hcju/isabelleps huggingface dataset:
mat_append_col_dims: fixes A :: "'b mat"   and b :: "'b vec" shows "mat_append_col A b \<in> carrier_mat (dim_row A) (dim_col A + 1)"
simps_1: fixes c :: "poly" shows "tmbound0 (CP c) = True"
assms: fixes n :: "nat" shows "f n \<in> borel_measurable M"
diff: fixes x :: "'a"   and y :: "'a" shows "(x - y) *s x = x *s x - y *s x"
eval_or_def: fixes \<phi> :: "tvl"   and \<psi> :: "tvl" shows "eval_or \<phi> \<psi> = (case (\<phi>, \<psi>) of (Yes, xa) \<Rightarrow> Yes | (No, Yes) \<Rightarrow> Yes | (No, No) \<Rightarrow> No | (No, Maybe) \<Rightarrow> Maybe | (Maybe, Yes) \<Rightarrow> Yes | (Maybe, _) \<Rightarrow> Maybe)"
set_map: fixes f :: "'f \<Rightarrow> 'd"   and v :: "'e \<Rightarrow> 'f" shows "range (f \<circ> v) = f ` range v"
matrix_to_iarray_Hermite_of_upt_row_i: fixes i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "i \<le> nrows A" shows "matrix_to_iarray (Hermite_of_upt_row_i A i ass res) = Hermite_of_upt_row_i_iarrays (matrix_to_iarray A) i ass res"
assms_4: shows "\<forall>x y z. snd (g x y) \<longrightarrow> snd (g y z) \<longrightarrow> snd (g x z)"
simps_2: fixes C :: "char list"   and e :: "expr" shows "fv (Cast C e) = fv e"
phis_in_\<alpha>n: fixes g :: "'a"   and n :: "'b"   and v :: "'e"   and vs :: "'e list" assumes "Mapping.lookup (phis g) (n, v) = Some vs" shows "n \<in> set (\<alpha>n g)"
induct: fixes x1 :: "'a set"   and x2 :: "nat"   and x3 :: "'a \<Rightarrow> 'a"   and P :: "'a set \<Rightarrow> nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" assumes "swapidseq_ext x1 x2 x3"   and "P {} 0 id"   and "\<And>S n p a. \<lbrakk>swapidseq_ext S n p; P S n p; a \<notin> S\<rbrakk> \<Longrightarrow> P (insert a S) n p"   and "\<And>S n p a b. \<lbrakk>swapidseq_ext S n p; P S n p; a \<noteq> b\<rbrakk> \<Longrightarrow> P (insert a (insert b S)) (Suc n) (Fun.swap a b id \<circ> p)" shows "P x1 x2 x3"
FL_distinguished_bounded_support: fixes F :: "'effect fs_set"   and x :: "('idx, 'pred, 'act, 'effect) formula"   and P :: "'state"   and Q :: "'state" assumes "x \<in> \<A>[F]"   and "x distinguishes P from Q" obtains y :: "('idx, 'pred, 'act, 'effect) formula" where "y \<in> \<A>[F]" and "supp y \<subseteq> supp (F, P)" and "y distinguishes P from Q"
lemma0: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula" shows "\<A> \<Turnstile> turn_true \<A> F"
ev_g: shows "\<forall>\<^sub>F x in at_top. \<bar>1 - (1 - ln (b * x + hb * x / ln x powr (1 + e)) powr (- e / 2))\<bar> < 1"
is_lasso_prpl_def: fixes G :: "('e, 'f) gb_graph_rec_scheme"   and prpl :: "'e list \<times> 'e list" assumes "gb_graph G" shows "gb_graph.is_lasso_prpl G prpl \<equiv> graph.is_lasso_prpl_pre G prpl \<and> (\<forall>A\<in>gbg_F G. set (snd prpl) \<inter> A \<noteq> {})"
assms_3: shows "Y \<in>\<^sub>\<circ> Vset \<alpha>"
whisker_can_right_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u"   and "horizontal_homs.obj V src a"   and "bicategorical_language.Src src t = bicategorical_language.Prim\<^sub>0 a" shows "H (bicategory.can V H \<a> \<i> src trg u t) a = bicategory.can V H \<a> \<i> src trg (bicategorical_language.Hcomp u (bicategorical_language.Prim\<^sub>0 a)) (bicategorical_language.Hcomp t (bicategorical_language.Prim\<^sub>0 a))"
last_index_less: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "last_index xs x < n"
mu_bound_factor: shows "\<mu>_bound_row fs' (4 * bnd) i"
