hcju/isabelleps huggingface dataset:
tr_tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p: fixes A' :: "('fun, 'var) strand_step list"   and A :: "('fun, 'var) stateful_strand_step list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "A' \<in> set (tr A D)"   and "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p A"   and "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "\<forall>(t, s)\<in>set D. (fv t \<union> fv s) \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "\<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` set D. \<forall>t'\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` set D. (\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t'" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p A'"
preserves_arr: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.arr f" shows "arr (HoHV f)"
map_preserves_dom: fixes f :: "'c" assumes "Cop.arr f" shows "Cones.map (Cop.dom f) = S.dom (Cones.map f)"
asTerm_pick: fixes X :: "('f, 'g, 'varSort, 'var, 'h) qTerm set" assumes "good X" shows "asTerm (pick X) = X"
dualization_rew_3: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<not> \<box>[A]_v \<equiv> \<diamond>\<langle>\<not> A\<rangle>_v"
simps_1: fixes lens_get :: "'b \<Rightarrow> 'c"   and lens_put :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and more :: "'d" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
eq0_2: shows "of_nat (Suc k) * (c / of_nat (Suc k)) = c"
step_Stuck: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Stuck" shows "s' = Stuck"
tensor\<^sub>S_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_strict_monoidal_category C" shows "free_strict_monoidal_category.tensor\<^sub>S C f g \<equiv> free_monoidal_category.D C (elementary_monoidal_category.T (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C) (f, g))"
spanset_reduce_Cons: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list"   and m :: "'b" assumes "RModule R smult M"   and "set ms \<subseteq> M"   and "m \<in> R_scalar_mult.RSpan R smult ms" shows "R_scalar_mult.RSpan R smult (m # ms) = R_scalar_mult.RSpan R smult ms"
affine_hull_convex_Int_nonempty_interior: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "S \<inter> interior T \<noteq> {}" shows "affine hull (S \<inter> T) = affine hull S"
typings_lengths: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and es :: "exp list"   and Cs :: "nat list" assumes "CT;\<Gamma> \<turnstile>+ es : Cs" shows "length es = length Cs"
Group_Qg: fixes N :: "'a set" assumes "G \<triangleright> N" shows "Group (G / N)"
get_ancestors_si_def: fixes ptr :: "(_) object_ptr" shows "get_ancestors_si ptr = Heap_Error_Monad.bind (check_in_heap ptr) (\<lambda>_. Heap_Error_Monad.bind (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> return [] | Some shadow_root_ptr \<Rightarrow> Heap_Error_Monad.bind (get_host shadow_root_ptr) (\<lambda>host. get_ancestors_si (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)) | Some node_ptr \<Rightarrow> Heap_Error_Monad.bind (get_parent node_ptr) (\<lambda>parent_ptr_opt. case parent_ptr_opt of None \<Rightarrow> return [] | Some parent_ptr \<Rightarrow> get_ancestors_si parent_ptr)) (\<lambda>ancestors. return (ptr # ancestors)))"
interp_subseteq_Interp: fixes C :: "'a literal multiset" shows "interp C \<subseteq> Interp C"
le_imp_less_or_eq: fixes x :: "'a"   and y :: "'a" assumes "le x y" shows "lt x y \<or> x = y"
CANDs_imply_SNa_2: fixes x :: "name"   and M :: "trm"   and B :: "ty" assumes "x:M \<in> \<parallel>(B)\<parallel>" shows "SNa M"
proves_eq_transitive_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> s"   and "term_ok \<Theta> t"   and "term_ok \<Theta> u"   and "typ_of s = typ_of t"   and "typ_of t = typ_of u"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq s t"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq s u"
