hcju/isabelleps huggingface dataset:
f_accessible_prepend: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set"   and q' :: "'a"   and x :: "'a" assumes "q \<rightarrow> l qs \<in> \<delta>"   and "q' \<in> set qs"   and "x \<in> f_accessible \<delta> {q'}" shows "x \<in> f_accessible \<delta> {q}"
him_add: fixes x :: "octo"   and y :: "octo" shows "HIm (x + y) = HIm x + HIm y"
drop_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'e list" shows "drop n (take m xs) = take (m - n) (drop n xs)"
assume_def: fixes p :: "'b Assertion" shows "[\<cdot> p ] = {\<cdot> p } ^ o"
gram_schmidt_result_3: fixes ws :: "rat vec list"   and us :: "rat vec list" assumes "set ws \<subseteq> Rn"   and "distinct ws"   and "gs.lin_indpt (set ws)"   and "us = gram_schmidt n ws" shows "set us \<subseteq> Rn"
w_simps_3: shows "src h.chine = src (tab\<^sub>0 h)"
simps: fixes a :: "'a list"   and A :: "'a set" shows "(a \<in> lists A) = (a = [] \<or> (\<exists>a l. a = a # l \<and> a \<in> A \<and> l \<in> lists A))"
mono_def: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and f :: "'b::type \<Rightarrow> 'c::order" assumes "class.order (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "order.mono (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (f::'b::type \<Rightarrow> 'c::order) = (\<forall>(x::'b::type) y::'b::type. less_eq x y \<longrightarrow> f x \<le> f y)"
law_restrict_zero: fixes R :: "('c, 'd) ring_scheme" shows "\<zero>\<^bsub>law_restrict R\<^esub> = \<zero>\<^bsub>R\<^esub>"
M_initV_def: shows "M_initV \<equiv> M_empty"
simps: fixes rf :: "recf"   and vl :: "nat" shows "rec_map rf vl = map (\<lambda>i. Cn vl rf [recf.id vl i]) [0..<vl]"
is_nf_def: fixes N :: "nat"   and dflt :: "'b"   and f :: "nat \<Rightarrow> 'b"   and a :: "'b Heap.array" shows "is_nf N dflt f a \<equiv> \<exists>\<^sub>Al. a \<mapsto>\<^sub>a l * \<up> (length l = N \<and> (\<forall>i<N. l ! i = f i) \<and> (\<forall>i\<ge>N. f i = dflt))"
compatible_modes_def: fixes mdss :: "(Mode \<Rightarrow> 'Var set) list" shows "compatible_modes mdss = (\<forall>i x. i < length mdss \<longrightarrow> (x \<in> (mdss ! i) AsmNoReadOrWrite \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoReadOrWrite)) \<and> (x \<in> (mdss ! i) AsmNoWrite \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoWrite)))"
ends_in_segment_1: fixes a :: "'c"   and b :: "'c" shows "a \<in> {a--b}"
ss: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in># X"   and "y \<in># Y"   and "z \<in># Z"   and "(x, y) \<in> s"   and "(y, z) \<in> s" shows "(x, z) \<in> s"
simplify_non_redundant: fixes S :: "'at Literal set set" shows "non_redundant (simplify S)"
tyenv_eq_def: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2 \<equiv> \<forall>x. to_total \<Gamma> x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
transpose_max_length: fixes xs :: "'g list list" shows "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length (filter (\<lambda>x. x \<noteq> []) xs)"
codomains_comp: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" shows "D.codomains (g \<cdot>\<^sub>D f) = D.codomains g"
Diff_insert: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "A - insert a B = A - B - {a}"
pp_arc_expanded: fixes x :: "'a" shows "pp_arc x = (coreflexive (x * top * x\<^sup>T) \<and> coreflexive (x\<^sup>T * top * x) \<and> times_top_class.total (top * - - x))"
WT_resource_of_oracle: fixes s :: "'s_core'" assumes "IC2 s" shows "\<I>E \<oplus>\<^sub>\<I> (\<I>CA \<oplus>\<^sub>\<I> \<I>CU) \<turnstile>res RES (callee_of_core core2) s \<surd>"
