hcju/isabelleps huggingface dataset:
Ahyp: fixes \<epsilon> :: "real"   and \<tau> :: "real" assumes "0 < \<epsilon>"   and "\<tau> \<in> T"   and "cball \<tau> \<epsilon> \<subseteq> T" shows "bdd_above {\<parallel>A t\<parallel>\<^sub>o\<^sub>p |t. t \<in> cball \<tau> \<epsilon>}"
wordinterval_to_set_alt: fixes r :: "'b wordinterval" shows "wordinterval_to_set r = {x. wordinterval_element x r}"
rev_replicate: fixes n :: "nat"   and x :: "'g" shows "rev (replicate n x) = replicate n x"
downclosed_Fix: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "class.preorder less_eq less" shows "ord.downclosed_set less_eq X = (ord.downset_set less_eq X = X)"
wf_envs_P_simps_3: fixes \<sigma> :: "('a \<times> 'b) trace"   and j :: "nat"   and P :: "'a \<Rightarrow> nat option"   and P' :: "'a \<Rightarrow> nat option"   and V :: "'a \<Rightarrow> (nat \<Rightarrow> 'b set) option"   and db :: "('a, 'b set list) mapping" assumes "wf_envs \<sigma> j P P' V db" shows "rel_mapping (\<le>) P P'"
pd_antisym: fixes x :: "'n"   and y :: "'n" assumes "x pd\<rightarrow> y"   and "y pd\<rightarrow> x" shows "x = y"
preprocess'_unsat_indexD: fixes i :: "'a"   and ns :: "('a \<times> QDelta ns_constraint) list"   and j :: "nat" assumes "i \<in> set (UnsatIndices (preprocess' ns j))" shows "\<exists>c. poly c = 0 \<and> \<not> zero_satisfies c \<and> (i, c) \<in> set ns"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (call_instr instr s1)) \<and> t2 = snd (fst (call_instr instr s2))"
hyps_12: shows "A\<^sub>P \<sharp>* M"
intros_3: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "N \<in> sexp" shows "Scons M N \<in> sexp"
H: fixes a :: "'a"   and rs :: "'a llist" assumes "r = a ## rs"   and "a \<in> A"   and "rs \<in> A\<^sup>\<star>" shows "R"
simps_1: fixes r :: "real"   and cs :: "real list"   and xs :: "real list" shows "I\<^sub>R (Less r cs) xs = (r < \<langle>cs,xs\<rangle>)"
rep_eq: fixes x :: "('b, 'a) aezfun"   and xa :: "('b, 'a) aezfun" shows "aezfun (x + xa) = aezfun x + aezfun xa"
cSup_least: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 X :: "'c set"   and z :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x z" shows "less_eq (Sup X) z"
diff_submanifold: fixes S :: "'a set" assumes "open S" shows "diff k (src.charts_submanifold S) charts_eucl f"
isTrivialPhi_det: fixes g :: "'g"   and v :: "'val" assumes "trivial g v" shows "\<exists>!v'. isTrivialPhi g v v'"
cr_stream_def: shows "cr_stream \<equiv> \<lambda>x y. x = llist_of_stream y"
relation_of_inject: fixes x :: "('c, 'd) action"   and y :: "('c, 'd) action" shows "(action.relation_of x = action.relation_of y) = (x = y)"
simps_2: fixes f :: "'b simple_match \<Rightarrow> simple_action \<Rightarrow> 'a"   and x1 :: "'b simple_match"   and x2 :: "simple_action" shows "(case SimpleRule x1 x2 of SimpleRule x xa \<Rightarrow> f x xa) = f x1 x2"
mat_swaprows_def: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "swaprows k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then A $$ (l, j) else if l = i then A $$ (k, j) else A $$ (i, j))"
discI_2: fixes "term" :: "('a, 'b) Term.term"   and x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" assumes "term = Fun x21 x22" shows "is_Fun term"
tMap_eq_imp_tLength_eq: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace"   and f' :: "'c \<Rightarrow> 'a"   and t' :: "'c Trace" assumes "tMap f t = tMap f' t'" shows "tLength t = tLength t'"
Join_def: fixes A :: "'b set" shows "\<Squnion>A = (THE sup. is_Sup A sup)"
