hcju/isabelleps huggingface dataset:
out_reg_mod_def: fixes data_w32 :: "word32"   and win :: "'b word"   and ur :: "word5"   and state :: "'b sparc_state" shows "out_reg_mod data_w32 win ur state \<equiv> let state' = state\<lparr>user_reg := (user_reg state)(win := (user_reg state win)(ur := data_w32))\<rparr>; win' = pre_window win; ur' = ur + 16 in state'\<lparr>user_reg := (user_reg state')(win' := (user_reg state' win')(ur' := data_w32))\<rparr>"
conjugateI1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "f x \<sqinter> y = \<bottom>" shows "x \<sqinter> conjugate f y = \<bottom>"
closed_empty: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.closed open {}"
iso_rep: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.ide V r" shows "category.iso V (bicategory_of_spans.rep V H \<a> \<i> src trg r)"
simps: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lfilter P xs = (case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then LCons x (lfilter P xs') else lfilter P xs')"
power_divide: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.field divide inverse times one plus zero minus uminus" shows "power.power one times (divide a b) n = divide (power.power one times a n) (power.power one times b n)"
virt_to_phys_unchanged_sub1: fixes v1 :: "'b word"   and v2 :: "'b word"   and s1 :: "('c, 'd) sparc_state_scheme"   and va :: "word32"   and s2 :: "('e, 'f) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" assumes "(let context_table_entry = (v1 >> 11 << 11) OR (v2 AND 511 << 2) in Let (mem_context_val_w32 (word_of_int 9) (ucast context_table_entry) (mem s1)) (case_option None (\<lambda>lvl1_page_table. ptd_lookup va lvl1_page_table (mem s1) 1))) = (let context_table_entry = (v1 >> 11 << 11) OR (v2 AND 511 << 2) in Let (mem_context_val_w32 (word_of_int 9) (ucast context_table_entry) (mem s2)) (case_option None (\<lambda>lvl1_page_table. ptd_lookup va lvl1_page_table (mem s2) 1)))" shows "(let context_table_entry = (v1 >> 11 << 11) OR (v2 AND 511 << 2) in Let (mem_context_val_w32 (word_of_int 9) (ucast context_table_entry) ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None)))) (case_option None (\<lambda>lvl1_page_table. ptd_lookup va lvl1_page_table ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) 1))) = (let context_table_entry = (v1 >> 11 << 11) OR (v2 AND 511 << 2) in Let (mem_context_val_w32 (word_of_int 9) (ucast context_table_entry) ((mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None)))) (case_option None (\<lambda>lvl1_page_table. ptd_lookup va lvl1_page_table ((mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))) 1)))"
inverse_unique: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.inverse_arrows f g" shows "CC.inv f = g"
While: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and e' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and y :: "char list"   and c :: "com"   and e'' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" assumes "\<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> bval b s \<and> e' s = l y} c { e'' \<Down> \<lambda>l s. P l s \<and> e s \<le> l y}"   and "\<forall>l s. bval b s \<and> P l s \<longrightarrow> 1 + e' s + e'' s \<le> e s"   and "\<forall>l s. \<not> bval b s \<and> P l s \<longrightarrow> 1 \<le> e s"   and "y \<notin> support P" shows "\<turnstile>\<^sub>1 {P} WHILE b DO c { e \<Down> \<lambda>l s. P l s \<and> \<not> bval b s}"
normal_exponent_def: fixes x :: "('a, 'b) IEEE.float" shows "normal_exponent x = int (IEEE.exponent x) - int (bias TYPE(('a, 'b) IEEE.float)) - int LENGTH('b)"
thunksI: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp" assumes "map_of \<Gamma> x = Some e"   and "\<not> isVal e" shows "x \<in> thunks \<Gamma>"
isCont_tan: fixes x :: "'a" assumes "cos x \<noteq> (0::'a)" shows "isCont tan x"
is_extension: shows "\<forall>f. C.arr f \<longrightarrow> F (inclusion_of_generators f) = V f"