absolutely_integrable_spike: fixes T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on T"   and "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "g absolutely_integrable_on T"
EQP: shows "(eq, (=)) \<in> R \<rightarrow> R \<rightarrow> bool_rel"
primitive_root_unityD_2: fixes k :: "nat"   and x :: "'a" assumes "primitive_root_unity k x" shows "x ^ k = (1::'a)"
absdiff_infnorm: fixes x :: "'a"   and y :: "'a" shows "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
hyps_2: fixes p :: "ref" assumes "Dag p low high rt"   and "ordered rt var"   and "no \<in> set_of rt"   and "low no \<noteq> Null"   and "high no \<noteq> Null" shows "var (low no) < var no \<and> var (high no) < var no"
shows_prec_append: fixes shows_prec :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and shows_list :: "'a list \<Rightarrow> char list \<Rightarrow> char list"   and p :: "nat"   and x :: "'a"   and r :: "char list"   and s :: "char list" assumes "class.show shows_prec shows_list" shows "shows_prec p x (r @ s) = shows_prec p x r @ s"
ex1_def: shows "Semantics.ex1 \<equiv> \<lambda>F w. {x. F (\<nu>\<upsilon> x) dj w}"
poly_mapping_eqI: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and g :: "'a \<Rightarrow>\<^sub>0 'b" assumes "\<And>k. poly_mapping.lookup f k = poly_mapping.lookup g k" shows "f = g"
combine_is_t_des: fixes \<V> :: "'a set"   and \<B> :: "'a set multiset"   and \<k> :: "int"   and \<B>' :: "'a set multiset"   and \<t> :: "int"   and \<Lambda>\<^sub>t :: "int"   and \<Lambda>\<^sub>t' :: "int" assumes "two_t_designs_eq_points \<V> \<B> \<k> \<B>' \<t> \<Lambda>\<^sub>t \<Lambda>\<^sub>t'" shows "t_design (\<V> \<union> \<V>) (\<B> + \<B>') \<k> \<t> (\<Lambda>\<^sub>t + \<Lambda>\<^sub>t')"
cases: fixes trsys :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and \<tau>move :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'c" assumes "\<tau>trsys.silent_move trsys \<tau>move a1 a2" obtains s :: "'c" and s' :: "'c" and tl :: "'d" where "a1 = s" and "a2 = s'" and "trsys s tl s'" and "\<tau>move s tl s'"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "CharacterDataClass.known_ptr ptr = (ElementClass.known_ptr ptr \<or> is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
simps_3: fixes b :: "bool" shows "typeof (Bool b) = \<lfloor>Boolean\<rfloor>"
possafetyD: fixes A :: "'b set"   and P :: "'b llist set"   and t :: "'b llist" assumes "possafety A P"   and "t \<in> A\<^sup>\<spadesuit>"   and "\<And>r. r \<in> pfinpref A t \<Longrightarrow> \<exists>s\<in>A\<^sup>\<infinity>. r @@ s \<in> P" shows "t \<in> P"
u: fixes n :: "nat" shows "u n \<in> Gromov_boundary"
eval_cases_14: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e\<^sub>1;; e\<^sub>2,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains v :: "val" and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" where "P \<turnstile> \<langle>e\<^sub>1,s\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>" and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   | e :: "char list exp" where "e' = throw e" and "P \<turnstile> \<langle>e\<^sub>1,s\<rangle> \<Rightarrow> \<langle>throw e,s'\<rangle>"
multiplicity_uminus_right: fixes x :: "'b"   and y :: "'b" shows "multiplicity x (- y) = multiplicity x y"
subtype_reflexivity: fixes \<Gamma> :: "binding list"   and T :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "T closed_in \<Gamma>" shows "\<Gamma>\<turnstile>T<:T"
