hcju/isabelleps huggingface dataset:
con_rews_4: fixes a :: "'r"   and slist :: "[:'r:]" shows "(a :# slist = \<bottom>) = (a = \<bottom> \<or> slist = \<bottom>)"
prems_1: shows "cfg\<^sub>1 = (c\<^sub>1, s)"
atomize_eq: fixes x :: "'b"   and y :: "'b" shows "(x \<equiv> y) \<equiv> x = y"
exists_fmap_of_list: fixes m :: "('c, 'd) fmap" shows "\<exists>xs. fmap_of_list xs = m"
times_poly_i_def: fixes ops :: "'b arith_ops_record"   and f :: "'b list"   and g :: "'b list" shows "times_poly_i ops f g \<equiv> let n = length f; m = length g in if n \<le> karatsuba_lower_bound \<or> m \<le> karatsuba_lower_bound then if n \<le> m then foldr (\<lambda>a p. plus_poly_i ops (smult_i ops a g) (cCons_i ops (arith_ops_record.zero ops) p)) f [] else foldr (\<lambda>a p. plus_poly_i ops (smult_i ops a f) (cCons_i ops (arith_ops_record.zero ops) p)) g [] else if n \<le> m then karatsuba_main_i ops g m f n else karatsuba_main_i ops f n g m"
Lp_cases: fixes p :: "ennreal" obtains p2 :: "real" where "p = ennreal p2" and "0 < p2" and "0 < p"   | "p = 0"   | "p = \<infinity>"
Sup_insert: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and a :: "'e"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (insert a A) = sup a (Sup A)"
assms: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "f m \<le> f n"
Sum_splitlist_nonempty: fixes f :: "'d list \<Rightarrow> 'd list \<Rightarrow> 'c"   and xs :: "'d list" shows "\<Sum> {f ys zs |ys zs. xs = ys @ zs} = f [] xs + \<Sum> {f ys zs |ys zs. xs = ys @ zs \<and> ys \<noteq> []}"
sep_conj_empty: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_empty zero) = P"
star_pratt1_def: fixes x :: "pratt1" shows "x\<^sup>\<star> \<equiv> if x = \<bottom>\<^sub>1 \<or> x = [0]\<^sub>1 then [0]\<^sub>1 else \<top>\<^sub>1"
subdistl: fixes z :: "('a \<times> 'a set) set"   and x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "z \<parallel> x \<subseteq> z \<parallel> (x \<union> y)"
cs_split_shift_nocd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>"   and "k < n"   and "\<forall>j. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<longrightarrow> k \<le> j" shows "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> (n - k)"
closed_segment_subsetD: fixes X :: "'b"   and H :: "'b"   and S :: "'b set"   and x :: "real" assumes "closed_segment X (X + H) \<subseteq> S"   and "0 \<le> x"   and "x \<le> 1" shows "X + x *\<^sub>R H \<in> S"
is_locally_frac_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and V :: "'c set set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.is_locally_frac R addition multiplication zero s V \<equiv> \<exists>r f. r \<in> R \<and> f \<in> R \<and> (\<forall>\<qq>\<in>V. f \<notin> \<qq> \<and> s \<qq> = quotient_ring.frac (R\<setminus>\<qq>) R addition multiplication zero r f)"
div_eq_0_iff: fixes m :: "nat"   and n :: "nat" shows "(m div n = 0) = (m < n \<or> n = 0)"
r_div2: fixes n :: "nat" shows "eval r_div2 [n] \<down>= n div 2"
complex_Re_numeral: fixes v :: "num" shows "Re (numeral v) = numeral v"
abs_monotone: fixes c :: "chan"   and G :: "msg set"   and H :: "msg set" assumes "c \<in> local.abs G"   and "G \<subseteq> H" shows "c \<in> local.abs H"
prv_cnj_imp_scnj2: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (cnj \<phi> \<psi>) (scnj {\<phi>, \<psi>}))"
param: shows "((A ===> (=)) ===> A ===> (=)) F G"
\<tau>is\<rho>: shows "selectlike (loopfp\<tau> the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>))** I X) \<omega> (- fst (usubstappp \<sigma> U \<alpha>)) = selectlike (loopfp\<tau> \<alpha> (USubst.adjoint \<sigma> I \<omega>) X) \<omega> (- fst (usubstappp \<sigma> U \<alpha>))"
