hcju/isabelleps huggingface dataset:
converse_Times: fixes A :: "'b set"   and B :: "'a set" shows "(A \<times> B)\<inverse> = B \<times> A"
Id_on_domain: fixes A :: "'b set"   and f :: "('b \<times> 'c) set" shows "Domain (Id_on A O f) = A \<inter> Domain f"
assms_2: shows "approx \<equiv> \<Sum>k<n. 2 * y ^ (2 * k + 1) / real (2 * k + 1)"
guardedSeqSubst: fixes P :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list" assumes "guarded P"   and "wellFormedSubst \<sigma>" shows "guarded (P[<\<sigma>>])"
swaps_eq_nth_take_drop: fixes i :: "nat"   and xs :: "'c list" assumes "i < length xs" shows "swaps [0..<i] xs = xs ! i # take i xs @ drop (Suc i) xs"
IH_1: fixes V :: "nat set"   and \<sigma> :: "nat \<Rightarrow> val" assumes "randomfree e1"   and "free_vars e1 \<subseteq> V"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "return_val (expr_sem_rf \<sigma> e1) = expr_sem \<sigma> e1"
xk'_2: shows "(x, k) \<noteq> (x', k')"
quad_part_1_less: fixes var :: "nat"   and L :: "real list"   and p :: "real mpoly"   and deg :: "nat"   and D :: "real"   and a :: "real mpoly"   and A :: "real"   and b :: "real mpoly"   and B :: "real"   and d :: "real mpoly"   and C :: "real" assumes "var < length L"   and "MPoly_Type.degree p var = deg"   and "D \<noteq> 0"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = A"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = B"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = D" shows "aEval (Less p) (L[var := (A + B * C) / D]) = aEval (Less (quadratic_part_1 var a b d (Less p))) (L[var := C])"
subsumees_conv: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "subsumees subs = {v. \<exists>v'. (v, v') \<in> subs}"
min_union_assoc: fixes A :: "'b fset set"   and B :: "'b fset set"   and C :: "'b fset set" shows "(A \<union>\<^sub>m B) \<union>\<^sub>m C = A \<union>\<^sub>m B \<union>\<^sub>m C"
shiftr1_eq_div_2: fixes w :: "'b word" shows "shiftr1 w = w div 2"
iter_bind_iter: fixes m :: "nat"   and s :: "'a"   and n :: "nat" shows "fuse.iter m s \<bind> fuse.iter n = fuse.iter (m + n) s"
carrierE: fixes x :: "'a" assumes "x \<in> carrier" obtains c :: "('a, 'b) chart" where "c \<in> charts" and "x \<in> domain c"
whisker_left: fixes f :: "'a"   and \<nu> :: "'a"   and \<mu> :: "'a" assumes "ide f"   and "seq \<nu> \<mu>" shows "f \<star> \<nu> \<cdot> \<mu> = (f \<star> \<nu>) \<cdot> (f \<star> \<mu>)"
type_less_x_Collection_intro_3: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = OrderedSet \<rho>"   and "\<rho> \<le> \<sigma>" shows "\<tau> < Collection \<sigma>"
Int_space_eq1: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and x :: "'c set" assumes "semiring_of_sets \<Omega> M"   and "x \<in> M" shows "\<Omega> \<inter> x = x"
is_generalisation_of_guard_subset: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" assumes "is_generalisation_of t' t i r" shows "set (Guards t') \<subseteq> set (Guards t)"
max_ext_additive: fixes A :: "'k set"   and B :: "'k set"   and R :: "('k \<times> 'k) set"   and C :: "'k set"   and D :: "'k set" assumes "(A, B) \<in> max_ext R"   and "(C, D) \<in> max_ext R" shows "(A \<union> C, B \<union> D) \<in> max_ext R"
bij_betw_imp_bij_betw_Pow: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "bij_betw ((`) f) (Pow A) (Pow B)"
t0: shows "t0 \<in> set (queue_\<alpha> queue)"
sharing_consis: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "sharing_consistent \<S> \<O> sb"
finite_maxlen: fixes M :: "'g list set" assumes "finite M" shows "\<exists>n. \<forall>s\<in>M. length s < n"
the_inv_into_onto: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "the_inv_into A f ` f ` A = A"
t_lt_order: shows "\<t> \<le> sys2.\<v>"
closest_supchamberD_1: fixes X :: "'b set set"   and F :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "F \<in> X"   and "SimplicialComplex.maxsimp X D" shows "SimplicialComplex.maxsimp X (ChamberComplex.closest_supchamber X F D)"
