hcju/isabelleps huggingface dataset:
image_stretch_interval: fixes m :: "'c \<Rightarrow> real"   and a :: "'c"   and b :: "'c" shows "(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x \<bullet> k)) *\<^sub>R k) ` cbox a b = (if cbox a b = {} then {} else cbox (\<Sum>k\<in>Basis. min (m k * (a \<bullet> k)) (m k * (b \<bullet> k)) *\<^sub>R k) (\<Sum>k\<in>Basis. max (m k * (a \<bullet> k)) (m k * (b \<bullet> k)) *\<^sub>R k))"
normal_distributed_variance: fixes \<sigma> :: "real"   and X :: "'a \<Rightarrow> real"   and \<mu> :: "real" assumes "0 < \<sigma>"   and "distributed M lborel X (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x))" shows "expectation (\<lambda>x. (X x - expectation X)\<^sup>2) = \<sigma>\<^sup>2"
simps_3: fixes \<sigma> :: "'tp" shows "TE_arOf (Wit \<sigma>) = []"
\<Lambda>_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and g :: "'a" shows "\<Lambda> a b c g \<equiv> THE f. \<guillemotleft>f : a \<rightarrow> exp b c\<guillemotright> \<and> g = C (eval b c) (local.prod f b)"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>"   and "identity_arrow_of_spans C \<mu>"   and "identity_arrow_of_spans C \<nu>" shows "two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu>"
converse_rtrancl_induct2: fixes ax :: "'c"   and ay :: "'d"   and bx :: "'c"   and "by" :: "'d"   and r :: "(('c \<times> 'd) \<times> 'c \<times> 'd) set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "((ax, ay), bx, by) \<in> r\<^sup>*"   and "P bx by"   and "\<And>a b aa ba. \<lbrakk>((a, b), aa, ba) \<in> r; ((aa, ba), bx, by) \<in> r\<^sup>*; P aa ba\<rbrakk> \<Longrightarrow> P a b" shows "P ax ay"
adds_ppE: fixes t :: "'a"   and v :: "'t" assumes "t adds\<^sub>p v" obtains u :: "'t" where "v = t \<oplus> u"
of_nat_pull_out_2: fixes m :: "nat"   and n :: "nat" shows "of_nat m * of_nat n = of_nat (m * n)"
gim: shows "g ` (T - K) \<subseteq> rel_frontier U"
divide_less_0_1_iff: 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 less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide one a) zero = less a zero"
ctor_fold_o_map: fixes f :: "('g, 'f) F0 \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "ctor_fold_F f \<circ> map_F g = ctor_fold_F (f \<circ> map_pre_F g id)"
hd_append: fixes xs :: "'a list"   and ys :: "'a list" shows "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
lr_of_tran_sorted_descending: fixes r :: "(of_match_field, of_action list) flow_entry_match list"   and rt :: "32 routing_rule list"   and fw :: "32 simple_rule list"   and ifs :: "char list list" assumes "Inr r = lr_of_tran rt fw ifs" shows "sorted_descending (map ofe_prio r)"
elims_3: fixes x :: "'e \<times> ('a, 'f, 'g, 'h) SLin_state_scheme" assumes "\<not> P10 x" obtains s1 :: "'e" and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (\<not> initialized s2 \<longrightarrow> dstate s2 = \<bottom>)"
TESL_interp_absorb1: fixes \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" assumes "set \<Phi>\<^sub>1 \<subseteq> set \<Phi>\<^sub>2" shows "\<lbrakk>\<lbrakk> \<Phi>\<^sub>1 @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
exponential_distributed_min: fixes l :: "real"   and u :: "real"   and X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real" assumes "0 < l"   and "0 < u"   and "distributed M lborel X (\<lambda>x. ennreal (exponential_density l x))"   and "distributed M lborel Y (\<lambda>x. ennreal (exponential_density u x))"   and "indep_var borel X borel Y" shows "distributed M lborel (\<lambda>x. min (X x) (Y x)) (\<lambda>x. ennreal (exponential_density (l + u) x))"
