hcju/isabelleps huggingface dataset:
less_eq_mask_iff_take_bit_eq_self: fixes w :: "'a word"   and n :: "nat" shows "(w \<le> mask n) = (take_bit n w = w)"
hnr_bind: fixes \<Gamma> :: "assn"   and m' :: "'a Heap"   and \<Gamma>1 :: "assn"   and Rh :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres"   and f' :: "'a \<Rightarrow> 'c Heap"   and \<Gamma>2 :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and f :: "'b \<Rightarrow> 'd nres"   and \<Gamma>' :: "assn"   and Rx :: "'b \<Rightarrow> 'a \<Rightarrow> assn" assumes "hn_refine \<Gamma> m' \<Gamma>1 Rh m"   and "\<And>x x'. RETURN x \<le> m \<Longrightarrow> hn_refine (\<Gamma>1 * hn_ctxt Rh x x') (f' x') (\<Gamma>2 x x') R (f x)"   and "\<And>x x'. \<Gamma>2 x x' \<Longrightarrow>\<^sub>t \<Gamma>' * hn_ctxt Rx x x'" shows "hn_refine \<Gamma> (m' \<bind> f') \<Gamma>' R (m \<bind> f)"
intros: fixes t :: "lam"   and s :: "lam" shows " App t s \<guillemotright> t"
rep_non_empty_means_seq: fixes i :: "nat_int" shows "i \<noteq> bot \<longrightarrow> (\<exists>m n. m \<le> n \<and> {m..n} = Rep_nat_int i)"
add_2_eq_Suc': fixes n :: "nat" shows "n + 2 = Suc (Suc n)"
statically_complete_calculus_axioms: fixes M :: "'a literal multiset set" shows "statically_complete_calculus {{#}} (G_Inf M) (\<TTurnstile>e) (G.Red_I M) G.Red_F"
AAE: shows "\<alpha> (add_edge e d g) = (\<alpha> g)(e := enat d)"
Conj_EH_2: fixes A :: "fm"   and Ba :: "fm"   and A1 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A (insert Ba (insert A1 H)) \<turnstile> B" shows "insert A1 (insert (A AND Ba) H) \<turnstile> B"
splus_zero: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.splus pair_of_term term_of_pair (0::'b) v = v"
Rep_inverse: fixes x :: "'b word" shows "word_of_nat (unat x) = x"
add_le_imp_le_left: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "c + a \<subseteq># c + b" shows "a \<subseteq># b"
ide_in_hom: fixes a :: "'a \<times> 'a" shows "B.VV.ide a = B.VV.in_hom a a a"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 g f"
ex1: fixes X :: "'b program set"   and GG :: "'b program set" assumes "ex_prop X"   and "finite GG"   and "GG \<inter> X \<noteq> {}"   and "OK GG (\<lambda>G. G)" shows "(\<Squnion>G\<in>GG. G) \<in> X"
trms_of_deg_leq_closed: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "trms_of_deg_leq n f \<in> carrier P"
A_i_least_H: shows "A $ i $ (LEAST n. H $ ia $ n \<noteq> (0::'a)) = (0::'a)"
FOREACHci_rule: fixes S :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and P :: "'d \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<noteq> {}; it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>C\<^bsup>I\<^esup> S c f \<sigma>0 \<le> SPEC P"
integrable_const_bound: fixes f :: "real \<Rightarrow> 'a"   and B :: "real" assumes "AE x in M. norm (f x) \<le> B"   and "random_variable borel f" shows "integrable M f"
WHILET_def: shows "WHILE\<^sub>T \<equiv> iWHILET (\<bind>) RETURN"
assms_2: shows "f \<cdot> g = h \<cdot> k"
strict_mono_eq: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "strict_mono f" shows "(f x = f y) = (x = y)"
genby_lcoset_empty: fixes w :: "'b" shows "w +o \<langle>{}\<rangle> = {w}"
simps_3: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "successive P (x # y # xs) = (\<not> P y \<and> successive P xs \<or> \<not> P x \<and> successive P (y # xs))"
