hcju/isabelleps huggingface dataset:
f_Equiv_Exec_Stream_Acc_LocalState_set: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "'h list"   and i :: "nat" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "Equiv_Exec_stable_set A equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2"   and "set input \<subseteq> A"   and "i < length input" shows "equiv_states (f_Exec_Comp_Stream_Acc_LocalState k1 localState1 trans_fun1 (map input_fun1 input) c1 ! i) (f_Exec_Comp_Stream_Acc_LocalState k2 localState2 trans_fun2 (map input_fun2 input) c2 ! i)"
word_of_int_neg_1: shows "word_of_int (- 1) = - 1"
alphaAbs_qAbs_iff_alphaAbs_ex_distinct_qAFresh: fixes xs :: "'f"   and x :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_distinct_qAFresh xs x X xs' x' X'"
cardSuc_Card_order: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "Card_order (cardSuc r)"
sets_sup_measure_F: fixes I :: "'a measure set"   and M :: "'a measure" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> sets i = sets M" shows "sets (sup_measure.F id I) = sets M"
ponesubst_correct: fixes l :: "'a list"   and P1 :: "pol"   and M :: "mon"   and P2 :: "pol"   and P3 :: "pol" assumes "in_carrier l"   and "ponesubst P1 M P2 = Some P3"   and "Imon l M = Ipol l P2" shows "Ipol l P1 = Ipol l P3"
abs_split_def: fixes xs :: "('a \<times> 'b) list"   and x :: "'b" shows "abs_split xs x = (takeWhile (\<lambda>(uu_, s). s < x) xs, dropWhile (\<lambda>(uu_, s). s < x) xs)"
select_convs_2: fixes aActs :: "'c"   and aTrans :: "'b"   and more :: "'d" shows "aTrans \<lparr>aActs = aActs, aTrans = aTrans, \<dots> = more\<rparr> = aTrans"
Nil_in_nlists: fixes A :: "'c set"   and n :: "nat" shows "([] \<in> nlists A n) = (n = 0)"
hyps_3: fixes P :: "'a set set" assumes "partition_on A P" shows "card P \<le> card A"
les_def: fixes s :: "bdd"   and s' :: "bdd" shows "mi.les s s' \<equiv> \<forall>ni n. (ni, n) \<in> Rmi s \<longrightarrow> (ni, n) \<in> Rmi s'"
set_aux_def: fixes uu :: "set_impl" shows "set_aux uu = set"
invh_baliL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (baliL l a r)"
simps_2: fixes c :: "color"   and l :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c"   and r :: "('a, 'c) rbt" shows "rbt_sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> rbt_sorted l \<and> rbt_sorted r)"
cyc: shows "\<not> acyclicP (\<lambda>y x. y \<in># Y \<and> x \<in># Y \<and> gt y x)"
Sups_dual_def: shows "\<partial> \<circ> Sup = Inf \<circ> (`) \<partial>"
strict_order_transitive_acyclic: fixes x :: "'a" shows "strict_order x = (transitive x \<and> pd_kleene_allegory_class.acyclic x)"
prod_foldl_conv: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'c list"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and b :: "'b" shows "(foldl f a xs, foldl g b xs) = foldl (\<lambda>(a, b) x. (f a x, g b x)) (a, b) xs"
infinitesimal_linear': fixes B :: "real"   and C :: "real"   and At :: "atomUni"   and x :: "real" assumes "B \<noteq> 0" shows "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. aEvalUni At x) = evalUni (substInfinitesimalLinearUni B C At) x"
simps_3: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "find_closest_pair (c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va) = (let p\<^sub>1 = find_closest_bf p\<^sub>0 (take 7 (v # va)) in if dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then find_closest_pair (c\<^sub>0, c\<^sub>1) (v # va) else find_closest_pair (p\<^sub>0, p\<^sub>1) (v # va))"
subterm_subst_not_img_subterm: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'c) Term.term"   and \<I> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "t \<sqsubseteq> s \<cdot> \<I>"   and "\<not> (\<exists>w\<in>fv s. t \<sqsubseteq> \<I> w)" shows "\<exists>f T. Fun f T \<sqsubseteq> s \<and> t = Fun f T \<cdot> \<I>"
dom_h': fixes x :: "'v option"   and y :: "'v"   and n :: "nat" assumes "(x, y) \<notin> R n" shows "h' n x y = 0"
