hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and a2 :: "('e \<Rightarrow> real) set set"   and a3 :: "'d"   and a4 :: "('e \<Rightarrow> real) set"   and a5 :: "'d"   and a6 :: "('e \<Rightarrow> real) set" assumes "a1,a2 \<turnstile> \<langle>a3, a4\<rangle> \<leadsto> \<langle>a5, a6\<rangle>" obtains \<R> :: "('e \<Rightarrow> real) set set" and X :: "'e set" and k :: "'e \<Rightarrow> nat" and A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)" and R :: "('e \<Rightarrow> real) set" and R' :: "('e \<Rightarrow> real) set" and l :: "'d" where "a1 = Ab__" and "a2 = \<R>'__" and "a3 = lb__" and "a4 = R__" and "a5 = lb__" and "a6 = R'" and "\<R>'__ = {region Xa__ I r |I r. valid_region Xa__ ka__ I r}" and "valid_abstraction Ab__ Xa__ (\<lambda>x. real (ka__ x))" and "R__ \<in> \<R>'__" and "R' \<in> Succ \<R>'__ R__" and "R__ \<subseteq> \<lbrace>inv_of Ab__ lb__\<rbrace>" and "R' \<subseteq> \<lbrace>inv_of Ab__ lb__\<rbrace>"   | \<R> :: "('e \<Rightarrow> real) set set" and X :: "'e set" and k :: "'e \<Rightarrow> nat" and A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)" and l :: "'d" and g :: "('e, real) cconstraint" and a :: "'f" and r :: "'e list" and l' :: "'d" and R :: "('e \<Rightarrow> real) set" where "a1 = A" and "a2 = \<R>" and "a3 = l" and "a4 = R" and "a5 = l'" and "a6 = region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'}" and "\<R> = {region X I r |I r. valid_region X k I r}" and "valid_abstraction A X (\<lambda>x. real (k x))" and "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" and "R \<in> \<R>"
div_poly_def: fixes a :: "'a"   and p :: "'a poly" shows "div_poly a p = map_poly (\<lambda>c. c div a) p"
UPD_STATE_IDX_def: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int"   and x :: "char list"   and i :: "int"   and v :: "int" shows "UPD_STATE_IDX s x i v \<equiv> s(x := UPD_IDX (s x) i v)"
hom_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "g x \<in> carrier D"
component_of_trans: fixes F :: "'b program"   and G :: "'b program"   and H :: "'b program" assumes "F component_of G"   and "G component_of H" shows "F component_of H"
i1_length: shows "i - 1 < length (sorted_list_of_set I)"
Parallel_Strong_Soundness_aux: fixes Ts' :: "'b com"   and s :: "'b"   and Rs' :: "'b com"   and t :: "'b"   and Ts :: "('b ann_com option \<times> 'b set) list"   and Rs :: "('b ann_com option \<times> 'b set) list" assumes "(Ts', s) -P*\<rightarrow> (Rs', t)"   and "Ts' = Parallel Ts"   and "interfree Ts"   and "\<And>i. i < length Ts \<Longrightarrow> \<exists>c q. Ts ! i = (Some c, q) \<and> s \<in> pre c \<and> \<turnstile> c  q"   and "Rs' = Parallel Rs" shows "(\<forall>j<length Rs. if OG_Hoare.com (Rs ! j) = None then t \<in> OG_Hoare.post (Ts ! j) else t \<in> pre (the (OG_Hoare.com (Rs ! j))) \<and> \<turnstile> the (OG_Hoare.com (Rs ! j))  OG_Hoare.post (Ts ! j)) \<and> interfree Rs"
assms_2: fixes s' :: "'state" assumes "(c, s) \<rightarrow>t s'" shows "phi"
walk_edges_vertices_2: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "(v, w) \<in> edges_of_walk xs" shows "w \<in> set xs"
inv_locate: fixes as :: "nat"   and am :: "nat list"   and q :: "nat"   and aaa :: "cell list"   and xs :: "cell list"   and ires :: "cell list" assumes "inv_locate_b (as, am) (q, aaa, Bk # xs) ires"   and "\<exists>n. xs = Bk \<up> n" shows "inv_locate_a (as, am) (Suc q, Bk # aaa, xs) ires"