simps_1: fixes pat :: "[:'r:]"   and x :: "'r" shows "op2\<cdot>pat\<cdot>Null\<cdot>x = root2\<cdot>pat"
lens_plus_eq_left: fixes X :: "'d \<Longrightarrow> 'e"   and Z :: "'f \<Longrightarrow> 'e"   and Y :: "'g \<Longrightarrow> 'e" assumes "X \<bowtie> Z"   and "X \<approx>\<^sub>L Y" shows "X +\<^sub>L Z \<approx>\<^sub>L Y +\<^sub>L Z"
left_neutral: fixes a :: "'b" shows "(0::'b) + a = a"
SUP_upper2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and i :: "'d"   and A :: "'d set"   and u :: "'b"   and f :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq u (f i)" shows "less_eq u (Sup (f ` A))"
intros: fixes f :: "'b"   and A :: "'b set"   and ts :: "'b tree list" assumes "f \<in> A"   and "\<forall>t\<in>set ts. t \<in> trees A" shows "Node f ts \<in> trees A"
inorder_update: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c"   and y :: "'d" assumes "t \<in> T h"   and "sorted1 (inorder t)" shows "inorder (update x y t) = upd_list x y (inorder t)"
in_homE: fixes f :: "(bool, 'arr list) concrete_category.arr"   and a :: "(bool, 'arr list) concrete_category.arr"   and b :: "(bool, 'arr list) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "arr f" and "local.dom f = a" and "cod f = b"
iter_unfold: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_or_infinite_iteration.iter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) = nil \<sqinter> seq c (finite_or_infinite_iteration.iter seq nil c)"
list_rel_takeD: fixes a :: "'a list"   and b :: "'b list"   and R :: "('a \<times> 'b) set"   and n :: "nat"   and n' :: "nat" assumes "(a, b) \<in> \<langle>R\<rangle>list_rel"   and "(n, n') \<in> nat_rel" shows "(take n a, take n' b) \<in> \<langle>R\<rangle>list_rel"
fold_keys_def_alt: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('c, 'd) RBT.rbt"   and s :: "'b" shows "fold_keys f t s = fold f (RBT.keys t) s"
power_lb: fixes X :: "'a list set"   and k :: "nat"   and n :: "nat" shows "(\<forall>w\<in>X. k \<le> |w|) \<longrightarrow> (\<forall>w. w \<in> X\<^bsup>Suc n\<^esup> \<longrightarrow> n \<cdot> k \<le> |w|)"
useful_tautologies_9: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> (\<^bold>\<not>\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> \<psi>) in v]"
rep_number: fixes x :: "'a" assumes "x \<in> \<V>" shows "\<B> rep x = \<r>"
y: shows "y = (abs_int_poly (poly_uminus p), - r, - l)"
bounded_linear_scaleR_const: fixes g :: "'d \<Rightarrow> real"   and x :: "'e" assumes "bounded_linear g" shows "bounded_linear (\<lambda>xa. g xa *\<^sub>R x)"
cases: fixes a1 :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and a2 :: "('thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action) llist" assumes "h.mthr.if.mthr.Runs a1 a2" obtains s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" where "a1 = s__" and "a2 = LNil" and "\<And>tl s'. \<not> h.mthr.if.redT s__ tl s'"   | s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" and tls :: "('thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action) llist" and tl :: "'thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action" where "a1 = s" and "a2 = LCons tl tls" and "h.mthr.if.redT s tl s'" and "h.mthr.if.mthr.Runs s' tls"
intro: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc_same_face G M a"   and "bidel_arc_tail_conn G M a" shows "bidel_arc_same_face_tail_conn G M a"
residual_l_pure_induction: fixes x :: "'a" shows "(x \<leftarrow> x)\<^sup>\<star> \<le> x \<leftarrow> x"
in_pair_same: fixes p :: "'e \<times> 'e"   and A :: "'e set" shows "(p \<in>\<^sub>p A \<times> A) = (p \<in> A \<times> A)"
list_ex1_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "list_ex1 P xs = (\<exists>!x. x \<in> set xs \<and> P x)"
