hcju/isabelleps huggingface dataset:
is_nextElem_congs_eq: fixes ls :: "'a list"   and ms :: "'a list"   and x :: "'a"   and y :: "'a" assumes "ls \<cong> ms" shows "is_nextElem ls x y = is_nextElem ms x y"
cinsertCI: fixes a :: "'d"   and B :: "'d cset"   and b :: "'d" assumes "\<not> cin a B \<Longrightarrow> a = b" shows "cin a (cinsert b B)"
simps_2: fixes b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option"   and xs :: "'h list" shows "lookup (Env b es) xs = (case xs of [] \<Rightarrow> Some (Env b es) | y # ys \<Rightarrow> ??.Nested_Environment.lookup_option (es y) ys)"
DESTRimpl_rule: fixes s :: "'a"   and ni :: "'b"   and n :: "'c ifex" assumes "I s"   and "(ni, n) \<in> R s" shows "ospec (DESTRimpl ni s) (\<lambda>r. case n of Trueif \<Rightarrow> r = TD | Falseif \<Rightarrow> r = FD | IF v t e \<Rightarrow> \<exists>tn en. r = IFD v tn en \<and> (tn, t) \<in> R s \<and> (en, e) \<in> R s)"
sel_1: fixes x1 :: "'b \<Rightarrow> 'b set"   and x2 :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "R (Model x1 x2) = x1"
smult_assoc: fixes r :: "'r"   and s :: "'r"   and m :: "'m" assumes "r \<in> R"   and "s \<in> R"   and "m \<in> M" shows "r \<cdot> s \<cdot> m = (r * s) \<cdot> m"
intro: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "WeakControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
hyps: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option" shows "recognises_execution e s r t = recognises_prim e s r t"
vars0_def: shows "b.vars0 = vars1 \<union> vars2 \<union> vars3"
fold_congs_2: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "m r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "m_update f r = m_update f' r'"
fermat_two_squares: fixes p :: "nat" assumes "prime p"   and "[p \<noteq> 3] (mod 4)" shows "\<exists>a b. p = a\<^sup>2 + b\<^sup>2"
cos_zero_iff_int2: fixes x :: "real" shows "(cos x = 0) = (\<exists>xa. x = real_of_int xa * pi + pi / 2)"
min_set_iff: fixes x :: "'b"   and X :: "'b set" shows "(x \<in> min_set X) = (x \<in> X \<and> (\<forall>y\<in>X. y \<le> x \<longrightarrow> y = x))"
1: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau> (f $ Bv 0)) f"
HMA_T_def: fixes n :: "nat"   and uu :: "'c itself" shows "HMA_T n uu = (n = CARD('c))"
primary_ideal_ideal: fixes q :: "'a set" assumes "primary_ideal R q" shows "ideal R q"
is_igb_graph: shows "igb_graph (nba_igbg A)"
comp_closed: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f)"   and "B.arr (fst g) \<and> B.arr (snd g) \<and> src\<^sub>B (fst g) = trg\<^sub>B (snd g)"   and "B.VxV.cod f = B.VxV.dom g" shows "B.arr (fst (B.VxV.comp g f)) \<and> B.arr (snd (B.VxV.comp g f)) \<and> src\<^sub>B (fst (B.VxV.comp g f)) = trg\<^sub>B (snd (B.VxV.comp g f))"
perm_1: fixes pi :: "(name \<times> name) list"   and x2 :: "'d"   and x1 :: "'d" shows "pi \<bullet> x2\<lparr>x1\<rparr> = (pi \<bullet> x2)\<lparr>(pi \<bullet> x1)\<rparr>"
orthogonal_transformation_quat_congruence: fixes q :: "quat" assumes "norm q = 1" shows "orthogonal_transformation (\<lambda>x. HIm (Quaternions.cnj q * Hv x * q))"
wellFormedAppend: fixes \<sigma> :: "('d list \<times> 'e list) list"   and \<sigma>' :: "('d list \<times> 'e list) list" shows "wellFormedSubst (\<sigma> @ \<sigma>') = (wellFormedSubst \<sigma> \<and> wellFormedSubst \<sigma>')"
intro: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "SimIncrEnvironment_axioms jkbp envInit envAction envTrans envVal jview simf simRels simVal" shows "SimIncrEnvironment jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal"
complex_of_real_pos_iff: fixes y :: "real" shows "(0 < complex_of_real y) = (0 < y)"
seqE: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" obtains "B.arr f" and "B.arr g" and "B.dom g = B.cod f"
