session stringclasses 1
value | dependency listlengths 0 0 | context stringlengths 31 38.1k | proof stringlengths 31 38.1k | proof_state stringlengths 38 24.6M | statement stringlengths 22 5.17k | name stringlengths 13 17 | theory_name stringclasses 518
values | num_steps int64 1 963 |
|---|---|---|---|---|---|---|---|---|
[] | lemma mask_over_length:
"LENGTH('a) \<le> n \<Longrightarrow> mask n = (-1::'a::len word)" by (simp add: mask_eq_decr_exp) | lemma mask_over_length:
"LENGTH('a) \<le> n \<Longrightarrow> mask n = (-1::'a::len word)" by (simp add: mask_eq_decr_exp) | proof (prove)
goal (1 subgoal):
1. LENGTH('a) \<le> n \<Longrightarrow> mask n = - 1 | lemma mask_over_length:
"LENGTH('a) \<le> n \<Longrightarrow> mask n = (-1::'a::len word)" | unnamed_thy_213 | More_Word | 1 | |
[] | lemma Suc_2p_unat_mask:
"n < LENGTH('a) \<Longrightarrow> Suc (2 ^ n * k + unat (mask n :: 'a::len word)) = 2 ^ n * (k+1)" by (simp add: unat_mask) | lemma Suc_2p_unat_mask:
"n < LENGTH('a) \<Longrightarrow> Suc (2 ^ n * k + unat (mask n :: 'a::len word)) = 2 ^ n * (k+1)" by (simp add: unat_mask) | proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> Suc (2 ^ n * k + unat (mask n)) = 2 ^ n * (k + 1) | lemma Suc_2p_unat_mask:
"n < LENGTH('a) \<Longrightarrow> Suc (2 ^ n * k + unat (mask n :: 'a::len word)) = 2 ^ n * (k+1)" | unnamed_thy_214 | More_Word | 1 | |
[] | lemma sint_of_nat_ge_zero:
"x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) \<ge> 0" by (simp add: bit_iff_odd signed_of_nat) | lemma sint_of_nat_ge_zero:
"x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) \<ge> 0" by (simp add: bit_iff_odd signed_of_nat) | proof (prove)
goal (1 subgoal):
1. x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> 0 \<le> sint (word_of_nat x) | lemma sint_of_nat_ge_zero:
"x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) \<ge> 0" | unnamed_thy_215 | More_Word | 1 | |
[] | lemma int_eq_sint:
"x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) = int x" apply transfer apply (rule signed_take_bit_int_eq_self) apply simp_all apply (metis negative_zle numeral_power_eq_of_nat_cancel_iff) done | lemma int_eq_sint:
"x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) = int x" apply transfer apply (rule signed_take_bit_int_eq_self) apply simp_all apply (metis negative_zle numeral_power_eq_of_nat_cancel_iff) done | proof (prove)
goal (1 subgoal):
1. x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (word_of_nat x) = int x proof (prove)
goal (1 subgoal):
1. \<And>x. x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> signed_take_bit (LENGTH('a) - Suc 0) (int x) = int x proof (prove)
goal (2 subgoals):
1. \<And>x. x < 2 ^ (LENGTH('a) - 1) ... | lemma int_eq_sint:
"x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) = int x" | unnamed_thy_216 | More_Word | 5 | |
[] | lemma sint_of_nat_le:
"\<lbrakk> b < 2 ^ (LENGTH('a) - 1); a \<le> b \<rbrakk>
\<Longrightarrow> sint (of_nat a :: 'a :: len word) \<le> sint (of_nat b :: 'a :: len word)" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply transfer apply (subst signed_take_bit_eq_if_positive) apply (simp add: bit_simps) ... | lemma sint_of_nat_le:
"\<lbrakk> b < 2 ^ (LENGTH('a) - 1); a \<le> b \<rbrakk>
\<Longrightarrow> sint (of_nat a :: 'a :: len word) \<le> sint (of_nat b :: 'a :: len word)" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply transfer apply (subst signed_take_bit_eq_if_positive) apply (simp add: bit_simps) ... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>b < 2 ^ (LENGTH('a) - 1); a \<le> b\<rbrakk> \<Longrightarrow> sint (word_of_nat a) \<le> sint (word_of_nat b) proof (prove)
goal (2 subgoals):
1. \<lbrakk>b < 2 ^ (LENGTH('a) - 1); a \<le> b; LENGTH('a) = 0\<rbrakk> \<Longrightarrow> sint (word_of_nat a) \<le> sint (word_o... | lemma sint_of_nat_le:
"\<lbrakk> b < 2 ^ (LENGTH('a) - 1); a \<le> b \<rbrakk>
\<Longrightarrow> sint (of_nat a :: 'a :: len word) \<le> sint (of_nat b :: 'a :: len word)" | unnamed_thy_217 | More_Word | 11 | |
[] | lemma word_le_not_less:
"((b::'a::len word) \<le> a) = (\<not>(a < b))" by fastforce | lemma word_le_not_less:
"((b::'a::len word) \<le> a) = (\<not>(a < b))" by fastforce | proof (prove)
goal (1 subgoal):
1. (b \<le> a) = (\<not> a < b) | lemma word_le_not_less:
"((b::'a::len word) \<le> a) = (\<not>(a < b))" | unnamed_thy_218 | More_Word | 1 | |
[] | lemma less_is_non_zero_p1:
fixes a :: "'a :: len word"
shows "a < k \<Longrightarrow> a + 1 \<noteq> 0" apply (erule contrapos_pn) apply (drule max_word_wrap) apply (simp add: not_less) done | lemma less_is_non_zero_p1:
fixes a :: "'a :: len word"
shows "a < k \<Longrightarrow> a + 1 \<noteq> 0" apply (erule contrapos_pn) apply (drule max_word_wrap) apply (simp add: not_less) done | proof (prove)
goal (1 subgoal):
1. a < k \<Longrightarrow> a + 1 \<noteq> 0 proof (prove)
goal (1 subgoal):
1. a + 1 = 0 \<Longrightarrow> \<not> a < k proof (prove)
goal (1 subgoal):
1. a = - 1 \<Longrightarrow> \<not> a < k proof (prove)
goal:
No subgoals! | lemma less_is_non_zero_p1:
fixes a :: "'a :: len word"
shows "a < k \<Longrightarrow> a + 1 \<noteq> 0" | unnamed_thy_219 | More_Word | 4 | |
[] | lemma unat_add_lem':
"(unat x + unat y < 2 ^ LENGTH('a)) \<Longrightarrow>
(unat (x + y :: 'a :: len word) = unat x + unat y)" by (subst unat_add_lem[symmetric], assumption) | lemma unat_add_lem':
"(unat x + unat y < 2 ^ LENGTH('a)) \<Longrightarrow>
(unat (x + y :: 'a :: len word) = unat x + unat y)" by (subst unat_add_lem[symmetric], assumption) | proof (prove)
goal (1 subgoal):
1. unat x + unat y < 2 ^ LENGTH('a) \<Longrightarrow> unat (x + y) = unat x + unat y | lemma unat_add_lem':
"(unat x + unat y < 2 ^ LENGTH('a)) \<Longrightarrow>
(unat (x + y :: 'a :: len word) = unat x + unat y)" | unnamed_thy_220 | More_Word | 1 | |
[] | lemma word_less_two_pow_divD:
"\<lbrakk> (x :: 'a::len word) < 2 ^ n div 2 ^ m \<rbrakk>
\<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))" apply (cases "n < LENGTH('a)") apply (cases "m < LENGTH('a)") apply (simp add: word_less_nat_alt) apply (subst(asm) unat_word_ariths) apply (subst(asm) mod_less) apply (r... | lemma word_less_two_pow_divD:
"\<lbrakk> (x :: 'a::len word) < 2 ^ n div 2 ^ m \<rbrakk>
\<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))" apply (cases "n < LENGTH('a)") apply (cases "m < LENGTH('a)") apply (simp add: word_less_nat_alt) apply (subst(asm) unat_word_ariths) apply (subst(asm) mod_less) apply (r... | proof (prove)
goal (1 subgoal):
1. x < 2 ^ n div 2 ^ m \<Longrightarrow> m \<le> n \<and> x < 2 ^ (n - m) proof (prove)
goal (2 subgoals):
1. \<lbrakk>x < 2 ^ n div 2 ^ m; n < LENGTH('a)\<rbrakk> \<Longrightarrow> m \<le> n \<and> x < 2 ^ (n - m)
2. \<lbrakk>x < 2 ^ n div 2 ^ m; \<not> n < LENGTH('a)\<rbrakk> \<Long... | lemma word_less_two_pow_divD:
"\<lbrakk> (x :: 'a::len word) < 2 ^ n div 2 ^ m \<rbrakk>
\<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))" | unnamed_thy_222 | More_Word | 12 | |
[] | lemma of_nat_less_two_pow_div_set:
"\<lbrakk> n < LENGTH('a) \<rbrakk> \<Longrightarrow>
{x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
= of_nat ` {k. k < 2 ^ n div 2 ^ m}" apply (simp add: image_def) apply (safe dest!: word_less_two_pow_divD less_two_pow_divD
intro!: word_less_two_pow_divI) apply (... | lemma of_nat_less_two_pow_div_set:
"\<lbrakk> n < LENGTH('a) \<rbrakk> \<Longrightarrow>
{x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
= of_nat ` {k. k < 2 ^ n div 2 ^ m}" apply (simp add: image_def) apply (safe dest!: word_less_two_pow_divD less_two_pow_divD
intro!: word_less_two_pow_divI) apply (... | proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> {x. x < 2 ^ n div 2 ^ m} = word_of_nat ` {k. k < 2 ^ n div 2 ^ m} proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> {x. x < 2 ^ n div 2 ^ m} = {y. \<exists>x<2 ^ n div 2 ^ m. y = word_of_nat x} proof (prove)
goal (2 subgoals):
1. \... | lemma of_nat_less_two_pow_div_set:
"\<lbrakk> n < LENGTH('a) \<rbrakk> \<Longrightarrow>
{x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
= of_nat ` {k. k < 2 ^ n div 2 ^ m}" | unnamed_thy_223 | More_Word | 14 | |
[] | lemma ucast_less:
"LENGTH('b) < LENGTH('a) \<Longrightarrow>
(ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)" by transfer simp | lemma ucast_less:
"LENGTH('b) < LENGTH('a) \<Longrightarrow>
(ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)" by transfer simp | proof (prove)
goal (1 subgoal):
1. LENGTH('b) < LENGTH('a) \<Longrightarrow> ucast x < 2 ^ LENGTH('b) | lemma ucast_less:
"LENGTH('b) < LENGTH('a) \<Longrightarrow>
(ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)" | unnamed_thy_224 | More_Word | 1 | |
[] | lemma ucast_range_less:
"LENGTH('a :: len) < LENGTH('b :: len) \<Longrightarrow>
range (ucast :: 'a word \<Rightarrow> 'b word) = {x. x < 2 ^ len_of TYPE ('a)}" apply safe apply (erule ucast_less) apply (simp add: image_def) apply (rule_tac x="ucast x" in exI) apply (rule bit_word_eqI) apply (auto simp add: bit_si... | lemma ucast_range_less:
"LENGTH('a :: len) < LENGTH('b :: len) \<Longrightarrow>
range (ucast :: 'a word \<Rightarrow> 'b word) = {x. x < 2 ^ len_of TYPE ('a)}" apply safe apply (erule ucast_less) apply (simp add: image_def) apply (rule_tac x="ucast x" in exI) apply (rule bit_word_eqI) apply (auto simp add: bit_si... | proof (prove)
goal (1 subgoal):
1. LENGTH('a) < LENGTH('b) \<Longrightarrow> range ucast = {x. x < 2 ^ LENGTH('a)} proof (prove)
goal (2 subgoals):
1. \<And>x xa. \<lbrakk>LENGTH('a) < LENGTH('b); xa \<in> UNIV\<rbrakk> \<Longrightarrow> ucast xa < 2 ^ LENGTH('a)
2. \<And>x. \<lbrakk>LENGTH('a) < LENGTH('b); x < 2 ^... | lemma ucast_range_less:
"LENGTH('a :: len) < LENGTH('b :: len) \<Longrightarrow>
range (ucast :: 'a word \<Rightarrow> 'b word) = {x. x < 2 ^ len_of TYPE ('a)}" | unnamed_thy_225 | More_Word | 8 | |
[] | lemma word_power_less_diff:
"\<lbrakk>2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (LENGTH('a) - n)\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n)" apply (case_tac "m \<ge> LENGTH('a)") apply (simp add: power_overflow) apply (case_tac "n \<ge> LENGTH('a)") apply (simp add: power_overflow) apply (cases "n = 0") apply simp a... | lemma word_power_less_diff:
"\<lbrakk>2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (LENGTH('a) - n)\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n)" apply (case_tac "m \<ge> LENGTH('a)") apply (simp add: power_overflow) apply (case_tac "n \<ge> LENGTH('a)") apply (simp add: power_overflow) apply (cases "n = 0") apply simp a... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>2 ^ n * q < 2 ^ m; q < 2 ^ (LENGTH('a) - n)\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n) proof (prove)
goal (2 subgoals):
1. \<lbrakk>2 ^ n * q < 2 ^ m; q < 2 ^ (LENGTH('a) - n); LENGTH('a) \<le> m\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n)
2. \<lbrakk>2 ^ n * q < 2 ^ m; q <... | lemma word_power_less_diff:
"\<lbrakk>2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (LENGTH('a) - n)\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n)" | unnamed_thy_226 | More_Word | 15 | |
[] | lemma word_less_sub_1:
"x < (y :: 'a :: len word) \<Longrightarrow> x \<le> y - 1" by (fact word_le_minus_one_leq) | lemma word_less_sub_1:
"x < (y :: 'a :: len word) \<Longrightarrow> x \<le> y - 1" by (fact word_le_minus_one_leq) | proof (prove)
goal (1 subgoal):
1. x < y \<Longrightarrow> x \<le> y - 1 | lemma word_less_sub_1:
"x < (y :: 'a :: len word) \<Longrightarrow> x \<le> y - 1" | unnamed_thy_227 | More_Word | 1 | |
[] | lemma word_sub_mono2:
"\<lbrakk> a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d \<rbrakk> \<Longrightarrow> b \<le> (d :: 'a :: len word)" by (drule(1) word_sub_mono; simp) | lemma word_sub_mono2:
"\<lbrakk> a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d \<rbrakk> \<Longrightarrow> b \<le> (d :: 'a :: len word)" by (drule(1) word_sub_mono; simp) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d\<rbrakk> \<Longrightarrow> b \<le> d | lemma word_sub_mono2:
"\<lbrakk> a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d \<rbrakk> \<Longrightarrow> b \<le> (d :: 'a :: len word)" | unnamed_thy_228 | More_Word | 1 | |
[] | lemma word_not_le:
"(\<not> x \<le> (y :: 'a :: len word)) = (y < x)" by (fact not_le) | lemma word_not_le:
"(\<not> x \<le> (y :: 'a :: len word)) = (y < x)" by (fact not_le) | proof (prove)
goal (1 subgoal):
1. (\<not> x \<le> y) = (y < x) | lemma word_not_le:
"(\<not> x \<le> (y :: 'a :: len word)) = (y < x)" | unnamed_thy_229 | More_Word | 1 | |
[] | lemma word_subset_less:
"\<lbrakk> {x .. x + r - 1} \<subseteq> {y .. y + s - 1};
x \<le> x + r - 1; y \<le> y + (s :: 'a :: len word) - 1;
s \<noteq> 0 \<rbrakk>
\<Longrightarrow> r \<le> s" apply (frule subsetD[where c=x]) apply simp apply (drule subsetD[where c="x + r - 1"]) apply simp apply (clarsi... | lemma word_subset_less:
"\<lbrakk> {x .. x + r - 1} \<subseteq> {y .. y + s - 1};
x \<le> x + r - 1; y \<le> y + (s :: 'a :: len word) - 1;
s \<noteq> 0 \<rbrakk>
\<Longrightarrow> r \<le> s" apply (frule subsetD[where c=x]) apply simp apply (drule subsetD[where c="x + r - 1"]) apply simp apply (clarsi... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>{x..x + r - 1} \<subseteq> {y..y + s - 1}; x \<le> x + r - 1; y \<le> y + s - 1; s \<noteq> 0\<rbrakk> \<Longrightarrow> r \<le> s proof (prove)
goal (2 subgoals):
1. \<lbrakk>{x..x + r - 1} \<subseteq> {y..y + s - 1}; x \<le> x + r - 1; y \<le> y + s - 1; s \<noteq> 0\<rbr... | lemma word_subset_less:
"\<lbrakk> {x .. x + r - 1} \<subseteq> {y .. y + s - 1};
x \<le> x + r - 1; y \<le> y + (s :: 'a :: len word) - 1;
s \<noteq> 0 \<rbrakk>
\<Longrightarrow> r \<le> s" | unnamed_thy_230 | More_Word | 11 | |
[] | lemma uint_power_lower:
"n < LENGTH('a) \<Longrightarrow> uint (2 ^ n :: 'a :: len word) = (2 ^ n :: int)" by (rule uint_2p_alt) | lemma uint_power_lower:
"n < LENGTH('a) \<Longrightarrow> uint (2 ^ n :: 'a :: len word) = (2 ^ n :: int)" by (rule uint_2p_alt) | proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> uint (2 ^ n) = 2 ^ n | lemma uint_power_lower:
"n < LENGTH('a) \<Longrightarrow> uint (2 ^ n :: 'a :: len word) = (2 ^ n :: int)" | unnamed_thy_231 | More_Word | 1 | |
[] | lemma power_le_mono:
"\<lbrakk>2 ^ n \<le> (2::'a::len word) ^ m; n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> n \<le> m" apply (clarsimp simp add: le_less) apply safe apply (simp add: word_less_nat_alt) apply (simp only: uint_arith_simps(3)) apply (drule uint_power_lower)+ apply simp done | lemma power_le_mono:
"\<lbrakk>2 ^ n \<le> (2::'a::len word) ^ m; n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> n \<le> m" apply (clarsimp simp add: le_less) apply safe apply (simp add: word_less_nat_alt) apply (simp only: uint_arith_simps(3)) apply (drule uint_power_lower)+ apply simp done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>2 ^ n \<le> 2 ^ m; n < LENGTH('a); m < LENGTH('a)\<rbrakk> \<Longrightarrow> n \<le> m proof (prove)
goal (1 subgoal):
1. \<lbrakk>2 ^ n < 2 ^ m \<or> 2 ^ n = 2 ^ m; n < LENGTH('a); m < LENGTH('a); n \<noteq> m\<rbrakk> \<Longrightarrow> n < m proof (prove)
goal (2 subgoals... | lemma power_le_mono:
"\<lbrakk>2 ^ n \<le> (2::'a::len word) ^ m; n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> n \<le> m" | unnamed_thy_232 | More_Word | 7 | |
[] | lemma two_power_eq:
"\<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> ((2::'a::len word) ^ n = 2 ^ m) = (n = m)" apply safe apply (rule order_antisym) apply (simp add: power_le_mono[where 'a='a])+ done | lemma two_power_eq:
"\<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> ((2::'a::len word) ^ n = 2 ^ m) = (n = m)" apply safe apply (rule order_antisym) apply (simp add: power_le_mono[where 'a='a])+ done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk> \<Longrightarrow> (2 ^ n = 2 ^ m) = (n = m) proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < LENGTH('a); m < LENGTH('a); 2 ^ n = 2 ^ m\<rbrakk> \<Longrightarrow> n = m proof (prove)
goal (2 subgoals):
1. \<lbrakk>n < LENGTH('a); m < L... | lemma two_power_eq:
"\<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> ((2::'a::len word) ^ n = 2 ^ m) = (n = m)" | unnamed_thy_233 | More_Word | 4 | |
[] | lemma unat_less_helper:
"x < of_nat n \<Longrightarrow> unat x < n" apply (simp add: word_less_nat_alt) apply (erule order_less_le_trans) apply (simp add: take_bit_eq_mod unsigned_of_nat) done | lemma unat_less_helper:
"x < of_nat n \<Longrightarrow> unat x < n" apply (simp add: word_less_nat_alt) apply (erule order_less_le_trans) apply (simp add: take_bit_eq_mod unsigned_of_nat) done | proof (prove)
goal (1 subgoal):
1. x < word_of_nat n \<Longrightarrow> unat x < n proof (prove)
goal (1 subgoal):
1. unat x < unat (word_of_nat n) \<Longrightarrow> unat x < n proof (prove)
goal (1 subgoal):
1. unat (word_of_nat n) \<le> n proof (prove)
goal:
No subgoals! | lemma unat_less_helper:
"x < of_nat n \<Longrightarrow> unat x < n" | unnamed_thy_234 | More_Word | 4 | |
[] | lemma nat_uint_less_helper:
"nat (uint y) = z \<Longrightarrow> x < y \<Longrightarrow> nat (uint x) < z" apply (erule subst) apply (subst unat_eq_nat_uint [symmetric]) apply (subst unat_eq_nat_uint [symmetric]) by (simp add: unat_mono) | lemma nat_uint_less_helper:
"nat (uint y) = z \<Longrightarrow> x < y \<Longrightarrow> nat (uint x) < z" apply (erule subst) apply (subst unat_eq_nat_uint [symmetric]) apply (subst unat_eq_nat_uint [symmetric]) by (simp add: unat_mono) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>nat (uint y) = z; x < y\<rbrakk> \<Longrightarrow> nat (uint x) < z proof (prove)
goal (1 subgoal):
1. x < y \<Longrightarrow> nat (uint x) < nat (uint y) proof (prove)
goal (1 subgoal):
1. x < y \<Longrightarrow> unat x < nat (uint y) proof (prove)
goal (1 subgoal):
1. x... | lemma nat_uint_less_helper:
"nat (uint y) = z \<Longrightarrow> x < y \<Longrightarrow> nat (uint x) < z" | unnamed_thy_235 | More_Word | 4 | |
[] | lemma of_nat_inj:
"\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
(of_nat x = (of_nat y :: 'a :: len word)) = (x = y)" by (metis unat_of_nat_len) | lemma of_nat_inj:
"\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
(of_nat x = (of_nat y :: 'a :: len word)) = (x = y)" by (metis unat_of_nat_len) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> (word_of_nat x = word_of_nat y) = (x = y) | lemma of_nat_inj:
"\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
(of_nat x = (of_nat y :: 'a :: len word)) = (x = y)" | unnamed_thy_237 | More_Word | 1 | |
[] | lemma div_to_mult_word_lt:
"\<lbrakk> (x :: 'a :: len word) \<le> y div z \<rbrakk> \<Longrightarrow> x * z \<le> y" apply (cases "z = 0") apply simp apply (simp add: word_neq_0_conv) apply (rule order_trans) apply (erule(1) word_mult_le_mono1) apply (simp add: unat_div) apply (rule order_le_less_trans [OF div_mult_l... | lemma div_to_mult_word_lt:
"\<lbrakk> (x :: 'a :: len word) \<le> y div z \<rbrakk> \<Longrightarrow> x * z \<le> y" apply (cases "z = 0") apply simp apply (simp add: word_neq_0_conv) apply (rule order_trans) apply (erule(1) word_mult_le_mono1) apply (simp add: unat_div) apply (rule order_le_less_trans [OF div_mult_l... | proof (prove)
goal (1 subgoal):
1. x \<le> y div z \<Longrightarrow> x * z \<le> y proof (prove)
goal (2 subgoals):