hdomain_hunion: fixes f :: "hf"   and g :: "hf" shows "hdomain (f \<squnion> g) = hdomain f \<squnion> hdomain g"
card_of_bool: fixes a1 :: "'d"   and a2 :: "'d" assumes "a1 \<noteq> a2" shows "|UNIV| =o |{a1, a2}|"
is_valid_problem_sas_plus_then_strips_transformation_too_iii: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "list_all (is_valid_operator_strips (\<phi> \<Psi> )) ((\<phi> \<Psi> )\<^sub>\<O>)"
ideD_1: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.ide a" shows "VVV.arr a"
any_\<sigma>: fixes \<Sigma> :: "'b list"   and bs :: "bool list"   and n :: "nat" assumes "formula \<Sigma>"   and "length bs = n" shows "(formula.any \<Sigma>, bs) \<in> set (\<sigma> \<Sigma> n)"
lt_def: fixes c :: "int"   and t :: "Ferrack.num" shows "lt c t = (if c = 0 then Lt t else if 0 < c then Lt (CN 0 c t) else Gt (CN 0 (- c) (Neg t)))"
no_grey_refsD_3: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state"   and r :: "'g"   and p :: "'d process_name" assumes "no_grey_refs s" shows "r \<notin> ghost_honorary_grey (s p)"
endofunctor_H\<^sub>L: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b" assumes "weak_composition V H"   and "partial_magma.ide V g"   and "H g g \<noteq> partial_magma.null V" shows "endofunctor (subcategory.comp V (weak_composition.left V H g)) (weak_composition.H\<^sub>L H g)"
\<tau>diverge_into_\<tau>mredT: fixes t :: "'t"   and x :: "'x2"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "r2.\<tau>diverge t (x, shr s)"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "wset s t = None" shows "r2.mthr.\<tau>diverge s"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1))))))) PC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1))))))))))) \<and> t2 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2))))))) PC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))))))))))"
Rf_mono_on_substitutes: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" assumes "Basis.mono_on (Pow A) (Rf f)" shows "substitutes_on A f"
bidel_arc_head_isolated_axioms: shows "bidel_arc_head_isolated G M a"
one_zeroD: assumes "\<one> = \<zero>" shows "carrier R = {\<zero>}"
paths_Nil: fixes t :: "'a ttree" shows "[] \<in> paths t"
simps_1: fixes hsx :: "'b tree"   and x :: "'b"   and hsy :: "'b tree"   and y :: "'b"   and hs :: "'b tree" shows "pass\<^sub>1 \<langle>hsx, x, \<langle>hsy, y, hs\<rangle>\<rangle> = link \<langle>hsx, x, \<langle>hsy, y, pass\<^sub>1 hs\<rangle>\<rangle>"
step_outputs: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "transition"   and s' :: "nat"   and p :: "value option list"   and r' :: "nat \<Rightarrow>f value option" assumes "step e s r l i = Some (t, s', p, r')" shows "evaluate_outputs t i r = p"
induced_automorphism_morphism: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexEndomorphism X (OpposedThinChamberComplexFoldings.induced_automorphism X f g)"
rep_eq: fixes x :: "'a"   and xa :: "'b"   and xb :: "('a, 'b) alist" shows "alist.impl_of (DAList.update x xa xb) = AList.update x xa (alist.impl_of xb)"
share_all_until_volatile_write_append_Ghost\<^sub>s\<^sub>b': fixes sb :: "'b memref list"   and i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and \<S> :: "nat \<Rightarrow> bool option"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and \<D>' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "share_all_until_volatile_write ts \<S> = share_all_until_volatile_write (ts[i := (p', is', \<theta>', sb @ [Ghost\<^sub>s\<^sub>b A L R W], \<D>', \<O>', \<R>')]) \<S>"
