hcju/isabelleps huggingface dataset:
coboundedI1: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c \<le> a" shows "c \<le> a + b"
weighted_RP_model: fixes Sts :: "(('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat) llist"   and St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and I :: "'a set" assumes "full_chain (\<leadsto>\<^sub>w) Sts"   and "wrp.P_of_wstate (lhd Sts) = {}"   and "wrp.Q_of_wstate (lhd Sts) = {}"   and "St \<leadsto>\<^sub>w St'" shows "(I \<Turnstile>s grounding_of_clss (wrp.N_of_wstate St' \<union> wrp.P_of_wstate St' \<union> wrp.Q_of_wstate St')) = (I \<Turnstile>s grounding_of_clss (wrp.N_of_wstate St \<union> wrp.P_of_wstate St \<union> wrp.Q_of_wstate St))"
Dlist_def: fixes xs :: "'d list" shows "Dlist xs = Abs_dlist (remdups xs)"
simps: fixes M :: "('d, 'e, 'f) FSM"   and p :: "'f"   and xs :: "'d list"   and q :: "'f" shows "d_reaches M p xs q = (\<exists>tr ys. d_reached_by M p xs q tr ys)"
case_resource_conv_run_resource: fixes f :: "('h \<Rightarrow> ('i \<times> ('h, 'i) resource) spmf) \<Rightarrow> 'g"   and res :: "('h, 'i) resource" shows "(case res of Resource x \<Rightarrow> f x) = f (run_resource res)"
\<phi>_ide_simp: 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 e\<^sub>0 :: "'b"   and d\<^sub>0 :: "'b"   and \<eta>\<^sub>0 :: "'b"   and \<epsilon>\<^sub>0 :: "'b"   and e\<^sub>1 :: "'b"   and d\<^sub>1 :: "'b"   and \<eta>\<^sub>1 :: "'b"   and \<epsilon>\<^sub>1 :: "'b"   and f :: "'b" assumes "two_equivalences_in_bicategory V H \<a> \<i> src trg e\<^sub>0 d\<^sub>0 \<eta>\<^sub>0 \<epsilon>\<^sub>0 e\<^sub>1 d\<^sub>1 \<eta>\<^sub>1 \<epsilon>\<^sub>1"   and "horizontal_homs.in_hhom V src trg f (src e\<^sub>0) (src e\<^sub>1)"   and "partial_magma.ide V f" shows "two_equivalences_in_bicategory.\<phi> V H \<a> \<i> src trg e\<^sub>0 d\<^sub>0 \<eta>\<^sub>0 e\<^sub>1 d\<^sub>1 \<eta>\<^sub>1 f = V (H d\<^sub>1 (bicategory.\<a>' V H \<a> src trg e\<^sub>1 (H f d\<^sub>0) e\<^sub>0)) (V (\<a> d\<^sub>1 e\<^sub>1 (H (H f d\<^sub>0) e\<^sub>0)) (V (H (H d\<^sub>1 e\<^sub>1) (bicategory.\<a>' V H \<a> src trg f d\<^sub>0 e\<^sub>0)) (V (H \<eta>\<^sub>1 (H f \<eta>\<^sub>0)) (V (category.inv V (bicategory.lunit V H \<a> \<i> src trg (H f (src e\<^sub>0)))) (category.inv V (bicategory.runit V H \<a> \<i> src f))))))"
SOSou_iff_inverse_of_indRelRTPO_is_strong_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "((\<forall>S T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<longrightarrow> (\<exists>S'. S \<longmapsto>Source S' \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)) \<and> strong_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target) = strong_reduction_simulation ((indRelRTPO TRel)\<inverse>) (STCal Source Target)"
basis_in_iff_rank_of: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "X \<subseteq> \<E>" shows "basis_in \<E> X = (rank_of X = card X \<and> rank_of X = rank_of \<E>)"
higher_pderiv_monom: fixes m :: "nat"   and n :: "nat"   and c :: "int" assumes "m \<le> n + 1" shows "(pderiv ^^ m) (Polynomial.monom c n) = Polynomial.monom (pochhammer (int n - int m + 1) m * c) (n - m)"
rep_eq: fixes x :: "'c multiset"   and xa :: "'c multiset" shows "count (x + xa) = (\<lambda>a. count x a + count xa a)"
hyps_3: assumes "x \<in> lists (UNIV \<times> gens)"   and "f \<in> gens \<rightarrow> carrier G" shows "lift f x = lift f y"
hd_ps_p: shows "hd (sturm_squarefree p) = p'"
fset_inject: fixes x :: "'a fset"   and y :: "'a fset" shows "(fset x = fset y) = (x = y)"
abst_inv: fixes z :: "'b \<times> 'c" shows "repr (abst z) = z"
Cons_2: shows "separateAtoms (At # L) = (a, b, c, d)"