cd_ipd_is_cd: fixes k :: "nat"   and m :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and j :: "nat" assumes "k < m"   and "\<pi> m = ipd (\<pi> k)"   and "\<forall>n\<in>{k..<m}. \<pi> n \<noteq> ipd (\<pi> k)"   and "m cd\<^bsup>\<pi>\<^esup>\<rightarrow> j" shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
updates_simps_1: fixes vs :: "'d list"   and ps :: "('c \<times> 'd) list" shows "updates [] vs ps = ps"
msum_comm: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and N1 :: "'i set"   and N2 :: "'i set" assumes "module R M"   and "LinearCombinations.submodule R N1 M"   and "LinearCombinations.submodule R N2 M" shows "LinearCombinations.module.submodule_sum M N1 N2 = LinearCombinations.module.submodule_sum M N2 N1"
fvi_Suc_bound: fixes b :: "nat"   and \<phi> :: "'a MFOTL.formula"   and n :: "nat" assumes "\<forall>i\<in>MFOTL.fvi (Suc b) \<phi>. i < n" shows "\<forall>i\<in>MFOTL.fvi b \<phi>. i < Suc n"
kat_1_equiv: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y \<cdot> r x \<le> r z \<cdot> y) = (ar z \<cdot> y \<le> y \<cdot> ar x)"
returns_to_def: fixes P :: "'d set"   and x :: "'d" shows "returns_to P x = ((\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P) \<and> (\<exists>t>0. t \<in> existence_ivl0 x \<and> flow0 x t \<in> P))"
simps_1: fixes c :: "color"   and lt :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c" shows "rbt_max_opt (Branch c lt k v rbt.Empty) = k"
measure_of_eq: fixes A :: "'b set set"   and \<Omega> :: "'b set"   and \<mu> :: "'b set \<Rightarrow> ennreal"   and \<mu>' :: "'b set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>"   and "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a" shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
continuous_log: fixes f :: "complex \<Rightarrow> complex" assumes "simply_connected S"   and "continuous_on S f"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" shows "\<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = exp (g z))"
inv_char: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" shows "B.VV.inv f = B.VxV.inv f"
sset_sproduct: fixes s1 :: "'c stream"   and s2 :: "'d stream" shows "sset (sproduct s1 s2) = sset s1 \<times> sset s2"
arr_cod: fixes f :: "'c" assumes "C.arr f" shows "C.arr (C.cod f)"
cp_thm': fixes p :: "fm"   and d :: "int"   and bbs :: "bool list"   and bs :: "int list"   and i :: "int" assumes "iszlfm p"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d" shows "(\<exists>x. Ifm bbs (x # bs) p) = ((\<exists>j\<in>{1..d}. Ifm bbs (j # bs) (minusinf p)) \<or> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (i # bs) ` set (\<beta> p). Ifm bbs ((b + j) # bs) p))"
bn_fresh_Act: fixes \<alpha> :: "'c"   and x :: "('a, 'b, 'c) formula" shows "bn \<alpha> \<sharp>* Act \<alpha> x"
new_pseudodeterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and v :: "('a, 'b, 'c) val"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])" shows "substitution.revision_step subst r s s' = (\<exists>l. l \<notin> LID\<^sub>G s \<and> s' = s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)])))"
code: fixes gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv" shows "TRY gpv ELSE gpv' = GPV (map_spmf (map_generat id id (\<lambda>c input. case c input of Inl gpv \<Rightarrow> TRY gpv ELSE gpv' | Inr gpv' \<Rightarrow> gpv')) (TRY map_spmf (map_generat id id (map_fun id Inl)) (the_gpv gpv) ELSE map_spmf (map_generat id id (map_fun id Inr)) (the_gpv gpv')))"
count_mset: fixes xs :: "'d list"   and x :: "'d" shows "count (mset xs) x = length (filter ((=) x) xs)"
hlp1: shows "simple_rule_dtor \<circ> (\<lambda>(u, a, b). SimpleRule u (if a = Accept \<and> b = Accept then Accept else Drop)) = apsnd (\<lambda>(a, b). if a = Accept \<and> b = Accept then Accept else Drop)"
gR_rel_mult: fixes x :: "'a"   and y :: "'a" assumes "D x y" shows "x \<preceq>\<^sub>R x \<oplus> y"