sum_list_const_mult: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" assumes "class.semiring_0 plus zero times" shows "monoid_add.sum_list plus zero (map (\<lambda>x. times c (f x)) xs) = times c (monoid_add.sum_list plus zero (map f xs))"
convolution_eq_sum_over_supp_right: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "g \<in> aezfun_set" shows "convolution f g = (\<lambda>x. \<Sum>y\<in>supp g. f (x - y) * g y)"
partial_division_extend_interval: fixes p :: "'a set set"   and a :: "'a"   and b :: "'a" assumes "p division_of \<Union> p"   and "\<Union> p \<subseteq> cbox a b" obtains q :: "'a set set" where "p \<subseteq> q" and "q division_of cbox a b"
rep_eq: fixes x :: "complex"   and xa :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "(*\<^sub>V) (x *\<^sub>C xa) = (\<lambda>xb. x *\<^sub>C (xa *\<^sub>V xb))"
epi_hom_induced_inclusion: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and p :: "int" assumes "homotopic_with (\<lambda>x. True) X X id f"   and "f ` topspace X \<subseteq> S" shows "hom_induced p (subtopology X S) {} X {} id \<in> epi (homology_group p (subtopology X S)) (homology_group p X)"
WFF_def: fixes \<tau> :: "'d state \<times> 'd state" shows "WFF \<tau> = ((\<forall>x\<in>ran (heap (fst \<tau>)). \<lceil>heap (fst \<tau>) (oid_of x)\<rceil> = x) \<and> (\<forall>x\<in>ran (heap (snd \<tau>)). \<lceil>heap (snd \<tau>) (oid_of x)\<rceil> = x))"
II1: shows "II1 \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}"
FvarsT_Num: fixes n :: "nat" shows "FvarsT (Num n) = {}"
pre_bisim_def: fixes as :: "'a list"   and r :: "'a rexp"   and s :: "'a rexp" shows "pre_bisim as r s = (\<lambda>(ws, R). (r, s) \<in> R \<and> set ws \<subseteq> R \<and> (\<forall>(r, s)\<in>R. atoms r \<union> atoms s \<subseteq> set as) \<and> (\<forall>(r, s)\<in>R - set ws. nullable r = nullable s \<and> (\<forall>a\<in>set as. (nderiv a r, nderiv a s) \<in> R)))"
pp_dist_pp_sup: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (uminus (sup (uminus (uminus x)) (uminus (uminus y)))) = uminus (uminus (sup x y))"
ptensor_encode1_encode2: shows "partial_state.encode1 c.dims0 c.vars1' = partial_state.encode2 c.dims0 c.vars2'"
simps: fixes xs :: "int poly list" shows "create_factor_tree_simple xs = (let n = length xs in if n \<le> 1 then Factor_Leaf () (hd xs) else let i = n div 2; xs1 = take i xs; xs2 = drop i xs in Factor_Node () (create_factor_tree_simple xs1) (create_factor_tree_simple xs2))"
sq_mtx_inv_unique: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" assumes "A * B = 1"   and "B * A = 1" shows "A\<^sup>-\<^sup>1 = B"
atom_in_atom_image: fixes j :: "'a"   and V :: "'a set" shows "(atom j \<in> atom ` V) = (j \<in> V)"
is_reducible_lit_mono_cls: fixes C :: "'a literal list"   and C' :: "'a literal list"   and Ds :: "'a literal list list"   and L :: "'a literal" assumes "mset C \<subseteq># mset C'"   and "is_reducible_lit Ds C L" shows "is_reducible_lit Ds C' L"
assms_2: shows "\<bar>Sup S\<bar> \<noteq> \<infinity>"
amax_def: fixes x :: "ant"   and y :: "ant" shows "amax x y = (if x \<le> y then y else x)"
update_convs_2: fixes c_pts' :: "('a \<Rightarrow> 'b zmultiset) \<Rightarrow> 'a \<Rightarrow> 'b zmultiset"   and c_work :: "'a \<Rightarrow> 'b zmultiset"   and c_pts :: "'a \<Rightarrow> 'b zmultiset"   and c_imp :: "'a \<Rightarrow> 'b zmultiset"   and more :: "'c" shows "c_pts_update c_pts' \<lparr>c_work = c_work, c_pts = c_pts, c_imp = c_imp, \<dots> = more\<rparr> = \<lparr>c_work = c_work, c_pts = c_pts' c_pts, c_imp = c_imp, \<dots> = more\<rparr>"
Lli_def: fixes i :: "nat" shows "Lli i = (SOME x. (prems_of \<iota>F ! i, x) \<in> NL)"
copyp_Suc: fixes k :: "nat"   and x :: "'a" shows "Suc k \<otimes> x = k \<otimes> x \<squnion> x"
main_int_poly_factorization_def: fixes f :: "int poly" shows "local.main_int_poly_factorization f = (let df = degree f in if df = 1 then [f] else if \<bar>coeff f 0\<bar> < \<bar>coeff f df\<bar> then map reflect_poly (int_poly_factorization_algorithm alg (reflect_poly f)) else int_poly_factorization_algorithm alg f)"