isomorphic_free_groups_bases_finite: fixes i :: "(bool \<times> 'c) list \<Rightarrow> (bool \<times> 'd) list"   and X :: "'c set"   and Y :: "'d set" assumes "i \<in> Group.iso \<F>\<^bsub>X\<^esub> \<F>\<^bsub>Y\<^esub>"   and "finite X" shows "\<exists>f. bij_betw f X Y"
until_introR: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'"   and "\<psi> \<pi>l" shows "Shallow.until \<phi> \<psi> \<pi>l"
cod_dom: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.cod (hom'.dom f) = hom'.dom f"
neq_le_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order less_eq less"   and "a \<noteq> b"   and "less_eq a b" shows "less a b"
cut_ge_Max_not_empty: fixes I :: "'a set"   and t :: "'a" assumes "I \<noteq> {}"   and "finite I"   and "t \<le> Max I" shows "I \<down>\<ge> t \<noteq> {}"
one_vec_def: shows "1 \<equiv> \<chi>i. (1::'d)"
timpl_closure_FunI3: fixes T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and c :: "('c set \<times> 'c set) set"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and f :: "('a, 'b, 'c) prot_fun"   and g :: "('a, 'b, 'c) prot_fun" assumes "\<And>i. i < length T \<Longrightarrow> \<exists>u. (T ! i, u) \<in> timpl_closure' c \<and> (S ! i, u) \<in> timpl_closure' c"   and "length T = length S"   and "f = g \<or> (\<exists>a b d. (a, d) \<in> c \<and> (b, d) \<in> c \<and> f = Abs a \<and> g = Abs b)" shows "\<exists>h U. (Fun f T, Fun h U) \<in> timpl_closure' c \<and> (Fun g S, Fun h U) \<in> timpl_closure' c"
red_diff_rtrancl': fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "(red F)\<^sup>*\<^sup>* (p - q) r" obtains p' :: "'t \<Rightarrow>\<^sub>0 'c" and q' :: "'t \<Rightarrow>\<^sub>0 'c" where "(red F)\<^sup>*\<^sup>* p p'" and "(red F)\<^sup>*\<^sup>* q q'" and "r = p' - q'"
unit_closed: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "Group_Theory.monoid M composition unit" shows "unit \<in> M"
rep_eq: fixes x :: "'b fset"   and xa :: "'b fset" shows "fset (x |\<union>| xa) = fset x \<union> fset xa"
intro: fixes carrier :: "'b set" assumes "finite carrier" shows "finite_total_preorder_on_axioms carrier"
config_append: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list"   and xs :: "'a list"   and ys :: "'a list" shows "config\<^sub>p A s (xs @ ys) = Partial_Cost_Model.config' A (config\<^sub>p A s xs) ys"
axioms_1: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" assumes "composition_series G \<GG>" shows "normal_series G \<GG>"
paths_Nil: fixes t :: "'b ttree" shows "[] \<in> paths t"
cmp_components_are_iso: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.iso (\<Phi>\<^sub>F (f, g))"
neq_NilE: fixes l :: "'b list" assumes "l \<noteq> []" obtains x :: "'b" and xs :: "'b list" where "l = x # xs"
permutation_list_succ: fixes xs :: "'b list" assumes "distinct xs" shows "permutation (list_succ xs)"
Image_eq_UN: fixes r :: "('i \<times> 'h) set"   and B :: "'i set" shows "r `` B = (\<Union>y\<in>B. r `` {y})"
deg_pow0: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "p \<in> carrier P"   and "deg R p \<le> n" shows "deg R (p [^]\<^bsub>P\<^esub> m) = m * deg R p"
tendsto_cong: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and F :: "'d filter"   and c :: "'c" assumes "class.topological_space open"   and "\<forall>\<^sub>F x in F. f x = g x" shows "filterlim f (topological_space.nhds open c) F = filterlim g (topological_space.nhds open c) F"
pointwise: fixes x :: "'a" assumes "x \<in> space M" shows "(\<lambda>i. s i x) \<longlonglongrightarrow> f x"
Rep_column_of_matrix: fixes A :: "'b matrix"   and c :: "nat"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (column_of_matrix A c) j i = (if i = 0 then Rep_matrix A j c else (0::'b))"
