hcju/isabelleps huggingface dataset:
ex_extensionTr: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and H1 :: "'a set"   and h :: "'a"   and g :: "'a \<Rightarrow> 'g"   and x :: "'a" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "H1 \<subseteq> H"   and "h \<in> H"   and "h \<notin> H1"   and "g \<in> mHom R (mdl M (fgs R M H1)) N"   and "x \<in> fgs R M H1 \<minusplus> fgs R M {h}" shows "\<exists>k1\<in>fgs R M H1. \<exists>k2\<in>fgs R M {h}. x = k1 \<plusminus> k2 \<and> (THE y. \<exists>h1\<in>fgs R M H1. \<exists>h2\<in>fgs R M {h}. x = h1 \<plusminus> h2 \<and> y = g h1 \<plusminus>\<^bsub>N\<^esub> monofun R M N f H h h2) = g k1 \<plusminus>\<^bsub>N\<^esub> monofun R M N f H h k2"
rtrancl_imp_rtrancl_UN: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and I :: "('a \<times> 'a) set set" assumes "(x, y) \<in> r\<^sup>*"   and "r \<in> I" shows "(x, y) \<in> (\<Union>r\<in>I. r)\<^sup>*"
eq_iff: fixes a :: "'a word"   and b :: "'a word" shows "(a = b) = (b \<le>s a \<and> a \<le>s b)"
1: fixes I :: "nat set"   and n :: "nat" assumes "finite I \<and> card I = n" shows "has_products I"
simps_2: fixes t\<^sub>1 :: "Term_Class.term"   and t\<^sub>2 :: "Term_Class.term" shows "frees (t\<^sub>1 $ t\<^sub>2) = frees t\<^sub>1 |\<union>| frees t\<^sub>2"
nonpos_Ints_cases': fixes x :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0" obtains n :: "nat" where "x = - of_nat n"
lsided_le3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "lsd x" shows "x \<sqinter> y \<cdot> z \<le> x"
simps: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d" shows "Domainp r a = (\<exists>aa b. a = aa \<and> r aa b)"
preferred_alts_subset: fixes x :: "'alt" shows "preferred_alts (Pareto R) x \<subseteq> alts"
option_while_rule: fixes C :: "'c \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'c option"   and s :: "'c"   and s' :: "'c"   and I :: "'c \<Rightarrow> bool" assumes "option_while C B s = Some s'"   and "I s"   and "\<And>s s'. \<lbrakk>C s; I s; B s = Some s'\<rbrakk> \<Longrightarrow> I s'" shows "I s' \<and> \<not> C s'"
OrdmemD: fixes k :: "hf"   and j :: "hf" assumes "Ord k"   and "j \<^bold>\<in> k" shows "j \<le> k"
ideal_prod_zero: fixes I :: "'a set" assumes "ideal I R" shows "I \<cdot> {\<zero>} = {\<zero>}"
ennreal_top_neq_zero: shows "\<top> \<noteq> 0"
mult_im_sheaf_def: shows "imS.mult_im_sheaf \<equiv> \<lambda>V. \<cdot>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S_rexp_of': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi> = lang n (rexp_of' n \<phi>)"
diff_fract: fixes b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<noteq> (0::'b)"   and "d \<noteq> (0::'b)" shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
alw_coinduct: fixes X :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and \<phi> :: "'a stream \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<phi> x"   and "\<And>x. \<lbrakk>X x; \<not> alw \<phi> (stl x)\<rbrakk> \<Longrightarrow> X (stl x)" shows "alw \<phi> x"
measure_preserving_composition: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and g :: "'e \<Rightarrow> 'f"   and P :: "'f measure" assumes "f \<in> measure_preserving M N"   and "g \<in> measure_preserving N P" shows "(\<lambda>x. g (f x)) \<in> measure_preserving M P"
euler_char_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map" shows "pre_digraph_map.euler_char G M \<equiv> int (card (verts G)) - int (card (arcs G) div 2) + int (card (pre_digraph_map.face_cycle_sets G M))"