filterlim_at_top_nhds: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" assumes "filterlim f at_top F"   and "(f \<longlongrightarrow> c) F"   and "F \<noteq> bot" shows "False"
in_face_cycle_sets: fixes a :: "'b" assumes "a \<in> arcs G" shows "face_cycle_set a \<in> face_cycle_sets"
\<R>\<^sub>V_inv: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and v' :: "('a, 'b, 'c) val"   and v :: "('a, 'b, 'c) val" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<R>\<^sub>V (inv \<alpha>) (inv \<beta>) v' = v) = (\<R>\<^sub>V \<alpha> \<beta> v = v')"
\<beta>'_def: shows "\<beta>' = (\<lambda>x. if x \<in> X then \<beta> x else 0)"
least_unique: fixes x :: "'e"   and A :: "'e set"   and y :: "'e" assumes "least L x A"   and "least L y A" shows "x = y"
add_comm: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set" shows "x \<union> y = y \<union> x"
tObsC_def: shows "tObsC \<equiv> tMap (\<lambda>s. (envObsC (es s), pubActs s))"
butlast_drop: fixes n :: "nat"   and xs :: "'b list" shows "butlast (drop n xs) = drop n (butlast xs)"
shift1_invD_1: fixes d :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "shift1_inv d (qs, B)" shows "B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}"
AE_conj_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and M :: "'b measure" shows "(AE x in M. P x \<and> Q x) = (almost_everywhere M P \<and> almost_everywhere M Q)"
is_aligned_no_wrap''': fixes ptr :: "'b word"   and sz :: "nat"   and off :: "nat" assumes "is_aligned ptr sz"   and "sz < LENGTH('b)"   and "off < 2 ^ sz" shows "unat ptr + off < 2 ^ LENGTH('b)"
assign_Some1: fixes s :: "state"   and f :: "'a \<Rightarrow> state \<Rightarrow> state"   and v :: "'a" assumes "\<not> normal s" shows "assign f v s = s"
seq: fixes P :: "'b set"   and c1 :: "'b com"   and Q :: "'b set"   and c2 :: "'b com"   and R :: "'b set" assumes "\<turnstile> P c1 Q"   and "\<turnstile> Q c2 R" shows "\<turnstile> P c1; c2 R"
f_expand_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'d message_af list" shows "xs \<odot> k \<up> (n * k) = xs \<up> n \<odot> k"
exhaust: fixes y :: "(_) object_ptr" obtains x :: "'g" where "y = object_ptr.Ext x"
induced_modulus_dvd: fixes d :: "nat" assumes "principal.induced_modulus d" shows "d dvd n"
ord_spmf_None: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b spmf" shows "ord_spmf ord (return_pmf None) x"
CSP_Sup: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "is_CSP_process (\<lambda>A. A \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` A})"
psi_eq_nth_prefixes: fixes x :: "nat"   and j :: "nat"   and t :: "nat" assumes "x < e_length (local.prefixes j t)" shows "\<psi> j x \<down>= e_nth (local.prefixes j t) x"
dbm_regions: fixes S :: "('a \<Rightarrow> real) set"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "S = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n"   and "normalized M"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}"   and "[M]\<^bsub>v,n\<^esub> \<subseteq> V" shows "\<exists>U\<subseteq>\<R>. S = \<Union> U"
negs_As0'_subset_DA0': shows "negs (mset As0') \<subseteq># DA0'"
E_wf_2: shows "snd ` E \<subseteq> V"
dom_antidom: fixes x :: "'a" shows "d (a x) = a x"
b: shows "inj_on fb (Rep_zet b) \<and> fb ` Rep_zet b = explode zb"
rep_eq: fixes x :: "('d, 'c) fmap" shows "fset (fmran x) = ran (fmlookup x)"
simps_2: fixes v :: "'v"   and u :: "'v"   and us :: "'v list" shows "adj_path v (u # us) = (adjacent v u \<and> adj_path u us)"
rel_interior_eq: fixes s :: "'b set" shows "(rel_interior s = s) = openin (top_of_set (affine hull s)) s"
