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 and_mask_arith:
"w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: \<open>'a::len word\<close> by (rule bit_word_eqI)
(auto simp add: bit_simps word_size simp flip: push_bit_eq_mult drop_bit_eq_div) | lemma and_mask_arith:
"w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: \<open>'a::len word\<close> by (rule bit_word_eqI)
(auto simp add: bit_simps word_size simp flip: push_bit_eq_mult drop_bit_eq_div) | proof (prove)
goal (1 subgoal):
1. w AND mask n = w * 2 ^ (size w - n) div 2 ^ (size w - n) | lemma and_mask_arith:
"w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: \<open>'a::len word\<close> | unnamed_thy_107 | More_Word | 1 | |
[] | lemma and_mask_arith':
"0 < n \<Longrightarrow> w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: \<open>'a::len word\<close> by (rule and_mask_arith) | lemma and_mask_arith':
"0 < n \<Longrightarrow> w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: \<open>'a::len word\<close> by (rule and_mask_arith) | proof (prove)
goal (1 subgoal):
1. 0 < n \<Longrightarrow> w AND mask n = w * 2 ^ (size w - n) div 2 ^ (size w - n) | lemma and_mask_arith':
"0 < n \<Longrightarrow> w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: \<open>'a::len word\<close> | unnamed_thy_108 | More_Word | 1 | |
[] | lemma mask_2pm1: "mask n = 2 ^ n - (1 :: 'a::len word)" by (fact mask_eq_decr_exp) | lemma mask_2pm1: "mask n = 2 ^ n - (1 :: 'a::len word)" by (fact mask_eq_decr_exp) | proof (prove)
goal (1 subgoal):
1. mask n = 2 ^ n - 1 | lemma mask_2pm1: "mask n = 2 ^ n - (1 :: 'a::len word)" | unnamed_thy_109 | More_Word | 1 | |
[] | lemma word_and_mask_le_2pm1: "w AND mask n \<le> 2 ^ n - 1"
for w :: \<open>'a::len word\<close> by (simp add: mask_2pm1[symmetric] word_and_le1) | lemma word_and_mask_le_2pm1: "w AND mask n \<le> 2 ^ n - 1"
for w :: \<open>'a::len word\<close> by (simp add: mask_2pm1[symmetric] word_and_le1) | proof (prove)
goal (1 subgoal):
1. w AND mask n \<le> 2 ^ n - 1 | lemma word_and_mask_le_2pm1: "w AND mask n \<le> 2 ^ n - 1"
for w :: \<open>'a::len word\<close> | unnamed_thy_111 | More_Word | 1 | |
[] | lemma is_aligned_AND_less_0:
"u AND mask n = 0 \<Longrightarrow> v < 2^n \<Longrightarrow> u AND v = 0"
for u v :: \<open>'a::len word\<close> apply (drule less_mask_eq) apply (simp flip: take_bit_eq_mask) apply (simp add: bit_eq_iff) apply (auto simp add: bit_simps) done | lemma is_aligned_AND_less_0:
"u AND mask n = 0 \<Longrightarrow> v < 2^n \<Longrightarrow> u AND v = 0"
for u v :: \<open>'a::len word\<close> apply (drule less_mask_eq) apply (simp flip: take_bit_eq_mask) apply (simp add: bit_eq_iff) apply (auto simp add: bit_simps) done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>u AND mask n = 0; v < 2 ^ n\<rbrakk> \<Longrightarrow> u AND v = 0 proof (prove)
goal (1 subgoal):
1. \<lbrakk>u AND mask n = 0; v AND mask n = v\<rbrakk> \<Longrightarrow> u AND v = 0 proof (prove)
goal (1 subgoal):
1. \<lbrakk>take_bit n u = 0; take_bit n v = v\<rbrakk> ... | lemma is_aligned_AND_less_0:
"u AND mask n = 0 \<Longrightarrow> v < 2^n \<Longrightarrow> u AND v = 0"
for u v :: \<open>'a::len word\<close> | unnamed_thy_112 | More_Word | 5 | |
[] | lemma and_mask_eq_iff_le_mask:
\<open>w AND mask n = w \<longleftrightarrow> w \<le> mask n\<close>
for w :: \<open>'a::len word\<close> apply (simp flip: take_bit_eq_mask) apply (cases \<open>n \<ge> LENGTH('a)\<close>; transfer) apply (simp_all add: not_le min_def) apply (simp_all add: mask_eq_exp_minus_1) apply ... | lemma and_mask_eq_iff_le_mask:
\<open>w AND mask n = w \<longleftrightarrow> w \<le> mask n\<close>
for w :: \<open>'a::len word\<close> apply (simp flip: take_bit_eq_mask) apply (cases \<open>n \<ge> LENGTH('a)\<close>; transfer) apply (simp_all add: not_le min_def) apply (simp_all add: mask_eq_exp_minus_1) apply ... | proof (prove)
goal (1 subgoal):
1. (w AND mask n = w) = (w \<le> mask n) proof (prove)
goal (1 subgoal):
1. (take_bit n w = w) = (w \<le> mask n) proof (prove)
goal (2 subgoals):
1. \<And>n w. LENGTH('a) \<le> n \<Longrightarrow> (take_bit LENGTH('a) (take_bit (min LENGTH('a) n) w) = take_bit LENGTH('a) w) = (take_b... | lemma and_mask_eq_iff_le_mask:
\<open>w AND mask n = w \<longleftrightarrow> w \<le> mask n\<close>
for w :: \<open>'a::len word\<close> | unnamed_thy_113 | More_Word | 8 | |
[] | lemma less_eq_mask_iff_take_bit_eq_self:
\<open>w \<le> mask n \<longleftrightarrow> take_bit n w = w\<close>
for w :: \<open>'a::len word\<close> by (simp add: and_mask_eq_iff_le_mask take_bit_eq_mask) | lemma less_eq_mask_iff_take_bit_eq_self:
\<open>w \<le> mask n \<longleftrightarrow> take_bit n w = w\<close>
for w :: \<open>'a::len word\<close> by (simp add: and_mask_eq_iff_le_mask take_bit_eq_mask) | proof (prove)
goal (1 subgoal):
1. (w \<le> mask n) = (take_bit n w = w) | lemma less_eq_mask_iff_take_bit_eq_self:
\<open>w \<le> mask n \<longleftrightarrow> take_bit n w = w\<close>
for w :: \<open>'a::len word\<close> | unnamed_thy_114 | More_Word | 1 | |
[] | lemma NOT_eq:
"NOT (x :: 'a :: len word) = - x - 1" by (fact not_eq_complement) | lemma NOT_eq:
"NOT (x :: 'a :: len word) = - x - 1" by (fact not_eq_complement) | proof (prove)
goal (1 subgoal):
1. NOT x = - x - 1 | lemma NOT_eq:
"NOT (x :: 'a :: len word) = - x - 1" | unnamed_thy_115 | More_Word | 1 | |
[] | lemma NOT_mask: "NOT (mask n :: 'a::len word) = - (2 ^ n)" by (simp add : NOT_eq mask_2pm1) | lemma NOT_mask: "NOT (mask n :: 'a::len word) = - (2 ^ n)" by (simp add : NOT_eq mask_2pm1) | proof (prove)
goal (1 subgoal):
1. NOT (mask n) = - (2 ^ n) | lemma NOT_mask: "NOT (mask n :: 'a::len word) = - (2 ^ n)" | unnamed_thy_116 | More_Word | 1 | |
[] | lemma le_m1_iff_lt: "(x > (0 :: 'a :: len word)) = ((y \<le> x - 1) = (y < x))" by uint_arith | lemma le_m1_iff_lt: "(x > (0 :: 'a :: len word)) = ((y \<le> x - 1) = (y < x))" by uint_arith | proof (prove)
goal (1 subgoal):
1. (0 < x) = ((y \<le> x - 1) = (y < x)) | lemma le_m1_iff_lt: "(x > (0 :: 'a :: len word)) = ((y \<le> x - 1) = (y < x))" | unnamed_thy_117 | More_Word | 1 | |
[] | lemma gt0_iff_gem1:
\<open>0 < x \<longleftrightarrow> x - 1 < x\<close>
for x :: \<open>'a::len word\<close> by (metis add.right_neutral diff_add_cancel less_irrefl measure_unat unat_arith_simps(2) word_neq_0_conv word_sub_less_iff) | lemma gt0_iff_gem1:
\<open>0 < x \<longleftrightarrow> x - 1 < x\<close>
for x :: \<open>'a::len word\<close> by (metis add.right_neutral diff_add_cancel less_irrefl measure_unat unat_arith_simps(2) word_neq_0_conv word_sub_less_iff) | proof (prove)
goal (1 subgoal):
1. (0 < x) = (x - 1 < x) | lemma gt0_iff_gem1:
\<open>0 < x \<longleftrightarrow> x - 1 < x\<close>
for x :: \<open>'a::len word\<close> | unnamed_thy_118 | More_Word | 1 | |
[] | lemma power_2_ge_iff:
\<open>2 ^ n - (1 :: 'a::len word) < 2 ^ n \<longleftrightarrow> n < LENGTH('a)\<close> using gt0_iff_gem1 p2_gt_0 by blast | lemma power_2_ge_iff:
\<open>2 ^ n - (1 :: 'a::len word) < 2 ^ n \<longleftrightarrow> n < LENGTH('a)\<close> using gt0_iff_gem1 p2_gt_0 by blast | proof (prove)
goal (1 subgoal):
1. (2 ^ n - 1 < 2 ^ n) = (n < LENGTH('a)) proof (prove)
using this:
(0 < ?x) = (?x - 1 < ?x)
(0 < 2 ^ ?n) = (?n < LENGTH(?'a))
goal (1 subgoal):
1. (2 ^ n - 1 < 2 ^ n) = (n < LENGTH('a)) | lemma power_2_ge_iff:
\<open>2 ^ n - (1 :: 'a::len word) < 2 ^ n \<longleftrightarrow> n < LENGTH('a)\<close> | unnamed_thy_119 | More_Word | 2 | |
[] | lemma le_mask_iff_lt_2n:
"n < len_of TYPE ('a) = (((w :: 'a :: len word) \<le> mask n) = (w < 2 ^ n))" unfolding mask_2pm1 by (rule trans [OF p2_gt_0 [THEN sym] le_m1_iff_lt]) | lemma le_mask_iff_lt_2n:
"n < len_of TYPE ('a) = (((w :: 'a :: len word) \<le> mask n) = (w < 2 ^ n))" unfolding mask_2pm1 by (rule trans [OF p2_gt_0 [THEN sym] le_m1_iff_lt]) | proof (prove)
goal (1 subgoal):
1. (n < LENGTH('a)) = ((w \<le> mask n) = (w < 2 ^ n)) proof (prove)
goal (1 subgoal):
1. (n < LENGTH('a)) = ((w \<le> 2 ^ n - 1) = (w < 2 ^ n)) | lemma le_mask_iff_lt_2n:
"n < len_of TYPE ('a) = (((w :: 'a :: len word) \<le> mask n) = (w < 2 ^ n))" | unnamed_thy_120 | More_Word | 2 | |
[] | lemma mask_lt_2pn:
\<open>n < LENGTH('a) \<Longrightarrow> mask n < (2 :: 'a::len word) ^ n\<close> by (simp add: mask_eq_exp_minus_1 power_2_ge_iff) | lemma mask_lt_2pn:
\<open>n < LENGTH('a) \<Longrightarrow> mask n < (2 :: 'a::len word) ^ n\<close> by (simp add: mask_eq_exp_minus_1 power_2_ge_iff) | proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> mask n < 2 ^ n | lemma mask_lt_2pn:
\<open>n < LENGTH('a) \<Longrightarrow> mask n < (2 :: 'a::len word) ^ n\<close> | unnamed_thy_121 | More_Word | 1 | |
[] | lemma word_unat_power:
"(2 :: 'a :: len word) ^ n = of_nat (2 ^ n)" by simp | lemma word_unat_power:
"(2 :: 'a :: len word) ^ n = of_nat (2 ^ n)" by simp | proof (prove)
goal (1 subgoal):
1. 2 ^ n = word_of_nat (2 ^ n) | lemma word_unat_power:
"(2 :: 'a :: len word) ^ n = of_nat (2 ^ n)" | unnamed_thy_122 | More_Word | 1 | |
[] | lemma of_nat_mono_maybe:
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
shows "y < x \<Longrightarrow> 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 (erule order_less_trans [OF _ xlt]) apply (subst mod_less [OF xlt]) apply assumption done | lemma of_nat_mono_maybe:
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
shows "y < x \<Longrightarrow> 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 (erule order_less_trans [OF _ xlt]) apply (subst mod_less [OF xlt]) apply assumption done | proof (prove)
goal (1 subgoal):
1. y < x \<Longrightarrow> word_of_nat y < word_of_nat x proof (prove)
goal (1 subgoal):
1. y < x \<Longrightarrow> unat (word_of_nat y) < unat (word_of_nat x) proof (prove)
goal (1 subgoal):
1. y < x \<Longrightarrow> y mod 2 ^ LENGTH('a) < x mod 2 ^ LENGTH('a) proof (prove)
goal (2 ... | lemma of_nat_mono_maybe:
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
shows "y < x \<Longrightarrow> of_nat y < (of_nat x :: 'a :: len word)" | unnamed_thy_123 | More_Word | 7 | |
[] | lemma word_and_max_word:
fixes a::"'a::len word"
shows "x = - 1 \<Longrightarrow> a AND x = a" by simp | lemma word_and_max_word:
fixes a::"'a::len word"
shows "x = - 1 \<Longrightarrow> a AND x = a" by simp | proof (prove)
goal (1 subgoal):
1. x = - 1 \<Longrightarrow> a AND x = a | lemma word_and_max_word:
fixes a::"'a::len word"
shows "x = - 1 \<Longrightarrow> a AND x = a" | unnamed_thy_124 | More_Word | 1 | |
[] | lemma word_and_full_mask_simp:
\<open>x AND mask LENGTH('a) = x\<close> for x :: \<open>'a::len word\<close> by (simp add: bit_eq_iff bit_simps) | lemma word_and_full_mask_simp:
\<open>x AND mask LENGTH('a) = x\<close> for x :: \<open>'a::len word\<close> by (simp add: bit_eq_iff bit_simps) | proof (prove)
goal (1 subgoal):
1. x AND mask LENGTH('a) = x | lemma word_and_full_mask_simp:
\<open>x AND mask LENGTH('a) = x\<close> for x :: \<open>'a::len word\<close> | unnamed_thy_125 | More_Word | 1 | |
[] | lemma of_int_uint:
"of_int (uint x) = x" by (fact word_of_int_uint) | lemma of_int_uint:
"of_int (uint x) = x" by (fact word_of_int_uint) | proof (prove)
goal (1 subgoal):
1. word_of_int (uint x) = x | lemma of_int_uint:
"of_int (uint x) = x" | unnamed_thy_126 | More_Word | 1 | |
[] | lemma unat_mask_eq:
\<open>unat (mask n :: 'a::len word) = mask (min LENGTH('a) n)\<close> by transfer (simp add: nat_mask_eq) | lemma unat_mask_eq:
\<open>unat (mask n :: 'a::len word) = mask (min LENGTH('a) n)\<close> by transfer (simp add: nat_mask_eq) | proof (prove)
goal (1 subgoal):
1. unat (mask n) = mask (min LENGTH('a) n) | lemma unat_mask_eq:
\<open>unat (mask n :: 'a::len word) = mask (min LENGTH('a) n)\<close> | unnamed_thy_129 | More_Word | 1 | |
[] | lemma word_plus_mono_left:
fixes x :: "'a :: len word"
shows "\<lbrakk>y \<le> z; x \<le> x + z\<rbrakk> \<Longrightarrow> y + x \<le> z + x" by unat_arith | lemma word_plus_mono_left:
fixes x :: "'a :: len word"
shows "\<lbrakk>y \<le> z; x \<le> x + z\<rbrakk> \<Longrightarrow> y + x \<le> z + x" by unat_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>y \<le> z; x \<le> x + z\<rbrakk> \<Longrightarrow> y + x \<le> z + x | lemma word_plus_mono_left:
fixes x :: "'a :: len word"
shows "\<lbrakk>y \<le> z; x \<le> x + z\<rbrakk> \<Longrightarrow> y + x \<le> z + x" | unnamed_thy_130 | More_Word | 1 | |
[] | lemma less_Suc_unat_less_bound:
"n < Suc (unat (x :: 'a :: len word)) \<Longrightarrow> n < 2 ^ LENGTH('a)" by (auto elim!: order_less_le_trans intro: Suc_leI) | lemma less_Suc_unat_less_bound:
"n < Suc (unat (x :: 'a :: len word)) \<Longrightarrow> n < 2 ^ LENGTH('a)" by (auto elim!: order_less_le_trans intro: Suc_leI) | proof (prove)
goal (1 subgoal):
1. n < Suc (unat x) \<Longrightarrow> n < 2 ^ LENGTH('a) | lemma less_Suc_unat_less_bound:
"n < Suc (unat (x :: 'a :: len word)) \<Longrightarrow> n < 2 ^ LENGTH('a)" | unnamed_thy_131 | More_Word | 1 | |
[] | lemma up_ucast_inj:
"\<lbrakk> ucast x = (ucast y::'b::len word); LENGTH('a) \<le> len_of TYPE ('b) \<rbrakk> \<Longrightarrow> x = (y::'a::len word)" by transfer (simp add: min_def split: if_splits) | lemma up_ucast_inj:
"\<lbrakk> ucast x = (ucast y::'b::len word); LENGTH('a) \<le> len_of TYPE ('b) \<rbrakk> \<Longrightarrow> x = (y::'a::len word)" by transfer (simp add: min_def split: if_splits) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>ucast x = ucast y; LENGTH('a) \<le> LENGTH('b)\<rbrakk> \<Longrightarrow> x = y | lemma up_ucast_inj:
"\<lbrakk> ucast x = (ucast y::'b::len word); LENGTH('a) \<le> len_of TYPE ('b) \<rbrakk> \<Longrightarrow> x = (y::'a::len word)" | unnamed_thy_132 | More_Word | 1 | |
[] | lemma up_ucast_inj_eq:
"LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow> (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))" by (fastforce dest: up_ucast_inj) | lemma up_ucast_inj_eq:
"LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow> (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))" by (fastforce dest: up_ucast_inj) | proof (prove)
goal (1 subgoal):
1. LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> (ucast x = ucast y) = (x = y) | lemma up_ucast_inj_eq:
"LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow> (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))" | unnamed_thy_133 | More_Word | 1 | |
[] | lemma no_plus_overflow_neg:
"(x :: 'a :: len word) < -y \<Longrightarrow> x \<le> x + y" by (metis diff_minus_eq_add less_imp_le sub_wrap_lt) | lemma no_plus_overflow_neg:
"(x :: 'a :: len word) < -y \<Longrightarrow> x \<le> x + y" by (metis diff_minus_eq_add less_imp_le sub_wrap_lt) | proof (prove)
goal (1 subgoal):
1. x < - y \<Longrightarrow> x \<le> x + y | lemma no_plus_overflow_neg:
"(x :: 'a :: len word) < -y \<Longrightarrow> x \<le> x + y" | unnamed_thy_134 | More_Word | 1 | |
[] | lemma ucast_ucast_eq:
"\<lbrakk> ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) \<le> LENGTH('b);
LENGTH('b) \<le> LENGTH('c) \<rbrakk> \<Longrightarrow>
x = ucast y" for x :: "'a::len word" and y :: "'b::len word" apply transfer apply (cases \<open>LENGTH('c) = LENGTH('a)\<close>) apply (auto... | lemma ucast_ucast_eq:
"\<lbrakk> ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) \<le> LENGTH('b);
LENGTH('b) \<le> LENGTH('c) \<rbrakk> \<Longrightarrow>
x = ucast y" for x :: "'a::len word" and y :: "'b::len word" apply transfer apply (cases \<open>LENGTH('c) = LENGTH('a)\<close>) apply (auto... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>ucast x = ucast (ucast y); LENGTH('a) \<le> LENGTH('b); LENGTH('b) \<le> LENGTH('c)\<rbrakk> \<Longrightarrow> x = ucast y proof (prove)
goal (1 subgoal):
1. \<And>x y. \<lbrakk>take_bit LENGTH('c) (take_bit LENGTH('a) x) = take_bit LENGTH('c) (take_bit LENGTH('a) (take_bit... | lemma ucast_ucast_eq:
"\<lbrakk> ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) \<le> LENGTH('b);
LENGTH('b) \<le> LENGTH('c) \<rbrakk> \<Longrightarrow>
x = ucast y" for x :: "'a::len word" and y :: "'b::len word" | unnamed_thy_135 | More_Word | 4 | |
[] | lemma ucast_0_I:
"x = 0 \<Longrightarrow> ucast x = 0" by simp | lemma ucast_0_I:
"x = 0 \<Longrightarrow> ucast x = 0" by simp | proof (prove)
goal (1 subgoal):
1. x = 0 \<Longrightarrow> ucast x = 0 | lemma ucast_0_I:
"x = 0 \<Longrightarrow> ucast x = 0" | unnamed_thy_136 | More_Word | 1 | |
[] | lemma word_add_offset_less:
fixes x :: "'a :: len word"
assumes yv: "y < 2 ^ n"
and xv: "x < 2 ^ m"
and mnv: "sz < LENGTH('a :: len)"
and xv': "x < 2 ^ (LENGTH('a :: len) - n)"
and mn: "sz = m + n"
shows "x * 2 ^ n + y < 2 ^ sz" proof (subst mn) from mnv mn have nv: "n < LENGTH('a)" and m... | lemma word_add_offset_less:
fixes x :: "'a :: len word"
assumes yv: "y < 2 ^ n"
and xv: "x < 2 ^ m"
and mnv: "sz < LENGTH('a :: len)"
and xv': "x < 2 ^ (LENGTH('a :: len) - n)"
and mn: "sz = m + n"
shows "x * 2 ^ n + y < 2 ^ sz" proof (subst mn) from mnv mn have nv: "n < LENGTH('a)" and m... | proof (prove)
goal (1 subgoal):
1. x * 2 ^ n + y < 2 ^ sz proof (state)
goal (1 subgoal):
1. x * 2 ^ n + y < 2 ^ (m + n) proof (chain)
picking this:
sz < LENGTH('a)
sz = m + n proof (prove)
using this:
sz < LENGTH('a)
sz = m + n
goal (1 subgoal):
1. n < LENGTH('a) &&& m < LENGTH('a) proof (state)
this:
n < LENGTH('... | lemma word_add_offset_less:
fixes x :: "'a :: len word"
assumes yv: "y < 2 ^ n"
and xv: "x < 2 ^ m"
and mnv: "sz < LENGTH('a :: len)"
and xv': "x < 2 ^ (LENGTH('a :: len) - n)"
and mn: "sz = m + n"
shows "x * 2 ^ n + y < 2 ^ sz" | unnamed_thy_137 | More_Word | 28 | |
[] | lemma word_less_power_trans:
fixes n :: "'a :: len word"
assumes nv: "n < 2 ^ (m - k)"
and kv: "k \<le> m"
and mv: "m < len_of TYPE ('a)"
shows "2 ^ k * n < 2 ^ m" using nv kv mv apply - apply (subst word_less_nat_alt) apply (subst unat_word_ariths) apply (subst mod_less) apply simp apply (rule nat_le... | lemma word_less_power_trans:
fixes n :: "'a :: len word"
assumes nv: "n < 2 ^ (m - k)"
and kv: "k \<le> m"
and mv: "m < len_of TYPE ('a)"
shows "2 ^ k * n < 2 ^ m" using nv kv mv apply - apply (subst word_less_nat_alt) apply (subst unat_word_ariths) apply (subst mod_less) apply simp apply (rule nat_le... | proof (prove)
goal (1 subgoal):
1. 2 ^ k * n < 2 ^ m proof (prove)
using this:
n < 2 ^ (m - k)
k \<le> m
m < LENGTH('a)
goal (1 subgoal):
1. 2 ^ k * n < 2 ^ m proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> 2 ^ k * n < 2 ^ m proof (prove)
goal (1 sub... | lemma word_less_power_trans:
fixes n :: "'a :: len word"
assumes nv: "n < 2 ^ (m - k)"
and kv: "k \<le> m"
and mv: "m < len_of TYPE ('a)"
shows "2 ^ k * n < 2 ^ m" | unnamed_thy_138 | More_Word | 17 | |
[] | lemma word_less_power_trans2:
fixes n :: "'a::len word"
shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m" by (subst field_simps, rule word_less_power_trans) | lemma word_less_power_trans2:
fixes n :: "'a::len word"
shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m" by (subst field_simps, rule word_less_power_trans) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m | lemma word_less_power_trans2:
fixes n :: "'a::len word"
shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m" | unnamed_thy_139 | More_Word | 1 | |
[] | lemma Suc_unat_diff_1:
fixes x :: "'a :: len word"
assumes lt: "1 \<le> x"
shows "Suc (unat (x - 1)) = unat x" proof - have "0 < unat x" by (rule order_less_le_trans [where y = 1], simp, subst unat_1 [symmetric],
rule iffD1 [OF word_le_nat_alt lt]) then show ?thesis by ((subst unat_sub [OF lt])+, simp onl... | lemma Suc_unat_diff_1:
fixes x :: "'a :: len word"
assumes lt: "1 \<le> x"
shows "Suc (unat (x - 1)) = unat x" proof - have "0 < unat x" by (rule order_less_le_trans [where y = 1], simp, subst unat_1 [symmetric],
rule iffD1 [OF word_le_nat_alt lt]) then show ?thesis by ((subst unat_sub [OF lt])+, simp onl... | proof (prove)
goal (1 subgoal):
1. Suc (unat (x - 1)) = unat x proof (state)
goal (1 subgoal):
1. Suc (unat (x - 1)) = unat x proof (prove)
goal (1 subgoal):
1. 0 < unat x proof (state)
this:
0 < unat x
goal (1 subgoal):
1. Suc (unat (x - 1)) = unat x proof (chain)
picking this:
0 < unat x proof (prove)
using this... | lemma Suc_unat_diff_1:
fixes x :: "'a :: len word"
assumes lt: "1 \<le> x"
shows "Suc (unat (x - 1)) = unat x" | unnamed_thy_140 | More_Word | 7 | |
[] | lemma word_eq_unatI:
\<open>v = w\<close> if \<open>unat v = unat w\<close> using that by transfer (simp add: nat_eq_iff) | lemma word_eq_unatI:
\<open>v = w\<close> if \<open>unat v = unat w\<close> using that by transfer (simp add: nat_eq_iff) | proof (prove)
goal (1 subgoal):
1. v = w proof (prove)
using this:
unat v = unat w
goal (1 subgoal):
1. v = w | lemma word_eq_unatI:
\<open>v = w\<close> if \<open>unat v = unat w\<close> | unnamed_thy_141 | More_Word | 2 | |
[] | lemma word_mult_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k < j * k" proof - from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)" by (auto intro: order_less_subst2 simp: word_less_nat_alt elim: ... | lemma word_mult_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k < j * k" proof - from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)" by (auto intro: order_less_subst2 simp: word_less_nat_alt elim: ... | proof (prove)
goal (1 subgoal):
1. i * k < j * k proof (state)
goal (1 subgoal):
1. i * k < j * k proof (chain)
picking this:
i < j
unat j * unat k < 2 ^ LENGTH('a)
0 < k proof (prove)
using this:
i < j
unat j * unat k < 2 ^ LENGTH('a)
0 < k
goal (1 subgoal):
1. unat i * unat k < 2 ^ LENGTH('a) proof (state)
this:
... | lemma word_mult_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k < j * k" | unnamed_thy_143 | More_Word | 9 | |
[] | lemma word_mult_less_dest:
fixes i :: "'a :: len word"
assumes ij: "i * k < j * k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i < j" using uik ujk ij by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem] elim: mult_less_mono1) | lemma word_mult_less_dest:
fixes i :: "'a :: len word"
assumes ij: "i * k < j * k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i < j" using uik ujk ij by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem] elim: mult_less_mono1) | proof (prove)
goal (1 subgoal):
1. i < j proof (prove)
using this:
unat i * unat k < 2 ^ LENGTH('a)
unat j * unat k < 2 ^ LENGTH('a)
i * k < j * k
goal (1 subgoal):
1. i < j | lemma word_mult_less_dest:
fixes i :: "'a :: len word"
assumes ij: "i * k < j * k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i < j" | unnamed_thy_144 | More_Word | 2 | |
[] | lemma word_mult_less_cancel:
fixes k :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)" by (rule iffI [OF word_mult_less_dest [OF _ uik ujk] word_mult_less_mono1 [OF _ knz ujk]]) | lemma word_mult_less_cancel:
fixes k :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)" by (rule iffI [OF word_mult_less_dest [OF _ uik ujk] word_mult_less_mono1 [OF _ knz ujk]]) | proof (prove)
goal (1 subgoal):
1. (i * k < j * k) = (i < j) | lemma word_mult_less_cancel:
fixes k :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)" | unnamed_thy_145 | More_Word | 1 | |
[] | lemma Suc_div_unat_helper:
assumes szv: "sz < LENGTH('a :: len)"
and usszv: "us \<le> sz"
shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))" proof - note usv = order_le_less_trans [OF usszv szv] from usszv obtain q where qv: "sz = us + q" by (auto simp: le_iff_add) have "Suc (unat ((... | lemma Suc_div_unat_helper:
assumes szv: "sz < LENGTH('a :: len)"
and usszv: "us \<le> sz"
shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))" proof - note usv = order_le_less_trans [OF usszv szv] from usszv obtain q where qv: "sz = us + q" by (auto simp: le_iff_add) have "Suc (unat ((... | proof (prove)
goal (1 subgoal):
1. 2 ^ (sz - us) = Suc (unat ((2 ^ sz - 1) div 2 ^ us)) proof (state)
goal (1 subgoal):
1. 2 ^ (sz - us) = Suc (unat ((2 ^ sz - 1) div 2 ^ us)) proof (state)
this:
us < LENGTH('a)
goal (1 subgoal):
1. 2 ^ (sz - us) = Suc (unat ((2 ^ sz - 1) div 2 ^ us)) proof (chain)
picking this:
us... | lemma Suc_div_unat_helper:
assumes szv: "sz < LENGTH('a :: len)"
and usszv: "us \<le> sz"
shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))" | unnamed_thy_146 | More_Word | 27 | |
[] | lemma enum_word_nth_eq:
\<open>(Enum.enum :: 'a::len word list) ! n = word_of_nat n\<close>
if \<open>n < 2 ^ LENGTH('a)\<close>
for n using that by (simp add: enum_word_def) | lemma enum_word_nth_eq:
\<open>(Enum.enum :: 'a::len word list) ! n = word_of_nat n\<close>
if \<open>n < 2 ^ LENGTH('a)\<close>
for n using that by (simp add: enum_word_def) | proof (prove)
goal (1 subgoal):
1. enum_class.enum ! n = word_of_nat n proof (prove)
using this:
n < 2 ^ LENGTH('a)
goal (1 subgoal):
1. enum_class.enum ! n = word_of_nat n | lemma enum_word_nth_eq:
\<open>(Enum.enum :: 'a::len word list) ! n = word_of_nat n\<close>
if \<open>n < 2 ^ LENGTH('a)\<close>
for n | unnamed_thy_147 | More_Word | 2 | |
[] | lemma length_enum_word_eq:
\<open>length (Enum.enum :: 'a::len word list) = 2 ^ LENGTH('a)\<close> by (simp add: enum_word_def) | lemma length_enum_word_eq:
\<open>length (Enum.enum :: 'a::len word list) = 2 ^ LENGTH('a)\<close> by (simp add: enum_word_def) | proof (prove)
goal (1 subgoal):
1. length enum_class.enum = 2 ^ LENGTH('a) | lemma length_enum_word_eq:
\<open>length (Enum.enum :: 'a::len word list) = 2 ^ LENGTH('a)\<close> | unnamed_thy_148 | More_Word | 1 | |
[] | lemma unat_lt2p [iff]:
\<open>unat x < 2 ^ LENGTH('a)\<close> for x :: \<open>'a::len word\<close> by transfer simp | lemma unat_lt2p [iff]:
\<open>unat x < 2 ^ LENGTH('a)\<close> for x :: \<open>'a::len word\<close> by transfer simp | proof (prove)
goal (1 subgoal):
1. unat x < 2 ^ LENGTH('a) | lemma unat_lt2p [iff]:
\<open>unat x < 2 ^ LENGTH('a)\<close> for x :: \<open>'a::len word\<close> | unnamed_thy_149 | More_Word | 1 | |
[] | lemma Suc_unat_minus_one [simp]:
"x \<noteq> 0 \<Longrightarrow> Suc (unat (x - 1)) = unat x" by (metis Suc_diff_1 unat_gt_0 unat_minus_one) | lemma Suc_unat_minus_one [simp]:
"x \<noteq> 0 \<Longrightarrow> Suc (unat (x - 1)) = unat x" by (metis Suc_diff_1 unat_gt_0 unat_minus_one) | proof (prove)
goal (1 subgoal):
1. x \<noteq> 0 \<Longrightarrow> Suc (unat (x - 1)) = unat x | lemma Suc_unat_minus_one [simp]:
"x \<noteq> 0 \<Longrightarrow> Suc (unat (x - 1)) = unat x" | unnamed_thy_151 | More_Word | 1 | |
[] | lemma word_add_le_dest:
fixes i :: "'a :: len word"
assumes le: "i + k \<le> j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i \<le> j" using uik ujk le by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem] elim: add_le_mono1) | lemma word_add_le_dest:
fixes i :: "'a :: len word"
assumes le: "i + k \<le> j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i \<le> j" using uik ujk le by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem] elim: add_le_mono1) | proof (prove)