represented_generic_objects_in_state: fixes \<tau> :: "'d state \<times> 'd state"   and pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>'e\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> OclAllInstances_generic pre_post H->includes\<^sub>S\<^sub>e\<^sub>t(x)" shows "is_represented_in_state pre_post x H \<tau>"
Integral_zero_fun: fixes a :: "real"   and b :: "real" shows "Integral (a, b) (\<lambda>x. 0) 0"
simps_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and k :: "nat"   and w :: "'c" shows "smap_stoch_proc (Suc n) f k w = f k w # smap_stoch_proc n f (Suc k) w"
simps_11: fixes \<nu> :: "'b"   and v :: "'b" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<langle>v\<^bold>\<rangle> = \<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>v\<^bold>\<rangle>"
restrictPE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "(r |` A) a b" obtains "r a b" and "a \<in> A" and "b \<in> A"
defensive_strategy_Odd: fixes p :: "'a list"   and n :: "nat"   and \<sigma>\<^sub>e :: "'a list \<Rightarrow> 'a" assumes "\<not> winning_position_Even p" shows "\<not> winning_position_Even ((augment_list (joint_strategy \<sigma>\<^sub>e defensive_strategy_Odd) ^^ n) p)"
R3: fixes p :: "'a point" assumes "on_curve a b p" shows "P p (opp p) Infinity"
in_face_cycle_set_other: fixes S :: "'b set" assumes "S \<in> G.face_cycle_sets"   and "{a, a'} \<inter> S = {}" shows "S \<in> H.face_cycle_sets"
map_removeAll_inj: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and xs :: "'a list" assumes "inj f" shows "map f (removeAll x xs) = removeAll (f x) (map f xs)"
elims_1: fixes x :: "'a \<Rightarrow> bool"   and xa :: "'a stream"   and y :: "bool" assumes "holds x xa = y" obtains P :: "'a \<Rightarrow> bool" and xs :: "'a stream" where "x = P" and "xa = xs" and "y = P (shd xs)"
properfactor_cong_r: fixes x :: "'a"   and y :: "'a"   and y' :: "'a" assumes "properfactor G x y"   and "y \<sim> y'"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "y' \<in> carrier G" shows "properfactor G x y'"
inj_map_strong: fixes x :: "'c set['d]"   and xa :: "'c set['d]"   and f :: "'c \<Rightarrow> 'e"   and fa :: "'c \<Rightarrow> 'e" assumes "\<And>z za. \<lbrakk>z \<in> set_bset x; za \<in> set_bset xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_bset f x = map_bset fa xa" shows "x = xa"
store_buffer_execution_result_sequential_consistent': fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and x :: "'c"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and x' :: "'c"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> unit) list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(ts\<^sub>s\<^sub>b, m, x) \<Rightarrow>\<^sub>s\<^sub>b\<^sup>* (ts\<^sub>s\<^sub>b', m', x')"   and "empty_store_buffers ts\<^sub>s\<^sub>b'"   and "ts\<^sub>s\<^sub>b \<sim> ts "   and "initial\<^sub>v ts \<S>"   and "safe_reach_virtual_free_flowing (ts, m, \<S>)" shows "\<exists>ts' \<S>''. (ts, m, \<S>) \<Rightarrow>\<^sub>v\<^sup>* (ts', m', \<S>'') \<and> ts\<^sub>s\<^sub>b' \<sim> ts' "
existence_ivl_subset: fixes x0 :: "'a" shows "ex_ivl x0 \<subseteq> T"
list_all_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun (rel_fun A (=)) (rel_fun (list_all2 A) (=)) list_all list_all"
preserve: fixes a :: "(real mpoly \<times> nat) list"   and b :: "(real mpoly \<times> nat) list"   and L :: "(real mpoly \<times> nat) list"   and L' :: "(real mpoly \<times> nat) list" assumes "(a, b) = chophelper L L'" shows "a @ b = L' @ L"