nat_pow_pow: fixes x :: "'a mod_ring poly"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "(x [^]\<^bsub>R\<^esub> n) [^]\<^bsub>R\<^esub> m = x [^]\<^bsub>R\<^esub> (n * m)"
simps: fixes V :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set" shows "filterQuery V Q = Set.filter (\<lambda>(s, uu_). \<not> Set.is_empty (s \<inter> V)) Q"
helper: fixes n :: "nat" shows "\<Q> (length Doc + n) = \<PP>"
sync_assoc_oneside: fixes P :: "'a process"   and C :: "'a set"   and Q :: "'a process"   and S :: "'a process" shows "(P \<lbrakk>C\<rbrakk> Q \<lbrakk>C\<rbrakk> S) \<le> (P \<lbrakk>C\<rbrakk> (Q \<lbrakk>C\<rbrakk> S))"
select_result_I2: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = x"
Framer: fixes C :: "com"   and ps1 :: "char list \<Rightarrow> int option"   and m :: "nat"   and ps1' :: "char list \<Rightarrow> int option"   and ps2 :: "char list \<Rightarrow> int option" assumes "(C, ps1) \<Rightarrow>\<^sub>A m \<Down> ps1'"   and "ps1 ## ps2" shows "(C, ps1 + ps2) \<Rightarrow>\<^sub>A m \<Down> ps1' + ps2"
fmfilter_comm: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "fmfilter P (fmfilter Q m) = fmfilter Q (fmfilter P m)"
lsum_def: fixes f :: "'a \<Rightarrow> nat"   and al :: "'a list" shows "lsum f al \<equiv> sum_list (map f al)"
coeff_Var: fixes i :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (Var i) m = ((1::'b) when m = Poly_Mapping.single i 1)"
sel: fixes x :: "'a \<Rightarrow> bool" shows "pred.eval (pred.Pred x) = x"
fls_inverse_eq_0': fixes f :: "'a fls" assumes "inverse (fls_nth f (fls_subdegree f)) = (0::'a)" shows "inverse f = 0"
extendf_deriv_continuous: fixes f' :: "('sf + 'sz, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and i :: "'sf"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "dfree f'"   and "is_interp I" shows "continuous_on UNIV (\<lambda>x. Blinfun (extendf_deriv I i f' \<nu> x))"
distinct_9: fixes x1 :: "char list \<times> Unary_PCF.type"   and x61 :: "Unary_PCF.type"   and x62 :: "expr" shows "\<langle>x1\<rangle> \<noteq> \<Lambda>\<langle>x61\<rangle> x62"
N2H_nz: fixes n :: "nat"   and c :: "nat" assumes "0 < n" shows "N2H (Suc (Suc c)) n \<noteq> C []"
merges_pres_le_ub: fixes ts :: "'a list"   and ss :: "'a list"   and ps :: "(nat \<times> 'a) list" assumes "set ts \<subseteq> A"   and "set ss \<subseteq> A"   and "\<forall>(p, t)\<in>set ps. t \<sqsubseteq>\<^bsub>r\<^esub> ts ! p \<and> t \<in> A \<and> p < length ts"   and "ss [\<sqsubseteq>\<^bsub>r\<^esub>] ts" shows "merges f ps ss [\<sqsubseteq>\<^bsub>r\<^esub>] ts"
ps_def: shows "ps \<equiv> smods p (pderiv p * q)"
not_arg_cong_Inr: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "Inr x \<noteq> Inr y"
f_bound: fixes x :: "real" assumes "a \<le> x" shows "norm (f x) \<le> g' x"
simps_2: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" shows "\<Phi> \<langle>l, a, r\<rangle> = \<Phi> l + \<Phi> r + \<phi> \<langle>l, a, r\<rangle>"
hdomain_restr: fixes r :: "hf"   and x :: "hf" shows "hdomain (hrestrict r x) = hdomain r \<sqinter> x"
mon_ww_of_le_rem: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and w :: "('c, 'f) label list el_step list" shows "mon_ww fg (map le_rem_s w) = mon_loc fg w \<union> mon_env fg w"
finite_incidence_system_axioms: shows "finite_incidence_system \<V> (multiple_blocks n)"
is_path_simps_2: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "valid_graph G" shows "valid_graph.is_path G v [(v, w, v')] v' = ((v, w, v') \<in> edges G)"
frag_extend_def: fixes b :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and x :: "'d \<Rightarrow>\<^sub>0 int" shows "frag_extend b x \<equiv> \<Sum>i\<in>Poly_Mapping.keys x. frag_cmul (poly_mapping.lookup x i) (b i)"