goal (1 subgoal):
1. i \<le> j proof (prove)
using this:
unat i + unat k < 2 ^ LENGTH('a)
unat j + unat k < 2 ^ LENGTH('a)
i + k \<le> j + k
goal (1 subgoal):
1. i \<le> j | lemma word_add_le_dest:
fixes i :: "'a :: len word"
assumes le: "i + k \<le> j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i \<le> j" | unnamed_thy_152 | More_Word | 2 | |
[] | lemma word_add_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k \<le> j + k" proof - from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)" by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: add_le_mono1) then sh... | lemma word_add_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k \<le> j + k" proof - from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)" by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: add_le_mono1) then sh... | proof (prove)
goal (1 subgoal):
1. i + k \<le> j + k proof (state)
goal (1 subgoal):
1. i + k \<le> j + k proof (chain)
picking this:
i \<le> j
unat j + unat k < 2 ^ LENGTH('a) proof (prove)
using this:
i \<le> j
unat j + unat k < 2 ^ LENGTH('a)
goal (1 subgoal):
1. unat i + unat k < 2 ^ LENGTH('a) proof (state)
th... | lemma word_add_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k \<le> j + k" | unnamed_thy_153 | More_Word | 9 | |
[] | lemma word_add_le_mono2:
fixes i :: "'a :: len word"
shows "\<lbrakk>i \<le> j; unat j + unat k < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> k + i \<le> k + j" by (subst field_simps, subst field_simps, erule (1) word_add_le_mono1) | lemma word_add_le_mono2:
fixes i :: "'a :: len word"
shows "\<lbrakk>i \<le> j; unat j + unat k < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> k + i \<le> k + j" by (subst field_simps, subst field_simps, erule (1) word_add_le_mono1) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>i \<le> j; unat j + unat k < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> k + i \<le> k + j | lemma word_add_le_mono2:
fixes i :: "'a :: len word"
shows "\<lbrakk>i \<le> j; unat j + unat k < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> k + i \<le> k + j" | unnamed_thy_154 | More_Word | 1 | |
[] | lemma word_add_le_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k \<le> j + k) = (i \<le> j)" proof assume "i \<le> j" show "i + k \<le> j + k" by (rule word_add_le_mono1) fact+ next assume "i + k \<le> j... | lemma word_add_le_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k \<le> j + k) = (i \<le> j)" proof assume "i \<le> j" show "i + k \<le> j + k" by (rule word_add_le_mono1) fact+ next assume "i + k \<le> j... | proof (prove)
goal (1 subgoal):
1. (i + k \<le> j + k) = (i \<le> j) proof (state)
goal (2 subgoals):
1. i + k \<le> j + k \<Longrightarrow> i \<le> j
2. i \<le> j \<Longrightarrow> i + k \<le> j + k proof (state)
this:
i \<le> j
goal (2 subgoals):
1. i + k \<le> j + k \<Longrightarrow> i \<le> j
2. i \<le> j \<L... | lemma word_add_le_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k \<le> j + k) = (i \<le> j)" | unnamed_thy_155 | More_Word | 9 | |
[] | lemma word_add_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k < j + k" proof - from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)" by (auto elim: order_le_less_subst2 simp: word_less_nat_alt elim: add_less_mono1) then show... | lemma word_add_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k < j + k" proof - from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)" by (auto elim: order_le_less_subst2 simp: word_less_nat_alt elim: add_less_mono1) then show... | proof (prove)
goal (1 subgoal):
1. i + k < j + k proof (state)
goal (1 subgoal):
1. i + k < j + k proof (chain)
picking this:
i < j
unat j + unat k < 2 ^ LENGTH('a) proof (prove)
using this:
i < j
unat j + unat k < 2 ^ LENGTH('a)
goal (1 subgoal):
1. unat i + unat k < 2 ^ LENGTH('a) proof (state)
this:
unat i + una... | lemma word_add_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i + k < j + k" | unnamed_thy_156 | More_Word | 9 | |
[] | lemma word_add_less_dest:
fixes i :: "'a :: len word"
assumes le: "i + k < j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i < j" using uik ujk le by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem] elim: add_less_mono1) | lemma word_add_less_dest:
fixes i :: "'a :: len word"
assumes le: "i + k < j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i < j" using uik ujk le by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem] elim: add_less_mono1) | proof (prove)
goal (1 subgoal):
1. i < j proof (prove)
using this:
unat i + unat k < 2 ^ LENGTH('a)
unat j + unat k < 2 ^ LENGTH('a)
i + k < j + k
goal (1 subgoal):
1. i < j | lemma word_add_less_dest:
fixes i :: "'a :: len word"
assumes le: "i + k < j + k"
and uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "i < j" | unnamed_thy_157 | More_Word | 2 | |
[] | lemma word_add_less_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k < j + k) = (i < j)" proof assume "i < j" show "i + k < j + k" by (rule word_add_less_mono1) fact+ next assume "i + k < j + k" show "i < ... | lemma word_add_less_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k < j + k) = (i < j)" proof assume "i < j" show "i + k < j + k" by (rule word_add_less_mono1) fact+ next assume "i + k < j + k" show "i < ... | proof (prove)
goal (1 subgoal):
1. (i + k < j + k) = (i < j) proof (state)
goal (2 subgoals):
1. i + k < j + k \<Longrightarrow> i < j
2. i < j \<Longrightarrow> i + k < j + k proof (state)
this:
i < j
goal (2 subgoals):
1. i + k < j + k \<Longrightarrow> i < j
2. i < j \<Longrightarrow> i + k < j + k proof (prov... | lemma word_add_less_iff:
fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows "(i + k < j + k) = (i < j)" | unnamed_thy_158 | More_Word | 9 | |
[] | lemma word_mult_less_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)" using assms by (rule word_mult_less_cancel) | lemma word_mult_less_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)" using assms by (rule word_mult_less_cancel) | proof (prove)
goal (1 subgoal):
1. (i * k < j * k) = (i < j) proof (prove)
using this:
0 < k
unat i * unat k < 2 ^ LENGTH('a)
unat j * unat k < 2 ^ LENGTH('a)
goal (1 subgoal):
1. (i * k < j * k) = (i < j) | lemma word_mult_less_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)" | unnamed_thy_159 | More_Word | 2 | |
[] | lemma word_le_imp_diff_le:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> n - k \<le> m" by (auto simp: unat_sub word_le_nat_alt) | lemma word_le_imp_diff_le:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> n - k \<le> m" by (auto simp: unat_sub word_le_nat_alt) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>k \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> n - k \<le> m | lemma word_le_imp_diff_le:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> n - k \<le> m" | unnamed_thy_160 | More_Word | 1 | |
[] | lemma word_less_imp_diff_less:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n < m\<rbrakk> \<Longrightarrow> n - k < m" by (clarsimp simp: unat_sub word_less_nat_alt
intro!: less_imp_diff_less) | lemma word_less_imp_diff_less:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n < m\<rbrakk> \<Longrightarrow> n - k < m" by (clarsimp simp: unat_sub word_less_nat_alt
intro!: less_imp_diff_less) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>k \<le> n; n < m\<rbrakk> \<Longrightarrow> n - k < m | lemma word_less_imp_diff_less:
fixes n :: "'a::len word"
shows "\<lbrakk>k \<le> n; n < m\<rbrakk> \<Longrightarrow> n - k < m" | unnamed_thy_161 | More_Word | 1 | |
[] | lemma word_mult_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k \<le> j * k" proof - from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)" by (auto elim: order_le_less_subst2 simp: word_le_nat_alt ... | lemma word_mult_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k \<le> j * k" proof - from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)" by (auto elim: order_le_less_subst2 simp: word_le_nat_alt ... | proof (prove)
goal (1 subgoal):
1. i * k \<le> j * k proof (state)
goal (1 subgoal):
1. i * k \<le> j * k proof (chain)
picking this:
i \<le> j
unat j * unat k < 2 ^ LENGTH('a)
0 < k proof (prove)
using this:
i \<le> j
unat j * unat k < 2 ^ LENGTH('a)
0 < k
goal (1 subgoal):
1. unat i * unat k < 2 ^ LENGTH('a) proo... | lemma word_mult_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i \<le> j"
and knz: "0 < k"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "i * k \<le> j * k" | unnamed_thy_162 | More_Word | 9 | |
[] | lemma word_mult_le_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k \<le> j * k) = (i \<le> j)" proof assume "i \<le> j" show "i * k \<le> j * k" by (rule word_mult_le_mono1) fact+ n... | lemma word_mult_le_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k \<le> j * k) = (i \<le> j)" proof assume "i \<le> j" show "i * k \<le> j * k" by (rule word_mult_le_mono1) fact+ n... | proof (prove)
goal (1 subgoal):