gWlsAllDisj_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Recursion.model_scheme" shows "gWlsAllDisj MOD \<equiv> gWlsDisj MOD \<and> gWlsAbsDisj MOD"
prime_factorization_nat: fixes n :: "nat" assumes "0 < n" shows "n = (\<Prod>p\<in>prime_factors n. p ^ multiplicity p n)"
s: shows "((\<lambda>x. f x *\<^sub>R g x) has_integral P) (cbox a b)"
CPR_TAG_def: fixes y :: "assn"   and x :: "assn" shows "CPR_TAG y x \<equiv> True"
simps_2: fixes x :: "'c"   and d :: "'b"   and p :: "'c"   and q :: "'b"   and t :: "('c \<times> 'b) list" shows "assoc x d ((p, q) # t) = (if x = p then q else assoc x d t)"
monom_one: shows "monom 0 (1::'b) = 1"
induct: fixes P :: "(char list, char list, nat \<times> tid_or_exn) namespace \<Rightarrow> pat \<Rightarrow> v \<Rightarrow> (char list \<times> v) list \<Rightarrow> bool"   and a0 :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and a1 :: "pat"   and a2 :: "v"   and a3 :: "(char list \<times> v) list" assumes "\<And>cenv x v0 env. P cenv (Pvar x) v0 env"   and "\<And>cenv n ps n' t' vs env. (\<And>x2 x y init xa ya x3. \<lbrakk>cupcake_nsLookup cenv n = Some x2; (x, y) = x2; same_tid y t' \<and> length ps = x; same_ctor (n, y) (n', t'); xa \<in> set ps; ya \<in> set vs; init = Match x3\<rbrakk> \<Longrightarrow> P cenv xa ya x3) \<Longrightarrow> P cenv (Pcon (Some (Short n)) ps) (Conv (Some (n', t')) vs) env"   and "\<And>cenv uv_ uw_. P cenv Pany uv_ uw_"   and "\<And>cenv v uv_ uw_. P cenv (Plit v) uv_ uw_"   and "\<And>cenv va uv_ uw_. P cenv (Pcon None va) uv_ uw_"   and "\<And>cenv vc vd va uv_ uw_. P cenv (Pcon (Some (Long vc vd)) va) uv_ uw_"   and "\<And>cenv v va vb uw_. P cenv (Pcon v va) (Litv vb) uw_"   and "\<And>cenv v va vc uw_. P cenv (Pcon v va) (Conv None vc) uw_"   and "\<And>cenv v va vb vc vd uw_. P cenv (Pcon v va) (Closure vb vc vd) uw_"   and "\<And>cenv v va vb vc vd uw_. P cenv (Pcon v va) (Recclosure vb vc vd) uw_"   and "\<And>cenv v va vb uw_. P cenv (Pcon v va) (Loc vb) uw_"   and "\<And>cenv v va vb uw_. P cenv (Pcon v va) (Vectorv vb) uw_"   and "\<And>cenv v uv_ uw_. P cenv (Pref v) uv_ uw_"   and "\<And>cenv v va uv_ uw_. P cenv (Ptannot v va) uv_ uw_" shows "P a0 a1 a2 a3"
cut_ge_not_empty_imp: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>\<ge> t \<noteq> {}" shows "I \<noteq> {}"
rotate7: fixes A7 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A7 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H)))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H)))))) \<turnstile> B"
comp_ide_self: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "B.VV.comp a a = a"
hyp': shows "(l = PAodv-:6 \<or> l = PAodv-:7 \<or> l = PRrep-:0 \<or> l = PRrep-:1) \<and> sip (\<sigma> i) \<noteq> dip (\<sigma> i)"
ccpo_Sup_upper: fixes A :: "'a set"   and x :: "'a" assumes "Complete_Partial_Order.chain orda A"   and "x \<in> A" shows "orda x (luba A)"
nn_integral_divide: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x / c \<partial>M = integral\<^sup>N M f / c"
Clos_closure: fixes rel :: "(nat \<times> nat) set"   and n :: "nat"   and x :: "nat"   and y :: "nat" assumes "is_bound rel n" shows "((x, y) \<in> Clos rel n) = ((x, y) \<in> rel\<^sup>+)"
kills_e1: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>"
mult_periodic_arithmetic: fixes f :: "nat \<Rightarrow> complex"   and k :: "nat"   and g :: "nat \<Rightarrow> complex" assumes "periodic_arithmetic f k"   and "periodic_arithmetic g k" shows "periodic_arithmetic (\<lambda>n. f n * g n) k"