lfinite_ltl: fixes xs :: "'c llist" shows "lfinite (ltl xs) = lfinite xs"
Guard: fixes s :: "'a"   and g :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and t :: "('a, 'c) xstate"   and f :: "'c" assumes "s \<in> g"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow> t"
integral_cong_AE: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AEeq M f g" shows "integral\<^sup>L M f = integral\<^sup>L M g"
f_aggregate_take_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate (xs \<down> (n * k)) k ag = f_aggregate xs k ag \<down> n"
produce_type_type: fixes xs :: "checker_type"   and cons :: "checker_type"   and tm :: "t list" assumes "produce xs cons = Type tm" shows "\<exists>tn. xs = Type tn"
Ioc_inj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b = ord.greaterThanAtMost less_eq less c d) = (less_eq b a \<and> less_eq d c \<or> a = c \<and> b = d)"
pgalleryI_gallery: fixes xs :: "'a set list" assumes "folding_g.gallery xs"   and "distinct xs" shows "pgallery xs"
bind_returns_result_E3: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and y :: "'c"   and x :: "'d" assumes "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "pure f h" shows "h \<turnstile> g x \<rightarrow>\<^sub>r y"
infinite_Ord_omega: fixes \<alpha> :: "V" assumes "Ord \<alpha>"   and "infinite (elts \<alpha>)" shows "\<omega> \<le> \<alpha>"
assume_def: fixes p :: "'c"   and q :: "'c" shows "[. p .] q \<equiv> - p \<squnion> q"
simps_2: fixes h :: "real poly"   and T :: "real poly list" shows "check_all_const_deg (h # T) = (if degree h = 0 then check_all_const_deg T else False)"
inext_nth_surj_aux1: fixes I :: "nat set" shows "{x \<in> I. \<nexists>n. I \<rightarrow> n = x} = {}"
assms: shows "\<exists>p r i. j = nxt (make_full_observation drinks (Some 1) r p) i"
isomorphic_implies_same_tab_1: fixes r' :: "'a" assumes "(tab\<^sub>0 r)\<^sup>* \<star> tab\<^sub>1 s \<cong> r'" shows "p\<^sub>0 = tab\<^sub>0 r'"
WT: shows "\<Gamma> \<turnstile>g gpv \<bind> f \<surd>"
r_prepend_recfn: fixes r :: "recf"   and vs :: "nat list" assumes "recfn 1 r" shows "recfn 1 (r_prepend vs r)"
nEq_neq_conv: fixes comp :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c" assumes "comparator comp" shows "(comp x y \<noteq> Eq) = (x \<noteq> y)"
wordinterval_CIDR_split1_distinct2: fixes r :: "'b wordinterval"   and s :: "'b prefix_match"   and u :: "'b wordinterval" assumes "wordinterval_CIDR_split1 r = (Some s, u)" shows "wordinterval_empty (wordinterval_intersection (prefix_to_wordinterval s) u)"
assms_2: fixes a :: "'a" assumes "a \<in> A" shows "k a = \<Union> (k ` A)"
simps_2: shows "link \<langle>\<rangle> = \<langle>\<rangle>"
less_le_not_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
image_of_maximum: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and m :: "'c" assumes "mono f"   and "\<And>x. x \<in> M \<Longrightarrow> x \<le> m"   and "m \<in> M" shows "\<Squnion> (f ` M) = f m"
jordancurve_3: shows "frontier (outside (path_image (p +++ linepath a b))) = path_image (p +++ linepath a b)"
types_conf_conf_types_conf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and es :: "expr list"   and Ts :: "ty list"   and Ts' :: "ty list" assumes "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts"   and "conformable Ts Ts'" shows "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts'"
that_2: fixes x :: "'a" assumes "a < x"   and "x < i" shows "g x = f x"
locally_sound_mode_use_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and lc :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.locally_sound_mode_use eval lc = (\<forall>c' mds' mem'. ((c', mds'), mem') \<in> sifum_security.loc_reach eval lc \<longrightarrow> (\<forall>x. (x \<in> mds' GuarNoRead \<longrightarrow> sifum_security.doesnt_read eval c' x) \<and> (x \<in> mds' GuarNoWrite \<longrightarrow> sifum_security.doesnt_modify eval c' x)))"