lboundmin: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real" assumes "C \<subseteq> PR"   and "np \<le> card C + khl" shows "\<exists>i\<in>C. f i \<le> Min (reduce f PR)"
set_take_subset: fixes n :: "nat"   and xs :: "'d list" shows "set (xs \<down> n) \<subseteq> set xs"
col_index_of_mat_col: fixes v :: "'a Matrix.mat"   and i :: "nat" assumes "dim_col v = 1"   and "i < dim_row v" shows "Matrix.col v 0 $ i = v $$ (i, 0)"
fresher_trans: fixes x :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and y :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and z :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
strong_fermat_liar_def: fixes a :: "nat"   and n :: "nat" shows "strong_fermat_liar a n = (\<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)))"
countable_nat_UN: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M" shows "\<Union> (range A) \<in> sets M"
case_optionE: fixes P :: "bool"   and Q :: "'c \<Rightarrow> bool"   and x :: "'c option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'c" where "x = Some y" and "Q y"
prefix_append: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" shows "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
s_3: fixes C :: "real set" assumes "compact C"   and "C \<subseteq> {a..}" shows "\<forall>\<^sub>F i in sequentially. s i \<notin> C"
one_smult_vec: fixes v :: "'b vec" shows "(1::'b) \<cdot>\<^sub>v v = v"
l_set_val_get_shadow_root_axioms_def: fixes set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_shadow_root_axioms set_val_locs get_shadow_root_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
comp_ipresIGSubstAbs: fixes MOD :: "('u, 'v, 'varSort, 'sort, 'w, 'x, 'y, 'z, 'aa) model_scheme"   and h :: "'y \<Rightarrow> 'ab"   and MOD' :: "('ac, 'ad, 'varSort, 'sort, 'ae, 'x, 'ab, 'af, 'ag) model_scheme"   and hA :: "'z \<Rightarrow> 'af"   and h' :: "'ab \<Rightarrow> 'ah"   and hA' :: "'af \<Rightarrow> 'ai"   and MOD'' :: "('aj, 'ak, 'varSort, 'al, 'am, 'x, 'ah, 'ai, 'an) model_scheme" assumes "igWlsAbsIsInBar MOD"   and "ipresIGWls h MOD MOD'"   and "ipresIGWlsAbs hA MOD MOD'"   and "ipresIGSubstAbs h hA MOD MOD'"   and "ipresIGSubstAbs h' hA' MOD' MOD''" shows "ipresIGSubstAbs (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
vec_of_nonzero: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "spp_inv sp" shows "vec_of sp \<noteq> 0"
rec: fixes b :: "int" shows "f (b + 1) = f b + m"
eq_Max_iff: fixes A :: "'a word set"   and m :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(m = signed.Max A) = (m \<in> A \<and> (\<forall>a\<in>A. a \<le>s m))"
reduced_GB_D3: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "punit.is_reduced_GB G" shows "punit.is_monic_set G"
well_order_on_Well_order: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "well_order_on A r" shows "A = Field r \<and> Well_order r"
inf_llist_neq_llist_of_1: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "llist_of xs \<noteq> inf_llist f"
d_min: shows "\<forall>DD'. set_mset DD' \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD' + CC \<longrightarrow> I \<Turnstile> E) \<and> (\<forall>D'. D' \<in># DD' \<longrightarrow> D' < D) \<longrightarrow> DD \<le> DD'"
path_component_mem_2: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "path_component S x y" shows "y \<in> S"
fs_int_mu_d_Z: fixes j :: "nat"   and ii :: "nat" assumes "j \<le> ii"   and "ii < length fs" shows "rat_of_int (d fs (Suc j)) * gs.\<mu> ii j \<in> \<int>"
push_def: fixes x :: "('a \<times> 'b) \<times> 'c" shows "push x \<equiv> (fst (fst x), snd (fst x), snd x)"