1. \<lbrakk>x \<le> y div z; z = 0\<rbrakk> \<Longrightarrow> x * z \<le> y
2. \<lbrakk>x \<le> y div z; z \<noteq> 0\<rbrakk> \<Longrightarrow> x * z \<le> y proof (prove)
goal (1 subgoal):
1. \<lbra... | lemma div_to_mult_word_lt:
"\<lbrakk> (x :: 'a :: len word) \<le> y div z \<rbrakk> \<Longrightarrow> x * z \<le> y" | unnamed_thy_238 | More_Word | 10 | |
[] | lemma ucast_ucast_mask:
"(ucast :: 'a :: len word \<Rightarrow> 'b :: len word) (ucast x) = x AND mask (len_of TYPE ('a))" apply (simp flip: take_bit_eq_mask) apply transfer apply (simp add: ac_simps) done | lemma ucast_ucast_mask:
"(ucast :: 'a :: len word \<Rightarrow> 'b :: len word) (ucast x) = x AND mask (len_of TYPE ('a))" apply (simp flip: take_bit_eq_mask) apply transfer apply (simp add: ac_simps) done | proof (prove)
goal (1 subgoal):
1. ucast (ucast x) = x AND mask LENGTH('a) proof (prove)
goal (1 subgoal):
1. ucast (ucast x) = take_bit LENGTH('a) x proof (prove)
goal (1 subgoal):
1. \<And>x. take_bit LENGTH('b) (take_bit LENGTH('a) (take_bit LENGTH('b) x)) = take_bit LENGTH('b) (take_bit (min LENGTH('b) LENGTH('a... | lemma ucast_ucast_mask:
"(ucast :: 'a :: len word \<Rightarrow> 'b :: len word) (ucast x) = x AND mask (len_of TYPE ('a))" | unnamed_thy_239 | More_Word | 4 | |
[] | lemma ucast_ucast_len:
"\<lbrakk> x < 2 ^ LENGTH('b) \<rbrakk> \<Longrightarrow> ucast (ucast x::'b::len word) = (x::'a::len word)" apply (subst ucast_ucast_mask) apply (erule less_mask_eq) done | lemma ucast_ucast_len:
"\<lbrakk> x < 2 ^ LENGTH('b) \<rbrakk> \<Longrightarrow> ucast (ucast x::'b::len word) = (x::'a::len word)" apply (subst ucast_ucast_mask) apply (erule less_mask_eq) done | proof (prove)
goal (1 subgoal):
1. x < 2 ^ LENGTH('b) \<Longrightarrow> ucast (ucast x) = x proof (prove)
goal (1 subgoal):
1. x < 2 ^ LENGTH('b) \<Longrightarrow> x AND mask LENGTH('b) = x proof (prove)
goal:
No subgoals! | lemma ucast_ucast_len:
"\<lbrakk> x < 2 ^ LENGTH('b) \<rbrakk> \<Longrightarrow> ucast (ucast x::'b::len word) = (x::'a::len word)" | unnamed_thy_240 | More_Word | 3 | |
[] | lemma ucast_ucast_id:
"LENGTH('a) < LENGTH('b) \<Longrightarrow> ucast (ucast (x::'a::len word)::'b::len word) = x" by (auto intro: ucast_up_ucast_id simp: is_up_def source_size_def target_size_def word_size) | lemma ucast_ucast_id:
"LENGTH('a) < LENGTH('b) \<Longrightarrow> ucast (ucast (x::'a::len word)::'b::len word) = x" by (auto intro: ucast_up_ucast_id simp: is_up_def source_size_def target_size_def word_size) | proof (prove)
goal (1 subgoal):
1. LENGTH('a) < LENGTH('b) \<Longrightarrow> ucast (ucast x) = x | lemma ucast_ucast_id:
"LENGTH('a) < LENGTH('b) \<Longrightarrow> ucast (ucast (x::'a::len word)::'b::len word) = x" | unnamed_thy_241 | More_Word | 1 | |
[] | lemma unat_ucast:
"unat (ucast x :: ('a :: len) word) = unat x mod 2 ^ (LENGTH('a))" proof - have \<open>2 ^ LENGTH('a) = nat (2 ^ LENGTH('a))\<close> by simp moreover have \<open>unat (ucast x :: 'a word) = unat x mod nat (2 ^ LENGTH('a))\<close> by transfer (simp flip: nat_mod_distrib take_bit_eq_mod) ultimately sh... | lemma unat_ucast:
"unat (ucast x :: ('a :: len) word) = unat x mod 2 ^ (LENGTH('a))" proof - have \<open>2 ^ LENGTH('a) = nat (2 ^ LENGTH('a))\<close> by simp moreover have \<open>unat (ucast x :: 'a word) = unat x mod nat (2 ^ LENGTH('a))\<close> by transfer (simp flip: nat_mod_distrib take_bit_eq_mod) ultimately sh... | proof (prove)
goal (1 subgoal):
1. unat (ucast x) = unat x mod 2 ^ LENGTH('a) proof (state)
goal (1 subgoal):
1. unat (ucast x) = unat x mod 2 ^ LENGTH('a) proof (prove)
goal (1 subgoal):
1. 2 ^ LENGTH('a) = nat (2 ^ LENGTH('a)) proof (state)
this:
2 ^ LENGTH('a) = nat (2 ^ LENGTH('a))
goal (1 subgoal):
1. unat (u... | lemma unat_ucast:
"unat (ucast x :: ('a :: len) word) = unat x mod 2 ^ (LENGTH('a))" | unnamed_thy_242 | More_Word | 10 | |
[] | lemma ucast_less_ucast:
"LENGTH('a) \<le> LENGTH('b) \<Longrightarrow>
(ucast x < ((ucast (y :: 'a::len word)) :: 'b::len word)) = (x < y)" apply (simp add: word_less_nat_alt unat_ucast) apply (subst mod_less) apply(rule less_le_trans[OF unat_lt2p], simp) apply (subst mod_less) apply(rule less_le_trans[OF unat_lt2... | lemma ucast_less_ucast:
"LENGTH('a) \<le> LENGTH('b) \<Longrightarrow>
(ucast x < ((ucast (y :: 'a::len word)) :: 'b::len word)) = (x < y)" apply (simp add: word_less_nat_alt unat_ucast) apply (subst mod_less) apply(rule less_le_trans[OF unat_lt2p], simp) apply (subst mod_less) apply(rule less_le_trans[OF unat_lt2... | proof (prove)
goal (1 subgoal):
1. LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> (ucast x < ucast y) = (x < y) proof (prove)
goal (1 subgoal):
1. LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> (unat x mod 2 ^ LENGTH('b) < unat y mod 2 ^ LENGTH('b)) = (unat x < unat y) proof (prove)
goal (2 subgoals):
1. LENGTH('a) \<... | lemma ucast_less_ucast:
"LENGTH('a) \<le> LENGTH('b) \<Longrightarrow>
(ucast x < ((ucast (y :: 'a::len word)) :: 'b::len word)) = (x < y)" | unnamed_thy_243 | More_Word | 7 | |
[] | lemma unat_Suc2:
fixes n :: "'a :: len word"
shows
"n \<noteq> -1 \<Longrightarrow> unat (n + 1) = Suc (unat n)" apply (subst add.commute, rule unatSuc) apply (subst eq_diff_eq[symmetric], simp add: minus_equation_iff) done | lemma unat_Suc2:
fixes n :: "'a :: len word"
shows
"n \<noteq> -1 \<Longrightarrow> unat (n + 1) = Suc (unat n)" apply (subst add.commute, rule unatSuc) apply (subst eq_diff_eq[symmetric], simp add: minus_equation_iff) done | proof (prove)
goal (1 subgoal):
1. n \<noteq> - 1 \<Longrightarrow> unat (n + 1) = Suc (unat n) proof (prove)
goal (1 subgoal):
1. n \<noteq> - 1 \<Longrightarrow> 1 + n \<noteq> 0 proof (prove)
goal:
No subgoals! | lemma unat_Suc2:
fixes n :: "'a :: len word"
shows
"n \<noteq> -1 \<Longrightarrow> unat (n + 1) = Suc (unat n)" | unnamed_thy_244 | More_Word | 3 | |
[] | lemma word_div_1:
"(n :: 'a :: len word) div 1 = n" by (fact bits_div_by_1) | lemma word_div_1:
"(n :: 'a :: len word) div 1 = n" by (fact bits_div_by_1) | proof (prove)
goal (1 subgoal):
1. n div 1 = n | lemma word_div_1:
"(n :: 'a :: len word) div 1 = n" | unnamed_thy_245 | More_Word | 1 | |
[] | lemma word_minus_one_le:
"-1 \<le> (x :: 'a :: len word) = (x = -1)" by (fact word_order.extremum_unique) | lemma word_minus_one_le:
"-1 \<le> (x :: 'a :: len word) = (x = -1)" by (fact word_order.extremum_unique) | proof (prove)
goal (1 subgoal):
1. (- 1 \<le> x) = (x = - 1) | lemma word_minus_one_le:
"-1 \<le> (x :: 'a :: len word) = (x = -1)" | unnamed_thy_246 | More_Word | 1 | |
[] | lemma up_scast_inj:
"\<lbrakk> scast x = (scast y :: 'b :: len word); size x \<le> LENGTH('b) \<rbrakk> \<Longrightarrow> x = y" apply transfer apply (cases \<open>LENGTH('a)\<close>; simp) apply (metis order_refl take_bit_signed_take_bit take_bit_tightened) done | lemma up_scast_inj:
"\<lbrakk> scast x = (scast y :: 'b :: len word); size x \<le> LENGTH('b) \<rbrakk> \<Longrightarrow> x = y" apply transfer apply (cases \<open>LENGTH('a)\<close>; simp) apply (metis order_refl take_bit_signed_take_bit take_bit_tightened) done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>scast x = scast y; size x \<le> LENGTH('b)\<rbrakk> \<Longrightarrow> x = y proof (prove)
goal (1 subgoal):
1. \<And>x y. \<lbrakk>take_bit LENGTH('b) (signed_take_bit (LENGTH('a) - Suc 0) x) = take_bit LENGTH('b) (signed_take_bit (LENGTH('a) - Suc 0) y); LENGTH('a) \<le> L... | lemma up_scast_inj:
"\<lbrakk> scast x = (scast y :: 'b :: len word); size x \<le> LENGTH('b) \<rbrakk> \<Longrightarrow> x = y" | unnamed_thy_247 | More_Word | 4 | |
[] | lemma up_scast_inj_eq:
"LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow>
(scast x = (scast y::'b::len word)) = (x = (y::'a::len word))" by (fastforce dest: up_scast_inj simp: word_size) | lemma up_scast_inj_eq:
"LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow>
(scast x = (scast y::'b::len word)) = (x = (y::'a::len word))" by (fastforce dest: up_scast_inj simp: word_size) | proof (prove)
goal (1 subgoal):
1. LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> (scast x = scast y) = (x = y) | lemma up_scast_inj_eq:
"LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow>
(scast x = (scast y::'b::len word)) = (x = (y::'a::len word))" | unnamed_thy_248 | More_Word | 1 | |
[] | lemma word_le_add:
fixes x :: "'a :: len word"
shows "x \<le> y \<Longrightarrow> \<exists>n. y = x + of_nat n" by (rule exI [where x = "unat (y - x)"]) simp | lemma word_le_add:
fixes x :: "'a :: len word"
shows "x \<le> y \<Longrightarrow> \<exists>n. y = x + of_nat n" by (rule exI [where x = "unat (y - x)"]) simp | proof (prove)
goal (1 subgoal):
1. x \<le> y \<Longrightarrow> \<exists>n. y = x + word_of_nat n | lemma word_le_add:
fixes x :: "'a :: len word"
shows "x \<le> y \<Longrightarrow> \<exists>n. y = x + of_nat n" | unnamed_thy_249 | More_Word | 1 | |
[] | lemma word_plus_mcs_4':
"\<lbrakk>x + v \<le> x + w; x \<le> x + v\<rbrakk> \<Longrightarrow> v \<le> w" for x :: "'a::len word" by (rule word_plus_mcs_4; simp add: add.commute) | lemma word_plus_mcs_4':
"\<lbrakk>x + v \<le> x + w; x \<le> x + v\<rbrakk> \<Longrightarrow> v \<le> w" for x :: "'a::len word" by (rule word_plus_mcs_4; simp add: add.commute) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x + v \<le> x + w; x \<le> x + v\<rbrakk> \<Longrightarrow> v \<le> w | lemma word_plus_mcs_4':
"\<lbrakk>x + v \<le> x + w; x \<le> x + v\<rbrakk> \<Longrightarrow> v \<le> w" for x :: "'a::len word" | unnamed_thy_250 | More_Word | 1 | |
[] | lemma unat_eq_1:
\<open>unat x = Suc 0 \<longleftrightarrow> x = 1\<close> by (auto intro!: unsigned_word_eqI [where ?'a = nat]) | lemma unat_eq_1:
\<open>unat x = Suc 0 \<longleftrightarrow> x = 1\<close> by (auto intro!: unsigned_word_eqI [where ?'a = nat]) | proof (prove)
goal (1 subgoal):
1. (unat x = Suc 0) = (x = 1) | lemma unat_eq_1:
\<open>unat x = Suc 0 \<longleftrightarrow> x = 1\<close> | unnamed_thy_251 | More_Word | 1 | |
[] | lemma word_unat_Rep_inject1:
\<open>unat x = unat 1 \<longleftrightarrow> x = 1\<close> by (simp add: unat_eq_1) | lemma word_unat_Rep_inject1:
\<open>unat x = unat 1 \<longleftrightarrow> x = 1\<close> by (simp add: unat_eq_1) | proof (prove)
goal (1 subgoal):
1. (unat x = unat 1) = (x = 1) | lemma word_unat_Rep_inject1:
\<open>unat x = unat 1 \<longleftrightarrow> x = 1\<close> | unnamed_thy_252 | More_Word | 1 | |
[] | lemma and_not_mask_twice:
"(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
for w :: \<open>'a::len word\<close> by (rule bit_word_eqI) (auto simp add: bit_simps) | lemma and_not_mask_twice:
"(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
for w :: \<open>'a::len word\<close> by (rule bit_word_eqI) (auto simp add: bit_simps) | proof (prove)
goal (1 subgoal):
1. (w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n)) | lemma and_not_mask_twice:
"(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
for w :: \<open>'a::len word\<close> | unnamed_thy_253 | More_Word | 1 | |
[] | lemma word_less_cases:
"x < y \<Longrightarrow> x = y - 1 \<or> x < y - (1 ::'a::len word)" apply (drule word_less_sub_1) apply (drule order_le_imp_less_or_eq) apply auto done | lemma word_less_cases:
"x < y \<Longrightarrow> x = y - 1 \<or> x < y - (1 ::'a::len word)" apply (drule word_less_sub_1) apply (drule order_le_imp_less_or_eq) apply auto done | proof (prove)
goal (1 subgoal):
1. x < y \<Longrightarrow> x = y - 1 \<or> x < y - 1 proof (prove)
goal (1 subgoal):
1. x \<le> y - 1 \<Longrightarrow> x = y - 1 \<or> x < y - 1 proof (prove)
goal (1 subgoal):
1. x < y - 1 \<or> x = y - 1 \<Longrightarrow> x = y - 1 \<or> x < y - 1 proof (prove)
goal:
No subgoals! | lemma word_less_cases:
"x < y \<Longrightarrow> x = y - 1 \<or> x < y - (1 ::'a::len word)" | unnamed_thy_254 | More_Word | 4 | |
[] | lemma mask_and_mask:
"mask a AND mask b = (mask (min a b) :: 'a::len word)" by (simp flip: take_bit_eq_mask ac_simps) | lemma mask_and_mask:
"mask a AND mask b = (mask (min a b) :: 'a::len word)" by (simp flip: take_bit_eq_mask ac_simps) | proof (prove)
goal (1 subgoal):
1. mask a AND mask b = mask (min a b) | lemma mask_and_mask:
"mask a AND mask b = (mask (min a b) :: 'a::len word)" | unnamed_thy_255 | More_Word | 1 | |
[] | lemma mask_eq_0_eq_x:
"(x AND w = 0) = (x AND NOT w = x)"
for x w :: \<open>'a::len word\<close> using word_plus_and_or_coroll2[where x=x and w=w] by auto | lemma mask_eq_0_eq_x:
"(x AND w = 0) = (x AND NOT w = x)"
for x w :: \<open>'a::len word\<close> using word_plus_and_or_coroll2[where x=x and w=w] by auto | proof (prove)
goal (1 subgoal):
1. (x AND w = 0) = (x AND NOT w = x) proof (prove)
using this:
(x AND w) + (x AND NOT w) = x
goal (1 subgoal):
1. (x AND w = 0) = (x AND NOT w = x) | lemma mask_eq_0_eq_x:
"(x AND w = 0) = (x AND NOT w = x)"
for x w :: \<open>'a::len word\<close> | unnamed_thy_256 | More_Word | 2 | |
[] | lemma mask_eq_x_eq_0:
"(x AND w = x) = (x AND NOT w = 0)"
for x w :: \<open>'a::len word\<close> using word_plus_and_or_coroll2[where x=x and w=w] by auto | lemma mask_eq_x_eq_0:
"(x AND w = x) = (x AND NOT w = 0)"
for x w :: \<open>'a::len word\<close> using word_plus_and_or_coroll2[where x=x and w=w] by auto | proof (prove)
goal (1 subgoal):
1. (x AND w = x) = (x AND NOT w = 0) proof (prove)
using this:
(x AND w) + (x AND NOT w) = x
goal (1 subgoal):
1. (x AND w = x) = (x AND NOT w = 0) | lemma mask_eq_x_eq_0:
"(x AND w = x) = (x AND NOT w = 0)"
for x w :: \<open>'a::len word\<close> | unnamed_thy_257 | More_Word | 2 | |
[] | lemma compl_of_1: "NOT 1 = (-2 :: 'a :: len word)" by (fact not_one_eq) | lemma compl_of_1: "NOT 1 = (-2 :: 'a :: len word)" by (fact not_one_eq) | proof (prove)
goal (1 subgoal):
1. NOT 1 = - 2 | lemma compl_of_1: "NOT 1 = (-2 :: 'a :: len word)" | unnamed_thy_258 | More_Word | 1 | |
[] | lemma split_word_eq_on_mask:
"(x = y) = (x AND m = y AND m \<and> x AND NOT m = y AND NOT m)"
for x y m :: \<open>'a::len word\<close> apply transfer apply (simp add: bit_eq_iff) apply (auto simp add: bit_simps ac_simps) done | lemma split_word_eq_on_mask:
"(x = y) = (x AND m = y AND m \<and> x AND NOT m = y AND NOT m)"
for x y m :: \<open>'a::len word\<close> apply transfer apply (simp add: bit_eq_iff) apply (auto simp add: bit_simps ac_simps) done | proof (prove)
goal (1 subgoal):
1. (x = y) = (x AND m = y AND m \<and> x AND NOT m = y AND NOT m) proof (prove)
goal (1 subgoal):
1. \<And>x y m. (take_bit LENGTH('a) x = take_bit LENGTH('a) y) = (take_bit LENGTH('a) (x AND m) = take_bit LENGTH('a) (y AND m) \<and> take_bit LENGTH('a) (x AND NOT m) = take_bit LENGTH(... | lemma split_word_eq_on_mask:
"(x = y) = (x AND m = y AND m \<and> x AND NOT m = y AND NOT m)"
for x y m :: \<open>'a::len word\<close> | unnamed_thy_259 | More_Word | 4 | |
[] | lemma word_FF_is_mask:
"0xFF = (mask 8 :: 'a::len word)" by (simp add: mask_eq_decr_exp) | lemma word_FF_is_mask:
"0xFF = (mask 8 :: 'a::len word)" by (simp add: mask_eq_decr_exp) | proof (prove)
goal (1 subgoal):
1. 255 = mask 8 | lemma word_FF_is_mask:
"0xFF = (mask 8 :: 'a::len word)" | unnamed_thy_260 | More_Word | 1 | |
[] | lemma word_1FF_is_mask:
"0x1FF = (mask 9 :: 'a::len word)" by (simp add: mask_eq_decr_exp) | lemma word_1FF_is_mask:
"0x1FF = (mask 9 :: 'a::len word)" by (simp add: mask_eq_decr_exp) | proof (prove)
goal (1 subgoal):
1. 511 = mask 9 | lemma word_1FF_is_mask:
"0x1FF = (mask 9 :: 'a::len word)" | unnamed_thy_261 | More_Word | 1 | |
[] | lemma ucast_of_nat_small:
"x < 2 ^ LENGTH('a) \<Longrightarrow> ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)" apply transfer apply (auto simp add: take_bit_of_nat min_def not_le) apply (metis linorder_not_less min_def take_bit_nat_eq_self take_bit_take_bit) done | lemma ucast_of_nat_small:
"x < 2 ^ LENGTH('a) \<Longrightarrow> ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)" apply transfer apply (auto simp add: take_bit_of_nat min_def not_le) apply (metis linorder_not_less min_def take_bit_nat_eq_self take_bit_take_bit) done | proof (prove)
goal (1 subgoal):
1. x < 2 ^ LENGTH('a) \<Longrightarrow> ucast (word_of_nat x) = word_of_nat x proof (prove)
goal (1 subgoal):
1. \<And>x. x < 2 ^ LENGTH('a) \<Longrightarrow> take_bit LENGTH('b) (take_bit LENGTH('a) (int x)) = take_bit LENGTH('b) (int x) proof (prove)
goal (1 subgoal):
1. \<And>x. \<... | lemma ucast_of_nat_small:
"x < 2 ^ LENGTH('a) \<Longrightarrow> ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)" | unnamed_thy_262 | More_Word | 4 | |
[] | lemma word_le_make_less:
fixes x :: "'a :: len word"
shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))" apply safe apply (erule plus_one_helper2) apply (simp add: eq_diff_eq[symmetric]) done | lemma word_le_make_less:
fixes x :: "'a :: len word"
shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))" apply safe apply (erule plus_one_helper2) apply (simp add: eq_diff_eq[symmetric]) done | proof (prove)
goal (1 subgoal):
1. y \<noteq> - 1 \<Longrightarrow> (x \<le> y) = (x < y + 1) proof (prove)
goal (1 subgoal):
1. \<lbrakk>y \<noteq> - 1; x \<le> y\<rbrakk> \<Longrightarrow> x < y + 1 proof (prove)
goal (1 subgoal):
1. y \<noteq> - 1 \<Longrightarrow> y + 1 \<noteq> 0 proof (prove)
goal:
No subgoals... | lemma word_le_make_less:
fixes x :: "'a :: len word"
shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))" | unnamed_thy_263 | More_Word | 4 | |
[] | lemma word_to_1_set:
"{0 ..< (1 :: 'a :: len word)} = {0}" by fastforce | lemma word_to_1_set:
"{0 ..< (1 :: 'a :: len word)} = {0}" by fastforce | proof (prove)
goal (1 subgoal):
1. {0..<1} = {0} | lemma word_to_1_set:
"{0 ..< (1 :: 'a :: len word)} = {0}" | unnamed_thy_264 | More_Word | 1 | |
[] | lemma word_leq_minus_one_le:
fixes x :: "'a::len word"
shows "\<lbrakk>y \<noteq> 0; x \<le> y - 1 \<rbrakk> \<Longrightarrow> x < y" using le_m1_iff_lt word_neq_0_conv by blast | lemma word_leq_minus_one_le:
fixes x :: "'a::len word"
shows "\<lbrakk>y \<noteq> 0; x \<le> y - 1 \<rbrakk> \<Longrightarrow> x < y" using le_m1_iff_lt word_neq_0_conv by blast | proof (prove)
goal (1 subgoal):
1. \<lbrakk>y \<noteq> 0; x \<le> y - 1\<rbrakk> \<Longrightarrow> x < y proof (prove)
using this:
(0 < ?x) = ((?y \<le> ?x - 1) = (?y < ?x))
(?w \<noteq> 0) = (0 < ?w)
goal (1 subgoal):
1. \<lbrakk>y \<noteq> 0; x \<le> y - 1\<rbrakk> \<Longrightarrow> x < y | lemma word_leq_minus_one_le:
fixes x :: "'a::len word"
shows "\<lbrakk>y \<noteq> 0; x \<le> y - 1 \<rbrakk> \<Longrightarrow> x < y" | unnamed_thy_265 | More_Word | 2 | |
[] | lemma word_count_from_top:
"n \<noteq> 0 \<Longrightarrow> {0 ..< n :: 'a :: len word} = {0 ..< n - 1} \<union> {n - 1}" apply (rule set_eqI, rule iffI) apply simp apply (drule word_le_minus_one_leq) apply (rule disjCI) apply simp apply simp apply (erule word_leq_minus_one_le) apply fastforce done | lemma word_count_from_top:
"n \<noteq> 0 \<Longrightarrow> {0 ..< n :: 'a :: len word} = {0 ..< n - 1} \<union> {n - 1}" apply (rule set_eqI, rule iffI) apply simp apply (drule word_le_minus_one_leq) apply (rule disjCI) apply simp apply simp apply (erule word_leq_minus_one_le) apply fastforce done | proof (prove)
goal (1 subgoal):
1. n \<noteq> 0 \<Longrightarrow> {0..<n} = {0..<n - 1} \<union> {n - 1} proof (prove)
goal (2 subgoals):
1. \<And>x. \<lbrakk>n \<noteq> 0; x \<in> {0..<n}\<rbrakk> \<Longrightarrow> x \<in> {0..<n - 1} \<union> {n - 1}