invar2_init: shows "prim_invar2 initQ init\<pi>"
Trans_msg: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "isTrans ev" shows "msgs c i = msgs c' i"
not_is_app_imp_strip_com_imp_unchanged: fixes t :: "Core.term"   and ts :: "Core.term list" assumes "\<not> is_app t" shows "strip_comb_imp (t, ts) = (t, ts)"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and s :: "char list \<Rightarrow> int" shows "postQ (C\<^sub>1;; C\<^sub>2) s = postQ C\<^sub>2 (postQ C\<^sub>1 s)"
results_gpv_stop_SomeD: fixes x :: "'h"   and \<I> :: "('i, 'j) \<I>"   and gpv :: "('h, 'i, 'j) gpv" assumes "Some x \<in> results_gpv (stop_\<I> \<I>) (gpv_stop gpv)" shows "x \<in> results_gpv \<I> gpv"
fmap_maybe_simps_1: fixes f :: "'d \<rightarrow> 'c" shows "fmap\<cdot>f\<cdot>\<bottom> = \<bottom>"
condition1_part2: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k a (Suc k) A"   and "a < b"   and "(if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) = nrows A"   and "\<forall>m>mod_type_class.from_nat (nrows A). A $ m $ mod_type_class.from_nat k = (0::'d)" shows "is_zero_row_upt_k b (Suc k) A"
T_Tinv_of_set: fixes A :: "'a set" assumes "I.invertible_qmpt"   and "A \<in> events" shows "Tinv -` (I.Tinv -` A \<inter> space M) \<inter> space M = A"
alpha_residual_eqvt: fixes r1 :: "'b \<times> 'c"   and r2 :: "'b \<times> 'c"   and p :: "perm" assumes "alpha_residual r1 r2" shows "alpha_residual (p \<bullet> r1) (p \<bullet> r2)"
match_subst_correctness0: fixes t :: "term"   and u :: "'b" assumes "linear t" shows "case match t u of None \<Rightarrow> \<forall>env. subst (convert_term t) env \<noteq> u | Some env \<Rightarrow> subst (convert_term t) env = u"
setdist_eq_0_closedin: fixes U :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "closedin (top_of_set U) S"   and "x \<in> U" shows "(setdist {x} S = 0) = (S = {} \<or> x \<in> S)"
AB: shows "((\<V>, A), \<V>\<^sub>B, B) \<in> polys_rel_full"
map_eq_Cons_conv: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and y :: "'g"   and ys :: "'g list" shows "(map f xs = y # ys) = (\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys)"
deg: shows "\<forall>d\<in>outcomes. b \<succeq>[\<R>] return_pmf d"
has_limits_preserved_by_isomorphism: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and J' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "J'J.A.has_limits_of_shape J"   and "isomorphic_categories J J'" shows "J'J.A.has_limits_of_shape J'"
wt_lbv_wt_step: fixes cert :: "(ty list \<times> ty err list) option err list" assumes "wt_lbv P C b Ts T\<^sub>r mxs mxl\<^sub>0 xt cert is" shows "\<exists>\<tau>s\<in>list (length is) A. wt_step r Err step \<tau>s \<and> OK first \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! 0"
diff_add_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a - b + c = a + c - b"
fin_dim_span: fixes A :: "'a vec set" assumes "finite A"   and "A \<subseteq> carrier V" shows "vectorspace.fin_dim class_ring (vs (local.span A))"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'a list" assumes "n < |xs|" shows "hd (drop n xs) = xs ! n"
finite_representation: fixes basis :: "'d set"   and v :: "'d" shows "finite {b. crepresentation basis v b \<noteq> 0}"
two_comparisons_into_case_order_4: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and Q :: "'b"   and P :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if le_of_comp comp x y then if lt_of_comp comp x y then Q else P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
