hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "('a, 'b, 'c, 'd) formula" shows "Rep_formula (FL_Formula.Not x) = Not\<^sub>\<alpha> (Rep_formula x)"
list_emb_eq_length_induct: fixes xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "length xs = length ys"   and "list_emb P xs ys"   and "Q [] []"   and "\<And>x y xs ys. \<lbrakk>P x y; list_emb P xs ys; Q xs ys\<rbrakk> \<Longrightarrow> Q (x # xs) (y # ys)" shows "Q xs ys"
cases: fixes a :: "'s Trace"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol" assumes "a \<in> jpTraces jp" obtains s :: "'s" where "a = tInit s__" and "s__ \<in> set envInit"   | t :: "'s Trace" and eact :: "'eAct" and aact :: "'a \<Rightarrow> 'aAct" and s :: "'s" where "a = t \<leadsto> s" and "t \<in> jpTraces jp" and "eact \<in> set (envAction (tLast t))" and "\<And>a. aact a \<in> set (actJP jp t a)" and "s = envTrans eact aact (tLast t)"
simps_15: fixes v :: "nat" shows "\<infinity>\<^sub>2 +\<^sub>2 [Suc v]\<^sub>2 = \<infinity>\<^sub>2"
elims_1: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "bool" assumes "same_level_path_aux x xa = y" obtains cs :: "'edge list" where "x = cs__" and "xa = []" and "y"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> same_level_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> False | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> same_level_path_aux cs' as | _ \<Rightarrow> same_level_path_aux cs as)"
last_ConsR: fixes xs :: "'c list"   and x :: "'c" assumes "xs \<noteq> []" shows "last (x # xs) = last xs"
resChainI: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi"   and lst :: "name list" assumes "eqvt Rel"   and "\<And>P Q a. (P, Q) \<in> Rel \<Longrightarrow> (<\<nu>a>P, <\<nu>a>Q) \<in> Rel"   and "P \<leadsto>\<^sup>^<Rel> Q" shows "resChain lst P \<leadsto>\<^sup>^<Rel> resChain lst Q"
maxchain_imp_chain: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and C :: "'c set" assumes "pred_on.maxchain A P C" shows "pred_on.chain A P C"
assms_2: shows "(i, j) \<in> set (arcs a a xs)"
insert_stays_forest_means_not_connected: fixes u :: "'a"   and v :: "'a"   and E :: "'a uprod set" assumes "forest (insert (Upair u v) E)"   and "Upair u v \<notin> E"   and "u \<noteq> v" shows "(u, v) \<notin> uconnected E"
simps_10: fixes m :: "nat"   and p :: "fm"   and q :: "fm" shows "bound m (Imp p q) = (bound m p \<and> bound m q)"
simps_24: fixes x1 :: "'b"   and x91 :: "'b local.term"   and x92 :: "'b local.term"   and x93 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<a>\<^bold>[x91, x92, x93\<^bold>]"
f2r_eta_pres: shows "\<R> \<eta> = Id"
deutsch_algo_eval_is_sum_mod_2: shows "deutsch_algo_eval = real ((f 0 + f 1) mod 2)"
T_ge_len: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s :: "'b list" assumes "length as = length rs" shows "length rs \<le> T s rs as"
elims_2: fixes x :: "nat"   and xa :: "('a, 'b) Node" assumes "FingerTreeStruc.is_leveln_node x xa" obtains uu_ :: "'a" and uv_ :: "'b" where "x = 0" and "xa = Tip uu_ uv_"   | n :: "nat" and uw_ :: "'b" and n1 :: "('a, 'b) Node" and n2 :: "('a, 'b) Node" where "x = Suc n__" and "xa = Node2 uw_ n1__ n2__" and "FingerTreeStruc.is_leveln_node n__ n1__ \<and> FingerTreeStruc.is_leveln_node n__ n2__"   | n :: "nat" and ux_ :: "'b" and n1 :: "('a, 'b) Node" and n2 :: "('a, 'b) Node" and n3 :: "('a, 'b) Node" where "x = Suc n" and "xa = Node3 ux_ n1 n2 n3" and "FingerTreeStruc.is_leveln_node n n1 \<and> FingerTreeStruc.is_leveln_node n n2 \<and> FingerTreeStruc.is_leveln_node n n3"
singleton_set_def: fixes X :: "'c set" shows "singleton_set X \<equiv> {Y. \<exists>x\<in>X. Y = {x}}"