iterate_M_concat: fixes h :: "'a"   and xs :: "('a, 'b, 'c) prog list"   and h' :: "'a"   and ys :: "('a, 'b, 'c) prog list"   and h'' :: "'a" assumes "h \<turnstile> iterate_M xs \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> iterate_M ys \<rightarrow>\<^sub>h h''" shows "h \<turnstile> iterate_M (xs @ ys) \<rightarrow>\<^sub>h h''"
simps_1: shows "size_tree_tm \<langle>\<rangle> =1 return 0"
small_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> o[F](g1)"   and "f2 \<in> o[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> o[F](\<lambda>x. g1 x * g2 x)"
simps_6: fixes b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "Language.normalize (While b c) = While b (Language.normalize c)"
l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, 'w option) prog) \<Rightarrow> ((_) heap, exception, 'w option) prog" shows "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order \<equiv> first_in_tree_order = l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order to_tree_order"
at_pi_rev: fixes pi :: "('a \<times> 'a) list"   and x :: "'a" assumes "at TYPE('a)" shows "pi \<bullet> rev pi \<bullet> x = x"
map_tree_primrec: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c tree" shows "map_tree f t = rec_tree \<langle>\<rangle> (\<lambda>l a r l'. Node l' (f a)) t"
not_fresh_nat_of: fixes a :: "atom" shows "\<not> a \<sharp> nat_of"
det_def: fixes A :: "'b mat" shows "Determinant.det A = (if dim_row A = dim_col A then \<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (i, p i)) else (0::'b))"
l11_44_2_a: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.Lt Bet Cong B A B C" shows "Tarski_neutral_dimensionless.LtA Bet Cong B C A B A C"
ik_append: fixes S :: "('d, 'e) strand_step list"   and S' :: "('d, 'e) strand_step list" shows "ik\<^sub>s\<^sub>t (S @ S') = ik\<^sub>s\<^sub>t S \<union> ik\<^sub>s\<^sub>t S'"
simps_1: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'b" shows "prg \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = prg"
strong_rev_children: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.tree_invar t" shows "BinomialHeapStruc.invar (rev (BinomialHeapStruc.children t))"
arrI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "C'.in_hom f a b" shows "C'.arr f"
in_qtableI: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and x :: "'a option list" assumes "qtable n A P Q X"   and "wf_tuple n A x"   and "P x"   and "Q x" shows "x \<in> X"
mvmult_fa_def: fixes n :: "nat"   and m :: "nat"   and AS :: "floatarith list"   and B :: "floatarith list" shows "mvmult_fa n m AS B = map (\<lambda>i. inner_floatariths (map (\<lambda>j. AS ! (i * m + j)) [0..<m]) (map ((!) B) [0..<m])) [0..<n]"
gc_marking_reaches_Mutate: fixes x :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and y :: "'e"   and p :: "'f process_name"   and ws :: "('g, 'h, 'e) mem_store_action list"   and r :: "'e"   and f :: "'g"   and opt_r' :: "'e option"   and m :: "'f" assumes "\<forall>y. (x reaches y) s \<longrightarrow> obj_at (\<lambda>s. True) y s"   and "(x reaches y) (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := opt_r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>))"   and "sys_mem_store_buffers (mutator m) s = mw_Mutate r f opt_r' # ws"   and "valid_refs_inv s" shows "obj_at (\<lambda>s. True) y s"
Red1LAss: fixes V :: "nat"   and l :: "'addr val list"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and v :: "'addr val"   and h :: "'heap" assumes "V < length l" shows "uf,P,t \<turnstile>1 \<langle>V:=Val v,(h, l)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>unit,(h, l[V := v])\<rangle>"
discrete_topology_def: fixes U :: "'c set" shows "discrete_topology U \<equiv> topology (\<lambda>S. S \<subseteq> U)"
seq_INF_distrib_UNIV: fixes c :: "'a::refinement_lattice"   and d :: "'b::type \<Rightarrow> 'a::refinement_lattice" shows "(c::'a::refinement_lattice) ; \<Sqinter> range (d::'b::type \<Rightarrow> 'a::refinement_lattice) = (\<Sqinter>x::'b::type. c ; d x)"