rtrancl_refl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" shows "rtranclpR r a a"
sks_prob: fixes p :: "PROB" shows "p = prob (sKs p)"
lookup_unit_vec: fixes j :: "nat"   and n :: "nat"   and i :: "nat" assumes "j < n" shows "lookup (unit_vec n i) [j] = (if i = j then 1::'b else (0::'b))"
simps_5: fixes r1 :: "'b rexp"   and r2 :: "'b rexp"   and c :: "'b"   and v1 :: "'b val"   and v2 :: "'b val" shows "injval (Times r1 r2) c (val.Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P (NOT p); P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q; P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. P (Dvd i a)"   and "\<And>i a. P (NDvd i a)"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (And p q))"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Or p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Imp p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q); P (NOT p); P q\<rbrakk> \<Longrightarrow> P (NOT (Iff p q))"   and "\<And>p. P p \<Longrightarrow> P (NOT (NOT p))"   and "P (NOT T)"   and "P (NOT F)"   and "\<And>a. P (Ge a) \<Longrightarrow> P (NOT (Lt a))"   and "\<And>a. P (Gt a) \<Longrightarrow> P (NOT (Le a))"   and "\<And>a. P (Le a) \<Longrightarrow> P (NOT (Gt a))"   and "\<And>a. P (Lt a) \<Longrightarrow> P (NOT (Ge a))"   and "\<And>a. P (NEq a) \<Longrightarrow> P (NOT (Eq a))"   and "\<And>a. P (Eq a) \<Longrightarrow> P (NOT (NEq a))"   and "\<And>i a. P (NDvd i a) \<Longrightarrow> P (NOT (Dvd i a))"   and "\<And>i a. P (Dvd i a) \<Longrightarrow> P (NOT (NDvd i a))"   and "P T"   and "P F"   and "\<And>va. P (NOT (E va))"   and "\<And>va. P (NOT (A va))"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
x': fixes s :: "real" assumes "s \<in> {t..u}" shows "(x has_vector_derivative f s (x s)) (at s within {t..u})"
product_topology_def: fixes T :: "'d \<Rightarrow> 'e topology"   and I :: "'d set" shows "product_topology T I = topology_generated_by {Pi\<^sub>E I X |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
induct: fixes P1 :: "('b, 'c, 'a) hp \<Rightarrow> bool"   and P2 :: "('b, 'c, 'a) formula \<Rightarrow> bool"   and formula :: "('b, 'c, 'a) formula" assumes "\<And>x. P1 ($\<alpha> x)"   and "\<And>x1 x2. P1 (x1 := x2)"   and "\<And>x1 x2. P1 (DiffAssign x1 x2)"   and "\<And>x. P2 x \<Longrightarrow> P1 (? x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (EvolveODE x1 x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<union>\<union> x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 ;; x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 x**"   and "\<And>x1 x2. P2 (Geq x1 x2)"   and "\<And>x1 x2. P2 ($\<phi> x1 x2)"   and "\<And>x. P2 x \<Longrightarrow> P2 (! x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 && x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Exists x1 x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (\<langle> x1 \<rangle> x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (InContext x1 x2)" shows "P2 formula"
open_except_def: fixes x :: "'c"   and y :: "'c" shows "open_except x y = (if x = y then {} else SOME A. open A \<and> x \<in> A \<and> y \<notin> A)"
Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and A :: "'e set"   and Q :: "'e set" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnExpr A) Throw Q, A"
YFtorMor: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "YFtor C ## f = YFtorNT C f"