gencount: shows "\<forall>i. I = {i} \<longrightarrow> x \<notin> Xout i \<and> x \<notin> Xin i"
setdist_gt_0_closedin: fixes U :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "closedin (top_of_set U) S"   and "x \<in> U"   and "S \<noteq> {}"   and "x \<notin> S" shows "0 < setdist {x} S"
sum_pos: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> (0::'d) < f i" shows "(0::'d) < sum f I"
assms_1: shows "\<not> hb (i, Add i e1) (ir, Rem is e2)"
assms: fixes x :: "Value"   and y :: "C \<rightarrow> CValue" assumes "x \<triangleleft>\<triangleright> y\<cdot>C\<^sup>\<infinity>" shows "f\<cdot>x \<triangleleft>\<triangleright> g\<cdot>y\<cdot>C\<^sup>\<infinity>"
config_config: fixes A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and init :: "'b list"   and qs :: "'b list" shows "set (fst (config A init qs)) = set init \<and> distinct (fst (config A init qs)) = distinct init \<and> length (fst (config A init qs)) = length init"
jozsa_algo_result: shows "jozsa_algo = Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))"
N_of_norm_1: fixes f :: "'c" shows "eNorm N_of_norm f = ennreal (norm f)"
subst_atm_mset_lists_empty: fixes \<sigma>s :: "'s list" shows "[] \<cdot>\<cdot>aml \<sigma>s = []"
eval_Mn: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list" assumes "recfn (length xs) (Mn n f)" shows "eval (Mn n f) xs = (if \<exists>z. eval f (z # xs) \<down>= 0 \<and> (\<forall>y<z. eval f (y # xs) \<down>) then Some (LEAST z. eval f (z # xs) \<down>= 0 \<and> (\<forall>y<z. eval f (y # xs) \<down>)) else None)"
PChain: shows "P \<Longrightarrow>\<^sub>\<tau> P'"
swap_def: fixes i :: "'b"   and j :: "'b"   and x :: "'b" shows "swap i j x = (if x = i then j else if x = j then i else x)"
list_nth: fixes i :: "nat"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "i < length (list s n)" shows "list s n ! i = s i"
mult_smult_left: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "smult a p * q = smult a (p * q)"
subtype_correct: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and t :: "'d set"   and P :: "'d set"   and t' :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.subtype ev\<^sub>B t P t' = (\<forall>mem. sifum_lang_no_dma.pred ev\<^sub>B P mem \<longrightarrow> sifum_types.type_max ev\<^sub>B t mem \<le> sifum_types.type_max ev\<^sub>B t' mem)"
le_int_or: fixes y :: "int"   and x :: "int" assumes "bin_sign y = 0" shows "x \<le> x OR y"
dist_diff_1: fixes a :: "'a"   and b :: "'a" shows "dist a (a - b) = norm b"
return_def: fixes a :: "'d" shows "pure a = State (Pair a)"
set_alls2: fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "set (alls2 A B a b) = {((x, s), y, t). length x = length a \<and> length y = length b \<and> (\<forall>i<length a. x ! i \<le> A) \<and> (\<forall>j<length b. y ! j \<le> B) \<and> s = a \<bullet> x \<and> t = b \<bullet> y}"
strip_eq_If: fixes C :: "'a acom"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com" shows "(strip C = IF b THEN c1 ELSE c2) = (\<exists>P1 P2 C1 C2 Q. C = IF b THEN {P1} C1 ELSE {P2} C2 {Q} \<and> strip C1 = c1 \<and> strip C2 = c2)"
\<eta>_def: shows "FG.\<eta> \<equiv> transformation_by_components.map (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map FG.\<eta>o"
set_map_1: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and f3 :: "'f \<Rightarrow> 'c"   and v :: "('d, 'e, 'f) val" shows "RID\<^sub>V (map_val f1 f2 f3 v) = f1 ` RID\<^sub>V v"
lex_sum_list: fixes xs :: "(real \<times> real) list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> Counterclockwise_2D_Arbitrary.lex x 0" shows "Counterclockwise_2D_Arbitrary.lex (sum_list xs) 0"
mem_Sigma_iff: fixes a :: "'f"   and b :: "'g"   and A :: "'f set"   and B :: "'f \<Rightarrow> 'g set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