nat_of_le_len_of: fixes n :: "'b sat" shows "nat_of n \<le> LENGTH('b)"
discI_1: fixes llist :: "'b llist" assumes "llist = <>" shows "lnull llist"
tr_filter_conc: fixes a :: "'c list"   and b :: "'c list"   and cs :: "'c set" shows "tr_filter (a @ b) cs = tr_filter a cs @ tr_filter b cs"
nodes_trans: fixes p :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and q :: "nat \<times> 'a" assumes "p \<in> gnodes A w V"   and "q \<in> greachable A w V p" shows "q \<in> gnodes A w V"
image_domain_eq: fixes c :: "('c, 'b) chart" shows "apply_chart c ` domain c = codomain c"
build_succ_aux2: fixes nds :: "('a, 'b) node_scheme set" assumes "expand_init \<notin> name ` nds"   and "\<And>nd. nd \<in> nds \<Longrightarrow> incoming nd \<subseteq> insert expand_init (name ` nds)" shows "(\<lambda>qn. dflt_None_set {qn'. \<exists>q'. q' \<in> nds \<and> qn' = name q' \<and> qn \<in> incoming q' \<and> qn \<noteq> expand_init}) = (\<lambda>qn. dflt_None_set (succ_of_E (rename_E name {(q, q'). q \<in> nds \<and> q' \<in> nds \<and> name q \<in> incoming q'}) qn))"
assms_4: shows "prv (cnj \<phi> (scnj F))"
has_field_map_of_fields: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "P \<turnstile> C has F:T (fm) in D" shows "map_of (TypeRel.fields P C) (F, D) = \<lfloor>(T, fm)\<rfloor>"
add_existing_point: fixes p :: "'a" assumes "p \<in> \<V>'" shows "sys2.add_point p = \<V>'"
prod_DFS_def: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list"   and n :: "nat" shows "prod_DFS A B n \<equiv> wf_dfa A n \<and> wf_dfa B n"
resultant_as_nonzero_poly: fixes p :: "'a poly"   and q :: "'a poly" assumes "0 < degree p"   and "0 < degree q" shows "\<exists>p' q'. degree p' < degree q \<and> degree q' < degree p \<and> [:resultant p q:] = p' * p + q' * q \<and> p' \<noteq> 0 \<and> q' \<noteq> 0"
arcs_predecessor: fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j xs)"   and "a \<noteq> i" shows "\<exists>c. (c, a) \<in> set (arcs i j xs)"
im_has_bottom: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetMap less_eq less less_eq' less' P f"   and "ordering.has_bottom less_eq P" shows "ordering.has_bottom less_eq' (f ` P)"
assms: fixes a :: "'rows"   and b :: "'cols" shows "mod_type_class.to_nat a = mod_type_class.to_nat b"
weak_reduction_bisimulations_impl_inverse_is_simulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_simulation (Rel\<inverse>) Cal"
op_custom_empty_def: shows "op_hm_empty = op_map_empty"
prems_5: assumes "snd b \<noteq> 0"   and "punit.lt (snd b) adds t"   and "\<not> (t - punit.lt (snd b)) \<oplus> fst b \<prec>\<^sub>t u" shows "thesis"
simps_5: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n\<^sub>s :: "nat"   and n' :: "nat" assumes "Option.is_none (devBC t n nid (n\<^sub>s + Suc n'))" shows "devExt t n nid n\<^sub>s (Suc n') = devExt t n nid n\<^sub>s n'"
vsubsetI: fixes a :: "V"   and b :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> a \<Longrightarrow> x \<in>\<^sub>\<circ> b" shows "a \<subseteq>\<^sub>\<circ> b"
composites_are_arrows_1: shows "hseq (trg f) (trg f \<star> f)"
vec_to_iarray_nth': fixes A :: "('d, 'e) vec"   and i :: "'e" shows "vec_to_iarray A !! mod_type_class.to_nat i = A $ i"
other_net_tree_ips_par_right: fixes U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'd"   and \<sigma>' :: "nat \<Rightarrow> 'd" assumes "other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) \<sigma> \<sigma>'"   and "\<And>\<xi>. U \<xi> \<xi>" shows "other U (net_tree_ips p\<^sub>2) \<sigma> \<sigma>'"