i: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list" assumes "spectral_radius_1_jnf_max A n m lam n_as" shows "spectral_radius_1_jnf_max.i n m n_as \<in> spectral_radius_1_jnf_max.I n m n_as"
nth_rotate: fixes n :: "nat"   and xs :: "'g list"   and m :: "nat" assumes "n < length xs" shows "rotate m xs ! n = xs ! ((m + n) mod length xs)"
set_valued_functor_axioms: shows "set_valued_functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a)"
card_eq_dim: fixes B :: "('b, 'c) vec set"   and V :: "('b, 'c) vec set" assumes "B \<subseteq> V"   and "card B = vec.dim V"   and "finite B" shows "vec.independent B = (V \<subseteq> vec.span B)"
fps_exp_def: fixes x :: "'a" shows "fps_exp x = Abs_fps (\<lambda>n. x ^ n / of_nat (fact n))"
Ana_Fu_cases_2: fixes f :: "('fun, 'atom, 'sets) prot_fun"   and T :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and K :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and g :: "'fun"   and K' :: "(('fun, 'atom, 'sets) prot_fun, nat) Term.term list"   and M' :: "nat list" assumes "Ana (Fun f T) = (K, M)"   and "f = Fu g"   and "Ana\<^sub>f g = (K', M')" shows "(K, M) = (K' \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t (!) T, map ((!) T) M') \<or> (K, M) = ([], [])"
Fract_add_one: fixes n :: "int"   and m :: "int" assumes "n \<noteq> 0" shows "Fract (m + n) n = Fract m n + 1"
not_mem_eq_mem_not: fixes a :: "'a"   and A :: "'a set" shows "(a \<notin> A) = (a \<in> - A)"
image_dual: shows "\<P> uminus \<circ> \<P> uminus = id"
curryI: fixes f :: "'b \<times> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "f (a, b)" shows "curry f a b"
b_dvd_a1: shows "gcd a1 a2 dvd a1"
subset_add_mset_notin_subset_mset: fixes A :: "'a multiset"   and b :: "'a"   and B :: "'a multiset" assumes "A \<subseteq># add_mset b B"   and "b \<notin># A" shows "A \<subseteq># B"
permutes_comp_preserves_cyclic2: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and B :: "'b set" assumes "f permutes A"   and "cyclic_on g C"   and "A \<inter> B = {}"   and "C \<subseteq> B" shows "cyclic_on (f \<circ> g) C"
is_wt_instance_of_condD': fixes t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "wf\<^sub>t\<^sub>r\<^sub>m s"   and "is_wt_instance_of_cond \<Gamma> t s" shows "\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = s \<cdot> \<delta>"
fps_tan_def: fixes c :: "'a" shows "fps_tan c = fps_sin c / fps_cos c"
beforeM: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = body in D" shows "compP\<^sub>2 P,D,M,0 \<rhd> compE\<^sub>2 body @ [Return]"
w_decomp: shows "w = (acs @ b # ds) \<bullet>\<cdot> (ufvs @ y # bvs)"
elims: fixes x :: "variable"   and xa :: "fml option"   and y :: "fml option" assumes "Existso x xa = y" obtains xb :: "variable" and \<phi> :: "fml" where "x = xb__" and "xa = Afml \<phi>" and "y = Afml (Exists xb__ \<phi>)"   | xb :: "variable" where "x = xb" and "xa = undeff" and "y = undeff"
image_constant_conv: fixes c :: "'b"   and A :: "'c set" shows "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"
adjoint_dotsubstt: fixes \<theta> :: "trm"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real" shows "USubst.adjoint (dotsubstt \<theta>) I \<omega> = repc I dotid (term_sem I \<theta> \<omega>)"
L7: fixes k :: "nat"   and xs :: "'b list" assumes "k < length xs" shows "map ((!) xs) [0..<k + 1] = take (k + 1) xs"
args_to_set_subset_Times: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)" shows "args_to_set (gs, bs, ps) = fst ` set gs \<union> fst ` set bs"
r_div_eq: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e"   and b :: "'e" assumes "Group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "a \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> b = \<one>\<^bsub>G\<^esub>" shows "a = b"