extf_ext_compat_cons: fixes f :: "'s" shows "ext_compat_cons (extf f)"
a_in: shows "a \<in> dom (share_all_until_volatile_write ts (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>t) \<S>))"
simps: fixes f :: "'c \<Rightarrow> 'b val"   and v :: "'c" shows "F_LEFT f v = val.Left (f v)"
independentD: fixes s :: "'d set"   and t :: "'d set"   and u :: "'d \<Rightarrow> complex"   and v :: "'d" assumes "cindependent s"   and "finite t"   and "t \<subseteq> s"   and "(\<Sum>v\<in>t. u v *\<^sub>C v) = (0::'d)"   and "v \<in> t" shows "u v = 0"
le_funD: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "f \<sqsubseteq> g" shows "f x \<sqsubseteq> g x"
SUP_le_iff: fixes u :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(u \<le> \<Sqinter> (f ` A)) = (\<forall>i\<in>A. u \<le> f i)"
rsa_proof_assm: fixes n :: "nat" shows "rsa_base (prime_set n)"
sort_lt_var_def: fixes uu :: "'a"   and uv :: "'b" shows "sort_lt_var uu uv \<equiv> |UNIV| <o |UNIV|"
ideD_2: fixes a :: "'s" assumes "ide a" shows "local.dom a = a"
cases: fixes a :: "('St, 'U, 'C) state list" assumes "wffp a" obtains s :: "('St, 'U, 'C) state" where "a = [s__]"   | s' :: "('St, 'U, 'C) state" and s :: "('St, 'U, 'C) state" and sl :: "('St, 'U, 'C) state list" where "a = s # s' # sl" and "s' \<in> \<delta> s" and "wffp (s' # sl)"
enc_reflects_pred_iff_indRelR_reflects_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_reflects_pred Pred = rel_reflects_pred indRelR Pred"
continuous_sin: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "continuous F f" shows "continuous F (\<lambda>x. sin (f x))"
a_get_disconnected_nodes_def: fixes document_ptr :: "(_) document_ptr" shows "a_get_disconnected_nodes document_ptr = get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes"
exactly_result: fixes x :: "char list"   and s :: "char list"   and y :: "char list"   and r :: "char list" assumes "exactly x s = Inr (y, r)" shows "\<exists>w. s = x @ w @ r \<and> y = x"
inv_approx_preserv2: fixes E :: "'b set set"   and S :: "'b set"   and X :: "'b set"   and s :: "nat"   and x :: "'b" assumes "Graph_E E"   and "Graph_E.inv_approx E S X s"   and "x \<in> Graph_E.vertices E - X" shows "Graph_E.inv_approx E (insert x S) (X \<union> Graph_E.neighbors E x \<union> {x}) (max (card (Graph_E.neighbors E x \<union> {x} - X)) s)"
reversed_arden_helper: fixes X :: "'b list set"   and A :: "'b list set"   and B :: "'b list set"   and n :: "nat" assumes "X = X @@ A \<union> B" shows "X = X @@ A ^^ Suc n \<union> (\<Union>m\<le>n. B @@ A ^^ m)"
ik\<^sub>s\<^sub>t_assignment_rhs\<^sub>s\<^sub>t_wfrestrictedvars_subset: fixes A :: "('b, 'a) strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>s\<^sub>t A) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t A"
enumerate_0: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" assumes "class.wellorder less_eq less" shows "wellorder.enumerate less_eq S 0 = ord.Least less_eq (\<lambda>n. n \<in> S)"
liminf_ereal_cminus: fixes c :: "ereal"   and f :: "nat \<Rightarrow> ereal" assumes "c \<noteq> - \<infinity>" shows "liminf (\<lambda>x. c - f x) = c - limsup f"
comp_apply: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "(f \<circ> g) x = f (g x)"