inv\<^sub>1E_3: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "B\<^sub>1 \<notin> P\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2"
less_eq_mod_ring_def: fixes x :: "'b mod_ring"   and y :: "'b mod_ring" shows "(x \<le> y) = (to_int_mod_ring x \<le> to_int_mod_ring y)"
merge_eq_Cons_imp: fixes cmp :: "'b comparator"   and xs :: "'b list"   and ys :: "'b list"   and z :: "'b"   and zs :: "'b list" assumes "Sorting_Algorithms.merge cmp xs ys = z # zs" shows "xs \<noteq> [] \<and> z = hd xs \<or> ys \<noteq> [] \<and> z = hd ys"
r_s_def: fixes s :: "nat \<Rightarrow> nat option" assumes "r1_bc s" shows "r1_bc.r_s s \<equiv> SOME rs. recfn 1 rs \<and> Partial_Recursive.total rs \<and> s = (\<lambda>x. eval rs [x])"
inf_bot_left: fixes x :: "'a upred" shows "(false \<and> x) = false"
dg_GRPH_Dom_app: fixes x :: "V"   and \<alpha> :: "V" assumes "x \<in>\<^sub>\<circ> all_dghms \<alpha>" shows "dg_GRPH \<alpha>\<lparr>Dom\<rparr>\<lparr>x\<rparr> = x\<lparr>HomDom\<rparr>"
full_SetCompr_eq: fixes f :: "'b \<Rightarrow> 'a" shows "{u. \<exists>x. u = f x} = range f"
exec_step_ind_Init_Prepared_Object: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and sh' :: "sheap"   and Cs :: "char list list"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "sh' = sh(C \<mapsto> (fst (the (sh C)), Processing))"   and "C = Object" shows "exec_step_ind (StepC C Cs) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Called (C # Cs)) # frs, sh')"
subst_trm_rename: fixes y :: "vrs"   and t :: "trm"   and x :: "vrs"   and u :: "trm" assumes "y \<sharp> t" shows "([(y, x)] \<bullet> t)[y \<mapsto> u] = t[x \<mapsto> u]"
substT_suc: fixes t1 :: "'trm"   and t :: "'trm"   and x :: "'var" assumes "t1 \<in> trm"   and "t \<in> trm"   and "x \<in> var" shows "substT (suc t1) t x = suc (substT t1 t x)"
is_bisimulation_def: fixes as :: "nat list"   and ps :: "(nat rexp \<times> nat rexp) list" shows "is_bisimulation as ps = (\<forall>(r, s)\<in>set ps. atoms r \<union> atoms s \<subseteq> set as \<and> nullable r = nullable s \<and> (\<forall>a\<in>set as. (nderiv a r, nderiv a s) \<in> set ps))"
assms_3: shows "\<forall>C\<in>H. atom i' \<sharp> C"
l_def: fixes n :: "nat" shows "l n = fst (bisect n)"
b: fixes j :: "nat" assumes "j < n" shows "b \<noteq> of_nat j"
seq_compactE: fixes S :: "'c set"   and f :: "nat \<Rightarrow> 'c" assumes "seq_compact S"   and "\<forall>n. f n \<in> S" obtains l :: "'c" and r :: "nat \<Rightarrow> nat" where "l \<in> S" and "strict_mono r" and "(f \<circ> r) \<longlonglongrightarrow> l"
r_le: shows "r < prod_list (nths (dims (dematricize rmodes A ds)) rmodes)"
linearI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>R b) = r *\<^sub>R f b" shows "linear f"
word_gt_0_no: fixes y :: "num" shows "(0 < numeral y) = (0 \<noteq> numeral y)"
fg_qmodule: fixes N :: "'a set" assumes "M fgover R"   and "submodule R M N" shows "M /\<^sub>m N fgover R"
rcos_const_imp_mem: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme"   and i :: "'c" assumes "ideal I R"   and "i \<in> carrier R"   and "I +>\<^bsub>R\<^esub> i = I" shows "i \<in> I"
less_numeral_extra_3: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
tok_id_pubk_prik_rel: fixes n :: "nat"   and m :: "nat" assumes "n \<in> bad_tok_prik" shows "s\<^sub>0 \<Turnstile> tok_id_pubk_prik n m"
no_outstanding_volatile_Write\<^sub>s\<^sub>b_acquired: fixes sb :: "'b memref list"   and A :: "nat set" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "acquired False sb A = {}"
count_append: fixes a :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "PermutationLemmas.count a (xs @ ys) = PermutationLemmas.count a xs + PermutationLemmas.count a ys"
fixed_points_def: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set" assumes "group_action G \<phi> M" shows "group_action.fixed_points G \<phi> M = {m \<in> M. carrier G \<subseteq> group_action.stabilizer G \<phi> m}"