pbval_extend: fixes ps1 :: "char list \<Rightarrow> int option"   and ps2 :: "char list \<Rightarrow> int option"   and b :: "bexp" assumes "ps1 ## ps2"   and "vars b \<subseteq> dom ps1" shows "pbval b (ps1 + ps2) = pbval b ps1"
iTrigger_False: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(P t'. t' \<T> t I. False) = (I = {})"
collapse_1: fixes literal :: "'a literal" assumes "sign literal" shows "Pos (get_pred literal) (get_terms literal) = literal"
Glb_greatest: fixes b :: "'a"   and i :: "'i"   and A :: "'a set" assumes "b \<in> L i"   and "\<forall>a\<in>A. b \<le> a" shows "b \<le> Glb i A"
finite_state_imp_restriction_finite_1: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a" assumes "finite (RID\<^sub>G s)" shows "finite (RID\<^sub>G (s(r := None)))"
finite_conversep: fixes r :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "finite {(x, y). r\<inverse>\<inverse> x y} = finite {(x, y). r x y}"
coprime_exp_mod: fixes lu :: "int"   and n :: "nat" assumes "comm_monoid_mult_class.coprime lu p"   and "n \<noteq> 0" shows "lu mod p ^ n \<noteq> 0"
sel_1: fixes x1 :: "'c \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> 'c list \<times> 'e"   and x2 :: "'e \<Rightarrow> 'c list \<times> 'e" shows "subseqs_foldr (Sublists_Foldr_Impl x1 x2) = x1"
ms_closure_mult: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and ys :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "ms_closure P xs f basis"   and "ms_closure P ys g basis" shows "ms_closure P (times_ms_aux xs ys) (\<lambda>x. f x * g x) basis"
funionCI: fixes c :: "'b"   and B :: "'b fset"   and A :: "'b fset" assumes "c |\<notin>| B \<Longrightarrow> c |\<in>| A" shows "c |\<in>| A |\<union>| B"
A_events: fixes i :: "'i" assumes "i \<in> I" shows "A i \<in> events"
list_last: fixes xs :: "'b list" shows "xs = [] \<or> (\<exists>y ys. xs = ys @ [y])"
inner_eucl_of_list: fixes i :: "'b"   and xs :: "real list" assumes "i \<in> Basis"   and "length xs = DIM('b)" shows "i \<bullet> eucl_of_list xs = xs ! index Basis_list i"
zet_ext_eq: fixes A :: "'b zet"   and B :: "'b zet" shows "(A = B) = (\<forall>x. zin x A = zin x B)"
elims: fixes x :: "('c, 'd) fmap"   and xa :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and y :: "('c, 'd) fmap list" assumes "state_list x xa = y" obtains s :: "('c, 'd) fmap" where "x = s__" and "xa = []" and "y = [s__]"   | s :: "('c, 'd) fmap" and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" where "x = s" and "xa = a # as" and "y = s # state_list (state_succ s a) as"
h7b: assumes "Av < 0"   and "0 < Bv"   and "Bv\<^sup>2 * Cv \<le> Av\<^sup>2" shows "Av + Bv * sqrt Cv \<le> 0"
is_nnf_BigAnd: fixes ls :: "'b formula list" shows "is_nnf \<^bold>\<And>ls = (\<forall>l\<in>set ls. is_nnf l)"
assms_2: shows "fst (rep_nat_term u) = fst (rep_nat_term v)"
total_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "Restricted_Predicates.total_on P A = (\<forall>x\<in>A. \<forall>y\<in>A. x = y \<or> P x y \<or> P y x)"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_mult" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_mult" and "vreal_mult\<lparr>a\<rparr> = b"
PivotAndUpdate_axioms: shows "PivotAndUpdate eq_idx_for_lvar pivot_and_update"
le_sequentially: fixes F :: "nat filter" shows "(F \<le> sequentially) = (\<forall>N. eventually ((\<le>) N) F)"
extremum_uniqueI: fixes a :: "'b" assumes "order.greater_eq bot a" shows "a = bot"
inj_ring_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "inj_ring_hom hom \<equiv> Ring_Hom.ring_hom hom \<and> zero_hom_0 hom"
locally_faithful_def: shows "G.locally_faithful \<equiv> \<forall>f g \<mu> \<mu>'. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> \<guillemotleft>\<mu>' : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> G \<mu> = G \<mu>' \<longrightarrow> \<mu> = \<mu>'"
gt_compat_fun: fixes t :: "('s, 'v) tm"   and t' :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary t"   and "local.kbo_std.gt t' t" shows "local.kbo_std.gt (App s t') (App s t)"