hcju/isabelleps huggingface dataset:
interchange_ik_not_zero: shows "interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n) $ mod_type_class.from_nat i $ mod_type_class.from_nat k \<noteq> (0::'a)"
cases: fixes A :: "('a, 'b) nba"   and a1 :: "('a \<times> 'b) stream"   and a2 :: "'b" assumes "run A a1 a2" obtains a :: "'a \<times> 'b" and p :: "'b" and r :: "('a \<times> 'b) stream" where "a1 = a ## r" and "a2 = p" and "fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p" and "run A r (snd a)"
wellformed_transaction_cases_3: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and l :: "'h strand_label"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step" assumes "wellformed_transaction T"   and "(l, x) \<in> set (transaction_checks T)" shows "(\<exists>t s. x = \<langle>t == s\<rangle>) \<or> (\<exists>t s. x = \<langle>t in s\<rangle>) \<or> (\<exists>X F G. x = \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>)"
expand_init_propag_on_create_graph: shows "create_graph \<phi> \<le> SPEC (\<lambda>nds. \<forall>nd\<in>nds. expand_init \<in> incoming nd \<longrightarrow> \<phi> \<in> old nd)"
exhaust: fixes y :: "mreal" obtains x1 :: "real" where "y = mreal x1"   | "y = MInfty"
candidates: shows "h \<turnstile> filter_M (\<lambda>document_ptr. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disconnected_nodes. return (node_ptr \<in> set disconnected_nodes))) (sorted_list_of_set (fset (document_ptr_kinds h))) \<rightarrow>\<^sub>r candidates"
append_assoc: fixes xs :: "'b List.list"   and ys :: "'b List.list"   and zs :: "'b List.list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
wffp_purgeIdle_Nil: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do sl"   and "GM_sec_model.purgeIdle sl = []" shows "\<exists>n st. 0 < n \<and> sl = replicate n (Idle st)"
sets_distr: fixes M :: "'c measure"   and N :: "'b measure"   and f :: "'c \<Rightarrow> 'b" shows "sets (distr M N f) = sets N"
hyps: shows "int v \<le> changes_itv_der d1 c (pderiv p) \<and> even (changes_itv_der d1 c (pderiv p) - int v)"
enumerate_aexp_inputs_not_empty: fixes a :: "vname aexp" shows "(enumerate_aexp_inputs a \<noteq> {}) = (\<exists>b c. enumerate_aexp_inputs a = set (b # c))"
swap_ident: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "X #[x \<and> x]_xs = X"
induct: fixes P :: "'a list list \<Rightarrow> bool"   and a0 :: "'a list list" assumes "P []"   and "\<And>c. P [c]"   and "\<And>c1 c2 cs. P (c2 # cs) \<Longrightarrow> P (c1 # c2 # cs)" shows "P a0"
hom_dvd: fixes p :: "int"   and q :: "int" assumes "p dvd q" shows "of_int p dvd of_int q"
assms_3: shows "\<guillemotleft>x' : local.map a' \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] d\<guillemotright>"
in_hom_def: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> domains f \<and> b \<in> codomains f"
rel_eq_onp: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool" shows "rel_sum (eq_onp P1) (eq_onp P2) = eq_onp (pred_sum P1 P2)"
swap_apply_1: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "Fun.swap a b f a = f b"
CondInv: fixes P :: "'d set"   and Q :: "'d set"   and b :: "'d set"   and P\<^sub>1 :: "'d set"   and P\<^sub>2 :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and A :: "'d set"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "P \<subseteq> Q"   and "Q \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> P\<^sub>2)}"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Cond b c\<^sub>1 c\<^sub>2 Q,A"