content_iff: fixes x :: "'a"   and p :: "'a poly" shows "(x dvd content p) = (\<forall>c\<in>set (coeffs p). x dvd c)"
twz: fixes t :: "real" shows "(1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
trace_matrix_sum_linear: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and d :: "nat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d" shows "trace (matrix_sum d f n) = (\<Sum>k = 0..<n. trace (f k))"
closedin_Euclidean_space: fixes n :: "nat" shows "closedin (powertop_real UNIV) (topspace (Euclidean_space n))"
func_one: shows "1 = (\<lambda>x. 1::'b)"
intro: fixes G_\<alpha> :: "'a \<Rightarrow> 'b \<times> 'b \<Rightarrow> enat"   and G_invar :: "'a \<Rightarrow> bool"   and G_succ :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and G_empty :: "'a"   and G_add :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> nat option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat" assumes "Dijkstra_Impl_Adts G_\<alpha> G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin" shows "Dijkstra_Impl_Defs G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin G_\<alpha>"
equal_nat_int_def: fixes i :: "nat_int"   and j :: "nat_int" shows "equal_class.equal i j \<equiv> i \<le> j \<and> j \<le> i"
p2s_conj_hom: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "p2s (P \<sqinter> Q) = p2s P \<inter> p2s Q"
one_monom_inv_sum_var_inv: fixes x :: "'b" shows "sum_var 1 x = 0"
star_power_slide: fixes k :: "nat"   and n :: "nat"   and x :: "'a" assumes "k \<le> n" shows "x\<^bsup>k\<^esup> \<cdot> (x\<^bsup>n\<^esup>)\<^sup>\<star> = (x\<^bsup>n\<^esup>)\<^sup>\<star> \<cdot> x\<^bsup>k\<^esup>"
IH: shows "psi (Discrete.sqrt n) = psi_even n"
T_eta: fixes \<sigma> :: "ty"   and \<eta> :: "val set list" shows "{v. \<exists>v'. v' \<in> T \<sigma> \<eta> \<and> v \<sqsubseteq> v' \<and> v \<noteq> Wrong} = T \<sigma> \<eta>"
dehomogenize_times: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "dehomogenize x (p * q) = dehomogenize x p * dehomogenize x q"
map_in_bicategory_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c" shows "map_in_bicategory V H \<a> \<i> src trg f \<equiv> bicategory V H \<a> \<i> src trg \<and> chosen_right_adjoints V H \<a> \<i> src trg \<and> map_in_bicategory_axioms V H \<a> \<i> src trg f"
closure_of_subset_topspace: fixes X :: "'b topology"   and S :: "'b set" shows "X closure_of S \<subseteq> topspace X"
inner_loop_refine: fixes succ :: "'c set"   and succi :: "'c set"   and ui :: "'d"   and u :: "'d"   and PREDi :: "'c \<Rightarrow> 'd option"   and PRED :: "'c \<Rightarrow> 'd option"   and Ni :: "'c set"   and N :: "'c set"   and dst :: "'c" assumes "finite succ"   and "(succi, succ) \<in> Id"   and "(ui, u) \<in> Id"   and "(PREDi, PRED) \<in> Id"   and "(Ni, N) \<in> Id" shows "inner_loop dst succi ui PREDi Ni \<le> \<Down> Id (add_succ_spec dst succ u PRED N)"
simps_6: fixes pi :: "(name \<times> name) list"   and x :: "'a" shows "pi \<bullet> rev pi \<bullet> x = x"
preserves_reflects_arr: fixes f :: "'a" shows "\<tau>.B.arr (\<nu>\<tau>.map f) = \<tau>.A.arr f"
cmp_simps'_1: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "D.arr (GoH.cmp (\<mu>, \<nu>))"
prv_eql_subst_trm_eqv: fixes x :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "prv (eql t1 t2)" shows "prv (eqv (subst \<phi> t1 x) (subst \<phi> t2 x))"
WT_restrict_gpv: fixes \<I> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv" shows "\<I> \<turnstile>g restrict_gpv \<I> gpv \<surd>"
prefix_of_pmap_\<Gamma>: fixes \<pi> :: "'c prefix"   and \<sigma> :: "'c trace"   and f :: "'c set \<Rightarrow> 'd set" assumes "prefix_of \<pi> \<sigma>" shows "prefix_of (pmap_\<Gamma> f \<pi>) (map_\<Gamma> f \<sigma>)"
sdrop_all: fixes xs :: "[:'r:]"   and i :: "Integer" assumes "le\<cdot>(slength\<cdot>xs)\<cdot>i = TT" shows "sdrop\<cdot>i\<cdot>xs = [::]"
sprem_true: shows "I_of N \<Turnstile> set (side_prems_of \<iota>)"
