hcju/isabelleps huggingface dataset:
cf_h_pol_coeff: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "pol_coeff S (n, f)" shows "cf_h h (n, f) = (n, cmp h f)"
slim_arcs_def: shows "slim_arcs \<equiv> \<Union>(uu_, p, uu_)\<in>slim_paths. set p"
coplanar_trivial: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "Coplanar A A B C"
and_one_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and a one = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
monad_alt_axioms_def: fixes bind :: "'d \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd"   and alt :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "monad_alt_axioms bind alt \<equiv> (\<forall>m1 m2 m3. alt (alt m1 m2) m3 = alt m1 (alt m2 m3)) \<and> (\<forall>m m' f. bind (alt m m') f = alt (bind m f) (bind m' f))"
equality: fixes r :: "('b, 'c) pair_pre_digraph_scheme"   and r' :: "('b, 'c) pair_pre_digraph_scheme" assumes "pverts r = pverts r'"   and "parcs r = parcs r'"   and "pair_pre_digraph.more r = pair_pre_digraph.more r'" shows "r = r'"
PQ: shows "P * P' * (Q' * Q) = 1\<^sub>m n"
bijective: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and H :: "'b set" assumes "GroupIso G T H" shows "bij_betw T G H"
UP\<^sub>0_def: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B" shows "strictified_bicategory.UP\<^sub>0 V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B a \<equiv> if horizontal_homs.obj V\<^sub>B src\<^sub>B a then concrete_category.MkArr (bicategorical_language.Prim\<^sub>0 a) (bicategorical_language.Prim\<^sub>0 a) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (bicategorical_language.Prim\<^sub>0 a)) else partial_magma.null (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B))"
map_nth_append_upt: fixes xs :: "'c list"   and a :: "nat"   and ys :: "'c list"   and b :: "nat" assumes "length xs \<le> a" shows "map ((!) (xs @ ys)) [a..<b] = map ((!) ys) [a - length xs..<b - length xs]"
Gcd_set_eq_fold: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and xs :: "'b list" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd (set xs) = fold gcd xs zero"
closedin_closed_Int: fixes S :: "'b set"   and U :: "'b set" assumes "closed S" shows "closedin (top_of_set U) (U \<inter> S)"
assms_1: fixes P :: "'state"   and Q :: "'state" assumes "P \<sim>\<cdot> Q" shows "P \<Turnstile> x = Q \<Turnstile> x"
imirror_not_empty: fixes I :: "nat set" assumes "I \<noteq> {}" shows "imirror I \<noteq> {}"
l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf \<equiv> type_wf = ShadowRootClass.type_wf"
VEnd_endpow: fixes n :: "nat" shows "VEnd (endpow n)"
f_is_g: fixes b :: "'b"   and a :: "'a"   and c :: "'a" assumes "b \<in> s"   and "f' a = f b (f' c)" shows "g' a = g b (g' c)"
funcset_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<forall>x\<in>A. f x = g x" shows "f = g"
mrg_left_vwb_lens: shows "vwb_lens mrg_left"
icheck_curr_inits': fixes es :: "'c exp list"   and e :: "'c exp"   and P :: "'d prog"   and C :: "char list" assumes "ss_exps es = \<lfloor>e\<rfloor>"   and "icheck P C e" shows "curr_inits P es = \<lfloor>C\<rfloor>"
proj2_abs_rep: fixes x :: "proj2" shows "proj2_abs (proj2_rep x) = x"
fds_nth_mult: fixes f :: "'b fds"   and g :: "'b fds" shows "($) (f * g) = ($) f \<star> ($) g"
inf_idem: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "class.semilattice_inf inf less_eq less" shows "inf x x = x"
z: shows "z / complex_of_real (cmod z) = Complex (cos \<psi>) (sin \<psi>)"
asymp_equiv_trans: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g"   and "g \<sim>[F] h" shows "f \<sim>[F] h"
Int_UNIV: fixes A :: "'d set"   and B :: "'d set" shows "(A \<inter> B = UNIV) = (A = UNIV \<and> B = UNIV)"
set_pmf_eq': fixes p :: "'c pmf" shows "set_pmf p = {x. 0 < pmf p x}"
shift_def: fixes lab :: "'c list \<Rightarrow> 'd"   and k :: "'c" shows "BNF_Greatest_Fixpoint.shift lab k = (\<lambda>kl. lab (k # kl))"
rep_eq: fixes x :: "'e \<Rightarrow> 'g"   and xa :: "('f, 'g) uexpr" shows "\<lbrakk>x \<dagger> xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>xa\<rbrakk>\<^sub>e (x b))"