sub: fixes \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P\<^sub>1 :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>\<^sub>1' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P\<^sub>1' :: "'BExp set"   and \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> 'BExp set option"   and P\<^sub>2 :: "'BExp set"   and \<Gamma>\<^sub>2' :: "'Var \<Rightarrow> 'BExp set option"   and P\<^sub>2' :: "'BExp set" assumes "\<turnstile> \<Gamma>\<^sub>1,\<S>,P\<^sub>1 {c} \<Gamma>\<^sub>1',\<S>',P\<^sub>1'"   and "\<Gamma>\<^sub>2 =:\<^sub>P\<^sub>2 \<Gamma>\<^sub>1"   and "\<forall>mds. tyenv_wellformed mds \<Gamma>\<^sub>2 \<S> P\<^sub>2 \<longrightarrow> tyenv_wellformed mds \<Gamma>\<^sub>1 \<S> P\<^sub>1"   and "\<forall>mds. tyenv_wellformed mds \<Gamma>\<^sub>1' \<S>' P\<^sub>1' \<longrightarrow> tyenv_wellformed mds \<Gamma>\<^sub>2' \<S>' P\<^sub>2'"   and "\<Gamma>\<^sub>1' =:\<^sub>P\<^sub>1' \<Gamma>\<^sub>2'"   and "P\<^sub>2 \<turnstile> P\<^sub>1"   and "P\<^sub>1' \<turnstile> P\<^sub>2'" shows "\<turnstile> \<Gamma>\<^sub>2,\<S>,P\<^sub>2 {c} \<Gamma>\<^sub>2',\<S>',P\<^sub>2'"
source: fixes S1 :: "'procS"   and S2 :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "(S1, S2) \<in> SRel" shows "SourceTerm S1 \<R>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> SourceTerm S2"
length_dropWhile_le: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (dropWhile P xs) \<le> length xs"
lnth_LCons': fixes x :: "'d"   and xs :: "'d llist"   and n :: "nat" shows "lnth (LCons x xs) n = (if n = 0 then x else lnth xs (n - 1))"
ConE_2: fixes p :: "'a pfm"   and q :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<and>\<^sub>! q)" shows "\<turnstile>\<^sub>! q"
enn2real_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "\<And>i. i \<in> I \<Longrightarrow> f i < top" shows "enn2real (sum f I) = sum (enn2real \<circ> f) I"
DEFL_eq_error: shows "DEFL('a\<cdot>'b error) = DEFL('b\<^sub>\<bottom> \<oplus> 'a\<^sub>\<bottom>)"
arr_dom_iff_arr: fixes f :: "'c" shows "C.arr (C.dom f) = C.arr f"
F1set3_JF2set_incl_JF1set: fixes a' :: "'d JF2"   and a :: "'d JF1" assumes "a' \<in> F1set3 (dtor1 a)" shows "JF2set a' \<subseteq> JF1set a"
intra_sum_SDG_path_rev_induct: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "n is-ns\<rightarrow>\<^sub>d* n'"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n ns n' n''. \<lbrakk>n s\<longrightarrow>\<^bsub>cd\<^esub> n''; n'' is-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'\<rbrakk> \<Longrightarrow> P n (n # ns) n'"   and "\<And>n ns n' V n''. \<lbrakk>n s-V\<rightarrow>\<^sub>d\<^sub>d n''; n \<noteq> n''; n'' is-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'\<rbrakk> \<Longrightarrow> P n (n # ns) n'"   and "\<And>n ns n' p n''. \<lbrakk>n s-p\<rightarrow>\<^bsub>sum\<^esub> n''; n'' is-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'\<rbrakk> \<Longrightarrow> P n (n # ns) n'" shows "P n ns n'"
j: shows "prv (neg (cnj (PP (Var xx)) (PP (encF N (Var xx)))))"
conc_trans_additional_5: fixes A :: "'c nres"   and B :: "'c nres"   and C :: "'c nres" assumes "A \<le> \<Down> Id B"   and "B \<le> C" shows "A \<le> C"
distinct_1: fixes x2 :: "'a" shows "\<bottom> \<noteq> Some x2"
linear: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<in> o[F](h2) \<or> h2 \<in> o[F](h1) \<or> h1 \<in> \<Theta>[F](h2)"
finsum_closed: fixes f :: "'f \<Rightarrow> 'c"   and A :: "'f set" assumes "f \<in> A \<rightarrow> carrier M" shows "finsum M f A \<in> carrier M"
load_word_mem_8_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and address :: "word32" assumes "low_equal s1 s2" shows "load_word_mem s1 address 8 = load_word_mem s2 address 8"
E_inV: fixes e :: "'edge" assumes "e \<in> E" shows "vertices e \<subseteq> V"
dot_lmul_matrix: fixes x :: "(real, 'e) vec"   and A :: "((real, 'd) vec, 'e) vec"   and y :: "(real, 'd) vec" shows "(x v* A) \<bullet> y = x \<bullet> (A *v y)"
detA: shows "(\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i)))) = Determinant.det (submatrix A UNIV (g ` {0..<n}))"