case_rews_3: fixes a :: "'a"   and slist :: "[:'a:]"   and f1 :: "'b"   and f2 :: "'a \<rightarrow> [:'a:] \<rightarrow> 'b" assumes "a \<noteq> \<bottom>"   and "slist \<noteq> \<bottom>" shows "slist_case\<cdot>f1\<cdot>f2\<cdot>(a :# slist) = f2\<cdot>a\<cdot>slist"
arr_inv: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "partial_magma.arr C (category.inv C f)"
cinner_ket: fixes i :: "'c"   and j :: "'c" shows "ket i \<bullet>\<^sub>C ket j = (if i = j then 1 else 0)"
comp: shows "l\<^sub>1 = l\<^sub>2\<^sup>c"
is_final_\<R>\<^sub>u_is_final: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "is_final c\<^sub>1" shows "is_final c\<^sub>2"
lang_subset_lists: fixes n :: "nat"   and r :: "'a atom rexp" assumes "local.wf n r" shows "lang n r \<subseteq> lists ((set \<circ> \<sigma> \<Sigma>) n)"
pair_relator_halflist2_conv_rev_pair_relator_halflist: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "class.group_add minus plus zero uminus" shows "(if gcd_nat.greater_eq (monoid_add.add_order plus zero (plus s t)) 2 then alternating_list (monoid_add.add_order plus zero (plus s t)) s t else alternating_list (monoid_add.add_order plus zero (plus t s)) t s) = rev (alternating_list (monoid_add.add_order plus zero (plus t s)) t s)"
differentiable_const: fixes a :: "'b"   and F :: "'a filter" shows "(\<lambda>z. a) differentiable F"
le_neq_trans: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.order less_eq less"   and "less_eq a b"   and "a \<noteq> b" shows "less a b"
interior_of_closure_of: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of S = topspace X - X closure_of (topspace X - S)"
acc81: shows "Acc level1 sA81 = {sA91, sA92} \<union> Acc level1 sA91"
sys_block_sizes_in: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "int (card bl) \<in> sys1.sys_block_sizes"
M_def: fixes \<pi> :: "'a prefix" shows "M \<pi> = {(i, v). i < progress \<pi> \<and> wf_tuple nfv fv v \<and> (\<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> sat \<sigma> (map the v) i)}"
assms_6: shows "length xvec3 = residualLength cRs"
sees_method_idemp: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a option"   and D :: "String.literal" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "P \<turnstile> D sees M: Ts\<rightarrow>T = m in D"
hypnat_of_nat_less_whn: fixes n :: "nat" shows "hypnat_of_nat n < whn"
run_detI: fixes c :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b"   and \<sigma> :: "heap option"   and s :: "'b" assumes "run c (Some h) (Some h') r"   and "run c (Some h) \<sigma> s" shows "\<sigma> = Some h' \<and> r = s"
f: fixes i :: "nat"   and j :: "nat" shows "eval f [i, j] \<down>= singleton_encode i"
desargues_config_2D_non_collinear_P_3: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {B', C', P} = 3"
simps: fixes k :: "nat" shows "\<Q> k = natUnion (\<P> k)"
resChainFreshSet_1: fixes Xs :: "name set"   and xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "Xs \<sharp>* (\<lparr>\<nu>*xvec\<rparr>P) = (\<forall>x\<in>Xs. x mem xvec \<or> x \<sharp> P)"
valuate_ex: fixes x :: "'b"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" shows "(x \<in> valuate f) = (\<exists>e. (\<forall>i. e i \<in> {- 1..1}) \<and> x = f e)"
card3_eI: fixes S :: "'c set" assumes "3 \<le> card S" shows "\<exists>x y z. x \<noteq> y \<and> x \<noteq> z \<and> y \<noteq> z \<and> x \<in> S \<and> y \<in> S"
poly_of_const_consistent: fixes K :: "'a set" assumes "subring K R" shows "ring.poly_of_const (R\<lparr>carrier := K\<rparr>) = poly_of_const"
inf_eq_neg_sup: fixes a :: "'a"   and b :: "'a" shows "inf a b = - sup (- a) (- b)"
hta_lookup_f_correct_1: fixes f :: "'L" assumes "hta_has_idx_f H" shows "ls.\<alpha> (hta_lookup_f f H) = {r \<in> ls.\<alpha> \<delta>. rhsl r = f}"