Decrypt: fixes K :: "key"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "Key (invKey K) \<in> analz H" shows "X \<in> analz H"
inject_5: fixes x5 :: "('b, 'c) fm"   and y5 :: "('b, 'c) fm" shows "((\<^bold>\<diamond> x5) = (\<^bold>\<diamond> y5)) = (x5 = y5)"
preserves_inv: fixes f :: "'a" assumes "B.iso f" shows "src\<^sub>B (B.inv f) = B.inv (src\<^sub>B f)"
ran_dom_the: fixes y :: "'a"   and G :: "'b \<Rightarrow> 'a set option"   and x :: "'b" assumes "y \<notin> \<Union> (ran G)"   and "x \<in> dom G" shows "y \<notin> the (G x)"
before_in_irefl: fixes x :: "'a"   and xs :: "'a list" shows "x < x in xs = False"
h: shows "set (take_indices (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (find_nonzeros_from_input_vec (solve_for_lhs_R p (qs1 @ qs2) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (M_mat_R (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))))))) = set (characterize_consistent_signs_at_roots p (qs1 @ qs2))"
empty: shows "{G \<in> \<G>. card G = 0} = {}"
sum_case_Inr: fixes f :: "V \<Rightarrow> 'a"   and g :: "V \<Rightarrow> 'a"   and y :: "V" shows "sum_case f g (ZFC_Cardinals.Inr y) = g y"
Shuffle_Plus_ShuffleLR_PlusL: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Times_Plus (ShuffleLR (Plus r s) t) = Plus (Times_Plus (ShuffleLR r t)) (Times_Plus (ShuffleLR s t))"
dilated_prefix: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and c :: "clock" assumes "dilating f sub r" shows "{i. i \<le> f n \<and> hamlet (Rep_run r i c)} = f ` {i. i \<le> n \<and> hamlet (Rep_run sub i c)}"
simps_2: fixes x2 :: "'a" shows "SOURCE \<noteq> Inner x2"
Ensures_def: fixes A :: "'c set"   and B :: "'c set" shows "A Ensures B \<equiv> {F. F \<in> reachable F \<inter> A ensures B}"
real_roots_of_int_poly_2: fixes p :: "int poly" shows "distinct (real_roots_of_int_poly p)"
simps: fixes x :: "'b"   and l :: "'b"   and u :: "'b" shows "elem x (l, u) = (l < x \<and> x < u)"
Pi_UNIV: fixes A :: "'c set" shows "A \<rightarrow> UNIV = UNIV"
assms_2: shows "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> d : B"
Nml_HcompD_1: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "\<^bold>\<langle>un_Prim t\<^bold>\<rangle> = t"
axioms_2: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "\<Z>_srng_axioms \<alpha> \<SS>"
step_2: fixes gpv :: "('a, 'b, 'c) gpv" assumes "\<I>' \<turnstile>g gpv \<surd>" shows "expectation_gpv' gpv \<le> expectation_gpv 0 \<I> (\<lambda>_. y) (mk_lossless_gpv (responses_\<I> \<I>') x gpv)"
nth_concat_diff: fixes i1 :: "nat"   and xs :: "'a list list"   and i2 :: "nat" assumes "i1 < length (concat xs)"   and "i2 < length (concat xs)"   and "i1 \<noteq> i2" shows "\<exists>j1 k1 j2 k2. (j1, k1) \<noteq> (j2, k2) \<and> j1 < length xs \<and> j2 < length xs \<and> k1 < length (xs ! j1) \<and> k2 < length (xs ! j2) \<and> concat xs ! i1 = xs ! j1 ! k1 \<and> concat xs ! i2 = xs ! j2 ! k2"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case OK x2 of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = f2 x2"
epsclo_idem: fixes Q :: "hf set" shows "epsclo (epsclo Q) = epsclo Q"
simps_4: fixes delta :: "'e"   and inp :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option"   and xs' :: "'c"   and x' :: "'d" shows "(qOp delta inp binp #= qVar xs' x') = False"
bigo_abs5: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(g)" shows "(\<lambda>x. \<bar>f x\<bar>) \<in> O(g)"
subset_eq: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. s2p B x)"
dist_nodes_suc: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and v :: "nat"   and p :: "nat"   and q :: "nat" shows "dist_nodes M (Suc n) v p q = (\<exists>bs. is_alph v bs \<and> dist_nodes M n v (dfa_trans M p bs) (dfa_trans M q bs))"
finsum_def: fixes G :: "('c, 'd) ring_scheme" shows "finsum G = finprod (add_monoid G)"
l_distr: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
node\<^sub>i_inorder: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "split split" shows "inorder_up\<^sub>i (split.node\<^sub>i k ts t) = inorder (Node ts t)"