module_axioms_def: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme" shows "module_axioms R M \<equiv> ((\<forall>a x. a \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M) \<and> (\<forall>a b x. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> (a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x)) \<and> (\<forall>a x y. a \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> y \<in> carrier M \<longrightarrow> a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y) \<and> (\<forall>a b x. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> x \<in> carrier M \<longrightarrow> a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)) \<and> (\<forall>x. x \<in> carrier M \<longrightarrow> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = x)"
null_sets: shows "null_sets N \<subseteq> null_sets (distr M N f)"
set_reorder: fixes xs :: "'e list"   and y :: "'e"   and ys :: "'e list" shows "set (xs @ y # ys) = set (y # xs @ ys)"
fls_inverse_subdegree_base_nonzero: fixes f :: "'a fls" assumes "f \<noteq> 0"   and "inverse (fls_nth f (fls_subdegree f)) \<noteq> (0::'a)" shows "fls_nth (inverse f) (fls_subdegree (inverse f)) = inverse (fls_nth f (fls_subdegree f))"
cancel_hyp: shows "matr_option (dim_row (M_mat_R signs subsets)) (mat_inverse (M_mat_R signs subsets)) * M_mat_R signs subsets *\<^sub>v w_vec_R p qs signs = w_vec_R p qs signs"
over_def: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b" shows "f over a = f a"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs G" shows "arev a \<noteq> a"
assms_3: shows "\<^bold>|u\<^bold>| + \<^bold>|v\<^bold>| \<le> \<^bold>|w\<^bold>|"
ext_inject: fixes Obj :: "'a set"   and Mor :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c"   and Obj' :: "'a set"   and Mor' :: "'b set"   and Dom' :: "'b \<Rightarrow> 'a"   and Cod' :: "'b \<Rightarrow> 'a"   and Id' :: "'a \<Rightarrow> 'b"   and Comp' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more' :: "'c" shows "(\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = \<lparr>Obj = Obj', Mor = Mor', Dom = Dom', Cod = Cod', Id = Id', Comp = Comp', \<dots> = more'\<rparr>) = (Obj = Obj' \<and> Mor = Mor' \<and> Dom = Dom' \<and> Cod = Cod' \<and> Id = Id' \<and> Comp = Comp' \<and> more = more')"
hcomp_in_hhomE: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<nu> \<star> \<mu> : a \<rightarrow> c\<guillemotright>" obtains "arr \<mu>" and "arr \<nu>" and "src \<nu> = trg \<mu>" and "src \<mu> = a" and "trg \<nu> = c"
Scons_inject_lemma2: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and N' :: "('a, 'b) node set" assumes "Scons M N \<subseteq> Scons M' N'" shows "N \<subseteq> N'"
rel_sum_map2: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'e + 'g"   and f1 :: "'i \<Rightarrow> 'f"   and f2 :: "'j \<Rightarrow> 'h"   and y :: "'i + 'j" shows "rel_sum R1 R2 x (map_sum f1 f2 y) = rel_sum (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y"
om_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<omega> \<le> y\<^sup>\<omega>"
left_commute: fixes b :: "'g"   and a :: "'g"   and c :: "'g" shows "sup b (sup a c) = sup a (sup b c)"
B: shows "B = [Dec (Suc arity) (length fap + 5), Dec (Suc arity) (length fap + 3), Goto (Suc (length fap)), Inc arity, Goto 0]"
mat_gt_imp_mat_ge: fixes sd :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "mat_gt (\<succ>) sd A B" shows "A \<ge>\<^sub>m B"
conj_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(conj_PL p q \<in> Cn A) = (p \<in> Cn A \<and> q \<in> Cn A)"
poly_pos_less: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x<a. 0 < poly p x) = (poly_neg_inf p = 1 \<and> (\<forall>x<a. poly p x \<noteq> 0))"
af_simps_3: fixes \<phi> :: "'a ltln"   and x :: "'a set"   and xs :: "'a set list" shows "af (X\<^sub>n \<phi>) (x # xs) = af \<phi> xs"
nat_pow_mult: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "x [^] n \<otimes> x [^] m = x [^] (n + m)"
Chains_FieldD: fixes M :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a" assumes "M \<in> Chains r"   and "x \<in> M" shows "x \<in> Field r"