inside_def: fixes S :: "'a set" shows "inside S \<equiv> {x. x \<notin> S \<and> bounded (connected_component_set (- S) x)}"
linorder_axioms_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.linorder_axioms less_eq \<equiv> \<forall>x y. less_eq x y \<or> less_eq y x"
seqr_right_zero: fixes P :: "('d, 'f) urel" shows "P ;; false = false"
Strong_Soundness_aux_aux: fixes co :: "'b ann_com option"   and s :: "'b"   and co' :: "'b ann_com option"   and t :: "'b"   and c :: "'b ann_com"   and q :: "'b set" assumes "(co, s) -1\<rightarrow> (co', t)"   and "co = Some c"   and "s \<in> pre c"   and "\<turnstile> c  q" shows "if co' = None then t \<in> q else t \<in> pre (the co') \<and> \<turnstile> the co'  q"
strict_linear_order_def: fixes r :: "('b \<times> 'b) set" shows "strict_linear_order r = (strict_partial_order r \<and> total r)"
assms_2: shows "list_all2 P (nths xs (- A)) (nths ys (- A))"
g_finite: shows "value_flow \<Delta> g \<noteq> \<top>"
k_memb_def: fixes s :: "'rep"   and A :: "('ma, 'mt) AlgState" shows "k_memb s A \<equiv> isSome (lookup aOps (aActs A) s)"
simple_arg_pos_def: fixes rel :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) set"   and f :: "'b \<times> nat"   and i :: "nat" shows "simple_arg_pos rel f i \<equiv> \<forall>ts. i < snd f \<longrightarrow> length ts = snd f \<longrightarrow> (Fun (fst f) ts, ts ! i) \<in> rel"
emb_def: fixes ps :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" shows "Partial_Evaluation.emb ps s = (\<lambda>v. case ps v of None \<Rightarrow> s v | Some r \<Rightarrow> r)"
elims_3: fixes x :: "('a, 'b) lit" assumes "\<not> local.wtL x" obtains a :: "('a, 'b) atm" where "x = Pos a__" and "\<not> local.wtA a__"   | a :: "('a, 'b) atm" where "x = Neg a" and "\<not> local.wtA a"
order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a = a \<cdot> b)"
e0: shows "\<guillemotleft>e0 : local.dom e0 \<rightarrow> local.dom f0\<guillemotright>"
constrains_UNIV2: fixes F :: "'d program"   and A :: "'d set" shows "F \<in> A co UNIV"
A\<^sub>i\<^sub>v: fixes x :: "i"   and y :: "i"   and z :: "i" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
big_step_t_determ: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int"   and q :: "nat"   and u :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> p \<Down> t"   and "(c, s) \<Rightarrow> q \<Down> u" shows "u = t"
set_fmap_def: fixes F :: "'c \<Rightarrow> 'd option" shows "set_fmap F = {(x, y). x \<in> dom F \<and> F x = Some y}"
abs_eq: shows "0 = Abs_Integ (0, 0)"
closed_glob_consistent_def: fixes \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "closed_glob_consistent \<R> = (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R> \<longrightarrow> (\<forall>A. (\<forall>x. case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1 x \<noteq> v \<or> mem\<^sub>2 x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x \<noteq> dma mem\<^sub>1 x \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x = Low \<and> (x \<notin> mds AsmNoReadOrWrite \<or> x \<in> \<C>) \<longrightarrow> mem\<^sub>1 [\<parallel>\<^sub>1 A] x = mem\<^sub>2 [\<parallel>\<^sub>2 A] x) \<longrightarrow> (\<langle>c\<^sub>1, mds, mem\<^sub>1 [\<parallel>\<^sub>1 A]\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2 [\<parallel>\<^sub>2 A]\<rangle>) \<in> \<R>))"
chop_apps: fixes t :: "('c, 'd) tm"   and ts :: "('c, 'd) tm list" assumes "is_App t" shows "chop (apps t ts) = apps (chop t) ts"
assms_2: fixes u :: "nat \<Rightarrow> 'a" assumes "F (first s ## u)" shows "Q"
phi_def: fixes g :: "'g"   and v :: "'var \<times> 'node \<times> Def" shows "braun_ssa.phi g v = phis' g (braun_ssa.defNode g v, v)"