that_8: fixes \<Gamma>' :: "char list \<Rightarrow> secLevel option"   and T :: "secLevel"   and b :: "expr"   and c1 :: "com"   and c2 :: "com" assumes "x = \<Gamma>'"   and "xa = T"   and "xb = if (b) c1 else c2"   and "\<Gamma>' \<turnstile> b : T"   and "\<Gamma>',T \<turnstile> c1"   and "\<Gamma>',T \<turnstile> c2" shows "thesis"
product_prob_space_axioms_def: fixes M :: "'b \<Rightarrow> 'c measure" shows "product_prob_space_axioms M \<equiv> \<forall>i. prob_space (M i)"
wf_acc_iff: fixes r :: "('b \<times> 'b) set" shows "wf r = (\<forall>x. x \<in> Wellfounded.acc r)"
extract_SomeE: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and y :: "'c"   and zs :: "'c list" assumes "List.extract P xs = Some (ys, y, zs)" shows "xs = ys @ y # zs \<and> P y \<and> \<not> (\<exists>y\<in>set ys. P y)"
cases: fixes x :: "'c list" obtains xa :: "'c" where "x = [xa__]"   | xa :: "'c" and v :: "'c" and va :: "'c list" where "x = xa # v # va"   | "x = []"
arl_length_def: shows "arl_length \<equiv> \<lambda>(a, n). return n"
hyps_3: fixes xs :: "'a llist" assumes "ltl (lalternate xsa__) = lalternate xs" shows "lfinite xs"
can_coherence': fixes f :: "'c local.term set" assumes "can f" shows "mkcan (local.dom f) (cod f) = f"
Nml_HcompD_2: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "arr (un_Prim t)"
astDom_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.astDom problem \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> D"
cases: fixes a :: "('test, 'atom) com" assumes "discr0 a" obtains c :: "('test, 'atom) com" where "a = c" and "\<And>s c' s'. \<lbrakk>mustT c s; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> s \<approx> s' \<and> discr0 c'" and "\<And>s s'. \<lbrakk>mustT c s; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> s \<approx> s'"
greaterThan_insert: fixes n :: "'a" shows "insert n {n<..} = {n..}"
unat_ucast: fixes x :: "'c word" shows "unat (ucast x) = unat x mod 2 ^ LENGTH('b)"
g1: shows " t3 \<longrightarrow>\<^sub>1 t4"
while_relpower_induct: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and i :: "'e set"   and A :: "'e set"   and b :: "'e set"   and Q :: "'e set"   and c' :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate"   and x :: "'e"   and n :: "nat" assumes "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c i, A"   and "i \<inter> b \<subseteq> P"   and "i \<inter> - b \<subseteq> Q"   and "final (c', t)"   and "x \<in> i"   and "t \<notin> Fault ` F"   and "c' = Throw \<longrightarrow> t \<notin> Normal ` A"   and "\<Gamma>\<turnstile> (While b c, Normal x) \<rightarrow>\<^sup>nn (c', t)" shows "c' = Skip \<and> t \<in> Normal ` Q"
i_mult_Complex: fixes a :: "real"   and b :: "real" shows "\<i> * Complex a b = Complex (- b) a"
retract_of_contractible: fixes T :: "'a set"   and S :: "'a set" assumes "contractible T"   and "S retract_of T" shows "contractible S"
divide_nonpos_nonpos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less_eq y zero" shows "less_eq zero (divide x y)"
coprime_commute: fixes b :: "'b"   and a :: "'b" shows "algebraic_semidom_class.coprime b a = algebraic_semidom_class.coprime a b"
ax1: fixes P :: "Points"   and Q :: "Points" shows "\<exists>l. incid P l \<and> incid Q l"
ra_ia_rb_ib_rc_ic_def_5: shows "rc = Re c"
det_bezout_matrix: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'd) vec, 'c) vec"   and j :: "'d" assumes "is_bezout_ext bezout"   and "a < b"   and "A $ a $ j \<noteq> (0::'b)" shows "det (bezout_matrix A a b j bezout) = (1::'b)"
AE_pairwise: fixes F :: "'b set"   and R :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'c measure" assumes "countable F" shows "pairwise (\<lambda>A B. AE x in M. R x A B) F = (AE x in M. pairwise (R x) F)"