bind_return_distr': fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "M \<bind> (\<lambda>x. return N (f x)) = distr M N f"
simps_3: fixes zc' :: "int \<Rightarrow> int"   and xc :: "int"   and yc :: "int"   and zc :: "int"   and more :: "'a" shows "zc_update zc' \<lparr>xc = xc, yc = yc, zc = zc, \<dots> = more\<rparr> = \<lparr>xc = xc, yc = yc, zc = zc' zc, \<dots> = more\<rparr>"
Un_Diff2: fixes B :: "'a set"   and C :: "'a set"   and A :: "'a set" assumes "B \<inter> C = {}" shows "A \<union> B - C = A - C \<union> B"
rtrancl_on_refl: fixes a :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set" assumes "a \<in> F" shows "(a, a) \<in> rtrancl_on F r"
bin_abs_lem: fixes bin :: "int"   and w :: "int"   and b :: "bool" assumes "bin = w BIT b" shows "bin \<noteq> - 1 \<longrightarrow> bin \<noteq> 0 \<longrightarrow> nat \<bar>w\<bar> < nat \<bar>bin\<bar>"
wp1_eq: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1 p"
71997: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> (y \<squnion> top) \<squnion> z) = - (x \<squnion> z)"
valid_regions_distinct: fixes X :: "'b set"   and k :: "'b \<Rightarrow> nat"   and I :: "'b \<Rightarrow> intv"   and r :: "('b \<times> 'b) set"   and I' :: "'b \<Rightarrow> intv"   and r' :: "('b \<times> 'b) set"   and v :: "'b \<Rightarrow> real" assumes "valid_region X k I r"   and "valid_region X k I' r'"   and "v \<in> region X I r"   and "v \<in> region X I' r'" shows "region X I r = region X I' r'"
drest_succ_proj_eq_drest_succ: fixes a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and s :: "('a, 'b) fmap"   and vs :: "'a set" assumes "fst a \<subseteq>\<^sub>f s" shows "state_succ (fmrestrict_set vs s) (action_proj a vs) = fmrestrict_set vs (state_succ s a)"
IdNatTransCompDefCod: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans f" shows "f \<approx>>\<bullet> IdNatTrans (NTCod f)"
Abs_image: shows "Abs_bit1 ` {0..<1 + 2 * int CARD('a)} = UNIV"
select_convs_3: fixes ob :: "'c set"   and ar :: "'b set"   and dom :: "'b \<Rightarrow> 'c"   and cod :: "'b \<Rightarrow> 'c"   and id :: "'c \<Rightarrow> 'b"   and comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'d" shows "category.dom \<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr> = dom"
lookup_eq_zero_in_keys_contradict: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" assumes "lookup f k = (0::'b)" shows "k \<notin> keys f"
iterate_add_to_map_dj_correct: fixes \<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and upd :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'a"   and it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and M :: "'b \<Rightarrow> 'c option" assumes "map_update_dj \<alpha> invar upd"   and "invar m"   and "map_iterator it M"   and "dom M \<inter> dom (\<alpha> m) = {}" shows "\<alpha> (iterate_add_to_map m upd it) = \<alpha> m ++ M \<and> invar (iterate_add_to_map m upd it)"
no_outstanding_write_to_read_only_memory_axioms: shows "no_outstanding_write_to_read_only_memory \<S> ts"
coset_mult_assoc: fixes G :: "('a, 'b) monoid_scheme"   and M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Group.group G"   and "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "M #>\<^bsub>G\<^esub> g #>\<^bsub>G\<^esub> h = M #>\<^bsub>G\<^esub> g \<otimes>\<^bsub>G\<^esub> h"
partition_filter_conv: fixes f :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "partition f xs = (filter f xs, filter (Not \<circ> f) xs)"
IHR: shows "\<forall>q\<in>set_kdt r \<union> set ps - set (nearest_nbors n ps p r). dist (last (nearest_nbors n ps p r)) p \<le> dist q p"
pdec1'_def: fixes e :: "nat" shows "pdec1' e \<equiv> e - triangle (GREATEST y. triangle y \<le> e)"
compose_def: fixes A :: "'d set"   and g :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f" shows "g \<circ> f \<down> A = (\<lambda>x\<in>A. g (f x))"