antisym_conv3: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and y :: "'g"   and x :: "'g" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
msed_map_invR: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset"   and b :: "'a"   and N :: "'a multiset" assumes "image_mset f M = add_mset b N" shows "\<exists>M1 a. M = add_mset a M1 \<and> f a = b \<and> image_mset f M1 = N"
mat_mult_scalar: fixes A :: "(('e, 'h) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec"   and B :: "(('e, 'f) Finite_Cartesian_Product.vec, 'h) Finite_Cartesian_Product.vec" shows "A ** B = HMA_Connect.mk_mat (\<lambda>i j. scalar_product (row i A) (column j B))"
sparse_row_vector_abs: fixes v :: "(nat \<times> 'a) list" assumes "sorted_spvec v" shows "sparse_row_vector (abs_spvec v) = \<bar>sparse_row_vector v\<bar>"
proper_subterm_red_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.proper_subterm_red trm_ord sel t S \<sigma> = (\<exists>p s. subterm t p s \<and> p \<noteq> [] \<and> basic_superposition.trm_rep trm_ord sel (s \<lhd> \<sigma>) S \<noteq> s \<lhd> \<sigma>)"
fresh_lookup: fixes X :: "W.tvar"   and Y :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "X \<sharp> Y"   and "X \<sharp> \<theta>" shows "X \<sharp> lookup \<theta> Y"
compatible_sym: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.cpo below"   and "cpo.compatible below x y" shows "cpo.compatible below y x"
interval_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" shows "interval r a b = {x. (a, x) \<in> r \<and> (x, b) \<in> r}"
boundedI: fixes b :: "'e"   and a :: "'e"   and c :: "'e" assumes "b \<le> a"   and "c \<le> a" shows "sup b c \<le> a"
complex_add_divide_simps_5: fixes v :: "'a"   and b :: "complex"   and z :: "complex"   and w :: "'a" shows "v - (b / z) *\<^sub>C w = (if z = 0 then v else (z *\<^sub>C v - b *\<^sub>C w) /\<^sub>C z)"
PO_refines_implies_Range_init: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme" assumes "PO_refines R Ta Tc" shows "init Tc \<subseteq> Range R"
summable_iff_suminf_neq_top: fixes f :: "nat \<Rightarrow> real" assumes "\<And>n. 0 \<le> f n"   and "\<not> summable f" shows "(\<Sum>i. ennreal (f i)) = \<top>"
inf_ifte_distrib: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "ifte x a b \<sqinter> ifte x c d = ifte x (a \<sqinter> c) (b \<sqinter> d)"
least: fixes s :: "real" assumes "0 < s"   and "s < t" shows "flow0 x s \<notin> P"
assms_1: shows "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"
well\<^sub>h: shows "well_base\<^sub>h (i + 3) (encode (i + 2) 0 (local.goodstein i))"
One_leq_div: fixes a :: "nat"   and b :: "nat" assumes "a dvd b"   and "a < b" shows "1 < b div a"
preserves_reflects_arr: fixes f :: "'b" shows "arr (\<rr> f) = arr f"
payload_inductive_cases_6: fixes X :: "msg"   and Y :: "msg" assumes "Aenc X Y \<in> payload" obtains "X \<in> payload" and "Y \<in> payload"
intros_4: fixes xs :: "'c option list"   and X :: "'c option set"   and P :: "'c option process"   and Q :: "'c option process"   and Y :: "'c option set" assumes "(xs, X) \<in> seq_comp_failures P Q"   and "(xs, Y) \<in> seq_comp_failures P Q" shows "(xs, X \<union> Y) \<in> seq_comp_failures P Q"
lcoset_eq_reps_subset: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "order.greater_eq (a +o B) (a +o A)" shows "order.greater_eq B A"
gauss_jordan_inverse_one_direction_1: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat"   and nc :: "nat"   and B' :: "'c mat"   and b :: "'d" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n nc"   and "gauss_jordan A B = (1\<^sub>m n, B')" shows "A \<in> Units (ring_mat TYPE('c) n b)"