2. \<And>x. \<lbrakk>n \<noteq> 0; x \<in> {0..<n - 1} \<union> ... | lemma word_count_from_top:
"n \<noteq> 0 \<Longrightarrow> {0 ..< n :: 'a :: len word} = {0 ..< n - 1} \<union> {n - 1}" | unnamed_thy_266 | More_Word | 9 | |
[] | lemma word_minus_one_le_leq:
"\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: 'a :: len word)" apply (cases "x = 0") apply simp apply (simp add: word_less_nat_alt word_le_nat_alt) apply (subst(asm) unat_minus_one) apply (simp add: word_less_nat_alt) apply (cases "unat x") apply (simp add: unat_eq_zero) ... | lemma word_minus_one_le_leq:
"\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: 'a :: len word)" apply (cases "x = 0") apply simp apply (simp add: word_less_nat_alt word_le_nat_alt) apply (subst(asm) unat_minus_one) apply (simp add: word_less_nat_alt) apply (cases "unat x") apply (simp add: unat_eq_zero) ... | proof (prove)
goal (1 subgoal):
1. x - 1 < y \<Longrightarrow> x \<le> y proof (prove)
goal (2 subgoals):
1. \<lbrakk>x - 1 < y; x = 0\<rbrakk> \<Longrightarrow> x \<le> y
2. \<lbrakk>x - 1 < y; x \<noteq> 0\<rbrakk> \<Longrightarrow> x \<le> y proof (prove)
goal (1 subgoal):
1. \<lbrakk>x - 1 < y; x \<noteq> 0\<rb... | lemma word_minus_one_le_leq:
"\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: 'a :: len word)" | unnamed_thy_267 | More_Word | 9 | |
[] | lemma word_must_wrap:
"\<lbrakk> x \<le> n - 1; n \<le> x \<rbrakk> \<Longrightarrow> n = (0 :: 'a :: len word)" using dual_order.trans sub_wrap word_less_1 by blast | lemma word_must_wrap:
"\<lbrakk> x \<le> n - 1; n \<le> x \<rbrakk> \<Longrightarrow> n = (0 :: 'a :: len word)" using dual_order.trans sub_wrap word_less_1 by blast | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x \<le> n - 1; n \<le> x\<rbrakk> \<Longrightarrow> n = 0 proof (prove)
using this:
\<lbrakk>?b \<le> ?a; ?c \<le> ?b\<rbrakk> \<Longrightarrow> ?c \<le> ?a
(?x \<le> ?x - ?z) = (?z = 0 \<or> ?x < ?z)
(?x < 1) = (?x = 0)
goal (1 subgoal):
1. \<lbrakk>x \<le> n - 1; n \<le>... | lemma word_must_wrap:
"\<lbrakk> x \<le> n - 1; n \<le> x \<rbrakk> \<Longrightarrow> n = (0 :: 'a :: len word)" | unnamed_thy_268 | More_Word | 2 | |
[] | lemma range_subset_card:
"\<lbrakk> {a :: 'a :: len word .. b} \<subseteq> {c .. d}; b \<ge> a \<rbrakk> \<Longrightarrow> d \<ge> c \<and> d - c \<ge> b - a" using word_sub_le word_sub_mono by fastforce | lemma range_subset_card:
"\<lbrakk> {a :: 'a :: len word .. b} \<subseteq> {c .. d}; b \<ge> a \<rbrakk> \<Longrightarrow> d \<ge> c \<and> d - c \<ge> b - a" using word_sub_le word_sub_mono by fastforce | proof (prove)
goal (1 subgoal):
1. \<lbrakk>{a..b} \<subseteq> {c..d}; a \<le> b\<rbrakk> \<Longrightarrow> c \<le> d \<and> b - a \<le> d - c proof (prove)
using this:
?y \<le> ?x \<Longrightarrow> ?x - ?y \<le> ?x
\<lbrakk>?a \<le> ?c; ?d \<le> ?b; ?a - ?b \<le> ?a; ?c - ?d \<le> ?c\<rbrakk> \<Longrightarrow> ?a - ?... | lemma range_subset_card:
"\<lbrakk> {a :: 'a :: len word .. b} \<subseteq> {c .. d}; b \<ge> a \<rbrakk> \<Longrightarrow> d \<ge> c \<and> d - c \<ge> b - a" | unnamed_thy_269 | More_Word | 2 | |
[] | lemma word_power_mod_div:
fixes x :: "'a::len word"
shows "\<lbrakk> n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" apply (simp add: word_arith_nat_div unat_mod power_mod_div) apply (subst unat_arith_simps(3)) apply (subst unat_mod) apply (subst unat... | lemma word_power_mod_div:
fixes x :: "'a::len word"
shows "\<lbrakk> n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" apply (simp add: word_arith_nat_div unat_mod power_mod_div) apply (subst unat_arith_simps(3)) apply (subst unat_mod) apply (subst unat... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk> \<Longrightarrow> x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m) proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk> \<Longrightarrow> word_of_nat (unat x div 2 ^ m mod 2 ^ (n - m)) = word_of_nat (... | lemma word_power_mod_div:
fixes x :: "'a::len word"
shows "\<lbrakk> n < LENGTH('a); m < LENGTH('a)\<rbrakk>
\<Longrightarrow> x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" | unnamed_thy_271 | More_Word | 6 | |
[] | lemma word_range_minus_1':
fixes a :: "'a :: len word"
shows "a \<noteq> 0 \<Longrightarrow> {a - 1<..b} = {a..b}" by (simp add: greaterThanAtMost_def atLeastAtMost_def greaterThan_def atLeast_def less_1_simp) | lemma word_range_minus_1':
fixes a :: "'a :: len word"
shows "a \<noteq> 0 \<Longrightarrow> {a - 1<..b} = {a..b}" by (simp add: greaterThanAtMost_def atLeastAtMost_def greaterThan_def atLeast_def less_1_simp) | proof (prove)
goal (1 subgoal):
1. a \<noteq> 0 \<Longrightarrow> {a - 1<..b} = {a..b} | lemma word_range_minus_1':
fixes a :: "'a :: len word"
shows "a \<noteq> 0 \<Longrightarrow> {a - 1<..b} = {a..b}" | unnamed_thy_272 | More_Word | 1 | |
[] | lemma word_range_minus_1:
fixes a :: "'a :: len word"
shows "b \<noteq> 0 \<Longrightarrow> {a..b - 1} = {a..<b}" apply (simp add: atLeastLessThan_def atLeastAtMost_def atMost_def lessThan_def) apply (rule arg_cong [where f = "\<lambda>x. {a..} \<inter> x"]) apply rule apply clarsimp apply (erule contrapos_pp) appl... | lemma word_range_minus_1:
fixes a :: "'a :: len word"
shows "b \<noteq> 0 \<Longrightarrow> {a..b - 1} = {a..<b}" apply (simp add: atLeastLessThan_def atLeastAtMost_def atMost_def lessThan_def) apply (rule arg_cong [where f = "\<lambda>x. {a..} \<inter> x"]) apply rule apply clarsimp apply (erule contrapos_pp) appl... | proof (prove)
goal (1 subgoal):
1. b \<noteq> 0 \<Longrightarrow> {a..b - 1} = {a..<b} proof (prove)
goal (1 subgoal):
1. b \<noteq> 0 \<Longrightarrow> {a..} \<inter> {x. x \<le> b - 1} = {a..} \<inter> {x. x < b} proof (prove)
goal (1 subgoal):
1. b \<noteq> 0 \<Longrightarrow> {x. x \<le> b - 1} = {x. x < b} proo... | lemma word_range_minus_1:
fixes a :: "'a :: len word"
shows "b \<noteq> 0 \<Longrightarrow> {a..b - 1} = {a..<b}" | unnamed_thy_273 | More_Word | 10 | |
[] | lemma ucast_nat_def:
"of_nat (unat x) = (ucast :: 'a :: len word \<Rightarrow> 'b :: len word) x" by transfer simp | lemma ucast_nat_def:
"of_nat (unat x) = (ucast :: 'a :: len word \<Rightarrow> 'b :: len word) x" by transfer simp | proof (prove)
goal (1 subgoal):
1. word_of_nat (unat x) = ucast x | lemma ucast_nat_def:
"of_nat (unat x) = (ucast :: 'a :: len word \<Rightarrow> 'b :: len word) x" | unnamed_thy_274 | More_Word | 1 | |
[] | lemma overflow_plus_one_self:
"(1 + p \<le> p) = (p = (-1 :: 'a :: len word))" apply rule apply (rule ccontr) apply (drule plus_one_helper2) apply (rule notI) apply (drule arg_cong[where f="\<lambda>x. x - 1"]) apply simp apply (simp add: field_simps) apply simp done | lemma overflow_plus_one_self:
"(1 + p \<le> p) = (p = (-1 :: 'a :: len word))" apply rule apply (rule ccontr) apply (drule plus_one_helper2) apply (rule notI) apply (drule arg_cong[where f="\<lambda>x. x - 1"]) apply simp apply (simp add: field_simps) apply simp done | proof (prove)
goal (1 subgoal):
1. (1 + p \<le> p) = (p = - 1) proof (prove)
goal (2 subgoals):
1. 1 + p \<le> p \<Longrightarrow> p = - 1
2. p = - 1 \<Longrightarrow> 1 + p \<le> p proof (prove)
goal (2 subgoals):
1. \<lbrakk>1 + p \<le> p; p \<noteq> - 1\<rbrakk> \<Longrightarrow> False
2. p = - 1 \<Longrightarr... | lemma overflow_plus_one_self:
"(1 + p \<le> p) = (p = (-1 :: 'a :: len word))" | unnamed_thy_275 | More_Word | 9 | |
[] | lemma plus_1_less:
"(x + 1 \<le> (x :: 'a :: len word)) = (x = -1)" apply (rule iffI) apply (rule ccontr) apply (cut_tac plus_one_helper2[where x=x, OF order_refl]) apply simp apply clarsimp apply (drule arg_cong[where f="\<lambda>x. x - 1"]) apply simp apply simp done | lemma plus_1_less:
"(x + 1 \<le> (x :: 'a :: len word)) = (x = -1)" apply (rule iffI) apply (rule ccontr) apply (cut_tac plus_one_helper2[where x=x, OF order_refl]) apply simp apply clarsimp apply (drule arg_cong[where f="\<lambda>x. x - 1"]) apply simp apply simp done | proof (prove)
goal (1 subgoal):
1. (x + 1 \<le> x) = (x = - 1) proof (prove)
goal (2 subgoals):
1. x + 1 \<le> x \<Longrightarrow> x = - 1
2. x = - 1 \<Longrightarrow> x + 1 \<le> x proof (prove)
goal (2 subgoals):
1. \<lbrakk>x + 1 \<le> x; x \<noteq> - 1\<rbrakk> \<Longrightarrow> False
2. x = - 1 \<Longrightarr... | lemma plus_1_less:
"(x + 1 \<le> (x :: 'a :: len word)) = (x = -1)" | unnamed_thy_276 | More_Word | 9 | |
[] | lemma pos_mult_pos_ge:
"[|x > (0::int); n>=0 |] ==> n * x >= n*1" apply (simp only: mult_left_mono) done | lemma pos_mult_pos_ge:
"[|x > (0::int); n>=0 |] ==> n * x >= n*1" apply (simp only: mult_left_mono) done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>0 < x; 0 \<le> n\<rbrakk> \<Longrightarrow> n * 1 \<le> n * x proof (prove)
goal:
No subgoals! | lemma pos_mult_pos_ge:
"[|x > (0::int); n>=0 |] ==> n * x >= n*1" | unnamed_thy_277 | More_Word | 2 | |
[] | lemma word_plus_strict_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk>y < z; x \<le> x + z\<rbrakk> \<Longrightarrow> x + y < x + z" by unat_arith | lemma word_plus_strict_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk>y < z; x \<le> x + z\<rbrakk> \<Longrightarrow> x + y < x + z" by unat_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>y < z; x \<le> x + z\<rbrakk> \<Longrightarrow> x + y < x + z | lemma word_plus_strict_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk>y < z; x \<le> x + z\<rbrakk> \<Longrightarrow> x + y < x + z" | unnamed_thy_278 | More_Word | 1 | |
[] | lemma word_div_mult:
"0 < c \<Longrightarrow> a < b * c \<Longrightarrow> a div c < b" for a b c :: "'a::len word" by (rule classical)
(use div_to_mult_word_lt [of b a c] in
\<open>auto simp add: word_less_nat_alt word_le_nat_alt unat_div\<close>) | lemma word_div_mult:
"0 < c \<Longrightarrow> a < b * c \<Longrightarrow> a div c < b" for a b c :: "'a::len word" by (rule classical)
(use div_to_mult_word_lt [of b a c] in
\<open>auto simp add: word_less_nat_alt word_le_nat_alt unat_div\<close>) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>0 < c; a < b * c\<rbrakk> \<Longrightarrow> a div c < b | lemma word_div_mult:
"0 < c \<Longrightarrow> a < b * c \<Longrightarrow> a div c < b" for a b c :: "'a::len word" | unnamed_thy_279 | More_Word | 1 | |
[] | lemma word_less_power_trans_ofnat:
"\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk>
\<Longrightarrow> of_nat n * 2 ^ k < (2::'a::len word) ^ m" apply (subst mult.commute) apply (rule word_less_power_trans) apply (simp_all add: word_less_nat_alt unsigned_of_nat) using take_bit_nat_less_eq_self apply (r... | lemma word_less_power_trans_ofnat:
"\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk>
\<Longrightarrow> of_nat n * 2 ^ k < (2::'a::len word) ^ m" apply (subst mult.commute) apply (rule word_less_power_trans) apply (simp_all add: word_less_nat_alt unsigned_of_nat) using take_bit_nat_less_eq_self apply (r... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> word_of_nat n * 2 ^ k < 2 ^ m proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> 2 ^ k * word_of_nat n < 2 ^ m proof (prove)
goal (3 subgoals... | lemma word_less_power_trans_ofnat:
"\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk>
\<Longrightarrow> of_nat n * 2 ^ k < (2::'a::len word) ^ m" | unnamed_thy_280 | More_Word | 7 | |
[] | lemma word_1_le_power:
"n < LENGTH('a) \<Longrightarrow> (1 :: 'a :: len word) \<le> 2 ^ n" by (rule inc_le[where i=0, simplified], erule iffD2[OF p2_gt_0]) | lemma word_1_le_power:
"n < LENGTH('a) \<Longrightarrow> (1 :: 'a :: len word) \<le> 2 ^ n" by (rule inc_le[where i=0, simplified], erule iffD2[OF p2_gt_0]) | proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> 1 \<le> 2 ^ n | lemma word_1_le_power:
"n < LENGTH('a) \<Longrightarrow> (1 :: 'a :: len word) \<le> 2 ^ n" | unnamed_thy_281 | More_Word | 1 | |
[] | lemma unat_1_0:
"1 \<le> (x::'a::len word) = (0 < unat x)" by (auto simp add: word_le_nat_alt) | lemma unat_1_0:
"1 \<le> (x::'a::len word) = (0 < unat x)" by (auto simp add: word_le_nat_alt) | proof (prove)
goal (1 subgoal):
1. (1 \<le> x) = (0 < unat x) | lemma unat_1_0:
"1 \<le> (x::'a::len word) = (0 < unat x)" | unnamed_thy_282 | More_Word | 1 | |
[] | lemma x_less_2_0_1':
fixes x :: "'a::len word"
shows "\<lbrakk>LENGTH('a) \<noteq> 1; x < 2\<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1" apply (cases \<open>2 \<le> LENGTH('a)\<close>; simp) apply transfer apply clarsimp apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial
... | lemma x_less_2_0_1':
fixes x :: "'a::len word"
shows "\<lbrakk>LENGTH('a) \<noteq> 1; x < 2\<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1" apply (cases \<open>2 \<le> LENGTH('a)\<close>; simp) apply transfer apply clarsimp apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial
... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>LENGTH('a) \<noteq> 1; x < 2\<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1 proof (prove)
goal (1 subgoal):
1. \<lbrakk>x < 2; 2 \<le> LENGTH('a)\<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1 proof (prove)
goal (1 subgoal):
1. \<And>x. \<lbrakk>take_bit LENGTH('a) x < take_bi... | lemma x_less_2_0_1':
fixes x :: "'a::len word"
shows "\<lbrakk>LENGTH('a) \<noteq> 1; x < 2\<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1" | unnamed_thy_283 | More_Word | 5 | |
[] | lemma of_nat_n_less_equal_power_2:
"n < LENGTH('a::len) \<Longrightarrow> ((of_nat n)::'a word) < 2 ^ n" apply (induct n) apply clarsimp apply clarsimp apply (metis of_nat_power n_less_equal_power_2 of_nat_Suc power_Suc) done | lemma of_nat_n_less_equal_power_2:
"n < LENGTH('a::len) \<Longrightarrow> ((of_nat n)::'a word) < 2 ^ n" apply (induct n) apply clarsimp apply clarsimp apply (metis of_nat_power n_less_equal_power_2 of_nat_Suc power_Suc) done | proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> word_of_nat n < 2 ^ n proof (prove)
goal (2 subgoals):
1. 0 < LENGTH('a) \<Longrightarrow> word_of_nat 0 < 2 ^ 0
2. \<And>n. \<lbrakk>n < LENGTH('a) \<Longrightarrow> word_of_nat n < 2 ^ n; Suc n < LENGTH('a)\<rbrakk> \<Longrightarrow> word_of_nat (... | lemma of_nat_n_less_equal_power_2:
"n < LENGTH('a::len) \<Longrightarrow> ((of_nat n)::'a word) < 2 ^ n" | unnamed_thy_285 | More_Word | 5 | |
[] | lemma eq_mask_less:
fixes w :: "'a::len word"
assumes eqm: "w = w AND mask n"
and sz: "n < len_of TYPE ('a)"
shows "w < (2::'a word) ^ n" by (subst eqm, rule and_mask_less' [OF sz]) | lemma eq_mask_less:
fixes w :: "'a::len word"
assumes eqm: "w = w AND mask n"
and sz: "n < len_of TYPE ('a)"
shows "w < (2::'a word) ^ n" by (subst eqm, rule and_mask_less' [OF sz]) | proof (prove)
goal (1 subgoal):
1. w < 2 ^ n | lemma eq_mask_less:
fixes w :: "'a::len word"
assumes eqm: "w = w AND mask n"
and sz: "n < len_of TYPE ('a)"
shows "w < (2::'a word) ^ n" | unnamed_thy_286 | More_Word | 1 | |
[] | lemma of_nat_mono_maybe':
fixes Y :: "nat"
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
assumes ylt: "y < 2 ^ len_of TYPE ('a)"
shows "(y < x) = (of_nat y < (of_nat x :: 'a :: len word))" apply (subst word_less_nat_alt) apply (subst unat_of_nat)+ apply (subst mod_less) apply (rule ylt) apply (subst mod_less) apply... | lemma of_nat_mono_maybe':
fixes Y :: "nat"
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
assumes ylt: "y < 2 ^ len_of TYPE ('a)"
shows "(y < x) = (of_nat y < (of_nat x :: 'a :: len word))" apply (subst word_less_nat_alt) apply (subst unat_of_nat)+ apply (subst mod_less) apply (rule ylt) apply (subst mod_less) apply... | proof (prove)
goal (1 subgoal):
1. (y < x) = (word_of_nat y < word_of_nat x) proof (prove)
goal (1 subgoal):
1. (y < x) = (unat (word_of_nat y) < unat (word_of_nat x)) proof (prove)
goal (1 subgoal):
1. (y < x) = (y mod 2 ^ LENGTH('a) < x mod 2 ^ LENGTH('a)) proof (prove)
goal (2 subgoals):
1. y < 2 ^ LENGTH('a)
2... | lemma of_nat_mono_maybe':
fixes Y :: "nat"
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
assumes ylt: "y < 2 ^ len_of TYPE ('a)"
shows "(y < x) = (of_nat y < (of_nat x :: 'a :: len word))" | unnamed_thy_287 | More_Word | 8 | |
[] | lemma of_nat_mono_maybe_le:
"\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
(y \<le> x) = ((of_nat y :: 'a :: len word) \<le> of_nat x)" apply (clarsimp simp: le_less) apply (rule disj_cong) apply (rule of_nat_mono_maybe', assumption+) apply auto using of_nat_inj apply blast done | lemma of_nat_mono_maybe_le:
"\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
(y \<le> x) = ((of_nat y :: 'a :: len word) \<le> of_nat x)" apply (clarsimp simp: le_less) apply (rule disj_cong) apply (rule of_nat_mono_maybe', assumption+) apply auto using of_nat_inj apply blast done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> (y \<le> x) = (word_of_nat y \<le> word_of_nat x) proof (prove)
goal (1 subgoal):
1. \<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> (y < x \<or> y = x) = (word_of_nat y < word_of_n... | lemma of_nat_mono_maybe_le:
"\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
(y \<le> x) = ((of_nat y :: 'a :: len word) \<le> of_nat x)" | unnamed_thy_288 | More_Word | 7 | |
[] | lemma mask_AND_NOT_mask:
"(w AND NOT (mask n)) AND mask n = 0"
for w :: \<open>'a::len word\<close> by (rule bit_word_eqI) (simp add: bit_simps) | lemma mask_AND_NOT_mask:
"(w AND NOT (mask n)) AND mask n = 0"
for w :: \<open>'a::len word\<close> by (rule bit_word_eqI) (simp add: bit_simps) | proof (prove)
goal (1 subgoal):
1. (w AND NOT (mask n)) AND mask n = 0 | lemma mask_AND_NOT_mask:
"(w AND NOT (mask n)) AND mask n = 0"
for w :: \<open>'a::len word\<close> | unnamed_thy_289 | More_Word | 1 | |
[] | lemma AND_NOT_mask_plus_AND_mask_eq:
"(w AND NOT (mask n)) + (w AND mask n) = w"
for w :: \<open>'a::len word\<close> apply (subst disjunctive_add) apply (auto simp add: bit_simps) apply (rule bit_word_eqI) apply (auto simp add: bit_simps) done | lemma AND_NOT_mask_plus_AND_mask_eq:
"(w AND NOT (mask n)) + (w AND mask n) = w"
for w :: \<open>'a::len word\<close> apply (subst disjunctive_add) apply (auto simp add: bit_simps) apply (rule bit_word_eqI) apply (auto simp add: bit_simps) done | proof (prove)
goal (1 subgoal):
1. (w AND NOT (mask n)) + (w AND mask n) = w proof (prove)
goal (2 subgoals):
1. \<And>na. \<not> bit (w AND NOT (mask n)) na \<or> \<not> bit (w AND mask n) na
2. w AND NOT (mask n) OR w AND mask n = w proof (prove)
goal (1 subgoal):
1. w AND NOT (mask n) OR w AND mask n = w proof (... | lemma AND_NOT_mask_plus_AND_mask_eq:
"(w AND NOT (mask n)) + (w AND mask n) = w"
for w :: \<open>'a::len word\<close> | unnamed_thy_290 | More_Word | 5 | |
[] | lemma mask_eqI:
fixes x :: "'a :: len word"
assumes m1: "x AND mask n = y AND mask n"
and m2: "x AND NOT (mask n) = y AND NOT (mask n)"
shows "x = y" proof - have *: \<open>x = x AND mask n OR x AND NOT (mask n)\<close> for x :: \<open>'a word\<close> by (rule bit_word_eqI) (auto simp add: bit_simps) from a... | lemma mask_eqI:
fixes x :: "'a :: len word"
assumes m1: "x AND mask n = y AND mask n"
and m2: "x AND NOT (mask n) = y AND NOT (mask n)"
shows "x = y" proof - have *: \<open>x = x AND mask n OR x AND NOT (mask n)\<close> for x :: \<open>'a word\<close> by (rule bit_word_eqI) (auto simp add: bit_simps) from a... | proof (prove)
goal (1 subgoal):
1. x = y proof (state)
goal (1 subgoal):
1. x = y proof (prove)
goal (1 subgoal):
1. x = x AND mask n OR x AND NOT (mask n) proof (state)
this:
?x = ?x AND mask n OR ?x AND NOT (mask n)