image_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'd" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and point :: "'c point" shows "P (case point of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = ((point = Infinity \<longrightarrow> P f1) \<and> (\<forall>x21 x22. point = Point x21 x22 \<longrightarrow> P (f2 x21 x22)))"
dim_basis: fixes A :: "'a vec set" assumes "finite A"   and "AB.basis A" shows "AB.dim = card A"
CCfix_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "CCfix \<Gamma> = (\<Lambda> aeG. \<mu> x. ccBindsExtra \<Gamma>\<cdot>(fst aeG, x) \<squnion> snd aeG)"
division_segment_of_nat: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "division_segment (semiring_1.of_nat one plus zero n) = one"
WeakControlDependencePDG_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b" shows "WeakControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit \<equiv> StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit \<and> CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
mult_matrix_nm: fixes A :: "'b matrix"   and n :: "nat"   and B :: "'c matrix"   and m :: "nat"   and fadd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and fmul :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "ncols A \<le> n"   and "nrows B \<le> n"   and "ncols A \<le> m"   and "nrows B \<le> m"   and "fadd (0::'d) (0::'d) = (0::'d)"   and "fmul (0::'b) (0::'c) = (0::'d)" shows "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B"
simps_1: fixes dims :: "nat list" shows "state_sig.well_com dims SKIP = True"
cong_solve_coprime_nat: fixes a :: "nat"   and n :: "nat" assumes "coprime a n" shows "\<exists>x. [a * x = Suc 0] (mod n)"
HIm_Quat: fixes w :: "real"   and x :: "real"   and y :: "real"   and z :: "real" shows "HIm (Quat w x y z) = vector [x, y, z]"
spanE: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> phull B" obtains A :: "('b \<Rightarrow>\<^sub>0 'c) set" and q :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'c" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b \<cdot> b)"
real_of_ereal_minus: fixes a :: "ereal"   and b :: "ereal" shows "real_of_ereal (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real_of_ereal a - real_of_ereal b)"
remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "\<Sqinter> A = (if A - {x} = {} then x else x \<sqinter> \<Sqinter> (A - {x}))"
lhopital_right_at_top: fixes g :: "real \<Rightarrow> real"   and x :: "real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and y :: "real" assumes "filterlim g at_top (at_right x)"   and "\<forall>\<^sub>F x in at_right x. g' x \<noteq> 0"   and "\<forall>\<^sub>F x in at_right x. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_right x. (g has_real_derivative g' x) (at x)"   and "((\<lambda>x. f' x / g' x) \<longlongrightarrow> y) (at_right x)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> y) (at_right x)"
lucas_lehmer_ring_def: shows "lucas_lehmer_ring = \<lparr>carrier = UNIV, monoid.mult = lucas_lehmer_mult', one = (1, 0), ring.zero = (0, 0), add = lucas_lehmer_add'\<rparr>"
goal3_4: shows "pointermap_getmk (v, ni1, ni2) (dpm bdd) = (x1, dpm s')"
map_filter_M_pure_E: fixes h :: "'x"   and f :: "'aa \<Rightarrow> ('x, 'y, 'z option) prog"   and xs :: "'aa list"   and ys :: "'z list"   and y :: "'z" assumes "h \<turnstile> map_filter_M f xs \<rightarrow>\<^sub>r ys"   and "y \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (f x) h" obtains x :: "'aa" where "h \<turnstile> f x \<rightarrow>\<^sub>r Some y" and "x \<in> set xs"
subst_cls_lists_Cons: fixes C :: "'a literal multiset"   and Cs :: "'a literal multiset list"   and \<sigma> :: "'s"   and \<sigma>s :: "'s list" shows "(C # Cs) \<cdot>\<cdot>cl (\<sigma> # \<sigma>s) = C \<cdot> \<sigma> # Cs \<cdot>\<cdot>cl \<sigma>s"
f_cont: shows "continuous_on (Sigma T X) (\<lambda>(t, x). f t x)"