ir_valid_decomp2: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and R :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P c SKIP R \<and> ir_valid R SKIP c' Q" shows "ir_valid P c c' Q"
analz_insert_freshK3_bis: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list"   and authK' :: "nat" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "authK \<noteq> authK'"   and "authK' \<notin> range shrK"   and "evs \<in> kerbIV_gets" shows "(Key servK \<in> analz (insert (Key authK') (knows Spy evs))) = (servK = authK' \<or> Key servK \<in> analz (knows Spy evs))"
vp_mem_Vr_mem: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> vp K v" shows "x \<in> carrier (Vr K v)"
oarrivemsg_simps_1: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and R :: "nat set"   and m :: "'e" shows "oarrivemsg P \<sigma> (R:*cast(m)) = True"
rec_newleft2_def: shows "rec_newleft2 = Cn 2 rec_quo [recf.id 2 0, Cn 2 (constn 2) [recf.id 2 0]]"
C_restr_cong: fixes r :: "C"   and f :: "C \<rightarrow> 'a"   and g :: "C \<rightarrow> 'a" assumes "\<And>r'. r' \<sqsubseteq> r \<Longrightarrow> f\<cdot>r' = g\<cdot>r'" shows "f|\<^bsub>r\<^esub> = g|\<^bsub>r\<^esub>"
poly_normalize: fixes p :: "'a list" shows "poly (pnormalize p) = poly p"
lookup_update_with_trie: fixes ks :: "'d list"   and f :: "'c option \<Rightarrow> 'c"   and t :: "('d, 'c) trie"   and ks' :: "'d list" shows "lookup_trie (update_with_trie ks f t) ks' = (if ks' = ks then Some (f (lookup_trie t ks')) else lookup_trie t ks')"
intros: fixes state :: "State" assumes "getCn state \<noteq> 1 \<Longrightarrow> applyExplainUIP_dom (applyExplain (getCl state) state)" shows "applyExplainUIP_dom state"
isEmpty_correct: fixes s :: "('a, unit) HashMap.hashmap" assumes "hs.invar s" shows "hs.isEmpty s = (hs.\<alpha> s = {})"
fls_shift_eq0_iff: fixes m :: "int"   and f :: "'b fls" shows "(fls_shift m f = 0) = (f = 0)"
nth_Cons_Suc: fixes x :: "'e"   and xs :: "'e list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
group_left_neutral: fixes a :: "'a" shows "(0::'a) + a = a"
offs_next_cons_eq: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and x :: "'c"   and ns :: "nat list"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and xs :: "'c list"   and j :: "nat"   and ub :: "nat" assumes "index key x (length ns) mi ma = i"   and "i < length ns"   and "0 < offs_num (length ns) (x # xs) index key mi ma j"   and "offs_set_prev ns (x # xs) index key mi ma i = {} \<or> Max (offs_set_prev ns (x # xs) index key mi ma i) \<noteq> j" shows "offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma j = offs_next ns ub (x # xs) index key mi ma j"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (new_shadow_root_ptr, h')"   and "ptr \<noteq> cast (cast new_shadow_root_ptr)" shows "get ptr h = get ptr h'"
inf_sup_aci_2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup" shows "inf (inf x y) z = inf x (inf y z)"
a: shows "Lam [y']. ((y' \<leftrightarrow> y) \<bullet> e') = Lam [y]. e'"
generated_ring_Inter: fixes A :: "'a set set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> generated_ring" shows "\<Inter> A \<in> generated_ring"
poly_mult_append_zero_lcancel: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list"   and r :: "'a list" assumes "subring K R"   and "polynomial K p"   and "polynomial K q"   and "poly_mult (p @ [\<zero>]) q = r @ [\<zero>]" shows "poly_mult p q = r"