card_insert_le_m1: fixes n :: "nat"   and y :: "'q set"   and x :: "'q" assumes "0 < n"   and "card y \<le> n - 1" shows "card (insert x y) \<le> n"
awalkE: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v" obtains "set (G.awalk_verts u p) \<subseteq> verts G" and "set p \<subseteq> arcs G" and "G.cas u p v" and "G.awhd u p = u" and "G.awlast u p = v"
compseries_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" shows "compseries G n f = (tW_cmpser G n f \<and> (if n = 0 then f 0 = {\<one>\<^bsub>G\<^esub>} else \<forall>i\<le>n - 1. simple_Group (\<natural>\<^bsub>G\<^esub>f i / f (Suc i))))"
Step0: fixes g :: "'a" assumes "g \<in> carrier G" shows "(\<lambda>H\<in>{H. subgroup H G}. g <# H #> inv g) \<in> Bij {H. subgroup H G}"
commutes_with_unit: shows "image.additive.induced (kernel.additive.Class \<one>) = \<one>'"
mark_loop_mo_mark_loop_field_done_subseteq_hs_in_sync_locs: shows "obj_fields_marked_good_ref_locs \<subseteq> hs_in_sync_locs"
pth_1: fixes x :: "'c" shows "x \<equiv> 1 *\<^sub>R x"
cSup_upper2: fixes x :: "'a multiset"   and X :: "'a multiset set"   and y :: "'a multiset" assumes "x \<in> X"   and "y \<subseteq># x"   and "subset_mset.bdd_above X" shows "y \<subseteq># Sup X"
parallelI: fixes xs :: "'c list"   and ys :: "'c list" assumes "\<not> prefix xs ys"   and "\<not> prefix ys xs" shows "xs \<parallel> ys"
ideal_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> I" shows "b \<cdot> a \<in> I"
assms: shows "\<not> \<Gamma>\<turnstile> (c, s) \<rightarrow> \<dots>(\<infinity>)"
f_p_1: shows "\<forall>i<n. CAs0f i \<in> M"
agreementInit_def: fixes i :: "('p, 'v, 's) configuration"   and c :: "('p, 'v, 's) configuration" shows "agreementInit i c \<equiv> initial i \<and> reachable i c \<longrightarrow> (\<forall>v1. <\<bottom>, outM v1> \<in># msgs c \<longrightarrow> (\<forall>v2. <\<bottom>, outM v2> \<in># msgs c = (v2 = v1)))"
update_convs_2: fixes getM' :: "(Literal \<times> bool) list \<Rightarrow> (Literal \<times> bool) list"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getM_update getM' \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = \<lparr>getF = getF, getM = getM' getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr>"
RDirProd_list_zero': fixes R :: "('a, 'b) ring_scheme"   and Rs :: "('a, 'b) ring_scheme list" shows "\<zero>\<^bsub>RDirProd_list (R # Rs)\<^esub> = \<zero>\<^bsub>R\<^esub> # \<zero>\<^bsub>RDirProd_list Rs\<^esub>"
assms_2: shows "\<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>"
FIXvarFIX: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" shows "PhiWhileP A b = (\<lambda>\<Phi> (s, t). (b, A, \<Phi>, s, t) \<in> var)"
join_split_i: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'a list"   and i :: "nat" assumes "0 < n" shows "join_time (split_time s n) n i = s i"
YnY: fixes x :: "real" assumes "x \<in> space Omega" shows "(\<lambda>n. Y_seq n x) \<longlonglongrightarrow> Y x"
emeasure_T_observation_backward: fixes s :: "'a"   and o :: "'b"   and os :: "'b list" shows "emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>xs \<omega>'. \<omega> = xs @- \<omega>' \<and> map snd xs = os} = backward s os"
assms: shows "A *\<^sub>V x = e *\<^sub>C x"
h: shows "h xcp = \<lfloor>obj\<rfloor>"
all_A_relprime: fixes x :: "int" assumes "x \<in> A" shows "coprime x (int p)"
smc_Dom_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
intro: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c" assumes "diagram J C D"   and "empty_diagram_axioms J" shows "empty_diagram J C D"