intro: fixes M :: "'c measure" assumes "emeasure M (space M) = 1" shows "prob_space_axioms M"
simps_19: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and f3 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b"   and f4 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b"   and x41 :: "PreSimplyTyped.type"   and x42 :: "PreSimplyTyped.type" shows "(case TPair x41 x42 of TUnit \<Rightarrow> f1 | TVar x \<Rightarrow> f2 x | TArr x xa \<Rightarrow> f3 x xa | TPair x xa \<Rightarrow> f4 x xa) = f4 x41 x42"
not_in_I: fixes x :: "nat"   and y :: "nat" assumes "x \<notin> I" shows "eval psi [x, y] = eval r_phi [j\<^sub>2, y]"
linear_subspace_image: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and S :: "('a, 'b) vec set" assumes "linear (*s) (*s) f"   and "vec.subspace S" shows "vec.subspace (f ` S)"
assms_2: shows "tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>s\<^sub>t ` \<S>\<^sub>0))"
InitBlockThrow: fixes V :: "char list"   and T :: "ty"   and v :: "val"   and a :: "nat"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; Throw a},s,b\<rangle> \<rightarrow> \<langle>Throw a,s,b\<rangle>"
pow_mult_distrib: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and n :: "nat" assumes "Group.monoid G"   and "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> y) [^]\<^bsub>G\<^esub> n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> y [^]\<^bsub>G\<^esub> n"
prems_9: shows "preds (slice_kinds S (a # as)) s"
lfp_cs_unfold: fixes S :: "(char list \<Rightarrow> int) set"   and c :: "com" shows "Collecting.lfp (step S) c = step S (Collecting.lfp (step S) c)"
gcs_plus_lcs: fixes s :: "'a"   and t :: "'a" shows "gcs s t + lcs s t = s + t"
cc_restr_predCC: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and f :: "Arity \<rightarrow> CoCalls"   and n :: "Arity" shows "predCC S' f\<cdot>n G|` S = predCC (S' \<inter> S) (\<Lambda> n. f\<cdot>n G|` S)\<cdot>n"
simps_1: fixes y :: "real_min_top" shows "PInfty + y = y"
igOpIPresIGWls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gOpPresGWls MOD" shows "igOpIPresIGWls (fromMOD MOD)"
sset_def: fixes xs :: "[:'d:]" shows "sset xs = {x. slistmem x xs}"
wlp_measure_expand_m: fixes m :: "nat"   and n :: "nat"   and dims :: "nat list"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and P :: "complex mat" assumes "m \<le> n"   and "state_sig.well_com dims (Measure n M S)" shows "state_sig.wlp dims (Measure m M S) P = matrix_sum (state_sig.d dims) (\<lambda>k. adjoint (M k) * state_sig.wlp dims (S ! k) P * M k) m"
intros_2: fixes ds :: "'d set"   and a :: "'x set"   and x :: "'x set" assumes "COP.iterates_abovep ds a x" shows "COP.iterates_abovep ds a (cop_F ds x)"
Sum_Icc_nat: fixes m :: "nat"   and n :: "nat" shows "\<Sum> {m..n} = (n * (n + 1) - m * (m - 1)) div 2"
open_continuous_vimage': fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "open s"   and "continuous_on s f"   and "open B" shows "open (s \<inter> f -` B)"
IH: assumes "P \<turnstile> C sees_methods Mm" shows "\<exists>Mm' Mm2. P \<turnstile> C' sees_methods Mm' \<and> Mm' = Mm ++ Mm2 \<and> (\<forall>M m D. Mm2 M = \<lfloor>(m, D)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C)"
token_run_enter_final_states: fixes x :: "nat"   and n :: "nat" assumes "\<UU>.token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q}" shows "\<exists>m\<ge>x. \<UU>.token_run x m \<notin> {q. \<G> \<Turnstile>\<^sub>P Rep q} - {Abs (Unf\<^sub>G \<phi>)} \<and> \<UU>.token_run x (Suc m) \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q}"
l_get_dom_component_get_elements_by_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_elements_by_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
cmp_simps'_4: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.dom (\<Phi> (\<mu>, \<nu>)) = F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>)"
inv_o_cancel: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f" shows "inv f \<circ> f = id"