wffp_cases3: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl" obtains n :: "nat" and st :: "'St" where "0 < n__" and "sl = map Idle (replicate n__ st__)"   | st :: "'St" and u :: "'U" and c :: "'C" and sl1 :: "('St, 'U, 'C) state list" where "sl = State sta__ u__ c__ # sl1__" and "sl1__ \<noteq> [] \<Longrightarrow> wffp sl1__ \<and> hd sl1__ \<in> \<delta> (State sta__ u__ c__)"   | n :: "nat" and st :: "'St" and u :: "'U" and c :: "'C" and sl1 :: "('St, 'U, 'C) state list" where "0 < n" and "sl = map Idle (replicate n st) @ [State (do st u c) u c] @ sl1" and "sl1 \<noteq> [] \<Longrightarrow> wffp sl1 \<and> hd sl1 \<in> \<delta> (State (do st u c) u c)"
indexed_padd_def: fixes P :: "'d multiset \<Rightarrow> 'a"   and Q :: "'d multiset \<Rightarrow> 'a" shows "P \<Oplus> Q = (\<lambda>m. P m \<oplus> Q m)"
positively_invariant_left_inter: fixes C :: "'a set"   and D :: "'a set" assumes "positively_invariant C"   and "\<forall>x\<in>C \<inter> D. trapped_forward x D" shows "positively_invariant (C \<inter> D)"
shiftr_mask2: fixes n :: "nat"   and m :: "nat" assumes "n \<le> LENGTH('a)" shows "mask n >> m = mask (n - m)"
nat_to_set_def: fixes u :: "nat" shows "nat_to_set u \<equiv> {x. 2 ^ x \<le> u \<and> c_in x u = 1}"
normal_kernel: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h" shows "kernel G H h \<lhd> G"
synthetic_div_correct': fixes c :: "'b"   and p :: "'b poly" shows "[:- c, 1::'b:] * synthetic_div p c + [:poly p c:] = p"
gap_gr0: fixes i :: "nat"   and t :: "nat" shows "0 < gap i t"
assign_eval\<^sub>w_const\<^sub>C: fixes x :: "var\<^sub>C"   and c :: "nat"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat" shows "eval_abv\<^sub>C \<langle>x \<leftarrow> aexp\<^sub>C.Const c, mds, mem\<rangle>\<^sub>C \<langle>Stop, mds, mem(x := c)\<rangle>\<^sub>C"
simps_129: fixes x31 :: "poscheckvariant"   and x32 :: "('d, 'c) Term.term"   and x33 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>x31: x32 \<doteq> x33\<rangle> = fv x32 \<union> fv x33"
map_single: fixes c :: "'a"   and f :: "'a \<Rightarrow> 'b"   and x :: "'c" assumes "c = (0::'a) \<Longrightarrow> f (0::'a) = (0::'b)" shows "Poly_Mapping.map f (monomial c x) = monomial (f c) x"
ExERule: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "[\<^bold>\<exists>x. \<phi> x in v]" obtains x :: "'a" where "[\<phi> x in v]"
convergent_def: fixes X :: "nat \<Rightarrow> 'a" shows "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
recfun_aux_dom_alt: shows "recfun_aux_dom = local.dom"
hyps_3: assumes "s' \<notin> Fault ` F" shows "strip F \<Gamma>\<turnstile> \<langle>s,Normal b\<rangle> =n\<Rightarrow> s'"
frac_unique_iff: fixes x :: "'b"   and a :: "'b" shows "(frac x = a) = (x - a \<in> \<int> \<and> (0::'b) \<le> a \<and> a < (1::'b))"
higher_deriv_id: fixes n :: "nat"   and z :: "'a" shows "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1::'a else (0::'a))"
at_within_t1_space_avoid_finite: fixes I :: "'c set"   and x :: "'c"   and X :: "'c set" assumes "finite I"   and "x \<notin> I" shows "at x within X - I = at x within X"
inv2Inv_2: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "cinv\<^bsub>C\<^esub> f g" shows "Cinv\<^bsub>C\<^esub> f = g"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and ns :: "nat list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and mi :: "'d"   and ma :: "'d" shows "fill (x # xs) ns index key n mi ma = (let i = index key x (length ns) mi ma; ys = fill xs (ns[i := Suc (ns ! i)]) index key n mi ma in ys[ns ! i := Some x])"