poly_map_poly_eval_poly: fixes a :: "'a"   and p :: "'a poly poly" shows "poly (map_poly (\<lambda>p. poly p a) p) = eval_poly (\<lambda>p. poly p a) p"
dest_port_int_def: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> 'a" shows "dest_port x \<equiv> (snd \<circ> fst \<circ> snd \<circ> snd) x"
distinct_5: fixes x2 :: "'b" shows "ext.Val x2 \<noteq> ext.Top"
assms: shows "s' = snd (fst (store_sub1 instr rd s_val s)) \<and> get_S (cpu_reg_val PSR s) = 0"
k_\<Lambda>_PBD_axioms: shows "k_\<Lambda>_PBD \<V> \<B> \<Lambda> \<k>"
ord_eq_le_trans: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a = b"   and "c \<le> b" shows "c \<le> a"
cEnum_finite_2_def: shows "CENUM(Enum.finite_2) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
pair_in_Id_conv: fixes a :: "'g"   and b :: "'g" shows "((a, b) \<in> Id) = (a = b)"
replace_f: shows "\<forall>a. \<not> has_disc disc (replace_f a)"
silent_moves_called_node_in_slice1_hd_nodestack_in_slice1: fixes S :: "'node SDG_node set"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and nx :: "'node SDG_node"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,kind \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   and "CFG_node m' \<in> sum_SDG_slice1 nx"   and "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "ms = targetnodes rs"   and "valid_return_list rs m"   and "length rs = length cs" obtains as' :: "'edge list" and a :: "'edge" and as'' :: "'edge list" where "as = as' @ a # as''" and "\<exists>Q r p fs. kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" and "call_of_return_node (hd ms') (sourcenode a)" and "targetnode a -as''\<rightarrow>\<^bsub>sl\<^esub>* m'" and "CFG_node (sourcenode a) \<in> sum_SDG_slice1 nx"   | "ms' = ms"
assms: shows "vars_term_ms s \<subseteq># vars_term_ms t"
d_def_expl: fixes R :: "('a \<times> 'a set) set" shows "Multirelations.d R = R \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma>"
expand: fixes name :: "name"   and name' :: "name" assumes "as_string name = as_string name'" shows "name = name'"
tab_in_vhom': shows "\<guillemotleft>g.tab : tab\<^sub>1 g \<rightarrow> g \<star> tab\<^sub>0 g\<guillemotright>"
dghm_cn_cov_comp_ObjMap_app: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and a :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<GG>\<lparr>ObjMap\<rparr>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr>"
Vset_mono_strict: fixes m :: "hf"   and n :: "hf" assumes "Ord m"   and "n \<^bold>\<in> m" shows "Vset n < Vset m"
f0: fixes y :: "(real, 'n) vec" assumes "y \<in> T" shows "0 \<le> f y"
preserves_assoc_2: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "H \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] = \<Phi>\<^sub>H (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi>\<^sub>H (f, g) \<star>\<^sub>D H h) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, H g, H h] \<cdot>\<^sub>D (H f \<star>\<^sub>D D.inv (\<Phi>\<^sub>H (g, h))) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (f, g \<star>\<^sub>C h))"
elims_1: fixes x :: "atom"   and y :: "bool" assumes "is_Less x = y" obtains i :: "nat" and j :: "nat" where "x = Less i j" and "y"   | v :: "nat" and va :: "nat" where "x = Eq v va" and "\<not> y"
real_poly_mult: fixes p :: "'a poly"   and q :: "'a poly" assumes "set (coeffs p) \<subseteq> \<real>"   and "set (coeffs q) \<subseteq> \<real>" shows "set (coeffs (p * q)) \<subseteq> \<real>"
divide_quat_def: fixes x :: "quat"   and y :: "quat" shows "x div y = x * inverse y"
image_compactin: fixes X :: "'b topology"   and S :: "'b set"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "compactin X S"   and "continuous_map X Y f" shows "compactin Y (f ` S)"
inverse_mod_ring_def: fixes x :: "'b mod_ring" shows "inverse x = (if x = 0 then 0 else x ^ nat (int (CARD('b) - 2)))"
lookup_update': fixes ks :: "'c list"   and v :: "'d"   and t :: "('c, 'd) Trie.trie" shows "lookup_trie (update_trie ks v t) = lookup_trie t(ks \<mapsto> v)"