upper: fixes P :: "'a \<Rightarrow> bool" assumes "eventually P F" shows "Inf (f ` Collect P) \<le> y"
hyps: assumes "symmetric G"   and "vwalk xs G" shows "vwalk (rev xs) G"
sum_zero_eq_neg: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<oplus>\<^bsub>R\<^esub> y = \<zero>\<^bsub>R\<^esub>" shows "x = \<ominus>\<^bsub>R\<^esub> y"
in_lex_prod: fixes a :: "'c"   and b :: "'d"   and a' :: "'c"   and b' :: "'d"   and r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" shows "(((a, b), a', b') \<in> r <*lex*> s) = ((a, a') \<in> r \<or> a = a' \<and> (b, b') \<in> s)"
map_disc_iff: fixes f1 :: "'i \<Rightarrow> 'g"   and f2 :: "'j \<Rightarrow> 'h"   and a :: "('i, 'j) tllist" shows "is_TNil (tmap f1 f2 a) = is_TNil a"
lookup_default: fixes x :: "'d"   and d :: "'c"   and m :: "('d, 'c) mapping" shows "Mapping.lookup (Mapping.default x d m) x = Some (Mapping.lookup_default d m x)"
global_def: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" shows "global P \<equiv> \<lambda>(\<sigma>, uu_). P \<sigma>"
scaleC_1_right: fixes x :: "complex" shows "x *\<^sub>C (1::'b) = of_complex x"
not_times_prv_neg_eql_tms: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m * n \<noteq> k" shows "prv (neg (eql (tms (Num m) (Num n)) (Num k)))"
select_convs_3: fixes iso_verts :: "'e \<Rightarrow> 'd"   and iso_arcs :: "'f \<Rightarrow> 'c"   and iso_head :: "'c \<Rightarrow> 'd"   and iso_tail :: "'c \<Rightarrow> 'd"   and more :: "'g" shows "iso_head \<lparr>iso_verts = iso_verts, iso_arcs = iso_arcs, iso_head = iso_head, iso_tail = iso_tail, \<dots> = more\<rparr> = iso_head"
wf_\<Gamma>_def: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list" shows "iptables_bigstep_fixedbackground.wf_\<Gamma> \<Gamma> rs \<equiv> \<forall>rsg\<in>ran \<Gamma> \<union> {rs}. \<forall>r\<in>set rsg. \<forall>chain. get_action r = Call chain \<longrightarrow> \<Gamma> chain \<noteq> None"
b_less_1: fixes b :: "real" assumes "b \<in> set bs" shows "b < 1"
assms_4: shows "asc_fault_domain M2 M1 m"
cases: fixes x :: "'c list" obtains x :: "'c" and xs :: "'c list" where "xa__ = x # xs"   | "xa__ = []"
A8_NSources: shows "\<forall>C\<in>AbstrLevel level0. C \<noteq> sA9 \<and> C \<noteq> sA8 \<longrightarrow> sA8 \<notin> Sources level0 C"
gcd: shows "gcd (\<Prod>(a, i)\<in>as. a) (\<Sum>(a, i)\<in>as. (\<Prod>(b, j)\<in>as - {(a, i)}. b) * smult (of_nat (Suc i)) (pderiv a)) = 1"
A'_def: shows "A' = mult_row interchange_A i ((1::'a) / interchange_A $ i $ j)"
irbd_Sup_pres: shows "Sup_pres bd\<^sup>-\<^sub>\<R>"
LstSeq_iff: fixes s :: "hf"   and k :: "hf"   and y :: "hf" shows "LstSeq s k y = (succ k \<le> hdomain s \<and> \<langle>k, y\<rangle> \<^bold>\<in> s \<and> hfun_sigma_ord s)"
discounted_AE_cong: fixes X :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and n :: "nat"   and Y :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and N :: "'c measure"   and r :: "real" assumes "AEeq N (X n) (Y n)" shows "AEeq N (discounted_value r X n) (discounted_value r Y n)"
prob0_bell_fst: fixes v :: "complex Matrix.mat" assumes "v = |\<beta>\<^sub>0\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>0\<^sub>1\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>1\<rangle>" shows "prob0 2 v 0 = 1 / 2"
Join_commute: fixes F :: "'c program"   and G :: "'c program" shows "F \<squnion> G = G \<squnion> F"
