 465 |
fun add_simprocs procs thy = |
466 |
(Simplifier.change_simpset_of thy (fn ss => ss |
|
467 |
addsimprocs (map (fn (name, raw_ts, proc) => |
|
468 |
Simplifier.simproc thy name raw_ts proc)) procs); thy); |
|
469 |
fun add_solver name tac thy = |
|
470 |
(Simplifier.change_simpset_of thy (fn ss => ss addSolver |
|
471 |
(mk_solver name (K tac))); thy); |
|
21083 | 472 |
|
473 |
in |
|
21248 | 474 |
add_simprocs [ |
475 |
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), |
|
476 |
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) |
|
477 |
] |
|
478 |
#> add_solver "Trans_linear" Order_Tac.linear_tac |
|
479 |
#> add_solver "Trans_partial" Order_Tac.partial_tac |
|
480 |
(* Adding the transitivity reasoners also as safe solvers showed a slight |
|
481 |
speed up, but the reasoning strength appears to be not higher (at least |
|
482 |
no breaking of additional proofs in the entire HOL distribution, as |
|
483 |
of 5 March 2004, was observed). *) |
|
21083 | 484 |
end |
485 |
*} |
|
15524 | 486 |
|
487 |
||
21083 | 488 |
subsection {* Bounded quantifiers *} |
489 |
||
490 |
syntax |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
491 |
"_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
492 |
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
493 |
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
494 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 495 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
496 |
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
497 |
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
498 |
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
499 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 500 |
|
501 |
syntax (xsymbols) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
502 |
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
503 |
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
504 |
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
505 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 506 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
507 |
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
508 |
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
509 |
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
510 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 511 |
|
512 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
513 |
"_All_less" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
514 |
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
515 |
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
516 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 517 |
|
518 |
syntax (HTML output) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
519 |
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
520 |
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
521 |
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
522 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 523 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
524 |
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
525 |
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
526 |
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
527 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 528 |
|
529 |
translations |
|
530 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
531 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
532 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
533 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
534 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
535 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
536 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
537 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
538 |
||
539 |
print_translation {* |
|
540 |
let |
|
22916 | 541 |
val All_binder = Syntax.binder_name @{const_syntax All}; |
542 |
val Ex_binder = Syntax.binder_name @{const_syntax Ex}; |
|
22377 | 543 |
val impl = @{const_syntax "op -->"}; |
544 |
val conj = @{const_syntax "op &"}; |
|
22916 | 545 |
val less = @{const_syntax less}; |
546 |
val less_eq = @{const_syntax less_eq}; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
547 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
548 |
val trans = |
21524 | 549 |
[((All_binder, impl, less), ("_All_less", "_All_greater")), |
550 |
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), |
|
551 |
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), |
|
552 |
((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))]; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
553 |
|
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
554 |
fun matches_bound v t = |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
555 |
case t of (Const ("_bound", _) $ Free (v', _)) => (v = v') |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
556 |
| _ => false |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
557 |
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false) |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
558 |
fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
559 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
560 |
fun tr' q = (q, |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
561 |
fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] => |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
562 |
(case AList.lookup (op =) trans (q, c, d) of |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
563 |
NONE => raise Match |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
564 |
| SOME (l, g) => |
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
565 |
if matches_bound v t andalso not (contains_var v u) then mk v l u P |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
566 |
else if matches_bound v u andalso not (contains_var v t) then mk v g t P |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
567 |
else raise Match) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
568 |
| _ => raise Match); |
21524 | 569 |
in [tr' All_binder, tr' Ex_binder] end |
21083 | 570 |
*} |
571 |
||
572 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
573 |
subsection {* Transitivity reasoning *} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
574 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
575 |
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c" |
23212 | 576 |
by (rule subst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
577 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
578 |
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c" |
23212 | 579 |
by (rule ssubst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
580 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
581 |
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" |
23212 | 582 |
by (rule subst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
583 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
584 |
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" |
23212 | 585 |
by (rule ssubst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
586 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
587 |
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
588 |
(!!x y. x < y ==> f x < f y) ==> f a < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
589 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
590 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
591 |
assume "a < b" hence "f a < f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
592 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
593 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
594 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
595 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
596 |
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
597 |
(!!x y. x < y ==> f x < f y) ==> a < f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
598 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
599 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
600 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
601 |
also assume "b < c" hence "f b < f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