ide_u: shows "ide r\<^sub>0s\<^sub>1.p\<^sub>0"
simps_17: fixes x1 :: "'d" shows "set2_chain \<lbrace> x1 \<rbrace> = {}"
Rep_numeral: fixes w :: "num" shows "Rep_bit0 (numeral w) = numeral w mod int CARD('a bit0)"
llist_all2_LCons1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a llist"   and ys :: "'b llist" shows "llist_all2 P (LCons x xs) ys = (\<exists>y ys'. ys = LCons y ys' \<and> P x y \<and> llist_all2 P xs ys')"
IMPRIGHT_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set"   and Y :: "ctrm set"   and Z :: "ntrm set"   and U :: "ctrm set" shows "pi \<bullet> IMPRIGHT (A IMP B) X Y Z U = IMPRIGHT (A IMP B) (pi \<bullet> X) (pi \<bullet> Y) (pi \<bullet> Z) (pi \<bullet> U)"
strip_def: fixes F :: "'f set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option" shows "strip F \<Gamma> = (\<lambda>p. map_option (strip_guards F) (\<Gamma> p))"
finite_extension_incl_aux: fixes K :: "'a set"   and x :: "'a"   and xs :: "'a list" assumes "K \<subseteq> carrier R"   and "x \<in> carrier R"   and "set xs \<subseteq> carrier R" shows "finite_extension K xs \<subseteq> finite_extension K (x # xs)"
elims_1: fixes x :: "'d mat"   and y :: "bool" assumes "square_mat x = y" obtains A :: "'d mat" where "x = A" and "y = (dim_col A = dim_row A)"
assms_3: shows "put \<sigma> u = put \<rho> v"
sst_vars_append_subset_4: fixes B :: "('d, 'c) stateful_strand_step list"   and A :: "('d, 'c) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t B \<subseteq> bvars\<^sub>s\<^sub>s\<^sub>t (A @ B)"
axioms_2: fixes C :: "('d, 'e, 'f) Category_scheme" assumes "Category C" shows "Category_axioms C"
sl5: shows "abscissa C - abscissa D \<in> radical_sqrt"
N2H_inv: fixes n :: "C"   and b :: "nat" assumes "n \<in> hbase b" shows "N2H b (H2N b n) = n"
finite_fset: fixes S :: "'c fset" shows "finite (fset S)"
simps_6: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and v :: "'c match_expr"   and rs :: "'c rule list" shows "no_matching_Goto \<gamma> p (Rule v Reject # rs) = no_matching_Goto \<gamma> p rs"
d_3: shows "poincare_distance u w = poincare_distance u' w'"
Hintikkas_lemma: fixes S :: "'b formula set" assumes "Hintikka (S::'b formula set)" shows "sat (S::'b formula set)"
cauchyI: fixes X :: "nat \<Rightarrow> rat" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r" shows "Real.cauchy X"
scene_union_incompat: fixes X :: "'a scene"   and Y :: "'a scene" assumes "\<not> X ##\<^sub>S Y" shows "X \<squnion>\<^sub>S Y = \<bottom>\<^sub>S"
initState_Spy: shows "initState Spy = Key ` (invKey ` pubEK ` bad \<union> invKey ` pubSK ` bad \<union> range pubEK \<union> range pubSK)"
has_vderiv_on_def: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and S :: "real set" shows "(f has_vderiv_on f') S = (\<forall>x\<in>S. (f has_vector_derivative f' x) (at x within S))"
preprm_apply_injective: fixes \<pi> :: "('b \<times> 'b) list" shows "inj (preprm_apply \<pi>)"
times_eint_simps_3: fixes n :: "int" shows "\<infinity> * eint n = \<infinity>"
indoc_set: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and u :: "'a list" assumes "traces ind"   and "traces.indoc ind a u" shows "a \<in> set u"
Pos_atm_of_iff: fixes L :: "'b literal" shows "(Pos (atm_of L) = L) = is_pos L"
Singl: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and trn :: "('a, 'b, 'c) trans" assumes "IO_Automaton.validTrans step trn" shows "IO_Automaton.valid2 step [trn]"