smallest_accepting_rank\<^sub>\<R>_def: shows "smallest_accepting_rank\<^sub>\<R> \<equiv> if accept\<^sub>R (\<delta>\<^sub>\<R>, q\<^sub>\<R>, {(fail\<^sub>R \<union> merge\<^sub>R j, succeed\<^sub>R j) |j. j < max_rank}) w then Some (LEAST i. accepting_pair\<^sub>R \<delta>\<^sub>\<R> q\<^sub>\<R> (fail\<^sub>R \<union> merge\<^sub>R i, succeed\<^sub>R i) w) else None"
has_extend_witness: fixes n :: "nat"   and S :: "'b set"   and xs :: "'b list" assumes "has n S"   and "set xs \<subseteq> S"   and "length xs < n" shows "set xs \<subset> S"
S_def: shows "S = R \<union> (\<approx>\<^bsub>d\<^esub>)"
enum_bit1_def: shows "enum_class.enum = map (Abs_bit1' \<circ> int) [0..<CARD('a bit1)]"
reds_pres_closed: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z"   and "fv (\<Gamma>, e) \<subseteq> set L \<union> domA \<Gamma>" shows "fv (\<Delta>, z) \<subseteq> set L \<union> domA \<Delta>"
uncurry_preserves_transformations: fixes F :: "'a1 \<Rightarrow> ('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and G :: "'a1 \<Rightarrow> ('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and \<tau> :: "'a1 \<Rightarrow> ('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "natural_transformation (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F G \<tau>" shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (uncurry F) (uncurry G) (uncurry \<tau>)"
execute_parallel_operator_cons: fixes s :: "'c \<Rightarrow> bool option"   and op :: "'c strips_operator"   and ops :: "'c strips_operator list" shows "execute_parallel_operator s (op # ops) = execute_parallel_operator (s ++ map_of (effect_to_assignments op)) ops"
latestActNxtAct: fixes n :: "nat"   and tid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'\<ge>n. \<parallel>tid\<parallel>\<^bsub>t n'\<^esub>"   and "\<exists>n'. latestAct_cond tid t n n'" shows "\<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub> < \<langle>tid \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
smap_union_comm: fixes m1 :: "'c \<Rightarrow> 'd set"   and m2 :: "'c \<Rightarrow> 'd set" shows "m1 \<union>. m2 = m2 \<union>. m1"
simps: fixes s :: "tm"   and k :: "tm"   and y :: "tm" shows "LstSeqP s k y = OrdP k AND HDomain_Incl s (SUCC k) AND HFun_Sigma s AND HPair k y IN s"
step_store_ubx: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and \<tau> :: "Unboxed.type"   and x :: "'var"   and i :: "('dyn, 'ubx1, 'ubx2) unboxed"   and i' :: "'dyn"   and blob :: "('dyn, 'ubx1, 'ubx2) unboxed"   and d :: "'dyn"   and H :: "'henv"   and H' :: "'henv"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and st :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" assumes "Fubx_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = IStoreUbx \<tau> x"   and "cast_Dyn i = Some i'"   and "Subx.cast_and_box \<tau> blob = Some d"   and "heap_add H (x, i') d = H'" shows "Subx.step (State F H (Frame f pc (i # blob # \<Sigma>) # st)) (State F H' (Frame f (Suc pc) \<Sigma> # st))"
norm_optim_instr: fixes n :: "nat"   and x :: "('dyn, 'b, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and \<Sigma>1 :: "Unboxed.type option list"   and x' :: "('dyn, 'b, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and \<Sigma>2 :: "Unboxed.type option list" assumes "optim_instr n x \<Sigma>1 = Ok (x', \<Sigma>2)" shows "norm_instr x' = norm_instr x"