inject_3: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com"   and y41 :: "bexp"   and y42 :: "com"   and y43 :: "com" shows "(IF x41 THEN x42 ELSE x43 = IF y41 THEN y42 ELSE y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
substitute_cong_T: fixes T :: "'a set"   and T' :: "'a set"   and f :: "'a \<Rightarrow> 'a ttree" assumes "\<And>x. (x \<in> T) = (x \<in> T') \<or> f x = TTree.empty" shows "substitute f T = substitute f T'"
induct: fixes x :: "'a"   and S :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> group_closure S"   and "\<And>s. s \<in> insert (0::'a) S \<Longrightarrow> P s"   and "\<And>s t. \<lbrakk>s \<in> group_closure S; P s; t \<in> group_closure S; P t\<rbrakk> \<Longrightarrow> P (s - t)" shows "P x"
simps_34: fixes vd :: "'a"   and v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a" shows "(SKIP {vd} \<le> {v} WHILE va DO vb {vc}) = False"
select_convs_8: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "state_var \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = state_var"
P_def: shows "P \<equiv> \<lambda>(s', a'). s' |\<in>| (fst \<circ> tree.root) |`| cont t \<and> pred (s, a) r (s', a') \<and> inv (s', a')"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length (map of_int_hom.vec_hom fs)" shows "gs.d k = (\<Prod>j<k. \<parallel>gs.gso j\<parallel>\<^sup>2)"
Lambert_W'_leI: fixes y :: "real"   and x :: "real" assumes "y * exp y \<le> x"   and "x < 0" shows "Lambert_W' x \<le> y"
IP: fixes \<sigma> :: "'c"   and k :: "'d"   and it :: "'d set"   and v :: "'a" assumes "c \<sigma>"   and "k \<in> dom m - it"   and "invar v"   and "m k = Some (\<alpha> v)"   and "it \<subseteq> dom m"   and "I it \<sigma>" shows "I (insert k it) (f (k, v) \<sigma>)"
l_set_tag_name_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs \<and> l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
vfsequence_vrange_vcpower: fixes A :: "V" assumes "\<R>\<^sub>\<circ> xs \<subseteq>\<^sub>\<circ> A" shows "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x mod xa) = int_of_integer x mod int_of_integer xa"
vector_component_5: fixes x :: "('d, 'e) Finite_Cartesian_Product.vec"   and y :: "('d, 'e) Finite_Cartesian_Product.vec"   and i :: "'e" shows "(x - y) $h i = x $h i - y $h i"
simps_24: fixes vc :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Star vc) (Plus v va) = True"
Cons_eq_map_conv: fixes x :: "'b"   and xs :: "'b list"   and f :: "'c \<Rightarrow> 'b"   and ys :: "'c list" shows "(x # xs = map f ys) = (\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs)"
the_inv_into_into: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'b"   and B :: "'a set" assumes "inj_on f A"   and "x \<in> f ` A"   and "A \<subseteq> B" shows "the_inv_into A f x \<in> B"
filtration_def: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" shows "Filtration.filtration M F = ((\<forall>t. subalgebra M (F t)) \<and> (\<forall>s t. s \<le> t \<longrightarrow> subalgebra (F t) (F s)))"
span_zero: fixes S :: "'a set" shows "(0::'a) \<in> span S"
loose_bvar1_eta_norm': fixes t :: "Core.term"   and lev :: "nat" assumes "\<not> loose_bvar1 t lev" shows "\<not> loose_bvar1 (eta_norm t) lev"
d\<mu>_def: fixes n :: "nat"   and fs :: "int vec list"   and i :: "nat"   and j :: "nat" assumes "fs_int_indpt n fs" shows "fs_int_indpt.d\<mu> n fs i j = int_of_rat (rat_of_int (fs_int.d n fs (Suc j)) * gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)"
fvs_simp_4: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm" shows "fvs (Fn x T A) = fvs A - {x}"
rev_o_def: fixes x :: "ant" shows "rev_o x = - x"
map_le_def: fixes m\<^sub>1 :: "'g \<Rightarrow> 'h option"   and m\<^sub>2 :: "'g \<Rightarrow> 'h 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)"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "R.seq g f" shows "R.comp g f = g \<cdot> f"