dist_discrete_def: fixes n :: "'b discrete"   and m :: "'b discrete" shows "dist n m = (if n = m then 0 else 1)"
Contracts_axioms: shows "Contracts X5d X5h PX5d CX5h"
D_CAST: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ" assumes "(G, c, HighSec) \<in> Deriv" shows "(G, c, Sec (\<lambda>(s, t, \<beta>). s \<equiv>\<^sub>\<beta> t)) \<in> Deriv"
IH_2: fixes x :: "('a, 'b) rbt"   and y :: "('a \<times> 'b) list"   and x21 :: "'a \<times> 'b"   and x22 :: "('a \<times> 'b) list"   and xa :: "'a"   and ya :: "'b" assumes "\<not> (n = 0 \<or> n = 1)"   and "n mod 2 = 0"   and "(x, y) = rbtreeify_g (n div 2) kvs"   and "y = x21 # x22"   and "(xa, ya) = x21"   and "0 < n div 2"   and "n div 2 \<le> Suc (length x22)" shows "length (snd (rbtreeify_g (n div 2) x22)) = Suc (length x22) - n div 2"
fps_binomial_add_mult: fixes c :: "'a"   and d :: "'a" shows "fps_binomial (c + d) = fps_binomial c * fps_binomial d"
polyfun_Sum: fixes I :: "'a set"   and N :: "nat set"   and f :: "'a \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> polyfun N (f i)" shows "polyfun N (\<lambda>x. \<Sum>i\<in>I. f i x)"
less_pow2: fixes x :: "nat"   and a :: "nat" assumes "x < 2 ^ a" shows "int x < 2 ^ a"
add_positive_le_reduce2: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "positive B"   and "B + A \<le>\<^sub>L C" shows "A \<le>\<^sub>L C"
path_connectedin_subset_topspace: fixes X :: "'c topology"   and S :: "'c set" assumes "path_connectedin X S" shows "S \<subseteq> topspace X"
assms: fixes i :: "'b" shows "continuous_on S (\<lambda>x. f x i)"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = a \<squnion> b" shows "b \<sqsubseteq> a"
alignUp_distance: fixes q :: "'b word"   and sz :: "nat" shows "alignUp q sz - q \<le> mask sz"
is_chainD': fixes f :: "nat \<Rightarrow> 'a set"   and x :: "'a"   and m :: "nat"   and k :: "nat" assumes "is_chain f"   and "x \<in> f m"   and "m \<le> k" shows "x \<in> f k"
pred_it_def: fixes pred_list_it :: "'d \<Rightarrow> 'b \<Rightarrow> (('b \<times> 'c) list \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list) \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list"   and g :: "'d"   and v :: "'b" shows "graph_pred_it_defs.pred_it pred_list_it g v \<equiv> it_to_it (pred_list_it g v)"
WT_gpv_Pause: fixes \<Gamma> :: "('d, 'e) \<I>"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv" shows "\<Gamma> \<turnstile>g Generative_Probabilistic_Value.Pause out c \<surd> = (out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. \<Gamma> \<turnstile>g c input \<surd>))"
lt_monic: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "lt (monic p) = lt p"
fs_Ints: shows "\<parallel>fs ! i\<parallel>\<^sup>2 \<in> \<int>"
log: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m Log # rs \<Rightarrow>\<^sub>z t"
prod_join: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "x \<squnion> y = (fst x \<squnion> fst y, snd x \<squnion> snd y)"
p: shows "urel64 (pp - 2) (int (nat (p - 2)))"
assms_3: shows "n pd\<rightarrow> \<pi> 0"
k_def: shows "k = the (threshold \<psi> w \<G>)"
normalize_eq_0_iff: fixes a :: "'a" shows "(normalize a = (0::'a)) = (a = (0::'a))"
basis_exists: fixes V :: "'b set" obtains B :: "'b set" where "B \<subseteq> V" and "vs1.independent B" and "V \<subseteq> vs1.span B" and "card B = vs1.dim V"
inv_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y \<otimes> x = \<one>"   and "x \<otimes> y' = \<one>"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "y' \<in> carrier G" shows "y = y'"
in_hom_def: fixes f :: "'s"   and a :: "'s"   and b :: "'s" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> S.domains f \<and> b \<in> S.codomains f"
get_document_ptr_simp6: fixes document_ptr :: "(_) document_ptr"   and character_data_ptr :: "(_) character_data_ptr"   and f :: "'u option RCharacterData_scheme"   and h :: "(_) heap" shows "get document_ptr (put character_data_ptr f h) = get document_ptr h"
simps_1: fixes n :: "nat \<Rightarrow> nat" shows "inf_concat n 0 = (LEAST j. 0 < n j, 0)"