strong_fermat_liar_def: shows "local.strong_fermat_liar = (\<forall>k m. odd m \<longrightarrow> n - 1 = 2 ^ k * m \<longrightarrow> [a ^ m = 1] (mod n) \<or> (\<exists>i\<in>{0..<k}. [a ^ (2 ^ i * m) = n - 1] (mod n)))"
UF_def: shows "UF \<equiv> \<Union>C\<in>F. C - {a C}"
sets_return: fixes M :: "'a measure"   and x :: "'a" shows "sets (return M x) = sets M"
1: shows "\<exists>!h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<and> D.cones_map h (D.mkCone p0 p1) = D.mkCone p0' p1'"
c: fixes n :: "nat" shows "integral {c n..b} f = integral {a..b} (\<lambda>x. \<Sum>k = 1..n. if real k / real n \<le> g x then f x /\<^sub>R real n else 0)"
cexpr_sem_eq_on_vars: fixes e :: "cexpr"   and \<sigma> :: "nat \<Rightarrow> val"   and \<sigma>' :: "nat \<Rightarrow> val" assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<sigma> x = \<sigma>' x" shows "cexpr_sem \<sigma> e = cexpr_sem \<sigma>' e"
pochhammer_product': fixes z :: "'a"   and n :: "nat"   and m :: "nat" shows "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
update_with_tries_def: fixes ks :: "'c list"   and f :: "'d list \<Rightarrow> 'd list" shows "update_with_tries ks f = update_with_trie ks (\<lambda>vo. case vo of None \<Rightarrow> f [] | Some vs \<Rightarrow> f vs)"
ord_gen_is_group_order: fixes G :: "('c, 'd) monoid_scheme"   and gen :: "'c" assumes "Finite_And_Cyclic_Groups.cyclic_group G gen" shows "group.ord G gen = order G"
lec_const_def: fixes le_constraint :: "'a le_constraint" shows "lec_const le_constraint = (case le_constraint of Le_Constraint x1 x2 x3 \<Rightarrow> x3)"
simps_12: fixes v :: "fm"   and va :: "fm" shows "not (Imp v va) = NOT (Imp v va)"
Interleaves_suffix_one_fst: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "P x []" shows "xs \<cong> {ys, zs, \<lambda>w ws. P w (ws @ [x])} = xs @ [x] \<cong> {ys @ [x], zs, P}"
abs_eq: fixes xa :: "real"   and x :: "real" assumes "eq_onp (\<lambda>x. x \<in> {0..1}) xa xa"   and "eq_onp (\<lambda>x. x \<in> {0..1}) x x" shows "Linear_Aggregation_Algebras.unit.Abs_unit xa + Linear_Aggregation_Algebras.unit.Abs_unit x = Linear_Aggregation_Algebras.unit.Abs_unit (local.tl xa x)"
convergent_prod_altdef: fixes f :: "nat \<Rightarrow> 'e" shows "convergent_prod f = (\<exists>M L. (\<forall>n\<ge>M. f n \<noteq> (0::'e)) \<and> (\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> L \<and> L \<noteq> (0::'e))"
assms_3: shows "(red B)\<^sup>*\<^sup>* p q"
basic_trans_rules_19: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x < y"   and "y < z" shows "x < z"
adjust_not_in_span: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us"   and "w \<notin> local.span (set us)" shows "adjuster n w us + w \<notin> local.span (set us)"
le_iff_inf: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (min x y = x)"
fmap_of_list_simps_2: fixes k :: "'a"   and v :: "'b"   and kvs :: "('a \<times> 'b) list" shows "fmap_of_list ((k, v) # kvs) = fmupd k v (fmap_of_list kvs)"
when_simps_2: fixes a :: "'a" shows "(a when False) = (0::'a)"
ThrowRed: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow> \<langle>throw e',s'\<rangle>"
ComposeN: fixes T :: "('fun, 'var) Term.term list"   and f :: "'fun"   and M :: "('fun, 'var) Term.term set"   and steps :: "('fun, 'var) Term.term \<Rightarrow> nat" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> \<langle>M; steps t\<rangle> \<turnstile>\<^sub>n t" shows "\<langle>M; Suc (Max (insert 0 (steps ` set T)))\<rangle> \<turnstile>\<^sub>n Fun f T"
max_Max_commute: fixes A :: "'a set"   and m :: "'a"   and x :: "'a" assumes "finite A" shows "max (Max (insert m A)) x = max m (Max (insert x A))"