eq_vecI: fixes w :: "'b vec"   and v :: "'b vec" assumes "\<And>i. i < dim_vec w \<Longrightarrow> v $ i = w $ i"   and "dim_vec v = dim_vec w" shows "v = w"
rcosets: fixes G :: "'a set"   and H :: "'a set"   and g :: "'a" assumes "Group G"   and "Group H \<and> H \<subseteq> G"   and "g \<in> G" shows "Group.rcoset_rel G H `` {g} = H + {g}"
is_character_data_ptr_kind\<^sub>_cast: fixes character_data_ptr :: "(_) character_data_ptr" shows "is_character_data_ptr_kind (cast character_data_ptr)"
jkbpC_represents_uniquely: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'e Trace set" assumes "PreEnvironmentJView jkbp jview"   and "PreEnvironmentJView.represents jkbp envInit envAction envTrans envVal jview T" shows "T = PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"
simps_2: fixes f :: "'d"   and ss1 :: "('d, 'f) Term.term list"   and D :: "('d, 'f) ctxt"   and ss2 :: "('d, 'f) Term.term list"   and \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "More f ss1 D ss2 \<cdot>\<^sub>c \<sigma> = More f (map (\<lambda>t. t \<cdot> \<sigma>) ss1) (D \<cdot>\<^sub>c \<sigma>) (map (\<lambda>t. t \<cdot> \<sigma>) ss2)"
ord_refl: fixes t :: "'a \<Rightarrow> 'b" shows "ord t t"
overlap_eq: fixes x :: "i"   and y :: "i" assumes "M4" shows "O x y = (\<exists>z. P z x \<and> P z y)"
assms: fixes x :: "'a" assumes "x \<in> set_pmf \<mu>" shows "(f x = y1) = (g x = y2)"
Radj_repd3: fixes z :: "'sz"   and x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and r :: "real" assumes "z \<noteq> x"   and "z \<noteq> y" shows "Radj x y (repd \<nu> z r) = repd (Radj x y \<nu>) z r"
\<sigma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_exists: shows "\<exists>qmsg q. (qmsg, q) \<in> \<sigma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
openin_Int_closure_of_eq_empty: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "(S \<inter> X closure_of T = {}) = (S \<inter> T = {})"
is_class_is_subcls: fixes m :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list" assumes "wf_prog m P" shows "is_class P C = P \<turnstile> C \<preceq>\<^sup>* Object"
iT_cut_ge_3: fixes n :: "nat"   and d :: "nat"   and t :: "nat" shows "[n\<dots>,d] \<down>\<ge> t = (if t < n then [n\<dots>,d] else if t \<le> n + d then [t\<dots>,n + d - t] else {})"
resolvent_upon_correct: fixes P1 :: "'at Literal set"   and S :: "'at Literal set set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and A :: "'at" assumes "P1 \<in> S"   and "P2 \<in> S"   and "C = resolvent_upon P1 P2 A" shows "entails S C"
case_prod_beta': fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
Box_def: fixes \<alpha> :: "('a, 'b, 'c) hp"   and P :: "('a, 'b, 'c) formula" shows "([[\<alpha>]]P) = ! (\<langle> \<alpha> \<rangle> ! P)"
monoD: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a" assumes "mono f"   and "x \<le> y" shows "f x \<le> f y"
nae2da_def: fixes A :: "'c \<times> ('d option \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)" shows "nae2da A = ({start A}, \<lambda>a Q. \<Union> (next A (Some a) ` (eps A)\<^sup>* `` Q), \<lambda>Q. \<exists>p\<in>(eps A)\<^sup>* `` Q. fin A p)"
select_convs_2: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cMassF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cMassF"
is_unit_hom_if: fixes x :: "'a poly" assumes "poly_lift x dvd 1" shows "x dvd 1"
no_conflicts_def: fixes p :: "'d \<Rightarrow> 'e decision option"   and q :: "'d \<Rightarrow> 'e decision option" shows "no_conflicts p q = (dom p = dom q \<and> (\<forall>x\<in>dom p. case p x of \<lfloor>allow y\<rfloor> \<Rightarrow> \<exists>z. q x = \<lfloor>allow z\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> \<exists>z. q x = \<lfloor>deny z\<rfloor>))"
simps_2: fixes e :: "'b equation"   and s :: "('b \<times> 'b trm) list" shows "subst_lit (Neg e) s = Neg (subst_equation e s)"