tND: fixes p :: "('a set set, 'b) Combinators list"   and x :: "('a set set, 'b) Combinators"   and y :: "('a set set, 'b) Combinators"   and a :: "'a set set"   and b :: "'a set set"   and c :: "'a set set"   and d :: "'a set set" assumes "allNetsDistinct p"   and "x \<in> set p"   and "y \<in> set p"   and "a = srcNet x"   and "b = destNet x"   and "c = srcNet y"   and "d = destNet y"   and "a \<noteq> c \<or> b \<noteq> d"   and "singleCombinators [x]"   and "x \<noteq> DenyAll"   and "singleCombinators [y]"   and "y \<noteq> DenyAll" shows "twoNetsDistinct a b c d"
times_not_zero: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "p * q \<noteq> 0"
brec_cond_abs: fixes \<Sigma> :: "('a, 'b tree) HashMap.hashmap \<times> ('a list \<times> 'a list) \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap \<times> 'a option"   and Qi :: "'a set"   and \<delta> :: "('a, 'b) ta_rule set" assumes "\<Sigma> \<in> brec_invar Qi \<delta>" shows "brec_cond \<Sigma> = (brec_\<alpha> \<Sigma> \<in> brw_cond Qi)"
comm_monoid_axioms: shows "comm_monoid (+) (0::'f)"
simps: fixes i :: "Integer"   and xs :: "[:'a:]" shows "sdrop\<cdot>i\<cdot>xs = If le\<cdot>i\<cdot>0 then xs else (case xs of [::] \<Rightarrow> [::] | y :# ys \<Rightarrow> sdrop\<cdot>(i - 1)\<cdot>ys)"
converse_unfold: fixes r :: "('f \<times> 'e) set" shows "r\<inverse> = {(y, x). (x, y) \<in> r}"
\<A>1d_2: shows "(\<S>, \<A>\<^sub>d) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>1, \<A>1d)"
bisim_inv_red_other: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and t' :: "'b"   and x :: "'a"   and m1 :: "'c"   and xx :: "'g"   and m2 :: "'h"   and t :: "'b"   and x1 :: "'a"   and x2 :: "'g"   and x1' :: "'a"   and ta1 :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and x1'' :: "'a"   and m1' :: "'c"   and x2' :: "'g"   and ta2 :: "('d, 'b, 'g, 'h, 'e, 'f) thread_action"   and x2'' :: "'g"   and m2' :: "'h" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "bisim t' (x, m1) (xx, m2)"   and "bisim t (x1, m1) (x2, m2)"   and "(\<tau>trsys.silent_move (r1 t) \<tau>move1)\<^sup>*\<^sup>* (x1, m1) (x1', m1)"   and "r1 t (x1', m1) ta1 (x1'', m1')"   and "\<not> \<tau>move1 (x1', m1) ta1 (x1'', m1')"   and "(\<tau>trsys.silent_move (r2 t) \<tau>move2)\<^sup>*\<^sup>* (x2, m2) (x2', m2)"   and "r2 t (x2', m2) ta2 (x2'', m2')"   and "\<not> \<tau>move2 (x2', m2) ta2 (x2'', m2')"   and "bisim t (x1'', m1') (x2'', m2')"   and "ta_bisim bisim ta1 ta2" shows "bisim t' (x, m1') (xx, m2')"
rec: fixes f1 :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and a :: "'h"   and b :: "'i" shows "old.rec_prod f1 (a, b) = f1 a b"
Q_altdef: shows "Q = (\<Sum>k\<le>3. Polynomial.monom ((- 1) ^ (3 - k) * e (3 - k)) k)"
lists_le_len_fin: fixes P :: "'a set"   and n :: "nat" assumes "finite P" shows "finite (lists P \<inter> {l. length l \<le> n})"
invertible_lf: shows "invertible_lf (*s) (*s) f"
CFG_wf_axioms: shows "CFG_wf src trg knd lve NewEntry lDef lUse state_val"
closure_is_csubspace: fixes A :: "'d set" assumes "csubspace A" shows "csubspace (closure A)"
tSup_def: fixes b :: "'g" shows "tSup b \<equiv> map_fun ((`) (\<lambda>ys. (llist_of_tllist ys, terminal ys))) (\<lambda>(xs, a). tllist_of_llist a xs) (\<lambda>A. (lSup (fst ` A), flat_lub b (snd ` (A \<inter> {(xs, uu_). lfinite xs}))))"
abs_eq: fixes xc :: "'a"   and xb :: "'b \<Rightarrow> 'a pmf"   and xa :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'b" shows "cfg_corec xc xb xa x = abs_cfg (xc, corec_scheduler xb (\<lambda>x s. Inr (xa x s)) x)"
comp_arr_inv': fixes f :: "'a" assumes "R.iso f" shows "R.comp f (R.inv f) = R.cod f"