proj_stoch_proc_component_1: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and w :: "'c" assumes "k < n" shows "proj_stoch_proc f n w !! k = f k w"
step: shows "\<Gamma>\<turnstile> ([], ass @ css, s) \<rightarrow> (cs', ass' @ css, t)"
tendsto_asymp_equiv_cong: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "f \<sim>[F] g" shows "(f \<longlongrightarrow> c) F = (g \<longlongrightarrow> c) F"
delta_append: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> bool)"   and xs :: "'d list"   and ys :: "'d list"   and q :: "'c" shows "delta A (xs @ ys) q = delta A ys (delta A xs q)"
iexp_approx1: fixes x :: "real"   and n :: "nat" shows "cmod (iexp x - (\<Sum>k\<le>n. (\<i> * complex_of_real x) ^ k / fact k)) \<le> \<bar>x\<bar> ^ Suc n / fact (Suc n)"
pp_ub1: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ls :: "'c list"   and y :: "'c"   and x :: "'c" assumes "Semilat A r f"   and "set ls \<subseteq> A"   and "y \<in> A"   and "x \<in> set ls" shows "x \<sqsubseteq>\<^bsub>r\<^esub> ls \<Squnion>\<^bsub>f\<^esub> y"
complete_less_not: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c" assumes "complete A r"   and "hasw [x, y] A"   and "\<not> x \<^bsub>r\<^esub>\<prec> y" shows "y \<^bsub>r\<^esub>\<preceq> x"
depthTransition_3: fixes P :: "pi"   and P' :: "pi" assumes "hnf P"   and "P \<longmapsto> \<tau> \<prec> P'" shows "depth P' < depth P"
unfinished_dnodes'_def: fixes V :: "'v \<Rightarrow> 'a option" shows "unfinished_dnodes' V \<equiv> unfinished_dnodes (dom V)"
idx_spec: fixes xs :: "'a list list" shows "maxi xs = ((\<forall>i<length xs. xs ! i \<noteq> []) \<and> (\<forall>i<length xs - 1. incr (xs ! i) \<and> \<not> last (xs ! i) < hd (xs ! Suc i) \<or> decr (xs ! i) \<and> \<not> hd (xs ! Suc i) \<le> last (xs ! i)))"
closedin_closed_Int: fixes S :: "'c set"   and U :: "'c set" assumes "closed S" shows "closedin (top_of_set U) (U \<inter> S)"
inject_1: fixes x1 :: "name"   and y1 :: "name" shows "(InputS x1 = InputS y1) = (x1 = y1)"
map_of_insertmap_SomeD': fixes fs :: "('b \<times> 'a) list"   and F :: "'b"   and y :: "'a"   and D :: "'c" assumes "map_of fs F = \<lfloor>y\<rfloor>" shows "map_of (map (\<lambda>(F, y). (F, D, y)) fs) F = \<lfloor>(D, y)\<rfloor>"
LSM_inc_T: fixes M :: "('e, 'a, 'f) Module_scheme"   and T :: "'e set" assumes "R module M"   and "T \<subseteq> carrier M" shows "T \<subseteq> LSM\<^bsub>R\<^esub> M T"
assms: shows "\<forall>A\<in>\<A>. closed_csubspace A"
list_all2_weaken: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and f' :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "list_all2 f xs ys"   and "\<And>x y. \<lbrakk>(x, y) \<in> set (zip xs ys); f x y\<rbrakk> \<Longrightarrow> f' x y" shows "list_all2 f' xs ys"
ag_eq_diffzero: fixes A :: "('h, 'i) aGroup_scheme"   and a :: "'h"   and b :: "'h" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A" shows "(a = b) = (a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b = \<zero>\<^bsub>A\<^esub>)"
min_intervalI: fixes a :: "'a"   and A :: "'a interval"   and b :: "'a"   and B :: "'a interval" assumes "a \<in>\<^sub>i A"   and "b \<in>\<^sub>i B" shows "min a b \<in>\<^sub>i min_interval A B"
axioms_1: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" assumes "SN_order_pair S NS" shows "order_pair S NS"
insert_subset: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