7: shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>h \<rightarrow> w\<^sub>h'\<guillemotright> \<and> \<beta>\<^sub>h = tab\<^sub>1 h \<star> \<gamma> \<and> \<theta>\<^sub>h = \<theta>\<^sub>h' \<cdot> (tab\<^sub>0 h \<star> \<gamma>)"
uniform_measure_def: fixes M :: "'b measure"   and A :: "'b set" shows "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
prems_3: shows "ord_rec_Seq T G s' (succ k) y'"
act_unitl: fixes e :: "'a"   and p :: "'b" assumes "e \<in> E" shows "Dla e p \<and> \<alpha> e p = p"
coprime_mult_left_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "coprime (a * b) c = (coprime a c \<and> coprime b c)"
288: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- x \<sqinter> y) = top"
compile_F1: shows "compile_env (map_option arity \<circ> Finca_get F1) load_oracle F1 = Ok F2"
assms_3: shows "f \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
subset_refl: fixes A :: "'b set" shows "(A::'b set) \<subseteq> A"
Zfun_norm_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "Zfun (\<lambda>x. norm (f x)) F = Zfun f F"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and p :: "('psym, 'tp) epsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "GE.satA \<xi> (Pr p Tl) = eintP p (map (GE.int \<xi>) Tl)"
load_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {load_store_type LDSB, load_store_type LDUB, load_store_type LDUBA, load_store_type LDUH, load_store_type LD, load_store_type LDA, load_store_type LDD}" shows "snd (load_instr i s) = False"
plus_down: fixes prec :: "nat"   and x :: "real"   and y :: "real" shows "plus_down prec x y \<le> x + y"
atLeastLessThan_subset_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linorder less_eq less"   and "ord.atLeastLessThan less_eq less a b \<subseteq> ord.atLeastLessThan less_eq less c d" shows "less_eq b a \<or> less_eq c a \<and> less_eq b d"
fmupd_idem: fixes a :: "'c"   and x :: "'d"   and y :: "'d"   and m :: "('c, 'd) fmap" shows "m(a $$:= y, a $$:= x) = m(a $$:= x)"
Phi_def: fixes n :: "nat" shows "\<Phi> n = E (map_pmf (\<phi> n) (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) (take n qs)))"
prems_1: fixes i :: "nat" assumes "i \<le> Suc n" shows "ideal (I i) R"
intro: fixes ifex_of :: "'a \<Rightarrow> 'b ifex"   and val :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<And>b s. val_ifex (ifex_of b) s = val b s"   and "\<And>b. reduced (ifex_of b) {}" shows "reduced_bdt_checkers ifex_of val"
Greatest_equality: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq y x" shows "order.Greatest less_eq P = x"
flush_append_Ghost\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and m :: "nat \<Rightarrow> nat" shows "flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Ghost\<^sub>s\<^sub>b A L R W])) m = flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) m"
cases: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" assumes "star r a1 a2" obtains x :: "'a" where "a1 = x__" and "a2 = x__"   | x :: "'a" and y :: "'a" and z :: "'a" where "a1 = x" and "a2 = z" and "r x y" and "star r y z"
rt_fresh_as_fresherI: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "the (rt1 dip) \<sqsubseteq> the (rt2 dip)"   and "the (rt2 dip) \<sqsubseteq> the (rt1 dip)" shows "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
constVal_eval_elim: fixes e :: "expr"   and c :: "val"   and G :: "prog"   and s0 :: "st"   and v :: "val"   and s :: "state" assumes "constVal e = Some c"   and "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s" obtains "v = c" and "normal s"