1. (i * k \<le> j * k) = (i \<le> j) proof (state)
goal (2 subgoals):
1. i * k \<le> j * k \<Longrightarrow> i \<le> j
2. i \<le> j \<Longrightarrow> i * k \<le> j * k proof (state)
this:
i \<le> j
goal (2 subgoals):
1. i * k \<le> j * k \<Longrightarrow> i \<le> j
2. i \<le> j \<L... | lemma word_mult_le_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k \<le> j * k) = (i \<le> j)" | unnamed_thy_163 | More_Word | 14 | |
[] | lemma word_diff_less:
fixes n :: "'a :: len word"
shows "\<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> m - n < m" apply (subst word_less_nat_alt) apply (subst unat_sub) apply assumption apply (rule diff_less) apply (simp_all add: word_less_nat_alt) done | lemma word_diff_less:
fixes n :: "'a :: len word"
shows "\<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> m - n < m" apply (subst word_less_nat_alt) apply (subst unat_sub) apply assumption apply (rule diff_less) apply (simp_all add: word_less_nat_alt) done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> m - n < m proof (prove)
goal (1 subgoal):
1. \<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> unat (m - n) < unat m proof (prove)
goal (2 subgoals):
1. \<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow>... | lemma word_diff_less:
fixes n :: "'a :: len word"
shows "\<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> m - n < m" | unnamed_thy_164 | More_Word | 6 | |
[] | lemma word_add_increasing:
fixes x :: "'a :: len word"
shows "\<lbrakk> p + w \<le> x; p \<le> p + w \<rbrakk> \<Longrightarrow> p \<le> x" by unat_arith | lemma word_add_increasing:
fixes x :: "'a :: len word"
shows "\<lbrakk> p + w \<le> x; p \<le> p + w \<rbrakk> \<Longrightarrow> p \<le> x" by unat_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>p + w \<le> x; p \<le> p + w\<rbrakk> \<Longrightarrow> p \<le> x | lemma word_add_increasing:
fixes x :: "'a :: len word"
shows "\<lbrakk> p + w \<le> x; p \<le> p + w \<rbrakk> \<Longrightarrow> p \<le> x" | unnamed_thy_165 | More_Word | 1 | |
[] | lemma word_random:
fixes x :: "'a :: len word"
shows "\<lbrakk> p \<le> p + x'; x \<le> x' \<rbrakk> \<Longrightarrow> p \<le> p + x" by unat_arith | lemma word_random:
fixes x :: "'a :: len word"
shows "\<lbrakk> p \<le> p + x'; x \<le> x' \<rbrakk> \<Longrightarrow> p \<le> p + x" by unat_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>p \<le> p + x'; x \<le> x'\<rbrakk> \<Longrightarrow> p \<le> p + x | lemma word_random:
fixes x :: "'a :: len word"
shows "\<lbrakk> p \<le> p + x'; x \<le> x' \<rbrakk> \<Longrightarrow> p \<le> p + x" | unnamed_thy_166 | More_Word | 1 | |
[] | lemma word_sub_mono:
"\<lbrakk> a \<le> c; d \<le> b; a - b \<le> a; c - d \<le> c \<rbrakk>
\<Longrightarrow> (a - b) \<le> (c - d :: 'a :: len word)" by unat_arith | lemma word_sub_mono:
"\<lbrakk> a \<le> c; d \<le> b; a - b \<le> a; c - d \<le> c \<rbrakk>
\<Longrightarrow> (a - b) \<le> (c - d :: 'a :: len word)" by unat_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>a \<le> c; d \<le> b; a - b \<le> a; c - d \<le> c\<rbrakk> \<Longrightarrow> a - b \<le> c - d | lemma word_sub_mono:
"\<lbrakk> a \<le> c; d \<le> b; a - b \<le> a; c - d \<le> c \<rbrakk>
\<Longrightarrow> (a - b) \<le> (c - d :: 'a :: len word)" | unnamed_thy_167 | More_Word | 1 | |
[] | lemma power_not_zero:
"n < LENGTH('a::len) \<Longrightarrow> (2 :: 'a word) ^ n \<noteq> 0" by (metis p2_gt_0 word_neq_0_conv) | lemma power_not_zero:
"n < LENGTH('a::len) \<Longrightarrow> (2 :: 'a word) ^ n \<noteq> 0" by (metis p2_gt_0 word_neq_0_conv) | proof (prove)
goal (1 subgoal):
1. n < LENGTH('a) \<Longrightarrow> 2 ^ n \<noteq> 0 | lemma power_not_zero:
"n < LENGTH('a::len) \<Longrightarrow> (2 :: 'a word) ^ n \<noteq> 0" | unnamed_thy_168 | More_Word | 1 | |
[] | lemma word_gt_a_gt_0:
"a < n \<Longrightarrow> (0 :: 'a::len word) < n" apply (case_tac "n = 0") apply clarsimp apply (clarsimp simp: word_neq_0_conv) done | lemma word_gt_a_gt_0:
"a < n \<Longrightarrow> (0 :: 'a::len word) < n" apply (case_tac "n = 0") apply clarsimp apply (clarsimp simp: word_neq_0_conv) done | proof (prove)
goal (1 subgoal):
1. a < n \<Longrightarrow> 0 < n proof (prove)
goal (2 subgoals):
1. \<lbrakk>a < n; n = 0\<rbrakk> \<Longrightarrow> 0 < n
2. \<lbrakk>a < n; n \<noteq> 0\<rbrakk> \<Longrightarrow> 0 < n proof (prove)
goal (1 subgoal):
1. \<lbrakk>a < n; n \<noteq> 0\<rbrakk> \<Longrightarrow> 0 < ... | lemma word_gt_a_gt_0:
"a < n \<Longrightarrow> (0 :: 'a::len word) < n" | unnamed_thy_169 | More_Word | 4 | |
[] | lemma word_power_less_1 [simp]:
"sz < LENGTH('a::len) \<Longrightarrow> (2::'a word) ^ sz - 1 < 2 ^ sz" apply (simp add: word_less_nat_alt) apply (subst unat_minus_one) apply simp_all done | lemma word_power_less_1 [simp]:
"sz < LENGTH('a::len) \<Longrightarrow> (2::'a word) ^ sz - 1 < 2 ^ sz" apply (simp add: word_less_nat_alt) apply (subst unat_minus_one) apply simp_all done | proof (prove)
goal (1 subgoal):
1. sz < LENGTH('a) \<Longrightarrow> 2 ^ sz - 1 < 2 ^ sz proof (prove)
goal (1 subgoal):
1. sz < LENGTH('a) \<Longrightarrow> unat (2 ^ sz - 1) < 2 ^ sz proof (prove)
goal (2 subgoals):
1. sz < LENGTH('a) \<Longrightarrow> 2 ^ sz \<noteq> 0
2. sz < LENGTH('a) \<Longrightarrow> unat (... | lemma word_power_less_1 [simp]:
"sz < LENGTH('a::len) \<Longrightarrow> (2::'a word) ^ sz - 1 < 2 ^ sz" | unnamed_thy_170 | More_Word | 4 | |
[] | lemma word_sub_1_le:
"x \<noteq> 0 \<Longrightarrow> x - 1 \<le> (x :: 'a :: len word)" apply (subst no_ulen_sub) apply simp apply (cases "uint x = 0") apply (simp add: uint_0_iff) apply (insert uint_ge_0[where x=x]) apply arith done | lemma word_sub_1_le:
"x \<noteq> 0 \<Longrightarrow> x - 1 \<le> (x :: 'a :: len word)" apply (subst no_ulen_sub) apply simp apply (cases "uint x = 0") apply (simp add: uint_0_iff) apply (insert uint_ge_0[where x=x]) apply arith done | proof (prove)
goal (1 subgoal):
1. x \<noteq> 0 \<Longrightarrow> x - 1 \<le> x proof (prove)
goal (1 subgoal):
1. x \<noteq> 0 \<Longrightarrow> uint 1 \<le> uint x proof (prove)
goal (1 subgoal):
1. x \<noteq> 0 \<Longrightarrow> 1 \<le> uint x proof (prove)
goal (2 subgoals):
1. \<lbrakk>x \<noteq> 0; uint x = 0... | lemma word_sub_1_le:
"x \<noteq> 0 \<Longrightarrow> x - 1 \<le> (x :: 'a :: len word)" | unnamed_thy_171 | More_Word | 7 | |
[] | lemma push_bit_word_eq_nonzero:
\<open>push_bit n w \<noteq> 0\<close> if \<open>w < 2 ^ m\<close> \<open>m + n < LENGTH('a)\<close> \<open>w \<noteq> 0\<close>
for w :: \<open>'a::len word\<close> using that apply (simp only: word_neq_0_conv word_less_nat_alt
mod_0 unat_word_ariths
... | lemma push_bit_word_eq_nonzero:
\<open>push_bit n w \<noteq> 0\<close> if \<open>w < 2 ^ m\<close> \<open>m + n < LENGTH('a)\<close> \<open>w \<noteq> 0\<close>
for w :: \<open>'a::len word\<close> using that apply (simp only: word_neq_0_conv word_less_nat_alt
mod_0 unat_word_ariths
... | proof (prove)
goal (1 subgoal):
1. push_bit n w \<noteq> 0 proof (prove)
using this:
w < 2 ^ m
m + n < LENGTH('a)
w \<noteq> 0
goal (1 subgoal):
1. push_bit n w \<noteq> 0 proof (prove)
goal (1 subgoal):
1. \<lbrakk>unat w < 2 ^ m; m + n < LENGTH('a); 0 < unat w\<rbrakk> \<Longrightarrow> 0 < unat (push_bit n w) pr... | lemma push_bit_word_eq_nonzero:
\<open>push_bit n w \<noteq> 0\<close> if \<open>w < 2 ^ m\<close> \<open>m + n < LENGTH('a)\<close> \<open>w \<noteq> 0\<close>
for w :: \<open>'a::len word\<close> | unnamed_thy_172 | More_Word | 4 | |
[] | lemma unat_less_power:
fixes k :: "'a::len word"
assumes szv: "sz < LENGTH('a)"
and kv: "k < 2 ^ sz"
shows "unat k < 2 ^ sz" using szv unat_mono [OF kv] by simp | lemma unat_less_power:
fixes k :: "'a::len word"
assumes szv: "sz < LENGTH('a)"
and kv: "k < 2 ^ sz"
shows "unat k < 2 ^ sz" using szv unat_mono [OF kv] by simp | proof (prove)
goal (1 subgoal):
1. unat k < 2 ^ sz proof (prove)
using this:
sz < LENGTH('a)
unat k < unat (2 ^ sz)
goal (1 subgoal):
1. unat k < 2 ^ sz | lemma unat_less_power:
fixes k :: "'a::len word"
assumes szv: "sz < LENGTH('a)"
and kv: "k < 2 ^ sz"
shows "unat k < 2 ^ sz" | unnamed_thy_173 | More_Word | 2 | |
[] | lemma unat_mult_power_lem:
assumes kv: "k < 2 ^ (LENGTH('a::len) - sz)"
shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k" proof (cases \<open>sz < LENGTH('a)\<close>) case True with assms show ?thesis by (simp add: unat_word_ariths take_bit_eq_mod mod_simps unsigned_of_nat)
(simp add: take_bit... | lemma unat_mult_power_lem:
assumes kv: "k < 2 ^ (LENGTH('a::len) - sz)"
shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k" proof (cases \<open>sz < LENGTH('a)\<close>) case True with assms show ?thesis by (simp add: unat_word_ariths take_bit_eq_mod mod_simps unsigned_of_nat)
(simp add: take_bit... | proof (prove)
goal (1 subgoal):
1. unat (2 ^ sz * word_of_nat k) = 2 ^ sz * k proof (state)
goal (2 subgoals):
1. sz < LENGTH('a) \<Longrightarrow> unat (2 ^ sz * word_of_nat k) = 2 ^ sz * k