add: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
finite_INT: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and M :: "'e measure" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "\<Inter> (A ` I) \<in> sets M"
eval_simps_2: fixes f :: "'a E.term" assumes "E.Arr f" shows "EVAL (E.Src f) = src\<^sub>B (EVAL f)"
lfp_trans_unfold: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t u. \<lbrakk>le_trans t u; \<And>P. sound P \<Longrightarrow> sound (t P); \<And>P. sound P \<Longrightarrow> sound (u P)\<rbrakk> \<Longrightarrow> le_trans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. sound Q \<Longrightarrow> sound (t Q); sound P\<rbrakk> \<Longrightarrow> sound (T t P)"   and "le_trans (T v) v"   and "\<And>P. sound P \<Longrightarrow> sound (v P)" shows "equiv_trans (lfp_trans T) (T (lfp_trans T))"
ldropWhile_LCons: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist" shows "ldropWhile P (LCons x xs) = (if P x then ldropWhile P xs else LCons x xs)"
source_size_def: shows "source_size \<equiv> map_fun (map_fun Word.Word id) id (\<lambda>_. LENGTH('b))"
bnEqvt: fixes p :: "(name \<times> name) list"   and \<alpha> :: "'d action" shows "p \<bullet> bn \<alpha> = bn (p \<bullet> \<alpha>)"
dom_fmlookup: fixes m :: "('c, 'd) fmap" shows "dom (($$) m) = fmdom' m"
Basis_le_norm: fixes b :: "'a"   and x :: "'a" assumes "b \<in> Basis" shows "\<bar>x \<bullet> b\<bar> \<le> norm x"
dg_Arr_vempty_if_Obj_vempty: assumes "\<CC>\<lparr>Obj\<rparr> = []\<^sub>\<circ>" shows "\<CC>\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
seqI': fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VoV.in_hom f a b"   and "VoV.in_hom g b c" shows "VoV.seq g f"
UNION_singleton_eq_range: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(\<Union>x\<in>A. {f x}) = f ` A"
id: fixes i :: "nat" assumes "i < nr" shows "row m1 i = row m2 i"
k_mod_eq: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>p. prime p \<and> [p = 3] (mod 4) \<longrightarrow> P p) = (\<forall>k. prime (4 * k + 3) \<longrightarrow> P (4 * k + 3))"
cos_zero_iff: fixes x :: "real" shows "(cos x = 0) = ((\<exists>n. odd n \<and> x = real n * (pi / 2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi / 2))))"
degree_pdevs_of_ivl_le: fixes l :: "'b"   and u :: "'b" shows "degree (pdevs_of_ivl l u) \<le> DIM('b)"
strip_eq_SKIP: fixes c :: "'b acom" shows "(strip c = com.SKIP) = (\<exists>P. c = SKIP {P})"
mat_help: shows "M_mat_R [[1], [0], [- 1]] [([], []), ([0], []), ([], [0])] = mat_of_rows_list 3 [[1, 1, 1], [0, 1, 0], [1, 0, - 1]]"
foldl_LTLAnd_prop_entailment: fixes S :: "'b ltl set"   and i :: "'b ltl"   and xs :: "'b ltl list" shows "S \<Turnstile>\<^sub>P foldl LTLAnd i xs = (S \<Turnstile>\<^sub>P i \<and> (\<forall>y\<in>set xs. S \<Turnstile>\<^sub>P y))"
bd_ctsD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and M :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and b :: "real" assumes "bd_cts t"   and "\<And>i. M i \<tturnstile> M (Suc i)"   and "\<And>i. sound (M i)"   and "\<And>i. bounded_by b (M i)" shows "t (Sup_exp (range M)) = Sup_exp (range (t \<circ> M))"
sumlist_map_as_finsum: fixes f :: "'a \<Rightarrow> int vec"   and xs :: "'a list" assumes "f \<in> set xs \<rightarrow> carrier_vec n"   and "distinct xs" shows "M.sumlist (map f xs) = finsum V f (set xs)"
n_def: shows "n = (LEAST n. \<forall>i\<ge>n. i \<notin> J)"