smash_list_list_def: fixes s1 :: "('a list \<times> 'a list) list"   and s2 :: "('a list \<times> 'a list) list" shows "smash_list_list s1 s2 = concat (map (\<lambda>l1. map (\<lambda>l2. (fst l1 @ fst l2, snd l1 @ snd l2)) s2) s1)"
lappend_eq_LNil_iff: fixes xs :: "'a llist"   and ys :: "'a llist" shows "(xs @\<^sub>l ys = []\<^sub>l) = (xs = []\<^sub>l \<and> ys = []\<^sub>l)"
a_HashI_inv: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and v :: "Syntax.term"   and vP :: "Syntax.term"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> v, Hashed (hash \<lparr>vP\<rparr>) vP, Hash (hash \<lparr>vP\<rparr>) : AuthT \<tau>" obtains "{$$} \<turnstile> v, vP, \<lparr>vP\<rparr> : \<tau>" and "value v" and "value vP"
prime_divisor_exists: obtains q :: "gauss_int" where "prime q" and "prime_elem (gauss_cnj q)" and "0 < ReZ q" and "0 < ImZ q" and "of_nat p = q * gauss_cnj q" and "gauss_int_norm q = p"
group_add_hom_axioms: shows "group_add_hom hom"
x_axis: fixes x :: "complex" assumes "is_real x"   and "0 < Re x"   and "Re x < 1" shows "P (of_complex x)"
l_adopt_node_get_shadow_root_def: fixes adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_adopt_node_get_shadow_root adopt_node_locs get_shadow_root_locs \<equiv> \<forall>parent owner_document document_ptr h h' ptr'. \<forall>w\<in>adopt_node_locs parent owner_document document_ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
decisive7_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x y" shows "decisive swf A Is {j} x y"
cases: fixes a1 :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a2 :: "'a expr list"   and a3 :: "'a OCL_Types.type list" assumes "a1 \<turnstile>\<^sub>L a2 : a3" obtains \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "a1 = \<Gamma>__" and "a2 = []" and "a3 = []"   | \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" and expr :: "'a expr" and \<tau> :: "'a OCL_Types.type" and exprs :: "'a expr list" and \<pi> :: "'a OCL_Types.type list" where "a1 = \<Gamma>" and "a2 = expr # exprs" and "a3 = \<tau> # \<pi>" and "\<Gamma> \<turnstile>\<^sub>E expr : \<tau>" and "\<Gamma> \<turnstile>\<^sub>L exprs : \<pi>"
returnOk_bindE: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool" shows "returnOk x >>=E f = f x"
bij_betw_restrict_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" shows "bij_betw (restrict f A) A B = bij_betw f A B"
confluentpD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "confluentp r"   and "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* x z" shows "\<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>*\<^sup>* z u"
rep_eq: fixes x :: "'b nonzero set" shows "open x = open (Rep_nonzero ` x)"
assoc: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "inf (inf a b) c = inf a (inf b c)"
foundation25: fixes \<tau> :: "'a state \<times> 'a state"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P" shows "\<tau> \<Turnstile> P or Q"
while_bot_def: fixes b :: "'c upred"   and P :: "'c hrel" shows "while\<^sub>\<bottom> b do P od = (\<mu> X \<bullet> P ;; X \<triangleleft> b \<triangleright>\<^sub>r II)"
starset_insert: fixes x :: "'a"   and A :: "'a set" shows "*s* insert x A = insert (star_of x) (*s* A)"
conj_cancel_left: fixes x :: "'b" shows "NOT x AND x = (0::'b)"
b_pr0'_def: shows "b_pr0' = pr0 a b"