ldrop_all: fixes xs :: "'a llist"   and m :: "enat" assumes "llength xs \<le> m" shows "ldrop m xs = LNil"
image_Nsetn_card_pos: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "0 < card (f ` {i. i \<le> n})"
boundedI: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<le> a"   and "c \<le> a" shows "sup b c \<le> a"
A: shows "A = one_dim_iso A *\<^sub>C id_cblinfun"
WT_restD_rinit: fixes \<I>_adv :: "('r, 's) \<I>"   and \<I>_usr :: "('t, 'u) \<I>"   and I :: "'v \<Rightarrow> bool"   and rest :: "('v, 'w, 'r, 't, 's, 'u, 'v) rest_scheme" assumes "WT_rest \<I>_adv \<I>_usr I rest" shows "I (rinit rest)"
paint_inv2: fixes t :: "('a, 'b) rbt"   and c :: "color" assumes "inv2 t" shows "inv2 (paint c t)"
swapAbs_preserves_good: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "goodAbs A" shows "goodAbs (A $[x \<and> y]_xs)"
IH: shows "pick I (card {a \<in> I. a < i}) = (LEAST a. a \<in> I \<and> i \<le> a)"
arrI: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "f \<noteq> Null"   and "Dom f \<in> {A. |A| <o \<AA>}"   and "Cod f \<in> {A. |A| <o \<AA>}"   and "Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)" shows "arr f"
column_iarray_def: fixes k :: "nat"   and A :: "'d iarray iarray" shows "column_iarray k A = IArray.of_fun (\<lambda>m. A !! m !! k) (IArray.length A)"
mem_scale: fixes x :: "'a \<Rightarrow> real"   and a :: "real" assumes "x \<in> diff_fun_space" shows "a *\<^sub>R x \<in> diff_fun_space"
label_def: shows "label \<equiv> fst \<circ> snd"
append_ap_distrib: fixes fs :: "('e \<Rightarrow> 'd) list"   and gs :: "('e \<Rightarrow> 'd) list"   and xs :: "'e list" shows "(fs @ gs) \<diamondop> xs = fs \<diamondop> xs @ gs \<diamondop> xs"
return_pd: fixes x :: "'n" assumes "x \<in> local.nodes" shows "local.return pd\<rightarrow> x"
frac_1_eq: fixes x :: "'d" shows "frac (x + (1::'d)) = frac x"
r_rchild_eq_r_if_well_shaped: fixes t :: "'b tree" assumes "well_shaped t"   and "l t < r t" shows "r (rchild t) = r t"
pin: shows "p \<in># mset_set (mset_set ` P)"
hext_arrD: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and h' :: "'a"   and a :: "'b"   and T :: "ty"   and n :: "nat" assumes "heap_base.hext typeof_addr h h'"   and "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>" shows "typeof_addr h' a = \<lfloor>Array_type T n\<rfloor>"
divisor_sigma_prime_power_field: fixes p :: "nat"   and x :: "'b"   and k :: "nat" assumes "prime p" shows "divisor_sigma x (p ^ k) = (if nat_power p x = (1::'b) then of_nat (k + 1) else (nat_power p x ^ Suc k - (1::'b)) / (nat_power p x - (1::'b)))"
compute_rflow_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity" shows "compute_rflow cf \<equiv> assert (RGraph c s t cf) \<bind> (\<lambda>_. return (flow_of_cf cf))"
scale_sum_right: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "module scale" shows "scale a (sum f A) = (\<Sum>x\<in>A. scale a (f x))"
lipschitz_on_leI: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and L :: "real" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X; x \<le> y\<rbrakk> \<Longrightarrow> dist (f x) (f y) \<le> L * dist x y"   and "0 \<le> L" shows "L-lipschitz_on X f"
bij_betw_linorder_of_list: fixes A :: "'b set" assumes "finite A" shows "bij_betw linorder_of_list (permutations_of_set A) {R. linorder_on A R}"
that: fixes x :: "'c" assumes "x \<in> X" shows "f x \<in>\<^sub>i g x"
qFresh_imp_ex_qAFresh1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and z :: "'var" assumes "qGood X"   and "qFresh zs z X" shows "\<exists>X'. qGood X' \<and> X #= X' \<and> qAFresh zs z X'"
surjective: fixes r :: "('e, 'f, 'g) spec_scheme" shows "r = \<lparr>init = init r, trans = TS.trans r, obs = obs r, \<dots> = spec.more r\<rparr>"
1: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G" shows "\<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> b \<and> (f \<notin> F' \<longrightarrow> q f = 0))"
pid_init_eq: fixes gc :: "('c, 'ls, 'gs) global_config" shows "(gc \<in> init) = (gc = pidgc_\<alpha> pid_init)"