all_derived_clauses_are_finite: fixes C :: "'a eclause"   and S :: "'a eclause set" assumes "derivable_ecl C S"   and "\<forall>x\<in>S. finite (cl_ecl x)" shows "finite (cl_ecl C)"
less_eq_integer_code_1: shows "(0 \<le> 0) = True"
abs_conv_abscissa_restrict: fixes P :: "nat \<Rightarrow> bool"   and f :: "'a fds" shows "abs_conv_abscissa (fds_subseries P f) \<le> abs_conv_abscissa f"
reduce_system_matrix_subsets_helper_aux_R: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list"   and S :: "nat list"   and p :: "real poly" assumes "length signs \<le> length subsets"   and "\<forall>x. List.member S x \<longrightarrow> x < length subsets"   and "p \<noteq> 0" shows "alt_matrix_A_R signs (take_indices subsets S) = take_rows_from_matrix (alt_matrix_A_R signs subsets) S"
map_U_id: fixes x :: "('n, 'o, 'p, 'q, 'r, 's) U" shows "map_U id id id id id x = x"
coprime_mult: fixes a :: "real poly"   and b :: "real poly"   and c :: "real poly" assumes "algebraic_semidom_class.coprime a b"   and "algebraic_semidom_class.coprime a c" shows "algebraic_semidom_class.coprime a (b * c)"
P': shows "P' \<in> carrier_mat (n - 1) (n - 1)"
valuationContainsItsFalseClausesVariables: fixes clause :: "Literal list"   and valuation :: "Literal list" assumes "clauseFalse clause valuation" shows "vars clause \<subseteq> vars valuation"
cmod_cor_divide: fixes z :: "complex"   and k :: "real" shows "cmod (z / cor k) = cmod z / \<bar>k\<bar>"
nth_tl: fixes n :: "nat"   and xs :: "'g list" assumes "n < length (tl xs)" shows "tl xs ! n = xs ! Suc n"
arith_simps_16: shows "neg_numeral_class.sub num.One num.One = (0::'c)"
less_degree_imp: fixes n :: "nat"   and p :: "'a poly" assumes "n < degree p" shows "\<exists>i>n. coeff p i \<noteq> (0::'a)"
source_iso_src: fixes \<mu> :: "'a"   and a :: "'a" assumes "arr \<mu>"   and "a \<in> sources \<mu>" shows "a \<cong> src \<mu>"
h0: shows "\<forall>i<length (take_indices signs S). col (alt_matrix_A (take_indices signs S) subsets) i = col (take_cols_from_matrix (alt_matrix_A signs subsets) S) i"
Arctan_def_moebius: fixes z :: "complex" shows "Arctan z = \<i> / 2 * Ln (moebius (- \<i>) 1 \<i> 1 z)"
inject_4: fixes x4 :: "ltkey"   and y4 :: "ltkey" shows "(cLtK x4 = cLtK y4) = (x4 = y4)"
simps_4: fixes v :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "assignment_rhs\<^sub>s\<^sub>t (receive\<langle>v\<rangle>\<^sub>s\<^sub>t # S) = assignment_rhs\<^sub>s\<^sub>t S"
unique_periodic_arithmetic_extension: fixes k :: "nat"   and g :: "nat \<Rightarrow> complex"   and h :: "nat \<Rightarrow> complex"   and i :: "nat" assumes "0 < k"   and "\<forall>j<k. g j = h j"   and "periodic_arithmetic g k"   and "periodic_arithmetic h k" shows "g i = h i"
f_def: fixes x :: "nat \<Rightarrow>\<^sub>0 nat" shows "f x = MPoly_Type.monom (remove_key x' x) (MPoly_Type.coeff p x) * (if lookup x x' = 0 then 1 else Var x' ^ lookup x x')"
indRelRSTPO_to_SRel_and_TRel_1: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q" shows "\<forall>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<longrightarrow> (SP, SQ) \<in> SRel\<^sup>+"
simps: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "('a \<times> 'b) list"   and ko :: "'c" shows "oalist_raw.update_by_fun_gr_raw rep_key_order k f (xs, ko) = (comparator.update_by_fun_gr_pair (key_compare (rep_key_order ko)) k f xs, ko)"