goal (1 subgoal):
1. x = y proof (chain)
picking this:
x AND mask n = y AND mask n
x AND NOT (mas... | lemma mask_eqI:
fixes x :: "'a :: len word"
assumes m1: "x AND mask n = y AND mask n"
and m2: "x AND NOT (mask n) = y AND NOT (mask n)"
shows "x = y" | unnamed_thy_291 | More_Word | 7 | |
[] | lemma neq_0_no_wrap:
fixes x :: "'a :: len word"
shows "\<lbrakk> x \<le> x + y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0" by clarsimp | lemma neq_0_no_wrap:
fixes x :: "'a :: len word"
shows "\<lbrakk> x \<le> x + y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0" by clarsimp | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x \<le> x + y; x \<noteq> 0\<rbrakk> \<Longrightarrow> x + y \<noteq> 0 | lemma neq_0_no_wrap:
fixes x :: "'a :: len word"
shows "\<lbrakk> x \<le> x + y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0" | unnamed_thy_292 | More_Word | 1 | |
[] | lemma word_of_nat_le:
"n \<le> unat x \<Longrightarrow> of_nat n \<le> x" apply (simp add: word_le_nat_alt unat_of_nat) apply (erule order_trans[rotated]) apply (simp add: take_bit_eq_mod) done | lemma word_of_nat_le:
"n \<le> unat x \<Longrightarrow> of_nat n \<le> x" apply (simp add: word_le_nat_alt unat_of_nat) apply (erule order_trans[rotated]) apply (simp add: take_bit_eq_mod) done | proof (prove)
goal (1 subgoal):
1. n \<le> unat x \<Longrightarrow> word_of_nat n \<le> x proof (prove)
goal (1 subgoal):
1. n \<le> unat x \<Longrightarrow> n mod 2 ^ LENGTH('a) \<le> unat x proof (prove)
goal (1 subgoal):
1. n mod 2 ^ LENGTH('a) \<le> n proof (prove)
goal:
No subgoals! | lemma word_of_nat_le:
"n \<le> unat x \<Longrightarrow> of_nat n \<le> x" | unnamed_thy_294 | More_Word | 4 | |
[] | lemma word_unat_less_le:
"a \<le> of_nat b \<Longrightarrow> unat a \<le> b" by (metis eq_iff le_cases le_unat_uoi word_of_nat_le) | lemma word_unat_less_le:
"a \<le> of_nat b \<Longrightarrow> unat a \<le> b" by (metis eq_iff le_cases le_unat_uoi word_of_nat_le) | proof (prove)
goal (1 subgoal):
1. a \<le> word_of_nat b \<Longrightarrow> unat a \<le> b | lemma word_unat_less_le:
"a \<le> of_nat b \<Longrightarrow> unat a \<le> b" | unnamed_thy_295 | More_Word | 1 | |
[] | lemma mask_Suc_0 : "mask (Suc 0) = (1 :: 'a::len word)" by (simp add: mask_eq_decr_exp) | lemma mask_Suc_0 : "mask (Suc 0) = (1 :: 'a::len word)" by (simp add: mask_eq_decr_exp) | proof (prove)
goal (1 subgoal):
1. mask (Suc 0) = 1 | lemma mask_Suc_0 : "mask (Suc 0) = (1 :: 'a::len word)" | unnamed_thy_296 | More_Word | 1 | |
[] | lemma bool_mask':
fixes x :: "'a :: len word"
shows "2 < LENGTH('a) \<Longrightarrow> (0 < x AND 1) = (x AND 1 = 1)" by (simp add: and_one_eq mod_2_eq_odd) | lemma bool_mask':
fixes x :: "'a :: len word"
shows "2 < LENGTH('a) \<Longrightarrow> (0 < x AND 1) = (x AND 1 = 1)" by (simp add: and_one_eq mod_2_eq_odd) | proof (prove)
goal (1 subgoal):
1. 2 < LENGTH('a) \<Longrightarrow> (0 < x AND 1) = (x AND 1 = 1) | lemma bool_mask':
fixes x :: "'a :: len word"
shows "2 < LENGTH('a) \<Longrightarrow> (0 < x AND 1) = (x AND 1 = 1)" | unnamed_thy_297 | More_Word | 1 | |
[] | lemma ucast_ucast_add:
fixes x :: "'a :: len word"
fixes y :: "'b :: len word"
shows
"LENGTH('b) \<ge> LENGTH('a) \<Longrightarrow>
ucast (ucast x + y) = x + ucast y" apply transfer apply simp apply (subst (2) take_bit_add [symmetric]) apply (subst take_bit_add [symmetric]) apply simp done | lemma ucast_ucast_add:
fixes x :: "'a :: len word"
fixes y :: "'b :: len word"
shows
"LENGTH('b) \<ge> LENGTH('a) \<Longrightarrow>
ucast (ucast x + y) = x + ucast y" apply transfer apply simp apply (subst (2) take_bit_add [symmetric]) apply (subst take_bit_add [symmetric]) apply simp done | proof (prove)
goal (1 subgoal):
1. LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> ucast (ucast x + y) = x + ucast y proof (prove)
goal (1 subgoal):
1. \<And>x y. LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> take_bit LENGTH('a) (take_bit LENGTH('b) (take_bit LENGTH('a) x + y)) = take_bit LENGTH('a) (x + take_bit LENGT... | lemma ucast_ucast_add:
fixes x :: "'a :: len word"
fixes y :: "'b :: len word"
shows
"LENGTH('b) \<ge> LENGTH('a) \<Longrightarrow>
ucast (ucast x + y) = x + ucast y" | unnamed_thy_298 | More_Word | 6 | |
[] | lemma lt1_neq0:
fixes x :: "'a :: len word"
shows "(1 \<le> x) = (x \<noteq> 0)" by unat_arith | lemma lt1_neq0:
fixes x :: "'a :: len word"
shows "(1 \<le> x) = (x \<noteq> 0)" by unat_arith | proof (prove)
goal (1 subgoal):
1. (1 \<le> x) = (x \<noteq> 0) | lemma lt1_neq0:
fixes x :: "'a :: len word"
shows "(1 \<le> x) = (x \<noteq> 0)" | unnamed_thy_299 | More_Word | 1 | |
[] | lemma word_plus_one_nonzero:
fixes x :: "'a :: len word"
shows "\<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> x + 1 \<noteq> 0" apply (subst lt1_neq0 [symmetric]) apply (subst olen_add_eqv [symmetric]) apply (erule word_random) apply (simp add: lt1_neq0) done | lemma word_plus_one_nonzero:
fixes x :: "'a :: len word"
shows "\<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> x + 1 \<noteq> 0" apply (subst lt1_neq0 [symmetric]) apply (subst olen_add_eqv [symmetric]) apply (erule word_random) apply (simp add: lt1_neq0) done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> x + 1 \<noteq> 0 proof (prove)
goal (1 subgoal):
1. \<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 \<le> x + 1 proof (prove)
goal (1 subgoal):
1. \<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longr... | lemma word_plus_one_nonzero:
fixes x :: "'a :: len word"
shows "\<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> x + 1 \<noteq> 0" | unnamed_thy_300 | More_Word | 5 | |
[] | lemma word_sub_plus_one_nonzero:
fixes n :: "'a :: len word"
shows "\<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> (n - n') + 1 \<noteq> 0" apply (subst lt1_neq0 [symmetric]) apply (subst olen_add_eqv [symmetric]) apply (rule word_random [where x' = n']) apply simp apply (erule word_sub_le) apply (si... | lemma word_sub_plus_one_nonzero:
fixes n :: "'a :: len word"
shows "\<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> (n - n') + 1 \<noteq> 0" apply (subst lt1_neq0 [symmetric]) apply (subst olen_add_eqv [symmetric]) apply (rule word_random [where x' = n']) apply simp apply (erule word_sub_le) apply (si... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> n - n' + 1 \<noteq> 0 proof (prove)
goal (1 subgoal):
1. \<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> 1 \<le> n - n' + 1 proof (prove)
goal (1 subgoal):
1. \<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<L... | lemma word_sub_plus_one_nonzero:
fixes n :: "'a :: len word"
shows "\<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> (n - n') + 1 \<noteq> 0" | unnamed_thy_301 | More_Word | 7 | |
[] | lemma word_le_minus_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk> z \<le> y; y \<le> x; z \<le> x \<rbrakk> \<Longrightarrow> x - y \<le> x - z" apply (rule word_sub_mono) apply simp apply assumption apply (erule word_sub_le) apply (erule word_sub_le) done | lemma word_le_minus_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk> z \<le> y; y \<le> x; z \<le> x \<rbrakk> \<Longrightarrow> x - y \<le> x - z" apply (rule word_sub_mono) apply simp apply assumption apply (erule word_sub_le) apply (erule word_sub_le) done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>z \<le> y; y \<le> x; z \<le> x\<rbrakk> \<Longrightarrow> x - y \<le> x - z proof (prove)
goal (4 subgoals):
1. \<lbrakk>z \<le> y; y \<le> x; z \<le> x\<rbrakk> \<Longrightarrow> x \<le> x
2. \<lbrakk>z \<le> y; y \<le> x; z \<le> x\<rbrakk> \<Longrightarrow> z \<le> y
... | lemma word_le_minus_mono_right:
fixes x :: "'a :: len word"
shows "\<lbrakk> z \<le> y; y \<le> x; z \<le> x \<rbrakk> \<Longrightarrow> x - y \<le> x - z" | unnamed_thy_302 | More_Word | 6 | |
[] | lemma word_0_sle_from_less:
\<open>0 \<le>s x\<close> if \<open>x < 2 ^ (LENGTH('a) - 1)\<close> for x :: \<open>'a::len word\<close> using that apply transfer apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply (metis bit_take_bit_iff min_def nat_less_le not_less_eq take_bit_int_eq_self_iff take_bit_take_bi... | lemma word_0_sle_from_less:
\<open>0 \<le>s x\<close> if \<open>x < 2 ^ (LENGTH('a) - 1)\<close> for x :: \<open>'a::len word\<close> using that apply transfer apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply (metis bit_take_bit_iff min_def nat_less_le not_less_eq take_bit_int_eq_self_iff take_bit_take_bi... | proof (prove)
goal (1 subgoal):
1. 0 \<le>s x proof (prove)
using this:
x < 2 ^ (LENGTH('a) - 1)
goal (1 subgoal):
1. 0 \<le>s x proof (prove)
goal (1 subgoal):
1. \<And>x. take_bit LENGTH('a) x < take_bit LENGTH('a) (2 ^ (LENGTH('a) - 1)) \<Longrightarrow> signed_take_bit (LENGTH('a) - Suc 0) 0 \<le> signed_take_b... | lemma word_0_sle_from_less:
\<open>0 \<le>s x\<close> if \<open>x < 2 ^ (LENGTH('a) - 1)\<close> for x :: \<open>'a::len word\<close> | unnamed_thy_303 | More_Word | 6 | |
[] | lemma ucast_sub_ucast:
fixes x :: "'a::len word"
assumes "y \<le> x"
assumes T: "LENGTH('a) \<le> LENGTH('b)"
shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)" proof - from T have P: "unat x < 2 ^ LENGTH('b)" "unat y < 2 ^ LENGTH('b)" by (fastforce intro!: less_le_trans[OF unat_lt2p])+ then show ?thes... | lemma ucast_sub_ucast:
fixes x :: "'a::len word"
assumes "y \<le> x"
assumes T: "LENGTH('a) \<le> LENGTH('b)"
shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)" proof - from T have P: "unat x < 2 ^ LENGTH('b)" "unat y < 2 ^ LENGTH('b)" by (fastforce intro!: less_le_trans[OF unat_lt2p])+ then show ?thes... | proof (prove)
goal (1 subgoal):
1. ucast (x - y) = ucast x - ucast y proof (state)