rev_morph_arg_rev: fixes f :: "'c list \<Rightarrow> 'b list"   and u :: "'c list" shows "rev_morph f (rev u) = rev (f u)"
SAT_def: shows "SAT \<equiv> \<lambda>\<Gamma> f. {x. SATp \<Gamma> f x}"
per_drop_exp: fixes u :: "'b list"   and r :: "'b list"   and m :: "nat" assumes "u \<le>p r \<^sup>@ m\<^sup>\<omega>" shows "u \<le>p r\<^sup>\<omega>"
in_list_induct_helper: fixes xs :: "'a list"   and X :: "'a set"   and P :: "'a list \<Rightarrow> bool" assumes "set xs \<subseteq> X"   and "P []"   and "\<And>x xs. \<lbrakk>x \<in> X; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
comp_ide_self: fixes a :: "'c" assumes "C.ide a" shows "C a a = a"
prv_cnj_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp (cnj \<phi> \<chi>) \<psi>)" shows "prv (imp \<phi> (imp \<chi> \<psi>))"
assms_3: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < f i" shows "A $$ (i, j) = (0::'a)"
exp_first_two_terms: fixes x :: "'b" shows "exp x = (1::'b) + x + (\<Sum>n. x ^ (n + 2) /\<^sub>R fact (n + 2))"
grid_def: fixes t :: "nat \<Rightarrow> real" shows "grid t \<equiv> \<forall>i. t i \<le> t (Suc i)"
distinct_sorted_rev: fixes xs :: "'a list" assumes "Sorted_Less.sorted (rev xs)" shows "distinct xs"
simps_1: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat"   and y21 :: "'b"   and y22 :: "'b tree"   and y23 :: "'b tree"   and y24 :: "nat" shows "(MKT x21 x22 x23 x24 = MKT y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
natural_transformation_axioms: shows "natural_transformation B D (L \<circ> G) (M \<circ> H) (\<mu> \<circ> \<nu>)"
1: fixes f :: "nat \<Rightarrow> real" shows "remove_weights (insert_weights shared_weights (deep_model_l rs) f) = deep_model_l rs"
surjective: fixes r :: "('d, 'e) partial_object_scheme" shows "r = \<lparr>carrier = carrier r, \<dots> = partial_object.more r\<rparr>"
codomain_additive: fixes x :: "brick list"   and y :: "brick list" shows "codomain_block (x \<otimes> y) = codomain_block x + codomain_block y"
\<phi>\<^sub>0_in_hom_1: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"   and "ide f" shows "\<guillemotleft>(d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, f \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (f \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> f \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f] : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
assms_6: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and c :: "'a" assumes "\<And>c. P1 c scrut"   and "\<And>c. P1 c e1"   and "\<And>c. P1 c e2" shows "P1 c (scrut ? e1 : e2)"
uniformity_Abort: shows "uniformity = Filter.abstract_filter (\<lambda>u. Code.abort STR ''uniformity is not executable'' (\<lambda>u. uniformity))"
assms: fixes g :: "'a"   and g' :: "'a" shows "T (g + g') = T g + T g'"
defined_and_I: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x"   and "\<tau> \<Turnstile> \<delta> y" shows "\<tau> \<Turnstile> \<delta> (x and y)"
dense: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.time plus zero minus uminus less_eq less"   and "less x y" shows "\<exists>z. less x z \<and> less z y"
left_assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * (b * c) = a * b * c"
set_insert: fixes x :: "'a"   and A :: "'a set" assumes "s2p A x" obtains B :: "'a set" where "A = insert x B" and "x \<notin> B"
tObsC_abs_rel: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a"   and x :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and y :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and as :: "'a list" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t"   and "(x, y) \<in> (\<Union> (relations (spr_repMC (tObsC_abs t)) ` set as))\<^sup>+" shows "(x \<in> tObsC_abs t) = (y \<in> tObsC_abs t)"
cl_def: shows "cl \<equiv> \<lambda>ts. {(a, b). (a, b) \<in> ts\<^sup>+ \<and> a \<noteq> b}"
ESem_considers_fv: fixes e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho> f|` fv e\<^esub>"
enorm_ge0: fixes x :: "nat \<Rightarrow> real" shows "0 \<le> enorm x"