distinct_47: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "Stmt.If x51 x52 x53 \<noteq> Await x71 x72"
S4: fixes z :: "nat" shows "(c_snd z \<in> nat_to_ce_set (c_fst z)) = (\<exists>y. p z y = 0)"
cotangent_space_def: fixes p :: "'b" shows "dest.cotangent_space p = dual_space (dest.tangent_space p)"
count_mset_set': fixes A :: "'d set"   and x :: "'d" shows "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
preorder_on_axioms: shows "Order_Predicates.preorder_on alts' (Pareto R')"
x: shows "\<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)) \<leadsto>\<^sup>* s"
VSGeneralBlocks_def: shows "VSGeneralBlocks = opt_group \<circ> QE_dnf' opt_group (the_real_step_augment gen_qe) \<circ> VSLuckiestBlocks \<circ> opt_group"
fact_ineq: fixes n :: "nat"   and k :: "nat" assumes "1 \<le> n" shows "fact n + k \<le> fact (n + k)"
is_rbt_def: fixes t :: "('a, 'c) rbt" shows "is_rbt t = (inv1 t \<and> inv2 t \<and> color_of t = B \<and> rbt_sorted t)"
iwf_length_inPorts: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and fc :: "('b, 'd) graph_node \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'b fset \<times> 'b \<Rightarrow> bool"   and t :: "('b, 'd, 'c, 'a) itnode rose_tree"   and ent :: "'b fset \<times> 'b"   and "is" :: "nat list" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "Abstract_Task.iwf freshenLC renameLCs subst anyP antecedent consequent rules assumptions conclusions fc t ent"   and "is \<in> it_paths t" shows "length (iAnts (tree_at t is)) \<le> length (Abstract_Task.inPorts' anyP antecedent (iNodeOf (tree_at t is)))"
subst_rel_lift_seq: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and P :: "('b, 'd) urel"   and Q :: "('d, 'c) urel" shows "\<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> (P ;; Q) = \<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> P ;; Q"
compare_EQ_dest: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b"   and y :: "'b" assumes "class.Ord_linear eq compare"   and "compare\<cdot>x\<cdot>y = EQ" shows "x = y"
inf_bot_left: fixes x :: "'a" shows "\<bottom> \<sqinter> x = \<bottom>"
mapM_return_idx: fixes f :: "'c \<Rightarrow> 'a + 'b"   and xs :: "'c list"   and ys :: "'b list"   and i :: "nat" assumes "mapM f xs = Inr ys"   and "i < length xs" shows "\<exists>y. f (xs ! i) = Inr y \<and> ys ! i = y"
FORK_def: fixes N :: "nat"   and i :: "nat" assumes "DiningPhilosophers N" shows "DiningPhilosophers.FORK N i = (\<mu> x. picks i i \<rightarrow> putsdown i i \<rightarrow> x [+] picks ((i + 1) mod N) i \<rightarrow> putsdown ((i + 1) mod N) i \<rightarrow> x)"
assms_1: shows "possible_steps e s r (fst h) (snd h) = {|(s', t)|}"
V_def: shows "V = (\<lambda>c x. {n. \<exists>l\<in>{1..n}. u n x - u (n - l) x \<le> - c * real l})"
S2_xor_def: fixes B :: "bool \<times> bool"   and out :: "bool" shows "S2_xor B out = return_spmf B"
I: shows "lift (subst_bvs1' t (Suc j) (map (\<lambda>t. lift t 0) ss)) (Suc i) = subst_bvs1' (lift t (Suc i + length (map (\<lambda>t. lift t 0) ss))) (Suc j) (map (\<lambda>a. lift a (Suc i)) (map (\<lambda>t. lift t 0) ss))"
wf_imethdsD: fixes im :: "qtname \<times> mhead"   and G :: "prog"   and I :: "qtname"   and sig :: "sig" assumes "im \<in> DeclConcepts.imethds G I sig"   and "wf_prog G"   and "is_iface G I" shows "\<not> is_static im \<and> accmodi im = Public"
scaleRat_rat_def: fixes x :: "rat"   and y :: "rat" shows "x *R y = x * y"
h1: shows "Controller_L s (fin_inf_append [Zero] l) l z"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ntrm set"   and Y :: "ntrm set" shows "ORLEFT (B OR C) X Y = {z:OrL x.M y.N z |x y z M N. fin (OrL x.M y.N z) z \<and> x:M \<in> X \<and> y:N \<in> Y}"