get_child_nodes_locs_def: shows "get_child_nodes_locs = (\<lambda>ptr. (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) child_nodes)} else {}) \<union> (if is_document_ptr_kind ptr then {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) document_element)} else {}) \<union> {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
assms_1: shows "H \<turnstile> OrdP (Var i)"
min_absorb2: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "min x y = y"
gf: fixes x :: "'b" assumes "x \<in> T - S" shows "g x = f x"
Constrains_UN: fixes I :: "'b set"   and F :: "'c program"   and A :: "'b \<Rightarrow> 'c set"   and A' :: "'b \<Rightarrow> 'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i Co A' i" shows "F \<in> \<Union> (A ` I) Co \<Union> (A' ` I)"
intersection_of_def: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool" shows "P intersection_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Inter> \<U> = S"
subseqs_of_indices: fixes xs :: "'c list" shows "map (map ((!) xs)) (subseqs [0..<length xs]) = subseqs xs"
writes_bind_pure: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and S :: "('x, 'y, 'aa) prog set"   and g :: "'z \<Rightarrow> ('x, 'y, 'ab) prog"   and h' :: "'x" assumes "pure f h"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> writes S (g x) h h'" shows "writes S (Heap_Error_Monad.bind f g) h h'"
shift_zero_id: fixes c :: "nat"   and \<tau> :: "ty" shows "shift 0 c \<tau> = \<tau>"
add_right: fixes f :: "'a \<Rightarrow> real" assumes "\<And>x y. x \<succeq>[relation] y \<Longrightarrow> f y \<le> f x" shows "ordinal_utility carrier relation (\<lambda>x. u x + f x)"
empty: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state" shows "evaluate_decs ck mn env s [] (s, Rval (make_sem_env nsEmpty nsEmpty))"
diff_eq: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and x :: "'g"   and y :: "'g" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "minus x y = inf x (uminus y)"
map_add_upds: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and xs :: "'a list"   and ys :: "'b list" shows "m1 ++ m2(xs [\<mapsto>] ys) = (m1 ++ m2)(xs [\<mapsto>] ys)"
deg_pow: fixes p :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "p \<in> carrier P" shows "deg R (p [^]\<^bsub>P\<^esub> m) = m * deg R p"
induct: fixes P :: "'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "('b \<times> color) tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x l a uu_ r. \<lbrakk>\<lbrakk>cmp x a = LT; l \<noteq> \<langle>\<rangle> \<and> color l = Black\<rbrakk> \<Longrightarrow> P x l; \<lbrakk>cmp x a = LT; \<not> (l \<noteq> \<langle>\<rangle> \<and> color l = Black)\<rbrakk> \<Longrightarrow> P x l; \<lbrakk>cmp x a = GT; r \<noteq> \<langle>\<rangle> \<and> color r = Black\<rbrakk> \<Longrightarrow> P x r; \<lbrakk>cmp x a = GT; \<not> (r \<noteq> \<langle>\<rangle> \<and> color r = Black)\<rbrakk> \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x \<langle>l, (a, uu_), r\<rangle>" shows "P a0 a1"
prems_sub_ge_i: shows "set (prems_of \<iota>) \<subseteq> \<Inter> (lnth As ` {j. i \<le> j \<and> enat j < llength As})"
cSup_upper: fixes x :: "'a multiset"   and X :: "'a multiset set" assumes "x \<in> X"   and "subset_mset.bdd_above X" shows "x \<subseteq># Sup X"
sub_is_proper: assumes "\<U> \<noteq> \<V>" shows "proper_sub_incidence_system \<U> \<A> \<V> \<B>"
test_fst_k_def: fixes k :: "nat" shows "test_fst_k k = mat N N (\<lambda>(i, j). if i = j \<and> i < k then 1 else 0)"