semiring_0_cancel_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "semiring_0_cancel_ow U plus minus zero times \<equiv> semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero"
S_reduced_imp_distinct_lconjseq': fixes S :: "'b set"   and ss :: "'b list" assumes "PreCoxeterSystem S"   and "ss \<in> lists S"   and "\<not> distinct (lconjseq ss)" shows "\<not> reduced_word S ss"
map_bind_spmf: fixes f :: "'s \<Rightarrow> 'r"   and p :: "'t spmf"   and g :: "'t \<Rightarrow> 's spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
fun_type_inv_1: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and t :: "('a, 'b) Term.term"   and f :: "'a"   and T :: "('a, 'c) Term.term list" assumes "typed_model arity public Ana \<Gamma>"   and "\<Gamma> t = Fun f T" shows "0 < arity f"
ab_group_add_ow_Ball_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and mns :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and um :: "'b \<Rightarrow> 'b" shows "ab_group_add_ow S pls z mns um = (comm_monoid_add_ow S pls z \<and> (\<forall>a\<in>S. pls (um a) a = z) \<and> (\<forall>a\<in>S. \<forall>b\<in>S. mns a b = pls a (um b)) \<and> (\<forall>a\<in>S. um a \<in> S))"
nonbossy_Ch: fixes ds :: "'b set"   and \<phi> :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('c \<Rightarrow> 'a set \<Rightarrow> 'a set) \<Rightarrow> 'b set \<Rightarrow> 'a set"   and Pd' :: "'b \<Rightarrow> ('a \<times> 'a) set"   and Ch' :: "'c \<Rightarrow> 'a set \<Rightarrow> 'a set"   and h :: "'c"   and Ch'' :: "'a set \<Rightarrow> 'a set" assumes "Bossy''.nonbossy ds \<phi>"   and "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd' Ch'"   and "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd' (Ch'(h := Ch''))"   and "Bossy''.hX (\<phi> Pd' Ch' ds) h = Bossy''.hX (\<phi> Pd' (Ch'(h := Ch'')) ds) h" shows "\<phi> Pd' Ch' ds = \<phi> Pd' (Ch'(h := Ch'')) ds"
assms_2: fixes k :: "nat" assumes "f n < k \<and> k \<le> m" shows "\<nexists>k\<^sub>0. f k\<^sub>0 = k"
set_tries_eq_ran: fixes t :: "('b, 'a list) trie" shows "set_tries t = \<Union> (set ` ran (lookup_trie t))"
subspace_0: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale"   and "module.subspace scale S" shows "(0::'b) \<in> S"
monad_resumption_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and pause :: "'d \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_resumption return bind pause" shows "monad_resumption (return_state return) (bind_state bind) (pause_state pause)"
active_threads_correct_2: fixes s :: "('l, 't) locks \<times> ('m_t \<times> 'm) \<times> 'm_w \<times> 's_i" assumes "state_invar s" shows "thr'_invar (active_threads s)"
between_chain: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "ch {a, b, c}"
hom_char: fixes a :: "'a"   and b :: "'a" shows "Right.hom a b = hom a b \<inter> Collect (right (src f))"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', M\<langle>N\<rangle>.P, M\<langle>N\<rangle>.Q) \<in> Rel"
newton_sqrt_aux_simps_1: fixes x :: "nat"   and n :: "nat" assumes "(x + n div x) div 2 < x" shows "newton_sqrt_aux x n = newton_sqrt_aux ((x + n div x) div 2) n"
DBSpec_def: shows "DBSpec \<equiv> ISpec inp q1 mid \<and> ISpec mid q2 out \<and> \<box>[\<not> (Enq inp q1 mid \<and> Deq mid q2 out)]_vars"
Card_order_csum: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "Card_order (r1 +c r2)"