602 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
603 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
604 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
605 |
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
606 |
(!!x y. x <= y ==> f x <= f y) ==> f a < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
607 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
608 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
609 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
610 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
611 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
612 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
613 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
614 |
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
615 |
(!!x y. x < y ==> f x < f y) ==> a < f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
616 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
617 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
618 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
619 |
also assume "b < c" hence "f b < f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
620 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
621 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
622 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
623 |
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
624 |
(!!x y. x < y ==> f x < f y) ==> f a < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
625 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
626 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
627 |
assume "a < b" hence "f a < f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
628 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
629 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
630 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
631 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
632 |
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
633 |
(!!x y. x <= y ==> f x <= f y) ==> a < f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
634 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
635 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
636 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
637 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
638 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
639 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
640 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
641 |
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
642 |
(!!x y. x <= y ==> f x <= f y) ==> a <= f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
643 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
644 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
645 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
646 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
647 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
648 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
649 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
650 |
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
651 |
(!!x y. x <= y ==> f x <= f y) ==> f a <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
652 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
653 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
654 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
655 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
656 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
657 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
658 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
659 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
660 |
(!!x y. x <= y ==> f x <= f y) ==> f a <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
661 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
662 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
663 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
664 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
665 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
666 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
667 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
668 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
669 |
(!!x y. x <= y ==> f x <= f y) ==> a <= f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
670 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
671 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
672 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
673 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
674 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
675 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
676 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
677 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
678 |
(!!x y. x < y ==> f x < f y) ==> f a < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
679 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
680 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
681 |
assume "a < b" hence "f a < f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
682 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
683 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
684 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
685 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
686 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
687 |
(!!x y. x < y ==> f x < f y) ==> a < f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
688 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
689 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
690 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
691 |
also assume "b < c" hence "f b < f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
692 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
693 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
694 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
695 |
text {* |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
696 |
Note that this list of rules is in reverse order of priorities. |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
697 |
*} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
698 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
699 |
lemmas order_trans_rules [trans] = |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
700 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
701 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
702 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
703 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
704 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
705 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
706 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
707 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
708 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
709 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
710 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
711 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
712 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
713 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
714 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
715 |
mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
716 |
order_neq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
717 |
order_le_neq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
718 |
order_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
719 |
order_less_asym' |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
720 |
order_le_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
721 |
order_less_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
722 |
order_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
723 |
order_antisym |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
724 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
725 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
726 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
727 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
