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word-lib: add upward cast monotonicity lemmata

This commit is contained in:
Zoltan Kocsis 2019-12-12 14:52:38 +11:00
parent f2d1f5ada7
commit 43fc7e26d8
1 changed files with 40 additions and 0 deletions
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40
lib/Word_Lib/Word_Lemmas.thy
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@ -1436,6 +1436,13 @@ lemma unat_less_helper:
apply (simp add: unat_of_nat)
done
lemma nat_uint_less_helper:
"nat (uint y) = z \<Longrightarrow> x < y \<Longrightarrow> nat (uint x) < z"
apply (erule subst)
apply (subst unat_def [symmetric])
apply (subst unat_def [symmetric])
by (simp add: unat_mono)
lemma of_nat_0:
"\<lbrakk>of_nat n = (0::'a::len word); n < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> n = 0"
by (drule unat_of_nat_eq, simp)
@ -3433,6 +3440,39 @@ lemma ucast_mono_le':
\<Longrightarrow> UCAST('a \<rightarrow> 'b) x \<le> UCAST('a \<rightarrow> 'b) y"
by (auto simp: word_less_nat_alt intro: ucast_mono_le)
lemma ucast_up_mono:
"LENGTH('a :: len) \<le> LENGTH('b :: len) \<Longrightarrow> x < y \<Longrightarrow> UCAST('a \<rightarrow> 'b) x < UCAST('a \<rightarrow> 'b) y"
apply (simp add: ucast_nat_def [symmetric])
apply (rule of_nat_mono_maybe)
apply (subgoal_tac "unat y < 2 ^ LENGTH('a)")
apply (subgoal_tac "2 ^ LENGTH('a) \<le> 2 ^ LENGTH('b)")
apply (erule order_class.order.strict_trans2)
apply assumption
apply (erule power_increasing, simp)
apply (rule unat_lt2p)
apply (erule unat_mono)
done
lemma ucast_up_mono_le:
"LENGTH('a :: len) \<le> LENGTH('b :: len) \<Longrightarrow> x \<le> y \<Longrightarrow> UCAST('a \<rightarrow> 'b) x \<le> UCAST('a \<rightarrow> 'b) y"
apply (simp add: ucast_nat_def [symmetric])
apply (subst of_nat_mono_maybe_le[symmetric])
apply (subgoal_tac "unat y < 2 ^ LENGTH('a)")
apply (subgoal_tac "2 ^ LENGTH('a) \<le> 2 ^ LENGTH('b)")
apply (erule order_class.order.strict_trans2)
apply assumption
apply (erule power_increasing, simp)
apply (rule unat_lt2p)
apply (subgoal_tac "unat x < 2 ^ LENGTH('a)")
apply (subgoal_tac "2 ^ LENGTH('a) \<le> 2 ^ LENGTH('b)")
apply (erule order_class.order.strict_trans2)
apply assumption
apply (erule power_increasing, simp)
apply (rule unat_lt2p)
apply (subst word_le_nat_alt[symmetric])
apply assumption
done
lemma zero_sle_ucast_up:
"\<not> is_down (ucast :: 'a word \<Rightarrow> 'b signed word) \<Longrightarrow>
(0 <=s ((ucast (b::('a::len) word)) :: ('b::len) signed word))"