2. \<not> sz < LENGTH('a) \<Longrightarrow> unat (2 ^ sz * word_of_nat k) = 2 ^ sz * k proof (state)
this:
sz < LENGTH('a)
go... | lemma unat_mult_power_lem:
assumes kv: "k < 2 ^ (LENGTH('a::len) - sz)"
shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k" | unnamed_thy_174 | More_Word | 11 | |
[] | lemma word_plus_mcs_4:
"\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)" by uint_arith | lemma word_plus_mcs_4:
"\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)" by uint_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> w | lemma word_plus_mcs_4:
"\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)" | unnamed_thy_175 | More_Word | 1 | |
[] | lemma word_plus_mcs_3:
"\<lbrakk>v \<le> w; x \<le> w + x\<rbrakk> \<Longrightarrow> v + x \<le> w + (x::'a::len word)" by unat_arith | lemma word_plus_mcs_3:
"\<lbrakk>v \<le> w; x \<le> w + x\<rbrakk> \<Longrightarrow> v + x \<le> w + (x::'a::len word)" by unat_arith | proof (prove)
goal (1 subgoal):
1. \<lbrakk>v \<le> w; x \<le> w + x\<rbrakk> \<Longrightarrow> v + x \<le> w + x | lemma word_plus_mcs_3:
"\<lbrakk>v \<le> w; x \<le> w + x\<rbrakk> \<Longrightarrow> v + x \<le> w + (x::'a::len word)" | unnamed_thy_176 | More_Word | 1 | |
[] | lemma word_le_minus_one_leq:
"x < y \<Longrightarrow> x \<le> y - 1" for x :: "'a :: len word" by transfer (metis le_less_trans less_irrefl take_bit_decr_eq take_bit_nonnegative zle_diff1_eq) | lemma word_le_minus_one_leq:
"x < y \<Longrightarrow> x \<le> y - 1" for x :: "'a :: len word" by transfer (metis le_less_trans less_irrefl take_bit_decr_eq take_bit_nonnegative zle_diff1_eq) | proof (prove)
goal (1 subgoal):
1. x < y \<Longrightarrow> x \<le> y - 1 | lemma word_le_minus_one_leq:
"x < y \<Longrightarrow> x \<le> y - 1" for x :: "'a :: len word" | unnamed_thy_177 | More_Word | 1 | |
[] | lemma word_less_sub_le[simp]:
fixes x :: "'a :: len word"
assumes nv: "n < LENGTH('a)"
shows "(x \<le> 2 ^ n - 1) = (x < 2 ^ n)" using le_less_trans word_le_minus_one_leq nv power_2_ge_iff by blast | lemma word_less_sub_le[simp]:
fixes x :: "'a :: len word"
assumes nv: "n < LENGTH('a)"
shows "(x \<le> 2 ^ n - 1) = (x < 2 ^ n)" using le_less_trans word_le_minus_one_leq nv power_2_ge_iff by blast | proof (prove)
goal (1 subgoal):
1. (x \<le> 2 ^ n - 1) = (x < 2 ^ n) proof (prove)
using this:
\<lbrakk>?x \<le> ?y; ?y < ?z\<rbrakk> \<Longrightarrow> ?x < ?z
?x < ?y \<Longrightarrow> ?x \<le> ?y - 1
n < LENGTH('a)
(2 ^ ?n - 1 < 2 ^ ?n) = (?n < LENGTH(?'a))
goal (1 subgoal):
1. (x \<le> 2 ^ n - 1) = (x < 2 ^ n) | lemma word_less_sub_le[simp]:
fixes x :: "'a :: len word"
assumes nv: "n < LENGTH('a)"
shows "(x \<le> 2 ^ n - 1) = (x < 2 ^ n)" | unnamed_thy_178 | More_Word | 2 | |
[] | lemma unat_of_nat_len:
"x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x :: 'a::len word) = x" by (simp add: unsigned_of_nat take_bit_nat_eq_self_iff) | lemma unat_of_nat_len:
"x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x :: 'a::len word) = x" by (simp add: unsigned_of_nat take_bit_nat_eq_self_iff) | proof (prove)
goal (1 subgoal):
1. x < 2 ^ LENGTH('a) \<Longrightarrow> unat (word_of_nat x) = x | lemma unat_of_nat_len:
"x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x :: 'a::len word) = x" | unnamed_thy_179 | More_Word | 1 | |
[] | lemma unat_of_nat_eq:
"x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x ::'a::len word) = x" by (rule unat_of_nat_len) | lemma unat_of_nat_eq:
"x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x ::'a::len word) = x" by (rule unat_of_nat_len) | proof (prove)
goal (1 subgoal):
1. x < 2 ^ LENGTH('a) \<Longrightarrow> unat (word_of_nat x) = x | lemma unat_of_nat_eq:
"x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x ::'a::len word) = x" | unnamed_thy_180 | More_Word | 1 | |
[] | lemma unat_eq_of_nat:
"n < 2 ^ LENGTH('a) \<Longrightarrow> (unat (x :: 'a::len word) = n) = (x = of_nat n)" by transfer
(auto simp add: take_bit_of_nat nat_eq_iff take_bit_nat_eq_self_iff intro: sym) | lemma unat_eq_of_nat:
"n < 2 ^ LENGTH('a) \<Longrightarrow> (unat (x :: 'a::len word) = n) = (x = of_nat n)" by transfer
(auto simp add: take_bit_of_nat nat_eq_iff take_bit_nat_eq_self_iff intro: sym) | proof (prove)
goal (1 subgoal):
1. n < 2 ^ LENGTH('a) \<Longrightarrow> (unat x = n) = (x = word_of_nat n) | lemma unat_eq_of_nat:
"n < 2 ^ LENGTH('a) \<Longrightarrow> (unat (x :: 'a::len word) = n) = (x = of_nat n)" | unnamed_thy_181 | More_Word | 1 | |
[] | lemma alignUp_div_helper:
fixes a :: "'a::len word"
assumes kv: "k < 2 ^ (LENGTH('a) - n)"
and xk: "x = 2 ^ n * of_nat k"
and le: "a \<le> x"
and sz: "n < LENGTH('a)"
and anz: "a mod 2 ^ n \<noteq> 0"
shows "a div 2 ^ n < of_nat k" proof - have kn: "unat (of_nat k :: 'a word) * unat ((2::'a wo... | lemma alignUp_div_helper:
fixes a :: "'a::len word"
assumes kv: "k < 2 ^ (LENGTH('a) - n)"
and xk: "x = 2 ^ n * of_nat k"
and le: "a \<le> x"
and sz: "n < LENGTH('a)"
and anz: "a mod 2 ^ n \<noteq> 0"
shows "a div 2 ^ n < of_nat k" proof - have kn: "unat (of_nat k :: 'a word) * unat ((2::'a wo... | proof (prove)
goal (1 subgoal):
1. a div 2 ^ n < word_of_nat k proof (state)
goal (1 subgoal):
1. a div 2 ^ n < word_of_nat k proof (prove)
goal (1 subgoal):
1. unat (word_of_nat k) * unat (2 ^ n) < 2 ^ LENGTH('a) proof (prove)
using this:
x = 2 ^ n * word_of_nat k
k < 2 ^ (LENGTH('a) - n)
n < LENGTH('a)
goal (1 su... | lemma alignUp_div_helper:
fixes a :: "'a::len word"
assumes kv: "k < 2 ^ (LENGTH('a) - n)"
and xk: "x = 2 ^ n * of_nat k"
and le: "a \<le> x"
and sz: "n < LENGTH('a)"
and anz: "a mod 2 ^ n \<noteq> 0"
shows "a div 2 ^ n < of_nat k" | unnamed_thy_182 | More_Word | 48 | |
[] | lemma mask_out_sub_mask:
"(x AND NOT (mask n)) = x - (x AND (mask n))"
for x :: \<open>'a::len word\<close> by (fact and_not_eq_minus_and) | lemma mask_out_sub_mask:
"(x AND NOT (mask n)) = x - (x AND (mask n))"
for x :: \<open>'a::len word\<close> by (fact and_not_eq_minus_and) | proof (prove)
goal (1 subgoal):
1. x AND NOT (mask n) = x - (x AND mask n) | lemma mask_out_sub_mask:
"(x AND NOT (mask n)) = x - (x AND (mask n))"
for x :: \<open>'a::len word\<close> | unnamed_thy_183 | More_Word | 1 | |
[] | lemma subtract_mask:
"p - (p AND mask n) = (p AND NOT (mask n))"
"p - (p AND NOT (mask n)) = (p AND mask n)"
for p :: \<open>'a::len word\<close> by (auto simp: and_not_eq_minus_and) | lemma subtract_mask:
"p - (p AND mask n) = (p AND NOT (mask n))"
"p - (p AND NOT (mask n)) = (p AND mask n)"
for p :: \<open>'a::len word\<close> by (auto simp: and_not_eq_minus_and) | proof (prove)
goal (1 subgoal):
1. p - (p AND mask n) = p AND NOT (mask n) &&& p - (p AND NOT (mask n)) = p AND mask n | lemma subtract_mask:
"p - (p AND mask n) = (p AND NOT (mask n))"
"p - (p AND NOT (mask n)) = (p AND mask n)"
for p :: \<open>'a::len word\<close> | unnamed_thy_184 | More_Word | 1 | |
[] | lemma take_bit_word_eq_self_iff:
\<open>take_bit n w = w \<longleftrightarrow> n \<ge> LENGTH('a) \<or> w < 2 ^ n\<close>
for w :: \<open>'a::len word\<close> using take_bit_int_eq_self_iff [of n \<open>take_bit LENGTH('a) (uint w)\<close>] by (transfer fixing: n) auto | lemma take_bit_word_eq_self_iff:
\<open>take_bit n w = w \<longleftrightarrow> n \<ge> LENGTH('a) \<or> w < 2 ^ n\<close>
for w :: \<open>'a::len word\<close> using take_bit_int_eq_self_iff [of n \<open>take_bit LENGTH('a) (uint w)\<close>] by (transfer fixing: n) auto | proof (prove)
goal (1 subgoal):
1. (take_bit n w = w) = (LENGTH('a) \<le> n \<or> w < 2 ^ n) proof (prove)
using this:
(take_bit n (take_bit LENGTH('a) (uint w)) = take_bit LENGTH('a) (uint w)) = (0 \<le> take_bit LENGTH('a) (uint w) \<and> take_bit LENGTH('a) (uint w) < 2 ^ n)
goal (1 subgoal):
1. (take_bit n w = w... | lemma take_bit_word_eq_self_iff:
\<open>take_bit n w = w \<longleftrightarrow> n \<ge> LENGTH('a) \<or> w < 2 ^ n\<close>
for w :: \<open>'a::len word\<close> | unnamed_thy_185 | More_Word | 2 | |
[] | lemma word_power_increasing:
assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a)" "y < LENGTH('a)"
shows "x < y" using x using assms by transfer simp | lemma word_power_increasing:
assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a)" "y < LENGTH('a)"
shows "x < y" using x using assms by transfer simp | proof (prove)
goal (1 subgoal):
1. x < y proof (prove)
using this:
2 ^ x < 2 ^ y
x < LENGTH('a)
y < LENGTH('a)
goal (1 subgoal):
1. x < y proof (prove)
using this:
2 ^ x < 2 ^ y
x < LENGTH('a)
y < LENGTH('a)
2 ^ x < 2 ^ y
x < LENGTH('a)
y < LENGTH('a)
goal (1 subgoal):
1. x < y | lemma word_power_increasing:
assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a)" "y < LENGTH('a)"
shows "x < y" | unnamed_thy_186 | More_Word | 3 | |
[] | lemma plus_one_helper[elim!]:
"x < n + (1 :: 'a :: len word) \<Longrightarrow> x \<le> n" apply (simp add: word_less_nat_alt word_le_nat_alt field_simps) apply (case_tac "1 + n = 0") apply simp_all apply (subst(asm) unatSuc, assumption) apply arith done | lemma plus_one_helper[elim!]:
"x < n + (1 :: 'a :: len word) \<Longrightarrow> x \<le> n" apply (simp add: word_less_nat_alt word_le_nat_alt field_simps) apply (case_tac "1 + n = 0") apply simp_all apply (subst(asm) unatSuc, assumption) apply arith done | proof (prove)
goal (1 subgoal):
1. x < n + 1 \<Longrightarrow> x \<le> n proof (prove)
goal (1 subgoal):
1. unat x < unat (1 + n) \<Longrightarrow> unat x \<le> unat n proof (prove)
goal (2 subgoals):
1. \<lbrakk>unat x < unat (1 + n); 1 + n = 0\<rbrakk> \<Longrightarrow> unat x \<le> unat n
2. \<lbrakk>unat x < un... | lemma plus_one_helper[elim!]:
"x < n + (1 :: 'a :: len word) \<Longrightarrow> x \<le> n" | unnamed_thy_188 | More_Word | 6 | |
[] | lemma plus_one_helper2:
"\<lbrakk> x \<le> n; n + 1 \<noteq> 0 \<rbrakk> \<Longrightarrow> x < n + (1 :: 'a :: len word)" by (simp add: word_less_nat_alt word_le_nat_alt field_simps
unatSuc) | lemma plus_one_helper2:
"\<lbrakk> x \<le> n; n + 1 \<noteq> 0 \<rbrakk> \<Longrightarrow> x < n + (1 :: 'a :: len word)" by (simp add: word_less_nat_alt word_le_nat_alt field_simps
unatSuc) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x \<le> n; n + 1 \<noteq> 0\<rbrakk> \<Longrightarrow> x < n + 1 | lemma plus_one_helper2:
"\<lbrakk> x \<le> n; n + 1 \<noteq> 0 \<rbrakk> \<Longrightarrow> x < n + (1 :: 'a :: len word)" | unnamed_thy_189 | More_Word | 1 | |
[] | lemma less_x_plus_1:
"x \<noteq> - 1 \<Longrightarrow> (y < x + 1) = (y < x \<or> y = x)" for x :: "'a::len word" by (meson max_word_wrap plus_one_helper plus_one_helper2 word_le_less_eq) | lemma less_x_plus_1:
"x \<noteq> - 1 \<Longrightarrow> (y < x + 1) = (y < x \<or> y = x)" for x :: "'a::len word" by (meson max_word_wrap plus_one_helper plus_one_helper2 word_le_less_eq) | proof (prove)
goal (1 subgoal):
1. x \<noteq> - 1 \<Longrightarrow> (y < x + 1) = (y < x \<or> y = x) | lemma less_x_plus_1:
"x \<noteq> - 1 \<Longrightarrow> (y < x + 1) = (y < x \<or> y = x)" for x :: "'a::len word" | unnamed_thy_190 | More_Word | 1 | |
[] | lemma word_Suc_leq:
fixes k::"'a::len word" shows "k \<noteq> - 1 \<Longrightarrow> x < k + 1 \<longleftrightarrow> x \<le> k" using less_x_plus_1 word_le_less_eq by auto | lemma word_Suc_leq:
fixes k::"'a::len word" shows "k \<noteq> - 1 \<Longrightarrow> x < k + 1 \<longleftrightarrow> x \<le> k" using less_x_plus_1 word_le_less_eq by auto | proof (prove)
goal (1 subgoal):
1. k \<noteq> - 1 \<Longrightarrow> (x < k + 1) = (x \<le> k) proof (prove)
using this:
?x \<noteq> - 1 \<Longrightarrow> (?y < ?x + 1) = (?y < ?x \<or> ?y = ?x)
(?x \<le> ?y) = (?x = ?y \<or> ?x < ?y)
goal (1 subgoal):
1. k \<noteq> - 1 \<Longrightarrow> (x < k + 1) = (x \<le> k) | lemma word_Suc_leq:
fixes k::"'a::len word" shows "k \<noteq> - 1 \<Longrightarrow> x < k + 1 \<longleftrightarrow> x \<le> k" | unnamed_thy_191 | More_Word | 2 | |
[] | lemma word_Suc_le:
fixes k::"'a::len word" shows "x \<noteq> - 1 \<Longrightarrow> x + 1 \<le> k \<longleftrightarrow> x < k" by (meson not_less word_Suc_leq) | lemma word_Suc_le:
fixes k::"'a::len word" shows "x \<noteq> - 1 \<Longrightarrow> x + 1 \<le> k \<longleftrightarrow> x < k" by (meson not_less word_Suc_leq) | proof (prove)
goal (1 subgoal):
1. x \<noteq> - 1 \<Longrightarrow> (x + 1 \<le> k) = (x < k) | lemma word_Suc_le:
fixes k::"'a::len word" shows "x \<noteq> - 1 \<Longrightarrow> x + 1 \<le> k \<longleftrightarrow> x < k" | unnamed_thy_192 | More_Word | 1 | |
[] | lemma word_lessThan_Suc_atMost:
\<open>{..< k + 1} = {..k}\<close> if \<open>k \<noteq> - 1\<close> for k :: \<open>'a::len word\<close> using that by (simp add: lessThan_def atMost_def word_Suc_leq) | lemma word_lessThan_Suc_atMost:
\<open>{..< k + 1} = {..k}\<close> if \<open>k \<noteq> - 1\<close> for k :: \<open>'a::len word\<close> using that by (simp add: lessThan_def atMost_def word_Suc_leq) | proof (prove)
goal (1 subgoal):
1. {..<k + 1} = {..k} proof (prove)
using this:
k \<noteq> - 1
goal (1 subgoal):
1. {..<k + 1} = {..k} | lemma word_lessThan_Suc_atMost:
\<open>{..< k + 1} = {..k}\<close> if \<open>k \<noteq> - 1\<close> for k :: \<open>'a::len word\<close> | unnamed_thy_193 | More_Word | 2 | |
[] | lemma word_atLeastLessThan_Suc_atLeastAtMost:
\<open>{l ..< u + 1} = {l..u}\<close> if \<open>u \<noteq> - 1\<close> for l :: \<open>'a::len word\<close> using that by (simp add: atLeastAtMost_def atLeastLessThan_def word_lessThan_Suc_atMost) | lemma word_atLeastLessThan_Suc_atLeastAtMost:
\<open>{l ..< u + 1} = {l..u}\<close> if \<open>u \<noteq> - 1\<close> for l :: \<open>'a::len word\<close> using that by (simp add: atLeastAtMost_def atLeastLessThan_def word_lessThan_Suc_atMost) | proof (prove)
goal (1 subgoal):
1. {l..<u + 1} = {l..u} proof (prove)
using this:
u \<noteq> - 1
goal (1 subgoal):
1. {l..<u + 1} = {l..u} | lemma word_atLeastLessThan_Suc_atLeastAtMost:
\<open>{l ..< u + 1} = {l..u}\<close> if \<open>u \<noteq> - 1\<close> for l :: \<open>'a::len word\<close> | unnamed_thy_194 | More_Word | 2 | |
[] | lemma word_atLeastAtMost_Suc_greaterThanAtMost:
\<open>{m<..u} = {m + 1..u}\<close> if \<open>m \<noteq> - 1\<close> for m :: \<open>'a::len word\<close> using that by (simp add: greaterThanAtMost_def greaterThan_def atLeastAtMost_def atLeast_def word_Suc_le) | lemma word_atLeastAtMost_Suc_greaterThanAtMost:
\<open>{m<..u} = {m + 1..u}\<close> if \<open>m \<noteq> - 1\<close> for m :: \<open>'a::len word\<close> using that by (simp add: greaterThanAtMost_def greaterThan_def atLeastAtMost_def atLeast_def word_Suc_le) | proof (prove)
goal (1 subgoal):
1. {m<..u} = {m + 1..u} proof (prove)
using this:
m \<noteq> - 1
goal (1 subgoal):
1. {m<..u} = {m + 1..u} | lemma word_atLeastAtMost_Suc_greaterThanAtMost:
\<open>{m<..u} = {m + 1..u}\<close> if \<open>m \<noteq> - 1\<close> for m :: \<open>'a::len word\<close> | unnamed_thy_195 | More_Word | 2 | |
[] | lemma word_atLeastLessThan_Suc_atLeastAtMost_union:
fixes l::"'a::len word"
assumes "m \<noteq> - 1" and "l \<le> m" and "m \<le> u"
shows "{l..m} \<union> {m+1..u} = {l..u}" proof - from ivl_disj_un_two(8)[OF assms(2) assms(3)] have "{l..u} = {l..m} \<union> {m<..u}" by blast with assms show ?thesis by(simp add:... | lemma word_atLeastLessThan_Suc_atLeastAtMost_union:
fixes l::"'a::len word"
assumes "m \<noteq> - 1" and "l \<le> m" and "m \<le> u"
shows "{l..m} \<union> {m+1..u} = {l..u}" proof - from ivl_disj_un_two(8)[OF assms(2) assms(3)] have "{l..u} = {l..m} \<union> {m<..u}" by blast with assms show ?thesis by(simp add:... | proof (prove)
goal (1 subgoal):
1. {l..m} \<union> {m + 1..u} = {l..u} proof (state)
goal (1 subgoal):
1. {l..m} \<union> {m + 1..u} = {l..u} proof (chain)
picking this:
{l..m} \<union> {m<..u} = {l..u} proof (prove)
using this:
{l..m} \<union> {m<..u} = {l..u}
goal (1 subgoal):
1. {l..u} = {l..m} \<union> {m<..u} ... | lemma word_atLeastLessThan_Suc_atLeastAtMost_union:
fixes l::"'a::len word"
assumes "m \<noteq> - 1" and "l \<le> m" and "m \<le> u"
shows "{l..m} \<union> {m+1..u} = {l..u}" | unnamed_thy_196 | More_Word | 8 | |
[] | lemma max_word_less_eq_iff [simp]:
\<open>- 1 \<le> w \<longleftrightarrow> w = - 1\<close> for w :: \<open>'a::len word\<close> by (fact word_order.extremum_unique) | lemma max_word_less_eq_iff [simp]:
\<open>- 1 \<le> w \<longleftrightarrow> w = - 1\<close> for w :: \<open>'a::len word\<close> by (fact word_order.extremum_unique) | proof (prove)
goal (1 subgoal):
1. (- 1 \<le> w) = (w = - 1) | lemma max_word_less_eq_iff [simp]:
\<open>- 1 \<le> w \<longleftrightarrow> w = - 1\<close> for w :: \<open>'a::len word\<close> | unnamed_thy_197 | More_Word | 1 | |
[] | lemma word_or_zero:
"(a OR b = 0) = (a = 0 \<and> b = 0)"
for a b :: \<open>'a::len word\<close> by (fact or_eq_0_iff) | lemma word_or_zero:
"(a OR b = 0) = (a = 0 \<and> b = 0)"
for a b :: \<open>'a::len word\<close> by (fact or_eq_0_iff) | proof (prove)
goal (1 subgoal):
1. (a OR b = 0) = (a = 0 \<and> b = 0) | lemma word_or_zero:
"(a OR b = 0) = (a = 0 \<and> b = 0)"
for a b :: \<open>'a::len word\<close> | unnamed_thy_198 | More_Word | 1 | |
[] | lemma word_2p_mult_inc:
assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
assumes suc_n: "Suc n < LENGTH('a::len)"
shows "2^n < (2::'a::len word)^m" by (smt suc_n le_less_trans lessI nat_less_le nat_mult_less_cancel_disj p2_gt_0
power_Suc power_Suc unat_power_lower word_less_nat_alt x) | lemma word_2p_mult_inc:
assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
assumes suc_n: "Suc n < LENGTH('a::len)"
shows "2^n < (2::'a::len word)^m" by (smt suc_n le_less_trans lessI nat_less_le nat_mult_less_cancel_disj p2_gt_0
power_Suc power_Suc unat_power_lower word_less_nat_alt x) | proof (prove)
goal (1 subgoal):
1. 2 ^ n < 2 ^ m | lemma word_2p_mult_inc:
assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
assumes suc_n: "Suc n < LENGTH('a::len)"
shows "2^n < (2::'a::len word)^m" | unnamed_thy_199 | More_Word | 1 | |
[] | lemma power_overflow:
"n \<ge> LENGTH('a) \<Longrightarrow> 2 ^ n = (0 :: 'a::len word)" by simp | lemma power_overflow:
"n \<ge> LENGTH('a) \<Longrightarrow> 2 ^ n = (0 :: 'a::len word)" by simp | proof (prove)
goal (1 subgoal):
1. LENGTH('a) \<le> n \<Longrightarrow> 2 ^ n = 0 | lemma power_overflow:
"n \<ge> LENGTH('a) \<Longrightarrow> 2 ^ n = (0 :: 'a::len word)" | unnamed_thy_200 | More_Word | 1 | |
[] | lemma word_sint_1:
"sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)" by (fact signed_1) | lemma word_sint_1:
"sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)" by (fact signed_1) | proof (prove)
goal (1 subgoal):
1. sint 1 = (if LENGTH('a) = 1 then - 1 else 1) | lemma word_sint_1:
"sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)" | unnamed_thy_201 | More_Word | 1 | |
[] | lemma ucast_of_nat:
"is_down (ucast :: 'a :: len word \<Rightarrow> 'b :: len word)
\<Longrightarrow> ucast (of_nat n :: 'a word) = (of_nat n :: 'b word)" by transfer simp | lemma ucast_of_nat:
"is_down (ucast :: 'a :: len word \<Rightarrow> 'b :: len word)
\<Longrightarrow> ucast (of_nat n :: 'a word) = (of_nat n :: 'b word)" by transfer simp | proof (prove)
goal (1 subgoal):
1. is_down ucast \<Longrightarrow> ucast (word_of_nat n) = word_of_nat n | lemma ucast_of_nat:
"is_down (ucast :: 'a :: len word \<Rightarrow> 'b :: len word)
\<Longrightarrow> ucast (of_nat n :: 'a word) = (of_nat n :: 'b word)" | unnamed_thy_202 | More_Word | 1 | |
[] | lemma scast_1':
"(scast (1::'a::len word) :: 'b::len word) =
(word_of_int (signed_take_bit (LENGTH('a::len) - Suc 0) (1::int)))" by transfer simp | lemma scast_1':
"(scast (1::'a::len word) :: 'b::len word) =
(word_of_int (signed_take_bit (LENGTH('a::len) - Suc 0) (1::int)))" by transfer simp | proof (prove)
goal (1 subgoal):
1. scast 1 = word_of_int (signed_take_bit (LENGTH('a) - Suc 0) 1) | lemma scast_1':
"(scast (1::'a::len word) :: 'b::len word) =
(word_of_int (signed_take_bit (LENGTH('a::len) - Suc 0) (1::int)))" | unnamed_thy_203 | More_Word | 1 | |
[] | lemma scast_1:
"(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)" by (fact signed_1) | lemma scast_1:
"(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)" by (fact signed_1) | proof (prove)
goal (1 subgoal):
1. scast 1 = (if LENGTH('a) = 1 then - 1 else 1) | lemma scast_1:
"(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)" | unnamed_thy_204 | More_Word | 1 | |
[] | lemma unat_minus_one_word:
"unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1" by (simp add: mask_eq_exp_minus_1 unsigned_minus_1_eq_mask) | lemma unat_minus_one_word:
"unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1" by (simp add: mask_eq_exp_minus_1 unsigned_minus_1_eq_mask) | proof (prove)
goal (1 subgoal):
1. unat (- 1) = 2 ^ LENGTH('a) - 1 | lemma unat_minus_one_word:
"unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1" | unnamed_thy_205 | More_Word | 1 | |
[] | lemma two_power_increasing:
"\<lbrakk> n \<le> m; m < LENGTH('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ n \<le> 2 ^ m" by (simp add: word_le_nat_alt) | lemma two_power_increasing:
"\<lbrakk> n \<le> m; m < LENGTH('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ n \<le> 2 ^ m" by (simp add: word_le_nat_alt) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>n \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> 2 ^ n \<le> 2 ^ m | lemma two_power_increasing:
"\<lbrakk> n \<le> m; m < LENGTH('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ n \<le> 2 ^ m" | unnamed_thy_206 | More_Word | 1 | |
[] | lemma word_leq_le_minus_one:
"\<lbrakk> x \<le> y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x - 1 < (y :: 'a :: len word)" apply (simp add: word_less_nat_alt word_le_nat_alt) apply (subst unat_minus_one) apply assumption apply (cases "unat x") apply (simp add: unat_eq_zero) apply arith done | lemma word_leq_le_minus_one:
"\<lbrakk> x \<le> y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x - 1 < (y :: 'a :: len word)" apply (simp add: word_less_nat_alt word_le_nat_alt) apply (subst unat_minus_one) apply assumption apply (cases "unat x") apply (simp add: unat_eq_zero) apply arith done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>x \<le> y; x \<noteq> 0\<rbrakk> \<Longrightarrow> x - 1 < y proof (prove)
goal (1 subgoal):
1. \<lbrakk>unat x \<le> unat y; x \<noteq> 0\<rbrakk> \<Longrightarrow> unat (x - 1) < unat y proof (prove)
goal (2 subgoals):
1. \<lbrakk>unat x \<le> unat y; x \<noteq> 0\<rbrak... | lemma word_leq_le_minus_one:
"\<lbrakk> x \<le> y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x - 1 < (y :: 'a :: len word)" | unnamed_thy_207 | More_Word | 7 | |
[] | lemma neg_mask_combine:
"NOT(mask a) AND NOT(mask b) = NOT(mask (max a b) :: 'a::len word)" by (rule bit_word_eqI) (auto simp add: bit_simps) | lemma neg_mask_combine:
"NOT(mask a) AND NOT(mask b) = NOT(mask (max a b) :: 'a::len word)" by (rule bit_word_eqI) (auto simp add: bit_simps) | proof (prove)
goal (1 subgoal):
1. NOT (mask a) AND NOT (mask b) = NOT (mask (max a b)) | lemma neg_mask_combine:
"NOT(mask a) AND NOT(mask b) = NOT(mask (max a b) :: 'a::len word)" | unnamed_thy_208 | More_Word | 1 | |
[] | lemma neg_mask_twice:
"x AND NOT(mask n) AND NOT(mask m) = x AND NOT(mask (max n m))"
for x :: \<open>'a::len word\<close> by (rule bit_word_eqI) (auto simp add: bit_simps) | lemma neg_mask_twice:
"x AND NOT(mask n) AND NOT(mask m) = x AND NOT(mask (max n m))"
for x :: \<open>'a::len word\<close> by (rule bit_word_eqI) (auto simp add: bit_simps) | proof (prove)
goal (1 subgoal):
1. x AND NOT (mask n) AND NOT (mask m) = x AND NOT (mask (max n m)) | lemma neg_mask_twice:
"x AND NOT(mask n) AND NOT(mask m) = x AND NOT(mask (max n m))"
for x :: \<open>'a::len word\<close> | unnamed_thy_209 | More_Word | 1 | |
[] | lemma multiple_mask_trivia:
"n \<ge> m \<Longrightarrow> (x AND NOT(mask n)) + (x AND mask n AND NOT(mask m)) = x AND NOT(mask m)"
for x :: \<open>'a::len word\<close> apply (rule trans[rotated], rule_tac w="mask n" in word_plus_and_or_coroll2) apply (simp add: word_bw_assocs word_bw_comms word_bw_lcs neg_mask_twic... | lemma multiple_mask_trivia:
"n \<ge> m \<Longrightarrow> (x AND NOT(mask n)) + (x AND mask n AND NOT(mask m)) = x AND NOT(mask m)"
for x :: \<open>'a::len word\<close> apply (rule trans[rotated], rule_tac w="mask n" in word_plus_and_or_coroll2) apply (simp add: word_bw_assocs word_bw_comms word_bw_lcs neg_mask_twic... | proof (prove)
goal (1 subgoal):
1. m \<le> n \<Longrightarrow> (x AND NOT (mask n)) + (x AND mask n AND NOT (mask m)) = x AND NOT (mask m) proof (prove)
goal (1 subgoal):
1. m \<le> n \<Longrightarrow> (x AND NOT (mask n)) + (x AND mask n AND NOT (mask m)) = ((x AND NOT (mask m)) AND mask n) + ((x AND NOT (mask m)) A... | lemma multiple_mask_trivia:
"n \<ge> m \<Longrightarrow> (x AND NOT(mask n)) + (x AND mask n AND NOT(mask m)) = x AND NOT(mask m)"
for x :: \<open>'a::len word\<close> | unnamed_thy_210 | More_Word | 3 | |
[] | lemma word_of_nat_less:
"\<lbrakk> n < unat x \<rbrakk> \<Longrightarrow> of_nat n < x" apply (simp add: word_less_nat_alt) apply (erule order_le_less_trans[rotated]) apply (simp add: unsigned_of_nat take_bit_eq_mod) done | lemma word_of_nat_less:
"\<lbrakk> n < unat x \<rbrakk> \<Longrightarrow> of_nat n < x" apply (simp add: word_less_nat_alt) apply (erule order_le_less_trans[rotated]) apply (simp add: unsigned_of_nat take_bit_eq_mod) done | proof (prove)
goal (1 subgoal):
1. n < unat x \<Longrightarrow> word_of_nat n < x proof (prove)
goal (1 subgoal):
1. n < unat x \<Longrightarrow> unat (word_of_nat n) < unat x proof (prove)
goal (1 subgoal):
1. unat (word_of_nat n) \<le> n proof (prove)
goal:
No subgoals! | lemma word_of_nat_less:
"\<lbrakk> n < unat x \<rbrakk> \<Longrightarrow> of_nat n < x" | unnamed_thy_211 | More_Word | 4 | |
[] | lemma unat_mask:
"unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1" apply (subst min.commute) apply (simp add: mask_eq_decr_exp not_less min_def split: if_split_asm) apply (intro conjI impI) apply (simp add: unat_sub_if_size) apply (simp add: power_overflow word_size) apply (simp add: unat_sub_if_size)... | lemma unat_mask:
"unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1" apply (subst min.commute) apply (simp add: mask_eq_decr_exp not_less min_def split: if_split_asm) apply (intro conjI impI) apply (simp add: unat_sub_if_size) apply (simp add: power_overflow word_size) apply (simp add: unat_sub_if_size)... | proof (prove)
goal (1 subgoal):
1. unat (mask n) = 2 ^ min n LENGTH('a) - 1 proof (prove)
goal (1 subgoal):
1. unat (mask n) = 2 ^ min LENGTH('a) n - 1 proof (prove)
goal (1 subgoal):
1. (LENGTH('a) \<le> n \<longrightarrow> unat (2 ^ n - 1) = 2 ^ LENGTH('a) - Suc 0) \<and> (\<not> LENGTH('a) \<le> n \<longrightarro... | lemma unat_mask:
"unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1" | unnamed_thy_212 | More_Word | 7 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.