neutral: fixes \<sigma> :: "'alt \<Rightarrow> 'alt"   and R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R" shows "scf (permute_profile \<sigma> R) = \<sigma> ` scf R"
asymp_equiv_eventually_pos_iff: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "f \<sim>[F] g" shows "\<forall>\<^sub>F x in F. (0 < f x) = (0 < g x)"
R1_1: shows "pmf (sds R1) a = 1 / 2"
freshEnv_swapEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and y :: "'var"   and z :: "'var"   and x :: "'var" assumes "goodEnv rho"   and "freshEnv xs y rho"   and "freshEnv xs z rho" shows "rho &[y \<and> x]_xs &[z \<and> y]_xs = rho &[z \<and> x]_xs"
Rep_vec_impl: fixes x :: "'a vec_impl" shows "Rep_vec_impl x \<in> {(n, v). IArray.length v = n}"
rangew1: shows "sint (SCAST(32 \<rightarrow> 64) w1) \<in> {- (2 ^ 31)..2 ^ 31}"
pmf_le_1: fixes p :: "'b pmf"   and x :: "'b" shows "pmf p x \<le> 1"
tendsto_ennrealI: fixes f :: "'c \<Rightarrow> real"   and x :: "real"   and F :: "'c filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F"
inset: shows "p \<in> grid b ({d} \<union> ds)"
intros_3: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m1 :: "'c"   and m2 :: "'c"   and m3 :: "'c" assumes "network history msg_id"   and "network.hb history m1 m2"   and "network.hb history m2 m3" shows "network.hb history m1 m3"
poly_eval_cong: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and a :: "'c \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'd" assumes "p = q"   and "\<And>x. x \<in> indets q \<Longrightarrow> a x = b x" shows "poly_eval a p = poly_eval b q"
cases: fixes x :: "'b poly" obtains a :: "'b" and p :: "'b poly" where "x = pCons a p"
leq_max_sup': fixes m :: "nat"   and n :: "nat" shows "m \<le> n \<longrightarrow> maximum (Abs_nat_int {m..n}) = n"
eq_Eq_conv: fixes x :: "'b"   and y :: "'b" shows "(comparator_of x y = Eq) = (x = y)"
l_set_attribute_get_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_attribute_get_disconnected_nodes type_wf set_attribute set_attribute_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_attribute type_wf set_attribute set_attribute_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_attribute_get_disconnected_nodes_axioms set_attribute_locs get_disconnected_nodes_locs"
lt_eq_TT_iff: fixes compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" shows "(Ord.lt compare\<cdot>x\<cdot>y = TT) = (compare\<cdot>x\<cdot>y = LT)"
cem: assumes "m \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "e O m = m"
simps_2: fixes ps :: "bool list"   and b :: "bool"   and l :: "trieP"   and r :: "trieP" shows "abs_trieP (NdP ps b (l, r)) = prefix_trie ps (Nd b (abs_trieP l, abs_trieP r))"
sfis_unique: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real" assumes "measure_space M"   and "a \<in> sfis f M"   and "b \<in> sfis f M" shows "a = b"
mult_le_less_imp_less: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c < d"   and "(0::'a) < a"   and "(0::'a) \<le> c" shows "a * c < b * d"
eq_upred_def: fixes x :: "('c, 'b) uexpr"   and y :: "('c, 'b) uexpr" shows "(x =\<^sub>u y) = bop (=) x y"
contour_integral_bound_exists: fixes S :: "complex set"   and g :: "real \<Rightarrow> complex" assumes "open S"   and "valid_path g"   and "path_image g \<subseteq> S" shows "\<exists>L>0. \<forall>f B. f holomorphic_on S \<and> (\<forall>z\<in>S. cmod (f z) \<le> B) \<longrightarrow> cmod (contour_integral g f) \<le> L * B"
sprefI1': fixes u :: "'b list"   and z :: "'b list"   and v :: "'b list" assumes "u \<cdot> z = v"   and "z \<noteq> \<epsilon>" shows "u <p v"
hyperpow_add: fixes r :: "'b star"   and n :: "nat star"   and m :: "nat star" shows "r pow (n + m) = r pow n * r pow m"
prime_elem_dvd_multD: fixes p :: "'a"   and a :: "'a"   and b :: "'a" assumes "prime_elem p"   and "p dvd a * b" shows "p dvd a \<or> p dvd b"