le_ref1: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<le> Q" shows "D Q \<subseteq> D P"
assoc_naturality: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "D.arr \<tau>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<tau>" shows "\<a>\<^sub>D[D.cod \<mu>, D.cod \<nu>, D.cod \<tau>] \<cdot>\<^sub>D ((\<mu> \<star>\<^sub>D \<nu>) \<star>\<^sub>D \<tau>) = (\<mu> \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>) \<cdot>\<^sub>D \<a>\<^sub>D[D.dom \<mu>, D.dom \<nu>, D.dom \<tau>]"
S_def: shows "S \<equiv> \<lambda>n. bernoulli (2 * Suc N) / (real (2 * Suc N) * real n ^ (2 * Suc N))"
fmdrop_fmupd: fixes x :: "'c"   and y :: "'c"   and z :: "'d"   and m :: "('c, 'd) fmap" shows "fmdrop x (fmupd y z m) = (if x = y then fmdrop x m else fmupd y z (fmdrop x m))"
converse_lehmer_weak: fixes p :: "nat" assumes "prime p" shows "\<exists>a. [a ^ (p - 1) = 1] (mod p) \<and> (\<forall>x>0. x \<le> p - 2 \<longrightarrow> [a ^ x \<noteq> 1] (mod p)) \<and> 0 < a \<and> a < p"
ide_cod: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.ide (B.VV.cod f)"
alpha_trans_twice: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X'' :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X''' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood X \<or> qGood X' \<or> qGood X'' \<or> qGood X'''"   and "X #= X'"   and "X' #= X''"   and "X'' #= X'''" shows "X #= X'''"
eqp: shows "\<forall>A\<in>sets N. LINT w|M. max 0 (X w) * indicat_real (Y -` A \<inter> space M) w = LINT w|M. expl_cond_expect M Y (\<lambda>w. max 0 (X w)) w * indicat_real (Y -` A \<inter> space M) w"
finite_IN_g: fixes x :: "'v" assumes "x \<noteq> sink \<Delta>" shows "d_IN g x \<noteq> \<top>"
simps_1: fixes c :: "poly"   and p :: "poly" shows "Reflected_Multivariate_Polynomial.degree (CN c 0 p) = 1 + Reflected_Multivariate_Polynomial.degree p"
snd_conv: fixes x1 :: "'c"   and x2 :: "'b" shows "final_b (x1, x2) = x2"
int_transE: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wt T"   and "I.wtE \<xi>"   and "\<And>x. \<lbrakk>\<not> infTp (tpOfV x); x \<in> nv2T T\<rbrakk> \<Longrightarrow> \<exists>l. wtL l \<and> \<not> I.satL \<xi> l \<and> isGuard x l" shows "Ik.int (transE \<xi>) T = I.int \<xi> T"
iso_cancel_right: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b"   and f' :: "'b" assumes "category C"   and "category.iso C g"   and "C f g = C f' g"   and "partial_magma.arr C (C f g)"   and "category.iso C g" shows "f = f'"
integral_cong: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L N g"
sup_Sup: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "sup_prod x y = Sup_prod (insert x (\<eta> y))"
UP_DOWN: fixes t :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "t \<in> Univ" shows "UP (DOWN t) = t"
of_nat_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 1 = one"
less_BEState_ext_def: fixes x :: "('a, 'b, 'c, 'd) BEState_scheme"   and y :: "('a, 'b, 'c, 'd) BEState_scheme" shows "x < y \<equiv> es x < es y \<or> es x = es y \<and> (ps x < ps y \<or> ps x = ps y \<and> BEState.more x < BEState.more y)"
spr_simRels_def: shows "spr_simRels \<equiv> \<lambda>a. rel_ext (\<lambda>uu_. \<exists>U u V v. uu_ = ((U, u), V, v) \<and> U = V \<and> {u, v} \<subseteq> U \<and> envObs a u = envObs a v)"
linorder_linear: fixes x :: "'i"   and y :: "'i" shows "x \<le> y \<or> y \<le> x"
l4_ports_negate_one_nodisc: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and pt :: "ipt_l4_ports" assumes "\<forall>a. \<not> disc (C a)"   and "\<forall>a. \<not> disc (Prot a)" shows "\<not> has_disc disc (l4_ports_negate_one C pt)"
n: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and x :: "'a" assumes "R N M" shows "n N M x \<in> space (prob_algebra (stream_space (count_space UNIV)))"