smult_eigen_vector: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" assumes "eigen_vector A v x" shows "eigen_vector (a *k A) v (a * x)"
BotL: fixes \<Gamma> :: "'b formula multiset" shows "\<bottom>, \<Gamma> \<Rightarrow>\<^sub>n"
simps_2: fixes uv :: "'a" shows "is_Inr (Inl uv) = False"
strong_induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "'a \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<longmapsto> x2"   and "\<And>t t' u y. \<lbrakk>t \<longmapsto> t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y \<langle>t, u\<rangle> \<langle>t', u\<rangle>"   and "\<And>u u' t y. \<lbrakk>u \<longmapsto> u'; \<And>z. P z u u'\<rbrakk> \<Longrightarrow> P y \<langle>t, u\<rangle> \<langle>t, u'\<rangle>"   and "\<And>t t' x T y. \<lbrakk>x \<sharp> y; t \<longmapsto> t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (\<lambda>x:T. t) (\<lambda>x:T. t')"   and "\<And>t t' u y. \<lbrakk>t \<longmapsto> t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (t \<cdot> u) (t' \<cdot> u)"   and "\<And>u u' t y. \<lbrakk>u \<longmapsto> u'; \<And>z. P z u u'\<rbrakk> \<Longrightarrow> P y (t \<cdot> u) (t \<cdot> u')"   and "\<And>x u T t y. \<lbrakk>x \<sharp> y; x \<sharp> u\<rbrakk> \<Longrightarrow> P y ((\<lambda>x:T. t) \<cdot> u) (t[x\<mapsto>u])"   and "\<And>p t \<theta> u y. \<lbrakk>supp p \<sharp>* y; supp p \<sharp>* t; distinct (pat_vars p); \<turnstile> p \<rhd> t \<Rightarrow> \<theta>\<rbrakk> \<Longrightarrow> P y (LET p = t IN u) (\<theta>\<lparr>u\<rparr>)" shows "P z x1 x2"
sp_compile_fundef: fixes get_arity :: "'fun \<Rightarrow> nat option"   and load_oracle :: "nat \<Rightarrow> Unboxed.type option"   and fd1 :: "('b, 'k, 'fun, 'd, 'e) Inca.instr fundef"   and fd2 :: "('b, 'k, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef"   and F :: "'fun \<Rightarrow> 'l fundef option" assumes "compile_fundef get_arity load_oracle fd1 = Ok fd2"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "sp_fundef F fd2 (body fd2) = Ok [None]"
Some_image_these_eq: fixes A :: "'c option set" shows "Some ` Option.these A = {x \<in> A. x \<noteq> \<bottom>}"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map (F \<circ> G) \<epsilon>_inv.map"
uw\<theta>\<omega>: shows "ide w' \<and> \<guillemotleft>\<theta>' : f \<star> w' \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<rho>' : local.dom \<rho>' \<Rightarrow> r \<star> f'\<guillemotright>"
infinite_public_consts: shows "infinite {c. public c \<and> arity c = 0}"
F'_def: shows "F' = {f \<in> F''. \<Sum> {h \<in> hom_components (q' f). poly_deg h + poly_deg f = poly_deg p} * f \<noteq> 0}"
all_tree: shows "\<forall>n. diago n \<in> S"
split_block_2: fixes A :: "'a mat"   and sr1 :: "nat"   and sc1 :: "nat"   and A1 :: "'a mat"   and A2 :: "'a mat"   and A3 :: "'a mat"   and A4 :: "'a mat"   and sr2 :: "nat"   and sc2 :: "nat" assumes "split_block A sr1 sc1 = (A1, A2, A3, A4)"   and "dim_row A = sr1 + sr2"   and "dim_col A = sc1 + sc2" shows "A2 \<in> carrier_mat sr1 sc2"
strict_order_iff: fixes a :: "'b"   and b :: "'b" shows "(a < b) = (a = min a b \<and> a \<noteq> b)"
eta_reds_into_equiv: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t" shows "s \<leftrightarrow> t"
refined_partition_finite: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set" assumes "finite (UNIV // R1)"   and "R1 \<subseteq> R2"   and "equiv UNIV R1"   and "equiv UNIV R2" shows "finite (UNIV // R2)"
a: shows "stream_space_single (proj_stoch_proc X n) r \<in> sets (stream_space borel) \<and> stream_space_single (proj_stoch_proc X n) r \<inter> range (proj_stoch_proc X n) = {r}"