goal (1 subgoal):
1. ucast (x - y) = ucast x - ucast y proof (chain)
picking this:
LENGTH('a) \<le> LENGTH('b) proof (prove)
using this:
LENGTH('a) \<le> LENGTH('b)
goal (1 subgoal):
1. unat x < 2 ^ LENGTH('b) &&& unat y < 2 ^ LENGTH... | lemma ucast_sub_ucast:
fixes x :: "'a::len word"
assumes "y \<le> x"
assumes T: "LENGTH('a) \<le> LENGTH('b)"
shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)" | unnamed_thy_304 | More_Word | 8 | |
[] | lemma word_1_0:
"\<lbrakk>a + (1::('a::len) word) \<le> b; a < of_nat x\<rbrakk> \<Longrightarrow> a < b" apply transfer apply (subst (asm) take_bit_incr_eq) apply (auto simp add: diff_less_eq) using take_bit_int_less_exp le_less_trans by blast | lemma word_1_0:
"\<lbrakk>a + (1::('a::len) word) \<le> b; a < of_nat x\<rbrakk> \<Longrightarrow> a < b" apply transfer apply (subst (asm) take_bit_incr_eq) apply (auto simp add: diff_less_eq) using take_bit_int_less_exp le_less_trans by blast | proof (prove)
goal (1 subgoal):
1. \<lbrakk>a + 1 \<le> b; a < word_of_nat x\<rbrakk> \<Longrightarrow> a < b proof (prove)
goal (1 subgoal):
1. \<And>a b x. \<lbrakk>take_bit LENGTH('a) (a + 1) \<le> take_bit LENGTH('a) b; take_bit LENGTH('a) a < take_bit LENGTH('a) (int x)\<rbrakk> \<Longrightarrow> take_bit LENGTH... | lemma word_1_0:
"\<lbrakk>a + (1::('a::len) word) \<le> b; a < of_nat x\<rbrakk> \<Longrightarrow> a < b" | unnamed_thy_305 | More_Word | 5 | |
[] | lemma unat_of_nat_less:"\<lbrakk> a < b; unat b = c \<rbrakk> \<Longrightarrow> a < of_nat c" by fastforce | lemma unat_of_nat_less:"\<lbrakk> a < b; unat b = c \<rbrakk> \<Longrightarrow> a < of_nat c" by fastforce | proof (prove)
goal (1 subgoal):
1. \<lbrakk>a < b; unat b = c\<rbrakk> \<Longrightarrow> a < word_of_nat c | lemma unat_of_nat_less:"\<lbrakk> a < b; unat b = c \<rbrakk> \<Longrightarrow> a < of_nat c" | unnamed_thy_306 | More_Word | 1 | |
[] | lemma word_le_plus_1: "\<lbrakk> (y::('a::len) word) < y + n; a < n \<rbrakk> \<Longrightarrow> y + a \<le> y + a + 1" by unat_arith | lemma word_le_plus_1: "\<lbrakk> (y::('a::len) word) < y + n; a < n \<rbrakk> \<Longrightarrow> y + a \<le> y + a + 1" by unat_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>y < y + n; a < n\<rbrakk> \<Longrightarrow> y + a \<le> y + a + 1 | lemma word_le_plus_1: "\<lbrakk> (y::('a::len) word) < y + n; a < n \<rbrakk> \<Longrightarrow> y + a \<le> y + a + 1" | unnamed_thy_307 | More_Word | 1 | |
[] | lemma word_le_plus:"\<lbrakk>(a::('a::len) word) < a + b; c < b\<rbrakk> \<Longrightarrow> a \<le> a + c" by (metis order_less_imp_le word_random) | lemma word_le_plus:"\<lbrakk>(a::('a::len) word) < a + b; c < b\<rbrakk> \<Longrightarrow> a \<le> a + c" by (metis order_less_imp_le word_random) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>a < a + b; c < b\<rbrakk> \<Longrightarrow> a \<le> a + c | lemma word_le_plus:"\<lbrakk>(a::('a::len) word) < a + b; c < b\<rbrakk> \<Longrightarrow> a \<le> a + c" | unnamed_thy_308 | More_Word | 1 | |
[] | lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply transfer apply (simp only: flip: signed_take_bit_eq_iff_take_bit_eq) apply simp done | lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply transfer apply (simp only: flip: signed_take_bit_eq_iff_take_bit_eq) apply simp done | proof (prove)
goal (1 subgoal):
1. (sint x = - 1) = (x = - 1) proof (prove)
goal (2 subgoals):
1. LENGTH('a) = 0 \<Longrightarrow> (sint x = - 1) = (x = - 1)
2. \<And>nat. LENGTH('a) = Suc nat \<Longrightarrow> (sint x = - 1) = (x = - 1) proof (prove)
goal (1 subgoal):
1. \<And>nat. LENGTH('a) = Suc nat \<Longright... | lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)" | unnamed_thy_309 | More_Word | 6 | |
[] | lemma sint_0 [simp]: "(sint x = 0) = (x = 0)" by (fact signed_eq_0_iff) | lemma sint_0 [simp]: "(sint x = 0) = (x = 0)" by (fact signed_eq_0_iff) | proof (prove)
goal (1 subgoal):
1. (sint x = 0) = (x = 0) | lemma sint_0 [simp]: "(sint x = 0) = (x = 0)" | unnamed_thy_310 | More_Word | 1 | |
[] | lemma sint_1_cases:
P if \<open>\<lbrakk> len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 \<rbrakk> \<Longrightarrow> P\<close>
\<open>\<lbrakk> len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 \<rbrakk> \<Longrightarrow> P\<close>
\<open>\<lbrakk> len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 ... | lemma sint_1_cases:
P if \<open>\<lbrakk> len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 \<rbrakk> \<Longrightarrow> P\<close>
\<open>\<lbrakk> len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 \<rbrakk> \<Longrightarrow> P\<close>
\<open>\<lbrakk> len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 ... | proof (prove)
goal (1 subgoal):
1. P proof (state)
goal (2 subgoals):
1. LENGTH('a) = 1 \<Longrightarrow> P
2. LENGTH('a) \<noteq> 1 \<Longrightarrow> P proof (state)
this:
LENGTH('a) = 1
goal (2 subgoals):
1. LENGTH('a) = 1 \<Longrightarrow> P
2. LENGTH('a) \<noteq> 1 \<Longrightarrow> P proof (chain)
picking th... | lemma sint_1_cases:
P if \<open>\<lbrakk> len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 \<rbrakk> \<Longrightarrow> P\<close>
\<open>\<lbrakk> len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 \<rbrakk> \<Longrightarrow> P\<close>
\<open>\<lbrakk> len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 ... | unnamed_thy_311 | More_Word | 14 | |
[] | lemma sint_int_min:
"sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply transfer apply (simp add: signed_take_bit_int_eq_self) done | lemma sint_int_min:
"sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply transfer apply (simp add: signed_take_bit_int_eq_self) done | proof (prove)
goal (1 subgoal):
1. sint (- (2 ^ (LENGTH('a) - Suc 0))) = - (2 ^ (LENGTH('a) - Suc 0)) proof (prove)
goal (2 subgoals):
1. LENGTH('a) = 0 \<Longrightarrow> sint (- (2 ^ (LENGTH('a) - Suc 0))) = - (2 ^ (LENGTH('a) - Suc 0))
2. \<And>nat. LENGTH('a) = Suc nat \<Longrightarrow> sint (- (2 ^ (LENGTH('a) -... | lemma sint_int_min:
"sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))" | unnamed_thy_312 | More_Word | 5 | |
[] | lemma sint_int_max_plus_1:
"sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply (subst word_of_int_2p [symmetric]) apply (subst int_word_sint) apply simp done | lemma sint_int_max_plus_1:
"sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))" apply (cases \<open>LENGTH('a)\<close>) apply simp_all apply (subst word_of_int_2p [symmetric]) apply (subst int_word_sint) apply simp done | proof (prove)
goal (1 subgoal):
1. sint (2 ^ (LENGTH('a) - Suc 0)) = - (2 ^ (LENGTH('a) - Suc 0)) proof (prove)
goal (2 subgoals):
1. LENGTH('a) = 0 \<Longrightarrow> sint (2 ^ (LENGTH('a) - Suc 0)) = - (2 ^ (LENGTH('a) - Suc 0))
2. \<And>nat. LENGTH('a) = Suc nat \<Longrightarrow> sint (2 ^ (LENGTH('a) - Suc 0)) = ... | lemma sint_int_max_plus_1:
"sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))" | unnamed_thy_313 | More_Word | 6 | |
[] | lemma uint_range':
\<open>0 \<le> uint x \<and> uint x < 2 ^ LENGTH('a)\<close>
for x :: \<open>'a::len word\<close> by transfer simp | lemma uint_range':
\<open>0 \<le> uint x \<and> uint x < 2 ^ LENGTH('a)\<close>
for x :: \<open>'a::len word\<close> by transfer simp | proof (prove)
goal (1 subgoal):
1. 0 \<le> uint x \<and> uint x < 2 ^ LENGTH('a) | lemma uint_range':
\<open>0 \<le> uint x \<and> uint x < 2 ^ LENGTH('a)\<close>
for x :: \<open>'a::len word\<close> | unnamed_thy_314 | More_Word | 1 | |
[] | lemma sint_of_int_eq:
"\<lbrakk> - (2 ^ (LENGTH('a) - 1)) \<le> x; x < 2 ^ (LENGTH('a) - 1) \<rbrakk> \<Longrightarrow> sint (of_int x :: ('a::len) word) = x" by (simp add: signed_take_bit_int_eq_self signed_of_int) | lemma sint_of_int_eq:
"\<lbrakk> - (2 ^ (LENGTH('a) - 1)) \<le> x; x < 2 ^ (LENGTH('a) - 1) \<rbrakk> \<Longrightarrow> sint (of_int x :: ('a::len) word) = x" by (simp add: signed_take_bit_int_eq_self signed_of_int) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>- (2 ^ (LENGTH('a) - 1)) \<le> x; x < 2 ^ (LENGTH('a) - 1)\<rbrakk> \<Longrightarrow> sint (word_of_int x) = x | lemma sint_of_int_eq:
"\<lbrakk> - (2 ^ (LENGTH('a) - 1)) \<le> x; x < 2 ^ (LENGTH('a) - 1) \<rbrakk> \<Longrightarrow> sint (of_int x :: ('a::len) word) = x" | unnamed_thy_315 | More_Word | 1 | |
[] | lemma of_int_sint:
"of_int (sint a) = a" by simp | lemma of_int_sint:
"of_int (sint a) = a" by simp | proof (prove)
goal (1 subgoal):
1. word_of_int (sint a) = a | lemma of_int_sint:
"of_int (sint a) = a" | unnamed_thy_316 | More_Word | 1 | |
[] | lemma sint_ucast_eq_uint:
"\<lbrakk> \<not> is_down (ucast :: ('a::len word \<Rightarrow> 'b::len word)) \<rbrakk>
\<Longrightarrow> sint ((ucast :: ('a::len word \<Rightarrow> 'b::len word)) x) = uint x" apply transfer apply (simp add: signed_take_bit_take_bit) done | lemma sint_ucast_eq_uint:
"\<lbrakk> \<not> is_down (ucast :: ('a::len word \<Rightarrow> 'b::len word)) \<rbrakk>
\<Longrightarrow> sint ((ucast :: ('a::len word \<Rightarrow> 'b::len word)) x) = uint x" apply transfer apply (simp add: signed_take_bit_take_bit) done | proof (prove)
goal (1 subgoal):
1. \<not> is_down ucast \<Longrightarrow> sint (ucast x) = uint x proof (prove)
goal (1 subgoal):
1. \<And>x. \<not> LENGTH('b) \<le> LENGTH('a) \<Longrightarrow> signed_take_bit (LENGTH('b) - Suc 0) (take_bit LENGTH('a) x) = take_bit LENGTH('a) x proof (prove)
goal:
No subgoals! | lemma sint_ucast_eq_uint:
"\<lbrakk> \<not> is_down (ucast :: ('a::len word \<Rightarrow> 'b::len word)) \<rbrakk>
\<Longrightarrow> sint ((ucast :: ('a::len word \<Rightarrow> 'b::len word)) x) = uint x" | unnamed_thy_317 | More_Word | 3 |
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