wf_atable_subset: fixes n :: "nat"   and V :: "nat set"   and X :: "'b option list set"   and Y :: "'b option list set" assumes "table n V X"   and "Y \<subseteq> X" shows "table n V Y"
rep_eq: fixes x :: "('b, 'c) RBT.rbt" shows "RBT.keys x = RBT_Impl.keys (rbt.impl_of x)"
LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. \<parallel>X n - L\<parallel> < r)"
\<gamma>_ivl_def: fixes i :: "ivl" shows "\<gamma>_ivl i = (case i of I None None \<Rightarrow> UNIV | I None (Some h) \<Rightarrow> {..h} | I (Some l) None \<Rightarrow> {l..} | I (Some l) (Some h) \<Rightarrow> {l..h})"
is_dghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> (\<Z> \<alpha> \<and> vfsequence \<FF>) \<and> digraph \<alpha> \<AA> \<and> digraph \<alpha> \<BB> \<and> is_dghm_axioms \<AA> \<BB> \<FF>"
arr_Rel_def: shows "S = [S\<lparr>ArrVal\<rparr>, S\<lparr>ArrDom\<rparr>, S\<lparr>ArrCod\<rparr>]\<^sub>\<circ>"
WTCond: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty"   and T :: "ty" assumes "P,E \<turnstile> e :: Boolean"   and "P,E \<turnstile> e\<^sub>1 :: T\<^sub>1"   and "P,E \<turnstile> e\<^sub>2 :: T\<^sub>2"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<or> P \<turnstile> T\<^sub>2 \<le> T\<^sub>1"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<longrightarrow> T = T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T\<^sub>1 \<longrightarrow> T = T\<^sub>1" shows "P,E \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 :: T"
norm_square_vec_eq: fixes x :: "(real, 'a) vec" shows "(norm x)\<^sup>2 = (\<Sum>i\<in>UNIV. (x $ i)\<^sup>2)"
induct: fixes x1 :: "'a \<Rightarrow> 'a list"   and x2 :: "('a, 'b) Term.term"   and x3 :: "('a, 'b) Term.term"   and P :: "('a \<Rightarrow> 'a list) \<Rightarrow> ('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term \<Rightarrow> bool" assumes "term_variants_pred x1 x2 x3"   and "\<And>Pa x. P Pa (Var x) (Var x)"   and "\<And>T S Pa g f. \<lbrakk>length T = length S; \<And>i. i < length T \<Longrightarrow> term_variants_pred Pa (T ! i) (S ! i); \<And>i. i < length T \<Longrightarrow> P Pa (T ! i) (S ! i); g \<in> set (Pa f)\<rbrakk> \<Longrightarrow> P Pa (Fun f T) (Fun g S)"   and "\<And>T S Pa f. \<lbrakk>length T = length S; \<And>i. i < length T \<Longrightarrow> term_variants_pred Pa (T ! i) (S ! i); \<And>i. i < length T \<Longrightarrow> P Pa (T ! i) (S ! i)\<rbrakk> \<Longrightarrow> P Pa (Fun f T) (Fun f S)" shows "P x1 x2 x3"
eq_imp_below: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.po below"   and "x = y" shows "below x y"
conc_fun_SPEC: fixes R :: "('a \<times> 'b) set"   and \<Phi> :: "'b \<Rightarrow> bool" shows "\<Down> R (SPEC \<Phi>) = SPEC (\<lambda>y. \<exists>x. (y, x) \<in> R \<and> \<Phi> x)"
inf_right_idem: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<cdot> y = x \<cdot> y"
contour_integrable_on_shiftpath_eq: fixes g :: "real \<Rightarrow> complex"   and a :: "real"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g"   and "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "(f contour_integrable_on shiftpath a g) = (f contour_integrable_on g)"
vars0_def: shows "st.vars0 = {0..<Suc k} \<union> {Suc k}"
MKT_MKT: shows "t = MKT n (MKT ln ll lr lh) (MKT rn rl rr rh) h"
take_conc': fixes xs :: "int list" assumes "xs \<noteq> []" shows "\<exists>ys. xs = Challenge1A.take xs @ ys \<and> (ys \<noteq> [] \<longrightarrow> Challenge1A.take xs = take_incr xs \<and> \<not> last (take_incr xs) < hd ys \<or> Challenge1A.take xs = take_decr xs \<and> \<not> hd ys \<le> last (take_decr xs))"