one_plus_x_powr_Taylor2_bigo: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and p :: "real" assumes "(f \<longlongrightarrow> 0) F" shows "(\<lambda>x. (1 + f x) powr p - 1 - p * f x) \<in> O[F](\<lambda>x. (f x)\<^sup>2)"
interior_maximal: fixes T :: "'c set"   and S :: "'c set" assumes "T \<subseteq> S"   and "open T" shows "T \<subseteq> interior S"
ordLess_ctwo_cexp: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "r <o ctwo ^c r"
cond_unit_T: fixes P :: "('a, 'b) uexpr"   and Q :: "('a, 'b) uexpr" shows "P \<triangleleft> true \<triangleright> Q = P"
distinct_244: fixes x151 :: "floatarith"   and x152 :: "nat" shows "x151 ^\<^sub>e x152 \<noteq> Pi"
le_LNil: fixes s :: "'b llist" shows "(s \<le> LNil) = (s = LNil)"
ccSquare_def: fixes S :: "Vars.var set" shows "S\<^sup>2 = S G\<times> S"
prems_2: shows "slice_edges S (upd_cs cs (x # xs)) asx = asx'"
eq: shows "set_mset (\<Sum>b2\<in>#remove1_mset blv \<B>. mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2}) = {bl. bl \<subseteq> blv \<and> card bl = 2}"
eq: shows "carrier (reduced_homology_group (1 + int p) (nsphere (Suc p))) = f ` carrier (reduced_homology_group (int p) (nsphere p))"
l_inv_ex: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
polar_ex1: fixes y :: "real" assumes "0 < y" shows "\<exists>r a. x = r * cos a \<and> y = r * sin a"
cgenideal_self: fixes i :: "'a list" assumes "i \<in> carrier (K [X])" shows "i \<in> PIdl\<^bsub>K [X]\<^esub> i"
nonpos_linorder_cases: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less_eq a zero"   and "less_eq b zero" obtains "less a b"   | "a = b"   | "less b a"
assms_1: shows "\<forall>y\<in>set aaa. y \<le> Max (set a)"
lossless_spmf_def: fixes p :: "'h spmf" shows "lossless_spmf p = (weight_spmf p = 1)"
graphI: fixes x :: "'a"   and F :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "x \<in> F" shows "(x, f x) \<in> graph F f"
ag_pOp_closed: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c"   and y :: "'c" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y \<in> carrier A"
tree_type: fixes t :: "'c bro"   and h :: "nat" assumes "t \<in> T (h + 1)" shows "tree t \<in> B (h + 1) \<union> B h"
distinct_2: fixes freeRes' :: "freeRes"   and pi' :: "pi"   and subject :: "subject"   and name :: "name"   and pi :: "pi" shows "freeRes' \<prec> pi' \<noteq> subject\<guillemotleft>name\<guillemotright> \<prec> pi"
strict_trans1: fixes b :: "'a multiset"   and a :: "'a multiset"   and c :: "'a multiset" assumes "b \<subseteq># a"   and "c \<subset># b" shows "c \<subset># a"
linear_order_on_singleton: fixes x :: "'a" shows "linear_order_on {x} {(x, x)}"
sub1_makes_0s: fixes A :: "'a mat"   and l :: "nat"   and k :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat"   and i :: "nat" assumes "local.sub1 (A $$ (l, l)) k l (r, A) = (r', A')"   and "l < dim_row A"   and "l < dim_col A"   and "l < i"   and "i \<le> k + l"   and "k + l < dim_row A" shows "A' $$ (i, l) = (0::'a)"
ran_is_image: fixes M :: "'d \<Rightarrow> 'c option" shows "ran M = (the \<circ> M) ` dom M"
1: shows "integrable (lebesgue_on {- pi..pi}) (\<lambda>x. f x * (cos (x + x * real n) * cos (t + t * real n)))"
WHILE_Its: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "(WHILE b DO c, s) \<Rightarrow> t"   and "Q t" shows "\<exists>n. wpw b c n Q s"
valid_path_rev_aux_split_2: fixes cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list" assumes "valid_path_rev_aux cs (as @ as')" shows "valid_path_rev_aux (upd_rev_cs cs as') as"
degree_rebase_poly_eq: fixes p :: "'q mod_ring poly" shows "degree (#p) = degree p"
CLF_dual: shows "(dual cl, f) \<in> CLF_set"
neg_equal_iff_equal: fixes a :: "'a"   and b :: "'a" shows "(- a = - b) = (a = b)"
multiset_filter_mono: fixes A :: "'b multiset"   and B :: "'b multiset"   and f :: "'b \<Rightarrow> bool" assumes "A \<subseteq># B" shows "filter_mset f A \<subseteq># filter_mset f B"