False: shows "p\<^sub>1 \<noteq> opp p\<^sub>1"
assms: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "f m < f n"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter"   and x :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter" assumes "eq_onp (\<lambda>pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)) xa xa"   and "eq_onp (\<lambda>pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)) x x" shows "Abs_lifted_pair xa \<squnion> Abs_lifted_pair x = Abs_lifted_pair (\<lambda>f. triple.pairs_sup (xa f) (x f))"
set_integral_diff_1: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "set_integrable M A f"   and "set_integrable M A g" shows "set_integrable M A (\<lambda>x. f x - g x)"
exec_elim_cases_11: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Call p,s\<rangle> \<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | bdy :: "('e, 'd, 'f) com" and s :: "'e" where "sb__ = Normal sc__" and "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile> \<langle>bdy,Normal sc__\<rangle> \<Rightarrow> t"   | s :: "'e" where "sb__ = Normal sd__" and "t = Stuck" and "\<Gamma> p = None"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
div_Tr2: fixes r :: "nat"   and s :: "nat"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "l < s * r" shows "l div s \<le> r - Suc 0"
invar: fixes s :: "('c, 'd) RBT.rbt" shows "rm.invar s"
scaleR_right_mono_neg: fixes b :: "real"   and a :: "real"   and c :: "'a" assumes "b \<le> a"   and "c \<le> (0::'a)" shows "a *\<^sub>R c \<le> b *\<^sub>R c"
comp_simp: fixes f :: "(bool, unit) J.arr"   and g :: "(bool, unit) J.arr" assumes "J.seq f g" shows "J.COMP f g = f"
prm_set_def: fixes \<pi> :: "'b prm"   and S :: "'b set" shows "\<pi> {$} S \<equiv> ($) \<pi> ` S"
inject_11: fixes x111 :: "'a exp"   and x112 :: "'a exp"   and y111 :: "'a exp"   and y112 :: "'a exp" shows "(x111;; x112 = y111;; y112) = (x111 = y111 \<and> x112 = y112)"
absolutely_integrable_on_def: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "(f absolutely_integrable_on S) = (f integrable_on S \<and> (\<lambda>x. norm (f x)) integrable_on S)"
SReal_less_omega: fixes r :: "real star" assumes "r \<in> \<real>" shows "r < \<omega>"
simps_3: fixes x :: "quat" shows "Im2 (- x) = - Im2 x"
similar_mat_witD_1: fixes n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "P * Q = 1\<^sub>m n"
measure_measure_spmf_def: fixes p :: "'c spmf" shows "measure_measure_spmf p = Sigma_Algebra.measure (measure_spmf p)"
l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs"
ucast_ucast_len: fixes x :: "'b word" assumes "x < 2 ^ LENGTH('c)" shows "ucast (ucast x) = x"
rechar_take_cols: fixes A :: "'a mat"   and B :: "nat list" shows "take_cols_var A B = take_cols_from_matrix A B"
eq2: fixes i :: "nat" assumes "i < DIM('a)" shows "xs1 ! (d1s ! i) = xs2 ! (d2s ! i)"
Rep_formula_inject: fixes x :: "('b, 'c, 'd) formula"   and y :: "('b, 'c, 'd) formula" shows "(Rep_formula x = Rep_formula y) = (x = y)"
cong_sym_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = unique_euclidean_semiring.cong modulo c b a"
assms_3: shows "Abs_Tangle_Diagram x = Abs_Tangle_Diagram y"
bindE_def: fixes f :: "'c \<Rightarrow> (('d + 'f) \<times> 'c) set \<times> bool"   and g :: "'f \<Rightarrow> 'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool" shows "f >>=E g \<equiv> f >>= lift g"
direct_lim_is_ring: fixes U :: "'a set" assumes "U \<in> I" shows "ring carrier_direct_lim add_rel mult_rel \<lfloor>U, \<zero>\<^bsub>U\<^esub>\<rfloor> \<lfloor>U, \<one>\<^bsub>U\<^esub>\<rfloor>"
concurrent_eff: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message"   and pm :: "'a extended"   and l :: "nat"   and sm :: "'a extended"   and u :: "nat" assumes "integrate_insert_commute_insert M a s m"   and "idx s pm = Inr l"   and "idx s sm = Inr u" obtains d :: "'a extended list list" where "mapM (concurrent s l u) (substr s l u) = Inr d \<and> set (concat d) = InString ` I ` {x. integrate_insert_commute_insert.is_concurrent a s pm sm x}"
apply_cltn2_id: fixes p :: "proj2" shows "apply_cltn2 p cltn2_id = p"