constr_def: shows "constr = (\<lambda>D. map (\<lambda>G. (i, \<forall>X\<langle>\<or>\<noteq>: (F @ G)\<rangle>\<^sub>s\<^sub>t)) (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' (map snd (dbproj i D))))"
leading_Oc: fixes list :: "nat list"   and aa :: "nat" shows "\<exists>xs. <rev list @ [aa]> = Oc # xs"
\<omega>_simps_2: shows "src \<rho> = src w"
assms: shows "\<forall>\<^sub>F x in net. f x = g x"
diagonal_to_Smith_i_dvd_jj: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and xs :: "nat list"   and a :: "'c"   and b :: "'b" assumes "is_bezout_ext bezout"   and "i < min (nrows A) (ncols A)"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "to_nat a \<in> set xs"   and "to_nat a = to_nat b"   and "to_nat a \<noteq> i"   and "distinct xs" shows "diagonal_to_Smith_i xs A i bezout $ from_nat i $ from_nat i dvd diagonal_to_Smith_i xs A i bezout $ a $ b"
norm_scaleR: fixes a :: "real"   and x :: "'c" shows "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
non_empty_cycle_root_move_root_same_reachable: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "non_empty_cycle_root r x"   and "point q"   and "q \<le> x\<^sup>\<star> ; r" shows "x\<^sup>\<star> ; r = x\<^sup>\<star> ; q"
qAbs_pick_Abs: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "qAbs xs x (pick X) $= pick (Abs xs x X)"
that: fixes Y :: "'a list set" assumes "Y \<in> UNIV // \<approx>A"   and "Y \<cdot> {[c]} \<subseteq> X"   and "s \<in> Y" shows "thesis"
in_A: shows "set (map OK \<tau>s) \<subseteq> A"
cases: fixes a :: "'a \<Rightarrow> real"   and X :: "'a set"   and I :: "'a \<Rightarrow> intv"   and r :: "('a \<times> 'a) set" assumes "a \<in> region X I r" obtains u :: "'a \<Rightarrow> real" and X\<^sub>0 :: "'a set" where "a = u" and "\<forall>x\<in>X. 0 \<le> u x" and "\<forall>x\<in>X. intv_elem x u (I x)" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}" and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))"
case_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Lt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f2"
mem_ball_leI: fixes x :: "'b"   and y :: "'b"   and e :: "real"   and f :: "real" assumes "x \<in> ball y e"   and "e \<le> f" shows "x \<in> ball y f"
outsideOutside: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set" shows "Outside' Y (Outside' X P) = Outside' (X \<union> Y) P"
gr1_conv_Suc: fixes mr :: "nat" shows "(Suc 0 < mr) = (\<exists>nat. mr = Suc (Suc nat))"
pochhammer_rec': fixes z :: "'b"   and n :: "nat" shows "pochhammer z (Suc n) = (z + of_nat n) * pochhammer z n"
finite_conv_nat_seg_image: fixes A :: "'c set" shows "finite A = (\<exists>n f. A = f ` {i. i < n})"
simps: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and a1 :: "('a, 'c, 'b) configuration"   and a2 :: "'a set"   and a3 :: "('a, 'c, 'b) configuration" shows "asynchronousSystem.qReachable trans sends a1 a2 a3 = ((\<exists>cfg1 Q. a1 = cfg1 \<and> a2 = Q \<and> a3 = cfg1) \<or> (\<exists>cfg1 Q cfg2 msg cfg3. a1 = cfg1 \<and> a2 = Q \<and> a3 = cfg3 \<and> asynchronousSystem.qReachable trans sends cfg1 Q cfg2 \<and> asynchronousSystem.steps trans sends cfg2 msg cfg3 \<and> (\<exists>p\<in>Q. isReceiverOf p msg)))"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_subst': fixes t :: "('a, 'b) Term.term"   and S :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>s\<in>trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t S. \<exists>X. set X \<subseteq> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<and> t = s \<cdot> rm_vars (set X) \<delta>"