union_mset_add_mset_right: fixes A :: "'d multiset"   and a :: "'d"   and B :: "'d multiset" shows "A + add_mset a B = add_mset a (A + B)"
msg_fresh_1: fixes \<sigma> :: "nat \<Rightarrow> state"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "msg_fresh \<sigma> (Rreq hops dip dsn dsk oip osn sip) = (1 \<le> osn \<and> (sip \<noteq> oip \<longrightarrow> oip \<in> kD (rt (\<sigma> sip)) \<and> osn \<le> nsqn (rt (\<sigma> sip)) oip \<and> (nsqn (rt (\<sigma> sip)) oip = osn \<longrightarrow> the (dhops (rt (\<sigma> sip)) oip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) oip) = Aodv_Basic.inv)))"
orthogonal_complement_def: fixes S :: "'b set" shows "orthogonal_complement S = {x |x. \<forall>y\<in>S. \<langle>x, y\<rangle> = 0}"
retraction_def: fixes g :: "'d" shows "D.retraction g \<equiv> \<exists>f. D.ide (g \<cdot>\<^sub>D f)"
assms_2: shows "(g \<longlongrightarrow> m) F"
cont_D: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "((\<Squnion>i. Y i) [+] S) = (\<Squnion>i. Y i [+] S)"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (law_restrict R)"   and "y \<in> carrier (law_restrict R)" shows "x \<oplus>\<^bsub>law_restrict R\<^esub> y = y \<oplus>\<^bsub>law_restrict R\<^esub> x"
allNetsDistinct_def: fixes p :: "('b set set, 'c) Combinators list" shows "allNetsDistinct p = (\<forall>a b. a \<noteq> b \<and> a \<in> set (net_list p) \<and> b \<in> set (net_list p) \<longrightarrow> netsDistinct a b)"
eq_fst_iff: fixes a :: "'f"   and p :: "'f \<times> 'g" shows "(a = fst p) = (\<exists>b. p = (a, b))"
typeofNew_simps_2: fixes T :: "Arraytype"   and l :: "nat" shows "typeofNew (new_array T l) = ArrT T"
mutator_phase_inv_aux_case: fixes xga :: "'e"   and xa :: "hs_phase" shows "mut_m.mutator_phase_inv_aux xga xa = (case xa of hp_Idle \<Rightarrow> \<lambda>s. True | hp_IdleInit \<Rightarrow> no_black_refs | hp_InitMark \<Rightarrow> mut_m.marked_insertions xga | hp_Mark \<Rightarrow> \<lambda>s. mut_m.marked_insertions xga s \<and> mut_m.marked_deletions xga s | hp_IdleMarkSweep \<Rightarrow> \<lambda>s. mut_m.marked_insertions xga s \<and> mut_m.marked_deletions xga s \<and> mut_m.reachable_snapshot_inv xga s)"
rel_mset_Zero: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_mset R {#} {#}"
dsafe_Times_simps: fixes a :: "('d, 'e) trm"   and b :: "('d, 'e) trm" shows "dsafe (Times a b) = (dsafe a \<and> dsafe b)"
not_Some_eq: fixes x :: "'b option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
assms_1: shows "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1, y2). f y1 y2)"
subst_type: fixes \<Delta> :: "binding list"   and x :: "vrs"   and U :: "ty"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "ty"   and u :: "trm" assumes "\<Delta> @ VarB x U # \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> u : U" shows "\<Delta> @ \<Gamma> \<turnstile> t[x \<mapsto> u] : T"
a'_length: shows "a' < length (sorted_list_of_set I)"
s\<^sub>1: shows "s\<^sub>1 \<in> A"
PosInf: shows "PosInf = real_of_int (sint POS_INF)"
ring_ops_def: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "ring_ops ops R \<equiv> ((bi_unique R \<and> right_total R) \<and> R (arith_ops_record.zero ops) (0::'c) \<and> R (arith_ops_record.one ops) (1::'c) \<and> (R ===> R ===> R) (arith_ops_record.plus ops) (+)) \<and> ((R ===> R ===> R) (arith_ops_record.minus ops) (-) \<and> (R ===> R) (arith_ops_record.uminus ops) uminus_class.uminus) \<and> (R ===> R ===> R) (arith_ops_record.times ops) (*) \<and> (R ===> R ===> (=)) (=) (=) \<and> Domainp R = arith_ops_record.DP ops"