rep_eq: fixes x :: "'c fset"   and xa :: "('c, 'd) fmap" shows "fmlookup (fmdrop_fset x xa) = map_drop_set (fset x) (fmlookup xa)"
cases: fixes \<tau>move :: "('c, 'd, 'a, 'b, 'e, 'f) \<tau>moves"   and a1 :: "('c, 'd) locks \<times> (('d \<Rightarrow> ('a \<times> 'c \<Rightarrow>f nat) option) \<times> 'b) \<times> ('d \<Rightarrow> 'e wait_set_status option) \<times> 'd set"   and a2 :: "'d \<times> ('c, 'd, 'a, 'b, 'e, 'f) thread_action"   and a3 :: "('c, 'd) locks \<times> (('d \<Rightarrow> ('a \<times> 'c \<Rightarrow>f nat) option) \<times> 'b) \<times> ('d \<Rightarrow> 'e wait_set_status option) \<times> 'd set" assumes "\<tau>multithreaded.m\<tau>move \<tau>move a1 a2 a3" obtains s :: "('c, 'd) locks \<times> (('d \<Rightarrow> ('a \<times> 'c \<Rightarrow>f nat) option) \<times> 'b) \<times> ('d \<Rightarrow> 'e wait_set_status option) \<times> 'd set" and t :: "'d" and x :: "'a" and s' :: "('c, 'd) locks \<times> (('d \<Rightarrow> ('a \<times> 'c \<Rightarrow>f nat) option) \<times> 'b) \<times> ('d \<Rightarrow> 'e wait_set_status option) \<times> 'd set" and x' :: "'a" and ln' :: "'c \<Rightarrow>f nat" and ta :: "('c, 'd, 'a, 'b, 'e, 'f) thread_action" where "a1 = s" and "a2 = (t, ta)" and "a3 = s'" and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "thr s' t = \<lfloor>(x', ln')\<rfloor>" and "\<tau>move (x, shr s) ta (x', shr s')"
convert_extTA_simps_4: fixes f :: "'i \<Rightarrow> 'e"   and ta :: "('d, 'c, 'i, 'f, 'g, 'h) thread_action" shows "\<lbrace>convert_extTA f ta\<rbrace>\<^bsub>c\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>"
fvi_trm_plus: fixes x :: "nat"   and b :: "nat"   and c :: "nat"   and t :: "trm" shows "(x \<in> fvi_trm (b + c) t) = (x + c \<in> fvi_trm b t)"
Infinitesimal_ratio: fixes y :: "'a star"   and x :: "'a star" assumes "y \<noteq> 0"   and "y \<in> Infinitesimal"   and "x / y \<in> HFinite" shows "x \<in> Infinitesimal"
pq: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> set_pmf pq" shows "R x y"
in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "B.arr f"   and "B.dom f = a"   and "B.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>"
extensional_def: fixes A :: "'c set" shows "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
spied_subset: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set" assumes "s \<Turnstile> s'" shows "spied s \<subseteq> spied s'"
cases: fixes a1 :: "real"   and a2 :: "(real \<Rightarrow> real) list" assumes "is_expansion a1 a2" obtains c :: "real" where "a1 = c" and "a2 = []"
Suc: shows "map_poly Re (p ^ n) = map_poly Re p ^ n"
insert_into_member_list_def: fixes new_el :: "'b"   and Sets :: "'b set list"   and S :: "'b set" shows "insert_into_member_list new_el Sets S = (S \<union> {new_el}) # remove1 S Sets"
mult_diag: fixes A :: "('c, 'd) sq_matrix"   and x :: "'c" shows "A * diag x = x *\<^sub>S A"
fresh_vsubst_ident: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x1 :: "'var" assumes "good Y"   and "fresh xs x Y" shows "Y #[x1 // x]_xs = Y"
rsp: shows "((=) ===> (=) ===> (\<approx>abs_set)) Pair Pair"
ws'ps': shows "set (map snd ws' @ ps') = set (remdups' (map_prod post post) (filter (\<lambda>p. map_prod post post p \<notin> N) (map (\<lambda>a. (delta a r', delta a s')) (\<sigma> n)))) \<union> set (map snd ws @ ps)"
simps_2: fixes k :: "'a"   and k' :: "'a"   and v :: "'b"   and xs :: "('a \<times> 'b) list" shows "delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # delete_aux k xs)"
Al: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "\<not> ewp y"   and "x = y \<cdot> x + z" shows "x = y\<^sup>\<star> \<cdot> z"