linear: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
config_rand_length: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). length (fst x) = length init"
map_le_def: fixes m\<^sub>1 :: "'d \<Rightarrow> 'e option"   and m\<^sub>2 :: "'d \<Rightarrow> 'e option" shows "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (\<forall>a\<in>dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
llist_all2_lfiniteD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "llist_all2 P xs ys" shows "lfinite xs = lfinite ys"
neg_bounds_compare_3: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c >\<^sub>u\<^sub>b b" shows "c \<le>\<^sub>u\<^sub>b b"
im_of_lideal_is_lideal: fixes I :: "'c set"   and A :: "'c set"   and addA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multA :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroA :: "'c"   and oneA :: "'c"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set"   and addB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroB :: "'d"   and oneB :: "'d" assumes "lideal I A addA multA zeroA oneA"   and "ring_epimorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "lideal (f ` I) B addB multB zeroB oneB"
wf_acc_iff: fixes r :: "('g \<times> 'g) set" shows "wf r = (\<forall>x. x \<in> Wellfounded.acc r)"
helper: assumes "bv_to_int w1 + bv_to_int w2 \<noteq> 0" shows "0 < max (length w1) (length w2)"
subgroups_card_coprime_inter_card_one: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "coprime (card H) (card J)" shows "card (H \<inter> J) = 1"
r_cancel_one': fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and a :: "'e" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x = a \<otimes>\<^bsub>G\<^esub> x) = (a = \<one>\<^bsub>G\<^esub>)"
cross_ratio_correct_cltn2: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2"   and C :: "cltn2" assumes "cross_ratio_correct p q r s" shows "cross_ratio_correct (apply_cltn2 p C) (apply_cltn2 q C) (apply_cltn2 r C) (apply_cltn2 s C)"
product_in_first_factor: fixes x :: "i"   and y :: "i" assumes "CM"   and "O x y" shows "P (x \<otimes> y) x"
swap_swap: fixes p :: "'e \<times> 'f" shows "prod.swap (prod.swap p) = p"
beukers_nn_integral3_finite: shows "local.beukers_nn_integral3 < \<infinity>"
\<omega>_sum_Nil: shows "\<omega>_sum [] = 0"
valid_two_chain_def: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "valid_two_chain twoChain \<equiv> (\<forall>twoCube\<in>twoChain. valid_two_cube twoCube) \<and> pairwise (\<lambda>c1 c2. boundary c1 \<inter> boundary c2 = {}) twoChain \<and> inj_on cubeImage twoChain"
of_int_Real: fixes x :: "int" shows "real_of_int x = Real.Real (\<lambda>n. rat_of_int x)"
A'_1: shows "q' \<rightarrow> f qs' \<in> \<delta>"
pcp_seq_mono: fixes n :: "nat"   and m :: "nat"   and C :: "'b set set"   and S :: "'b set" assumes "(n::nat) \<le> (m::nat)" shows "pcp_seq (C::'b set set) (S::'b set) (n::nat) \<subseteq> pcp_seq C S (m::nat)"
dim_row_of_cjn_prod: fixes M :: "complex mat"   and N :: "complex mat" shows "dim_row (M\<^sup>\<star> * N\<^sup>\<star>) = dim_row M"
flatten_sigma_2_intersection_eq: fixes M :: "'b ltln set"   and \<phi> :: "'b ltln"   and M' :: "'b ltln set" assumes "M \<inter> subformulas\<^sub>\<mu> \<phi> = M'" shows "\<phi>[M']\<^sub>\<Sigma>\<^sub>2 = \<phi>[M]\<^sub>\<Sigma>\<^sub>2"
sum_order_add_mset: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c multiset"   and a :: "'c" assumes "\<And>x. (0::'a) \<le> f x" shows "\<Sum>\<^sub># (image_mset f A) \<le> \<Sum>\<^sub># (image_mset f (add_mset a A))"