HPhase1or2ReadElse_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2ReadElse s s' p d q"   and "HInv3 s" shows "HInv3 s'"
consec_un_min_max: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> Rep_nat_int (i \<squnion> j) = {minimum i..maximum j}"
mem_simps_2: fixes c :: "'a" shows "(c \<in> {}) = False"
inner_vec_def: fixes x :: "('a, 'cols) vec"   and y :: "('a, 'cols) vec" shows "inner_cols x y = (\<Sum>i\<in>UNIV. x $ i * cnj_class.cnj (y $ i))"
shadows_consts_frees: fixes t :: "'a" assumes "\<not> shadows_consts t" shows "fdisjnt all_consts (frees t)"
v_notin_Op: shows "State l (index (\<Pi>\<^sub>\<V>) v) \<notin> {Operator k (index (\<Pi>\<^sub>\<O>) op) |k op. k \<in> {0..<length \<pi>} \<and> op \<in> set (\<Pi>\<^sub>\<O>)}"
tpOfV: fixes \<sigma> :: "'tp" shows "infinite (tpOfV -` {\<sigma>})"
index_Fun_fv_subset: fixes i :: "nat"   and T :: "('c, 'd) Term.term list"   and f :: "'c" assumes "i < length T" shows "fv (T ! i) \<subseteq> fv (Fun f T)"
expandSC: fixes R :: "pi"   and P :: "pi"   and Q :: "pi" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q" shows "P \<parallel> Q \<sim> R"
apply_singleton_matrix: fixes f :: "'c \<Rightarrow> 'b"   and j :: "nat"   and i :: "nat"   and x :: "'c" assumes "f (0::'c) = (0::'b)" shows "apply_matrix f (singleton_matrix j i x) = singleton_matrix j i (f x)"
is_minimal_basisD1: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis B"   and "p \<in> B" shows "p \<noteq> 0"
brouwer_cube: fixes f :: "'a \<Rightarrow> 'a" assumes "continuous_on (cbox (0::'a) One) f"   and "f ` cbox (0::'a) One \<subseteq> cbox (0::'a) One" shows "\<exists>x\<in>cbox (0::'a) One. f x = x"
getEnv_ext: fixes rho :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and rho' :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "\<And>xs x. rho xs x = rho' xs x" shows "rho = rho'"
pairwise_finiteI: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite {b. \<exists>a. P a b}"   and "finite {a. \<exists>b. P a b}" shows "finite {(a, b). P a b}"
1: shows "Left.in_hom \<mu> (local.dom \<mu>) (cod \<mu>)"
consec_inter_empty: fixes i :: "nat_int"   and j :: "nat_int" assumes "consec i j" shows "i \<sqinter> j = bot"
KL_gt_0: fixes D :: "'a \<Rightarrow> real" assumes "prob_space (density M (\<lambda>x. ennreal (D x)))"   and "random_variable borel D"   and "AE x in M. 0 \<le> D x"   and "integrable M (\<lambda>x. D x * log b (D x))"   and "density M (\<lambda>x. ennreal (D x)) \<noteq> M" shows "0 < KL_divergence b M (density M (\<lambda>x. ennreal (D x)))"
mk_poly2_row_dim: fixes A :: "'b mat"   and d :: "nat"   and j :: "nat"   and pv :: "'b poly vec"   and i :: "nat" shows "dim_vec (mk_poly2_row A d j pv i) = dim_vec pv"
map_fset_image: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d fset" shows "fset (map_fset f S) = f ` fset S"
alw_ev_same2: fixes u :: "'c \<Rightarrow> real" shows "almost_everywhere (MDP.MC.T (repcs st cfg)) (alw (\<lambda>\<omega>. HLD (state -` snd -` {u}) \<omega> \<longrightarrow> ev (HLD {cfg. \<forall>cfg'\<in>set_pmf (K_cfg cfg). fst (state cfg') = fst (state cfg)}) \<omega>))"
closed_sum_left_subset: fixes B :: "'b set"   and A :: "'b set" assumes "(0::'b) \<in> B" shows "A \<subseteq> A +\<^sub>M B"
cod_mkIde: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>" shows "partial_magma.cod S (set_category_given_img.mkIde S img \<AA> A) = set_category_given_img.mkIde S img \<AA> A"