le_ASSERT_iff: fixes S :: "'b nres"   and \<Phi> :: "bool"   and S' :: "'b nres" shows "(S \<le> ASSERT \<Phi> \<bind> (\<lambda>_. S')) = (\<Phi> \<longrightarrow> S \<le> S')"
nn_integral_nonneg_infinite: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "\<not> integrable M f"   and "AE x in M. 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = \<infinity>"
bot_xor: fixes x :: "'b" shows "\<bottom> \<oplus> x = x"
order_gt_one: fixes G :: "('c, 'd) monoid_scheme" assumes "simple_group G" shows "1 < order G"
Ebackflowssimp: shows "backflows (filternew_flows_state \<T>) - (backflows (filternew_flows_state \<T>) - E) = E"
set_upt: fixes i :: "nat"   and j :: "nat" shows "set [i..<j] = {i..<j}"
bag_bound_V_empty: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and t :: "'g" assumes "TreeDecomposition G T bag"   and "V\<^bsub>G\<^esub> = {}"   and "t \<in> V\<^bsub>T\<^esub>" shows "card (bag t) = 0"
mpoly_induct: fixes P :: "'b mpoly \<Rightarrow> bool"   and p :: "'b mpoly" assumes "\<And>m a. P (monom m a)"   and "\<And>p1 p2 m a. \<lbrakk>P p1; P p2; p2 = monom m a; m \<notin> keys (mapping_of p1)\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"
Abs_Node_inverse: fixes y :: "(nat \<Rightarrow> 'f + nat) \<times> ('g + nat)" assumes "y \<in> Old_Datatype.Node" shows "Rep_Node (Abs_Node y) = y"
matrix_sum_shift_Suc: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and d :: "nat" assumes "\<And>k. k < Suc n \<Longrightarrow> f k \<in> carrier_mat d d" shows "matrix_sum d f (Suc n) = f 0 + matrix_sum d (\<lambda>k. f (Suc k)) n"
rwof_FAIL_imp_WHILE_FAIL: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and s :: "'c" assumes "rwof m0 cond step s"   and "cond s"   and "step s = FAIL" shows "m0 \<bind> WHILE cond step = FAIL"
card_nonzero: fixes A :: "'c set" assumes "finite A"   and "card A \<noteq> 0" shows "A \<noteq> {}"
length_sorted_list_of_set: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less" shows "length (linorder.sorted_list_of_set less_eq A) = card A"
path_empt: fixes SES :: "('c, 'd) SES_rec"   and s1 :: "'c" shows "s1 []\<Longrightarrow>\<^bsub>SES\<^esub> s1"
pred_eq_const2: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and c :: "'d" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "c \<in> space N \<Longrightarrow> {c} \<in> sets N" shows "pred M (\<lambda>x. c = f x)"
DiamondTwo: fixes cfg :: "('p, 'v, 's) configuration"   and Q :: "'p set"   and cfg1 :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message"   and cfg2 :: "('p, 'v, 's) configuration" assumes "qReachable cfg Q cfg1"   and "cfg \<turnstile> msg \<mapsto> cfg2"   and "\<exists>p\<in>Proc - Q. isReceiverOf p msg" shows "\<exists>cfg'. cfg1 \<turnstile> msg \<mapsto> cfg' \<and> qReachable cfg2 Q cfg'"
well\<^sub>h: shows "well_base\<^sub>h base M"
cdf_bounded_prob: fixes k :: "nat"   and x :: "real" shows "cdf (\<mu> k) x \<le> 1"
powsum_split_var6: fixes x :: "'a"   and k :: "nat"   and n :: "nat" shows "x\<^bsub>0\<^esub>\<^bsup>Suc k + Suc n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>Suc k\<^esup> + x\<^bsub>0 + Suc (Suc k)\<^esub>\<^bsup>n\<^esup>"
left_partI: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set"   and v :: "'c"   and u :: "'e"   and s :: "'e"   and t :: "'e" assumes "TreeDecomposition G T bag"   and "v \<in> bag u"   and "u \<in> Tree.left_tree T s t" shows "v \<in> TreeDecomposition.left_part T bag s t"
traces_def: fixes P :: "'c process" shows "traces P \<equiv> Domain (failures P)"
n_mult_glb': fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(!z \<le> !x \<and> !z \<le> !y) = (!z \<le> !x \<cdot> !y)"