cases: fixes a1 :: "meth_def"   and a2 :: "char list" assumes "method_name a1 a2" obtains cl :: "cl" and meth :: "char list" and vds :: "vd list" and meth_body :: "meth_body" where "a1 = meth_def_def (meth_sig_def cl meth vds) meth_body" and "a2 = meth"
head_in_verts: fixes G :: "nat graph_rec"   and e :: "nat \<times> nat" assumes "cfg_doms G"   and "e \<in> g_E G" shows "head G e \<in> set (g_V G)"
arr_dom: fixes f :: "'s" assumes "S.arr f" shows "S.arr (S.dom f)"
mapl_G_def: fixes l1 :: "'p \<Rightarrow> 'w"   and l2 :: "'q \<Rightarrow> 'x" shows "mapl_G l1 l2 = map_G l1 l2 id id id id"
intro: fixes M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b" assumes "\<And>a b. \<lbrakk>a \<in> M; b \<in> M\<rbrakk> \<Longrightarrow> composition a b \<in> M"   and "unit \<in> M"   and "\<And>a b c. \<lbrakk>a \<in> M; b \<in> M; c \<in> M\<rbrakk> \<Longrightarrow> composition (composition a b) c = composition a (composition b c)"   and "\<And>a. a \<in> M \<Longrightarrow> composition unit a = a"   and "\<And>a. a \<in> M \<Longrightarrow> composition a unit = a" shows "Group_Theory.monoid M composition unit"
bij_betw_iff_bijections: fixes f :: "'ag \<Rightarrow> 'ah"   and A :: "'ag set"   and B :: "'ah set" shows "bij_betw f A B = (\<exists>g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
eucl_rel_poly_by_0: fixes x :: "'b poly" shows "eucl_rel_poly x 0 (0, x)"
simps_2: fixes U :: "complex mat" shows "well_com (Utrans U) = (U \<in> carrier_mat d d \<and> unitary U)"
elem_to_ccField: fixes x :: "Vars.var"   and y :: "Vars.var"   and G :: "CoCalls" assumes "x--y\<in>G" shows "x \<in> ccField G \<and> y \<in> ccField G"
abs_eq_iff: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "(abs_int x = abs_int y) = (x \<approx> y)"
Par2B: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<Longrightarrow>a<\<nu>x> \<prec> P \<parallel> Q'"
integer_of_uint64_def: shows "integer_of_uint64 = integer_of_int \<circ> int_of_uint64"
converse_Rel_components_3: fixes T :: "V" shows "T\<inverse>\<^sub>R\<^sub>e\<^sub>l\<lparr>ArrCod\<rparr> = T\<lparr>ArrDom\<rparr>"
inducts: fixes e :: "expr"   and t :: "pdf_type"   and f :: "cexpr" assumes "e : t \<Rightarrow>\<^sub>c f" obtains "(\<lambda>_. UNIT) \<turnstile> e : t" and "free_vars e = {}" and "([], [], \<lambda>_. UNIT, CReal 1) \<turnstile>\<^sub>c e \<Rightarrow> f"
st_unique: fixes r :: "real star"   and x :: "real star" assumes "r \<in> \<real>"   and "r \<approx> x" shows "st x = r"
cs_length_one_2: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "length (cs\<^bsup>\<pi>\<^esup> i) = 1" shows "\<forall>k. \<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
h2: shows "freeIn var (Or (fm.Atom (Less (- A * B))) (fm.Atom (Neq (A\<^sup>2 - B\<^sup>2 * c))))"
nhds_ereal: fixes r :: "real" shows "nhds (ereal r) = filtermap ereal (nhds r)"
decompose_def: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'c) Term.term" shows "decompose s t = (case (s, t) of (Var x, b) \<Rightarrow> Map.empty x | (Fun f ss, Var x) \<Rightarrow> Map.empty x | (Fun f ss, Fun g ts) \<Rightarrow> if f = g then zip_option ss ts else None)"
lQuot_def: fixes x :: "'c"   and A :: "'c list set" shows "lQuot x A = {xs. x # xs \<in> A}"