sngl_if_invar: fixes l :: "('c \<times> nat) tree"   and a :: "'c"   and n :: "nat"   and r :: "('c \<times> nat) tree" assumes "invar \<langle>l, (a, n), r\<rangle>"   and "n = lvl r" shows "sngl r"
emp_act_var: fixes p :: "'b" shows "\<alpha> \<bottom> p = \<bottom>"
component_mult_def: fixes v :: "'b vec"   and w :: "'b vec" shows "component_mult v w = vec (min (dim_vec v) (dim_vec w)) (\<lambda>i. v $ i * w $ i)"
comm_semiring_1_cancel_def: 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" shows "class.comm_semiring_1_cancel plus minus zero times one \<equiv> (class.cancel_comm_monoid_add plus minus zero \<and> class.comm_monoid_mult times one) \<and> class.comm_semiring plus times \<and> class.zero_neq_one one zero \<and> class.comm_semiring_1_cancel_axioms minus times"
equivp_reflp_symp_transp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "equivp R = (reflp R \<and> symp R \<and> transp R)"
fun_upd_upd: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e"   and z :: "'e" shows "f(x := y, x := z) = f(x := z)"
vimage_restr_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T" shows "qmpt.vimage_restr M f A \<equiv> f -` (A \<inter> space M) \<inter> space M"
c_manifold_axioms: shows "c_manifold (charts_submanifold S) k"
smash_list_list_property_set: fixes l1 :: "('a list \<times> 'a list) list"   and l2 :: "('a list \<times> 'a list) list" shows "\<forall>elem. elem \<in> set (smash_list_list l1 l2) \<longrightarrow> (\<exists>elem1 elem2. elem1 \<in> set l1 \<and> elem2 \<in> set l2 \<and> elem = (fst elem1 @ fst elem2, snd elem1 @ snd elem2))"
ok_JN_iff1: fixes F :: "'c program"   and I :: "'d set"   and G :: "'d \<Rightarrow> 'c program" shows "F ok JOIN I G = (\<forall>i\<in>I. F ok G i)"
uminus_finite_3_def: fixes x :: "finite_3" shows "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>1 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2)"
assms_8: shows "isUIP (opposite (getCl state)) (getC state) (getM state)"
surjective: fixes r :: "('c, 'd, 'e) ipurge_rec_scheme" shows "r = \<lparr>Pol = Pol r, Map = Map r, Dom = Dom r, In = In r, Out = Out r, \<dots> = more r\<rparr>"
has_sgnx_unique: fixes F :: "real filter"   and f :: "real \<Rightarrow> real"   and c1 :: "real"   and c2 :: "real" assumes "F \<noteq> bot"   and "(f has_sgnx c1) F"   and "(f has_sgnx c2) F" shows "c1 = c2"
lookup_map_default': fixes x :: "'c"   and d :: "'b"   and f :: "'b \<Rightarrow> 'b"   and m :: "('c, 'b) mapping"   and y :: "'c" shows "Mapping.lookup (Mapping.map_default x d f m) y = (if x = y then Some (f (Mapping.lookup_default d m x)) else Mapping.lookup m y)"
vars_iff_subterm_or_eq: fixes x :: "'c"   and t :: "('d, 'c) Term.term" shows "(x \<in> fv t) = (Var x \<sqsubset> t \<or> Var x = t)"
ext_induct: fixes P :: "'b alpha_d_scheme \<Rightarrow> bool"   and r :: "'b alpha_d_scheme" assumes "\<And>ok more. P \<lparr>ok = ok, \<dots> = more\<rparr>" shows "P r"
IH: assumes "evaluate_match ck env s v0 pes err_v (s', bv)" shows "case bv of Rval va \<Rightarrow> \<exists>e env'. match_result env s v0 pes err_v = Rval (e, env') \<and> evaluate ck (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s e (s', Rval va) | Rerr err \<Rightarrow> match_result env s v0 pes err_v = Rerr err \<or> (\<exists>e env'. match_result env s v0 pes err_v = Rval (e, env') \<and> evaluate ck (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s e (s', Rerr err))"