elims_2: fixes x :: "'b trie"   and xa :: "'b list" assumes "isin x xa" obtains b :: "bool" and m :: "'b \<Rightarrow> 'b trie option" where "x = Nd b__ m__" and "xa = []" and "b__"   | b :: "bool" and m :: "'b \<Rightarrow> 'b trie option" and k :: "'b" and xs :: "'b list" where "x = Nd b m" and "xa = k # xs" and "case m k of None \<Rightarrow> False | Some t \<Rightarrow> isin t xs"
triangle_left: shows "Cop.H f \<epsilon> \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> Cop.H \<eta> f = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
idealprod_whole_r: fixes I :: "'a set" assumes "ideal R I" shows "I \<diamondsuit>\<^sub>r carrier R = I"
bools_of_nat_inverse: fixes k :: "nat"   and n :: "nat" shows "nat_of_bools (bools_of_nat k n) = n"
prob_space_axioms: fixes s :: "'a" shows "prob_space (K1.T s)"
arg: fixes rs :: "(name \<times> pterm) fset"   and u :: "pterm"   and u' :: "pterm"   and t :: "pterm" assumes "rs \<turnstile>\<^sub>p u \<longrightarrow> u'" shows "rs \<turnstile>\<^sub>p t $\<^sub>p u \<longrightarrow> t $\<^sub>p u'"
elims_2: fixes x :: "'a extB"   and xa :: "'a extB" assumes "x < xa" obtains x :: "'a extB" and y :: "'a extB" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
strict_trans1: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "prefix a b"   and "strict_prefix b c" shows "strict_prefix a c"
body: shows "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2),b\<^sub>2\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"
assms_1: shows "(r, s) \<in> R \<times> S"
cases: fixes a1 :: "'a list"   and a2 :: "'b sequent" assumes "(a1, a2) \<in> upRules" obtains c :: "'b sequent" and R :: "'b" and Fs :: "'b form list" and ps :: "'a list" where "a1 = ps__" and "a2 = c__" and "c__ \<equiv>  \<LM> Compound R Fs__  \<RM> \<Rightarrow>* Em" and "ps__ \<noteq> []"   | c :: "'b sequent" and F :: "'b" and Fs :: "'b form list" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "c \<equiv>  \<Empt> \<Rightarrow>* Compound F Fs" and "ps \<noteq> []"
A: shows "x * \<top> \<sqinter> (1::'a) = x"
conjugate_exponent_real_iff: fixes p :: "real"   and q :: "real" assumes "1 < p" shows "(q = conjugate_exponent p) = (1 / p + 1 / q = 1)"
pr_weak'_def: fixes f :: "'f \<times> nat"   and g :: "'f \<times> nat" shows "pr_weak' f g = (f \<notin> lConsts \<and> pr_weak f g \<or> g \<in> lConsts)"
cdifi: shows "d\<inverse> O f\<inverse> \<subseteq> d\<inverse>"
tensor_mat_add1: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m3 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)" shows "tensor_mat dims vars (m1 + m2) m3 = tensor_mat dims vars m1 m3 + tensor_mat dims vars m2 m3"
pseudo_proj_False_def: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat" assumes "infinite_coin_toss_space p M" shows "pseudo_proj_False n = (\<lambda>w. (stake n w @ [False]) @- sconst True)"
ignoreI: fixes \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and s :: "'a"   and a :: "'b"   and ou :: "'c"   and s' :: "'a"   and \<Delta> :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and vl' :: "'d"   and s1 :: "'e"   and vl1 :: "'f" assumes "\<not> \<gamma> (Trans s a ou s')"   and "\<Delta> s' vl' s1 vl1" shows "BD_Security.ignore \<gamma> \<Delta> s s1 vl1 a ou s' vl'"
simps: fixes \<delta> :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<delta>\<^sub>M :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<Sigma> :: "'b set set" shows "ltl_to_rabin_base_def.delta \<delta> \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma> = \<delta> \<times> \<Delta>\<^sub>\<times> (semi_mojmir_def.step \<Sigma> \<delta>\<^sub>M \<circ> q\<^sub>0\<^sub>M \<circ> theG)"