Abs_inverse: fixes y :: "'b" assumes "y \<in> A" shows "Rep (Abs y) = y"
sum_with_closed: fixes S :: "'d set"   and pls :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "comm_monoid_add_ow S pls z"   and "g ` A \<subseteq> S" shows "sum_with pls z g A \<in> S"
dual_upsets: shows "ord.upsets (\<lambda>x y. y \<le> x) = downsets"
less_Suc_eq: fixes m :: "nat"   and n :: "nat" shows "(m < Suc n) = (m < n \<or> m = n)"
max_coeff_bound_right: fixes xs :: "nat list"   and ys :: "nat list" assumes "(xs, ys) \<in> ba.Minimal_Solutions" shows "\<forall>x\<in>set xs. x \<le> maxne0 ys a"
simps_4: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list" shows "Mn_free (Cn n f gs) = (Ball (set gs) Mn_free \<and> Mn_free f)"
fansD: fixes subs :: "'b \<Rightarrow> 'b set"   and A :: "'b" assumes "fans subs" shows "finite (subs A)"
in_case: shows "{Var y IN k, Var z IN Var y, OrdP k} \<turnstile> Var z SUBS Var y"
sigma_sets_Un: fixes a :: "'c set"   and sp :: "'c set"   and A :: "'c set set"   and b :: "'c set" assumes "a \<in> sigma_sets sp A"   and "b \<in> sigma_sets sp A" shows "a \<union> b \<in> sigma_sets sp A"
finite_fmdom': fixes m :: "'d \<Rightarrow>\<^sub>F 'e" shows "finite (fmdom' m)"
arc_top_arc: fixes x :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])" shows "rectangle x"
termi_id: fixes n :: "nat"   and m :: "nat"   and xs :: "nat list" assumes "n < m"   and "length xs = m" shows "terminate (recf.id m n) xs"
yield\<^sub>C_def: fixes f :: "'d \<Rightarrow> 'e" shows "yield\<^sub>C f \<equiv> \<lambda>\<sigma>. Some (f \<sigma>, \<sigma>)"
length_operators: shows "length ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>) = Suc (length ast\<delta>)"
fresh_dom: fixes X :: "tyvrs"   and \<Gamma> :: "binding list" assumes "X \<sharp> \<Gamma>" shows "X \<sharp> ty_dom \<Gamma>"
pos_bounded: shows "\<exists>K>0. \<forall>x. norm (g' x) \<le> norm x * K"
periodic_simps_3: fixes x :: "'a"   and n :: "int" shows "sin (x + of_int n * ((2::'a) * of_real pi)) = sin x"
induct: fixes P :: "'b wordinterval \<Rightarrow> bool"   and a0 :: "'b wordinterval" assumes "\<And>s e. P (WordInterval s e)"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (RangeUnion A B)" shows "P a0"
vec_of_dim_0: fixes v :: "'b vec" shows "(dim_vec v = 0) = (v = 0\<^sub>v 0)"
h: shows "bij_betw h (path_components_of X) (path_components_of Y)"
assms: shows "s \<in> fst (simp r) \<rightarrow> v"
sup_ge1: fixes x :: "'k multiset"   and y :: "'k multiset" shows "x \<subseteq># x \<union># y"
ring_distrib1: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> (y \<plusminus>\<^bsub>R\<^esub> z) = x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<plusminus>\<^bsub>R\<^esub> x \<cdot>\<^sub>r\<^bsub>R\<^esub> z"
infsetsum_cmult_right: fixes c :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "c \<noteq> (0::'d) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. c * f x) = c * infsetsum f A"
rhs: shows "map_spmf (([^]) \<^bold>g) (sample_uniform (order G)) = spmf_of_set (carrier G)"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = vec n (\<lambda>j. if j = i then 1::'d else (0::'d))"
diff_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and d :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b"   and "less_eq d c" shows "less_eq (minus a c) (minus b d)"
per_perp: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "Per A B C" shows "A B Perp B C"
fin_E': shows "finite (edges (del_unEdge v e v' G))"
lebesgue_affine_measurable: fixes c :: "'c \<Rightarrow> real"   and t :: "'c" assumes "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0" shows "(\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)) \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
injectivity: fixes x :: "'b poly"   and y :: "'b poly" assumes "map_poly to_fract x = map_poly to_fract y" shows "x = y"