setConflictAnalysisClause_def: fixes clause :: "Literal list"   and state :: "State" shows "setConflictAnalysisClause clause state = (let oppM0 = oppositeLiteralList (elements (prefixToLevel 0 (getM state))); state' = state\<lparr>getC := remdups (list_diff clause oppM0)\<rparr> in countCurrentLevelLiterals (findLastAssertedLiteral state'))"
scale_right_distrib: fixes a :: "'b"   and x :: "'b"   and y :: "'b" shows "a * (x + y) = a * x + a * y"
below_set_def: fixes A :: "'b set"   and B :: "'b set" shows "(A \<sqsubseteq> B) = (A \<subseteq> B)"
assms: shows "(Ps,  \<Phi> \<Rightarrow>* \<Psi>) \<in> upRules"
C1: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel"
eq: shows "t \<^sup>@ i \<cdot> r \<cdot> u = t \<^sup>@ k"
assms: shows "fds_nth f' 1 = (0::'a)"
bottomD_1: fixes P :: "'b set" assumes "codomain.has_bottom P" shows "codomain.bottom P \<in> P"
A3: shows "((\<sigma>, p'), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"
simps: fixes a1 :: "val"   and a2 :: "val" shows "a1 \<sqsubseteq> a2 = ((\<exists>n. a1 = VNat n \<and> a2 = VNat n) \<or> (\<exists>t1 t2. a1 = VFun t1 \<and> a2 = VFun t2 \<and> t1 \<lesssim> t2))"
Nil_not_valid2: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a" shows "\<not> IO_Automaton.valid2 step []"
ide_src: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.ide (S.src \<mu>)"
subst_ground_ident_compose_1: fixes \<sigma> :: "'c \<Rightarrow> ('b, 'a) Term.term"   and x :: "'c"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv (\<sigma> x) = {}" shows "(\<sigma> \<circ>\<^sub>s \<theta>) x = \<sigma> x"
1: shows "to_nat (from_nat c) \<notin> set [i + 1..<min (nrows A) (ncols A)]"
that: fixes y :: "'a" assumes "y \<in> B"   and "setdist A B = infdist y A" shows "thesis"
simps_2: fixes x :: "'a list"   and xs :: "'a list list" shows "concat (x # xs) = x @ concat xs"
not_less_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and n :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
map_in_list_rel_conv: fixes l :: "'a list"   and l' :: "'b list"   and \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "((l, l') \<in> \<langle>br \<alpha> I\<rangle>list_rel) = ((\<forall>x\<in>set l. I x) \<and> l' = map \<alpha> l)"
norm_blinfun_id: shows "norm 1\<^sub>L = 1"
arrI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "Fop.Aop.in_hom f a b" shows "Fop.Aop.arr f"
vcard_finite_set: fixes A :: "V set" assumes "finite A" shows "vcard (set A) = ord_of_nat (card A)"
Bex_def_raw: shows "Bex \<equiv> \<lambda>A P. \<exists>x. x \<in> A \<and> P x"
nn_integral_mono_AE: fixes u :: "'a \<Rightarrow> ennreal"   and v :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
FB: shows "FB \<equiv> \<lambda>Bb Cc. four_block_mat Bb (0\<^sub>m (dim_row Bb) (dim_col Cc)) (0\<^sub>m (dim_row Cc) (dim_col Bb)) Cc"
Comp_def: fixes G :: "'a set"   and F :: "'a set" shows "Maps.Comp G F \<equiv> {h. is_iso_class F \<and> is_iso_class G \<and> (\<exists>f g. f \<in> F \<and> g \<in> G \<and> g \<star> f \<cong> h)}"
prems_1: shows "v \<notin> ifex_var_set t"
r_eq_def: shows "r_eq \<equiv> Cn 2 r_sign [Cn 2 r_add [r_sub, r_swap r_sub]]"
fract_collapse_3: fixes k :: "'b" shows "Fract k (0::'b) = 0"
mk_moebius_cmat_def: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" shows "mk_moebius_cmat a b c d = (let M = (a, b, c, d) in if mat_det M \<noteq> 0 then M else eye)"
SCSubset: fixes b :: "('a, 'b) Combinators list"   and a :: "('a, 'b) Combinators list" assumes "singleCombinators b"   and "set a \<subseteq> set b" shows "singleCombinators a"
bernoulli_pmf_1: shows "bernoulli_pmf 1 = return_pmf True"