728 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
729 |
|
21083 | 730 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
731 |
(* FIXME cleanup *) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
732 |
|
21083 | 733 |
text {* These support proving chains of decreasing inequalities |
734 |
a >= b >= c ... in Isar proofs. *} |
|
735 |
||
736 |
lemma xt1: |
|
737 |
"a = b ==> b > c ==> a > c" |
|
738 |
"a > b ==> b = c ==> a > c" |
|
739 |
"a = b ==> b >= c ==> a >= c" |
|
740 |
"a >= b ==> b = c ==> a >= c" |
|
741 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
742 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
743 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
744 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
745 |
"(a::'a::order) > b ==> b > a ==> ?P" |
|
746 |
"(x::'a::order) > y ==> y > z ==> x > z" |
|
747 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
748 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
749 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
|
750 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
|
751 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
752 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
753 |
by auto |
|
754 |
||
755 |
lemma xt2: |
|
756 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
757 |
by (subgoal_tac "f b >= f c", force, force) |
|
758 |
||
759 |
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
|
760 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
761 |
by (subgoal_tac "f a >= f b", force, force) |
|
762 |
||
763 |
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
|
764 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
|
765 |
by (subgoal_tac "f b >= f c", force, force) |
|
766 |
||
767 |
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
|
768 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
769 |
by (subgoal_tac "f a > f b", force, force) |
|
770 |
||
771 |
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> |
|
772 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
773 |
by (subgoal_tac "f b > f c", force, force) |
|
774 |
||
775 |
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
|
776 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
|
777 |
by (subgoal_tac "f a >= f b", force, force) |
|
778 |
||
779 |
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
|
780 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
781 |
by (subgoal_tac "f b > f c", force, force) |
|
782 |
||
783 |
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
|
784 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
785 |
by (subgoal_tac "f a > f b", force, force) |
|
786 |
||
787 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
|
788 |
||
789 |
(* |
|
790 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
|
791 |
for the wrong thing in an Isar proof. |
|
792 |
||
793 |
The extra transitivity rules can be used as follows: |
|
794 |
||
795 |
lemma "(a::'a::order) > z" |
|
796 |
proof - |
|
797 |
have "a >= b" (is "_ >= ?rhs") |
|
798 |
sorry |
|
799 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
800 |
sorry |
|
801 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
802 |
sorry |
|
803 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
804 |
sorry |
|
805 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
806 |
sorry |
|
807 |
also (xtrans) have "?rhs > z" |
|
808 |
sorry |
|
809 |
finally (xtrans) show ?thesis . |
|
810 |
qed |
|
811 |
||
812 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
813 |
leave out the "(xtrans)" above. |
|
814 |
*) |
|
815 |
||
21546 | 816 |
subsection {* Order on bool *} |
817 |
||
22886 | 818 |
instance bool :: order |
21546 | 819 |
le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q" |
820 |
less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q" |
|
22916 | 821 |
by intro_classes (auto simp add: le_bool_def less_bool_def) |
21546 | 822 |
|
823 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
23212 | 824 |
by (simp add: le_bool_def) |
21546 | 825 |
|
826 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
23212 | 827 |
by (simp add: le_bool_def) |
21546 | 828 |
|
829 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
23212 | 830 |
by (simp add: le_bool_def) |
21546 | 831 |
|
832 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
23212 | 833 |
by (simp add: le_bool_def) |
21546 | 834 |
|
22348 | 835 |
lemma [code func]: |
836 |
"False \<le> b \<longleftrightarrow> True" |
|
837 |
"True \<le> b \<longleftrightarrow> b" |
|
838 |
"False < b \<longleftrightarrow> b" |
|
839 |
"True < b \<longleftrightarrow> False" |
|
840 |
unfolding le_bool_def less_bool_def by simp_all |
|
841 |
||
22424 | 842 |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
843 |
subsection {* Monotonicity, syntactic least value operator and min/max *} |
21083 | 844 |
|
21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
845 |
locale mono = |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
846 |
fixes f |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
847 |
assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B" |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
848 |
|
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
849 |
lemmas monoI [intro?] = mono.intro |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
850 |
and monoD [dest?] = mono.mono |
21083 | 851 |
|
852 |
constdefs |
|
853 |
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10) |
|
854 |
"Least P == THE x. P x & (ALL y. P y --> x <= y)" |
|
855 |
-- {* We can no longer use LeastM because the latter requires Hilbert-AC. *} |
|
856 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
857 |
lemma LeastI2_order: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
858 |
"[| P (x::'a::order); |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
859 |
!!y. P y ==> x <= y; |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
860 |
!!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
861 |
==> Q (Least P)" |
23212 | 862 |
apply (unfold Least_def) |
863 |
apply (rule theI2) |
|
864 |
apply (blast intro: order_antisym)+ |
|
865 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
866 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
867 |
lemma Least_equality: |
23212 | 868 |
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k" |
869 |
apply (simp add: Least_def) |
|
870 |
apply (rule the_equality) |
|
871 |
apply (auto intro!: order_antisym) |
|
872 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
873 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
874 |
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" |
23212 | 875 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
876 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
877 |
lemma max_leastL: "(!!x. least <= x) ==> max least x = x" |
23212 | 878 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
879 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
880 |
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least" |
23212 | 881 |
apply (simp add: min_def) |
882 |
apply (blast intro: order_antisym) |
|
883 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
884 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
885 |
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x" |
23212 | 886 |
apply (simp add: max_def) |
887 |
apply (blast intro: order_antisym) |
|
888 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
889 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
890 |
lemma min_of_mono: |
23212 | 891 |
"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)" |
892 |
by (simp add: min_def) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
893 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
894 |
lemma max_of_mono: |
23212 | 895 |
"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)" |
896 |
by (simp add: max_def) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
897 |
|
22548 | 898 |
|
899 |
subsection {* legacy ML bindings *} |
|
21673 | 900 |
|
901 |
ML {* |
|
22548 | 902 |
val monoI = @{thm monoI}; |
22886 | 903 |
*} |
21673 | 904 |
|
15524 | 905 |
end |