Mem: fixes x :: "name" assumes "x mem xvec" shows "x \<in> supp N"
span_base: fixes a :: "('a, 'b) vec"   and S :: "('a, 'b) vec set" assumes "a \<in> S" shows "a \<in> vec.span S"
list_to_vec_def: fixes xs :: "'a list" shows "list_to_vec xs = (\<chi>i. xs ! mod_type_class.to_nat i)"
triangulize_def: fixes A :: "'a mat" shows "local.triangulize A = local.sub3 (dim_row A) (dim_row A) (1::'a, A)"
nonneg_cexprI: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<And>\<sigma>. \<sigma> \<in> space (state_measure V \<Gamma>) \<Longrightarrow> 0 \<le> extract_real (cexpr_sem \<sigma> e)" shows "nonneg_cexpr V \<Gamma> e"
distinct_32: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Stmt.If x51 x52 x53 \<noteq> x41 ;; x42"
that: fixes V :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "S \<subseteq> V"   and "open V"   and "continuous_on V g"   and "g ` V \<subseteq> T"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
path_component_eq_empty: fixes S :: "'b set"   and x :: "'b" shows "(path_component_set S x = {}) = (x \<notin> S)"
pos_less_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a < b /\<^sub>R c) = (c *\<^sub>R a < b)"
Min_less_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less (linorder.Min less_eq A) x = (\<exists>a\<in>A. less a x)"
iso_set_sym: fixes G :: "('e, 'f) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and H :: "('g, 'h) monoid_scheme" assumes "Group.group G"   and "h \<in> Group.iso G H" shows "inv_into (carrier G) h \<in> Group.iso H G"
vars_prod: fixes f :: "'b \<Rightarrow> 'a mpoly"   and A :: "'b set" shows "vars (prod f A) \<subseteq> (\<Union>x\<in>A. vars (f x))"
deg_ant_int: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "deg R S X p = ant (int (deg_n R S X p))"
exists_double_arc: fixes g :: "real \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "simple_path g"   and "pathfinish g = pathstart g"   and "a \<in> path_image g"   and "b \<in> path_image g"   and "a \<noteq> b" obtains u :: "real \<Rightarrow> 'a" and d :: "real \<Rightarrow> 'a" where "arc u" and "arc d" and "pathstart u = a" and "pathfinish u = b" and "pathstart d = b" and "pathfinish d = a" and "path_image u \<inter> path_image d = {a, b}" and "path_image u \<union> path_image d = path_image g"
times_mat_def: fixes A :: "'d mat"   and B :: "'d mat" shows "A * B \<equiv> Matrix.mat (dim_row A) (dim_col B) (\<lambda>(i, j). Matrix.row A i \<bullet> col B j)"
open_preimage_identity: fixes B :: "'a set set" assumes "is_zariski_open B" shows "identity Spec \<^sup>\<inverse> Spec B = B"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>L 'd" shows "blinfun_apply (comp3 x) = (o\<^sub>L) x"
rename_invariant: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set" assumes "bij h" shows "(rename h F \<in> invariant (h ` A)) = (F \<in> invariant A)"
\<T>_2: shows "\<Union> \<T> = \<Union> \<N>"
integrable_on_subinterval: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and a :: "'c"   and b :: "'c" assumes "f integrable_on S"   and "{a..b} \<subseteq> S" shows "f integrable_on {a..b}"
adjoint_clauses_1: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" assumes "linear f" shows "x \<bullet> adjoint f y = f x \<bullet> y"
spaceN_cmult: fixes x :: "'b"   and N :: "'b quasinorm"   and c :: "real" assumes "x \<in> space\<^sub>N N" shows "c *\<^sub>R x \<in> space\<^sub>N N"
P1: fixes m :: "nat" assumes "N < m" shows "s m k = r"