circle_cube_valid_two_cube: shows "valid_two_cube circle_cube"
a1: shows "Unifier (\<delta> \<circ>\<^sub>s \<alpha>) s t"
distinct_25: shows "cospan_shape.TT \<noteq> cospan_shape.AT"
infinite_Ico_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less" shows "infinite (ord.atLeastLessThan less_eq less a b) = less a b"
real_resource_def: fixes key_rest :: "('x, otp.key.event, 'p, 'r, 't, 'v, 'x) rest_scheme"   and auth_rest :: "('y, otp.sec.event, 'q, 's, 'u, 'w, 'y) rest_scheme" shows "one_time_pad.real_resource \<L> key_rest auth_rest \<equiv> attach_c1f22_c1f22 (CNV otp.enc_callee ()) (CNV otp.dec_callee ()) (otp.key.resource key_rest) (dh.auth.resource auth_rest)"
frechet_derivative_transform_within_open: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and X :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f differentiable at x"   and "open X"   and "x \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x" shows "frechet_derivative f (at x) = frechet_derivative g (at x)"
compare_double_eq: fixes x :: "double"   and y :: "double" shows "(compare_double x y = 0) = (Code_Double.is_nan x \<and> Code_Double.is_nan y \<or> fcompare_double x y = ccode.Eq)"
one_dvd: fixes a :: "'a" shows "(1::'a) dvd a"
subring_Ring: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme" assumes "Ring R"   and "Subring R S" shows "Ring S"
finiteUniqAndConsistentTrailsWithGivenVariableSet: fixes V :: "nat set" assumes "finite V" shows "finite {M. vars (elements M) = V \<and> uniq (elements M) \<and> consistent (elements M)}"
all_PiE_elements: fixes I :: "'f set"   and S :: "'f \<Rightarrow> 'g set"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "(\<forall>z\<in>Pi\<^sub>E I S. \<forall>i\<in>I. P i (z i)) = (Pi\<^sub>E I S = {} \<or> (\<forall>i\<in>I. \<forall>x\<in>S i. P i x))"
has_derivative_within: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> ((\<lambda>y. (1 / norm (y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> (0::'d)) (at x within s))"
Ring1: fixes R :: "'a set" assumes "R_scalar_mult R" shows "Ring1 R"
path_image_nonempty: fixes g :: "real \<Rightarrow> 'a" shows "path_image g \<noteq> {}"
identify_blocks_def: fixes A :: "'c mat"   and i :: "nat" shows "identify_blocks A i = identify_blocks_main A i []"
monom_mult_in_pmdl: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c"   and t :: "'a" assumes "b \<in> B" shows "monom_mult c t b \<in> pmdl B"
tab_in_vhom': shows "\<guillemotleft>tab : p\<^sub>1 \<Rightarrow> (k\<^sup>* \<star> h) \<star> p\<^sub>0\<guillemotright>"
add_implies_diff: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and c :: "'k"   and b :: "'k"   and a :: "'k" assumes "class.cancel_comm_monoid_add plus minus zero"   and "plus c b = a" shows "c = minus a b"
pos_def: fixes l :: "real list" shows "pos l = (if l = [] then False else \<forall>e. e \<in> set l \<longrightarrow> 0 < e)"
fix_eq: shows "P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>r\<^sub>p i,Up\<^sub>r\<^sub>p i\<rbrakk> = (\<Lambda> X. (\<lambda>s. Mprefix (Tr\<^sub>r\<^sub>p i s) (\<lambda>e. X (Up\<^sub>r\<^sub>p i s e))))\<cdot>(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>r\<^sub>p i,Up\<^sub>r\<^sub>p i\<rbrakk>)"
mono_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "Sup (f ` A) \<le> f (Sup A)"
rt_fresher_update_other: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "dip \<in> kD rt"   and "dip \<noteq> ip" shows "rt \<sqsubseteq>\<^bsub>dip\<^esub> update rt ip r"
OrR_inv: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and G :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F, G, \<Delta>"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "F.satL \<xi> (Neg a) = (\<not> F.satA \<xi> a)"