shadow_root_ptr_c: shows "h \<turnstile> get_dom_component (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr) \<rightarrow>\<^sub>r shadow_root_ptr_c"
bicategory_def: 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" shows "bicategory V H \<a> \<i> src trg \<equiv> (horizontal_composition V H src trg \<and> natural_isomorphism (subcategory.comp (product_category.comp V (product_category.comp V V)) (\<lambda>\<tau>\<mu>\<nu>. partial_magma.arr V (fst \<tau>\<mu>\<nu>) \<and> partial_magma.arr (subcategory.comp (product_category.comp V V) (\<lambda>\<mu>\<nu>. partial_magma.arr V (fst \<mu>\<nu>) \<and> partial_magma.arr V (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))) (snd \<tau>\<mu>\<nu>) \<and> src (fst \<tau>\<mu>\<nu>) = trg (fst (snd \<tau>\<mu>\<nu>)))) V (horizontal_composition.HoHV V H src trg) (horizontal_composition.HoVH V H src trg) (\<lambda>\<mu>\<nu>\<tau>. \<a> (fst \<mu>\<nu>\<tau>) (fst (snd \<mu>\<nu>\<tau>)) (snd (snd \<mu>\<nu>\<tau>)))) \<and> fully_faithful_functor V V (\<lambda>\<mu>. if partial_magma.arr V \<mu> then H (trg \<mu>) \<mu> else partial_magma.null V) \<and> fully_faithful_functor V V (\<lambda>\<mu>. if partial_magma.arr V \<mu> then H \<mu> (src \<mu>) else partial_magma.null V) \<and> bicategory_axioms V H \<a> \<i> src trg"
continuous_map_subtopology_eu: fixes S :: "'a set"   and T :: "'b set"   and h :: "'a \<Rightarrow> 'b" shows "continuous_map (top_of_set S) (top_of_set T) h = (continuous_on S h \<and> h ` S \<subseteq> T)"
linear_independent_extend: fixes B :: "('b, 'c) vec set"   and f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "vec.independent B" shows "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> (\<forall>x\<in>B. g x = f x)"
size_eq_0_iff_empty: fixes M :: "'b multiset" shows "(size M = 0) = (M = {#})"
snd_diag_fst: shows "snd \<circ> ((\<lambda>x. (x, x)) \<circ> fst) = fst"
simps: fixes a :: "('fun, 'var) Term.term" shows "public_ground_wf_term a = (\<exists>f T. a = Fun f T \<and> public f \<and> arity f = length T \<and> (\<forall>x. x \<in> set T \<longrightarrow> public_ground_wf_term x))"
abs_if: fixes abs :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and zero :: "'a"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.abs_if abs uminus zero less" shows "abs a = (if less a zero then uminus a else a)"
times_in_intervalI: fixes x :: "'b"   and X :: "'b interval"   and y :: "'b"   and Y :: "'b interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x * y \<in>\<^sub>i X * Y"
obj_is_weak_unit: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "prebicategory_with_homs V H \<a> src trg"   and "horizontal_homs.obj V src a" shows "weak_composition.weak_unit V H a"
IH: fixes x :: "pat"   and y :: "value"   and env :: "(name, value) fmap" assumes "name = name' \<and> length ps = length vs"   and "x \<in> set ps"   and "y \<in> set vs"   and "vmatch x y = Some env"   and "vwelldefined' y" shows "fmpred (\<lambda>_. vwelldefined') env"
either_above_arg2: fixes t' :: "'a ttree"   and t :: "'a ttree" shows "t' \<sqsubseteq> t \<oplus>\<oplus> t'"
p'_def: shows "p' \<equiv> p div gcd p (pderiv p)"
norm_blinfun: fixes f :: "'b \<Rightarrow>\<^sub>L 'a"   and x :: "'b" shows "norm (blinfun_apply f x) \<le> norm f * norm x"
ring_incl_imp_subring: fixes H :: "'a set" assumes "H \<subseteq> carrier R"   and "ring (R\<lparr>carrier := H\<rparr>)" shows "subring H R"