lappend_ltake_enat_ldropn: fixes n :: "nat"   and xs :: "'a llist" shows "lappend (ltake (enat n) xs) (ldropn n xs) = xs"
vec_nth_cases: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> UNIV" obtains x :: "('d, 'c) vec" where "y = ($) x"
rename_interpretation_def: fixes Sel :: "'at set"   and I :: "'at set" shows "rename_interpretation Sel I = {A \<in> I. A \<notin> Sel} \<union> {A. A \<notin> I \<and> A \<in> Sel}"
supset_circuit_imp_dep: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and C :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep"   and "indep_system.circuit carrier indep C \<and> C \<subseteq> X" shows "\<not> indep X"
ceiling_less_iff: fixes x :: "'d"   and z :: "int" shows "(\<lceil>x\<rceil> < z) = (x \<le> of_int z - (1::'d))"
Cx_le_step: fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "C * x \<le> bs ! i * x + (hs ! i) x"
Entry_target: fixes a :: "'f" assumes "valid_edge a"   and "targetnode a = Entry" shows "False"
matrix_left_invertible_ker: fixes A :: "(('c, 'e) vec, 'd) vec" shows "(\<exists>B. B ** A = mat (1::'c)) = (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
elims: fixes x :: "'a \<times> 'd \<Rightarrow> 'a \<times> 'b"   and xa :: "('a \<times> 'd) list \<times> 'c"   and y :: "('a \<times> 'b) list \<times> 'c" assumes "map_raw x xa = y" obtains f :: "'a \<times> 'd \<Rightarrow> 'a \<times> 'b" and xs :: "('a \<times> 'd) list" and ko :: "'c" where "x = f" and "xa = (xs, ko)" and "y = (map_pair f xs, ko)"
resNil: fixes x :: "name" shows "\<lparr>\<nu>x\<rparr>\<zero> \<sim> \<zero>"
typeI_and_typII_one_chains_have_common_subdiv: shows "common_boundary_sudivision_exists one_chain_typeI one_chain_typeII"
and_int_rec: shows "x BIT b AND y BIT c = of_bool (odd (x BIT b) \<and> odd (y BIT c)) + 2 * (x BIT b div 2 AND y BIT c div 2)"
drink_costs_money_aux: fixes j :: "state stream"   and drink :: "char list" assumes "\<exists>p r t. j = make_full_observation drinks (Some 1) r p t" shows "alw (\<lambda>xs. output_eq [Some (EFSM.Str drink)] (stl xs) \<longrightarrow> check_exp (Ge (V (Rg 2)) (L (Num 100))) xs) j"
0: shows "x \<notin> set_tree lb \<and> x \<notin> set_tree la"
is_chainD': fixes f :: "nat \<Rightarrow> 'b set"   and x :: "'b"   and m :: "nat"   and k :: "nat" assumes "is_chain f"   and "x \<in> f m"   and "m \<le> k" shows "x \<in> f k"
pos_neg_kauff_mat: shows "kauff_mat (basic [over] \<circ> basic [brick.under]) = kauff_mat (basic [vert, vert] \<circ> basic [vert, vert])"
repl_invariance: fixes v :: "'a"   and t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'c) Term.term"   and u :: "('b, 'c) Term.term" assumes "v \<notin> fv t" shows "t \<cdot> s(v := u) = t \<cdot> s"
replace6: fixes oldfs :: "'c list"   and x :: "'c"   and oldF :: "'c"   and newfs :: "'c list" assumes "distinct oldfs" shows "(x \<in> set (replace oldF newfs oldfs)) = ((x \<noteq> oldF \<or> oldF \<in> set newfs) \<and> (oldF \<in> set oldfs \<and> x \<in> set newfs \<or> x \<in> set oldfs))"
coinduct: fixes X :: "('c, 'd) resource \<Rightarrow> bool"   and x :: "('c, 'd) resource"   and \<I> :: "('c, 'd) \<I>" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>res. x = res \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (run_resource res x) \<longrightarrow> xa \<in> responses_\<I> \<I> x \<and> (X xb \<or> \<I> \<turnstile>res xb \<surd>))" shows "\<I> \<turnstile>res x \<surd>"