map_G_comp: fixes l1 :: "'aa \<Rightarrow> 'u"   and l2 :: "'ab \<Rightarrow> 'v"   and co1 :: "'ac \<Rightarrow> 'w"   and co2 :: "'ad \<Rightarrow> 'x"   and contra1 :: "'y \<Rightarrow> 'ae"   and contra2 :: "'z \<Rightarrow> 'af"   and l1' :: "'n \<Rightarrow> 'aa"   and l2' :: "'o \<Rightarrow> 'ab"   and co1' :: "'p \<Rightarrow> 'ac"   and co2' :: "'q \<Rightarrow> 'ad"   and contra1' :: "'ae \<Rightarrow> 'r"   and contra2' :: "'af \<Rightarrow> 's" shows "map_G l1 l2 co1 co2 contra1 contra2 \<circ> map_G l1' l2' co1' co2' contra1' contra2' = map_G (l1 \<circ> l1') (l2 \<circ> l2') (co1 \<circ> co1') (co2 \<circ> co2') (contra1' \<circ> contra1) (contra2' \<circ> contra2)"
bspec: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x::'b\<in>A::'b set. (P::'b \<Rightarrow> bool) x"   and "(x::'b) \<in> (A::'b set)" shows "(P::'b \<Rightarrow> bool) (x::'b)"
vimage2p_def: fixes f :: "'e \<Rightarrow> 'h"   and g :: "'f \<Rightarrow> 'i"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> 'g" shows "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
nz: fixes x :: "'a" assumes "x \<in> carrier" shows "b x \<noteq> (0::'g)"
assms_3: fixes x :: "'a" assumes "x \<in> R" shows "poly P x = (0::'a)"
h: fixes x :: "nat" shows "the (eval h [x]) = smn 1 (encode r_univuniv) [x]"
homotopic_with_imp_subset2: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "homotopic_with_canon P X Y f g" shows "g ` X \<subseteq> Y"
multlist_prime_pos: fixes a :: "'a"   and as :: "'a list" assumes "prime G a"   and "a \<in> carrier G"   and "set as \<subseteq> carrier G"   and "a divides foldr (\<otimes>) as \<one>" shows "\<exists>i<length as. a divides as ! i"
aa: shows "anyinternal (cutoff red (ds @ [Left]) T\<^sub>1) (\<lambda>p. \<not> red ((ds @ [Left]) @ p))"
vimage_image_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "f -` f ` A = {y. \<exists>x\<in>A. f x = f y}"
last_bin_last': fixes xs :: "bool list"   and w :: "int" assumes "0 < length xs" shows "last xs = odd (bl_to_bin_aux xs w)"
sublinear_intros_6: fixes x :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and y :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "sublinear (wp x)"   and "sublinear (wp y)" shows "sublinear (wp (x \<Sqinter> y))"
is_weakest_liberal_precondition_def: fixes dims :: "nat list"   and W :: "complex mat"   and S :: "com"   and P :: "complex mat" shows "state_sig.is_weakest_liberal_precondition dims W S P = (state_sig.is_quantum_predicate dims W \<and> state_sig.hoare_partial_correct dims W S P \<and> (\<forall>Q. state_sig.is_quantum_predicate dims Q \<longrightarrow> state_sig.hoare_partial_correct dims Q S P \<longrightarrow> Q \<le>\<^sub>L W))"
tendsto: fixes g :: "'c \<Rightarrow> 'a"   and a :: "'a"   and F :: "'c filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. h (g x)) \<longlongrightarrow> h a) F"
Prefixes_rev: fixes A :: "'a list set" shows "Prefixes (rev ` A) = rev ` Suffixes A"
smult_assoc2: fixes a :: "'a"   and x :: "'c"   and y :: "'c" assumes "a \<in> carrier R"   and "x \<in> carrier M"   and "y \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> x \<otimes>\<^bsub>M\<^esub> y = a \<odot>\<^bsub>M\<^esub> (x \<otimes>\<^bsub>M\<^esub> y)"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(deny x2 = deny y2) = (x2 = y2)"
diff_cancel: fixes a :: "'c" shows "a - a = (0::'c)"
timpl_closureton_is_timpl_closure: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "timpl_closure_set {t} TI = timpl_closure t TI"
mono_body_fixp: fixes F :: "('c \<Rightarrow> 'd Heap) \<Rightarrow> 'c \<Rightarrow> 'd Heap" assumes "\<And>x. mono_Heap (\<lambda>f. F f x)" shows "heap.fixp_fun F = F (heap.fixp_fun F)"
prc_stri_imp_nstri: fixes f :: "'f \<times> nat"   and g :: "'f \<times> nat" assumes "fst (prc f g)" shows "snd (prc f g)"