hcju/isabelleps huggingface dataset:
assms_1: shows "zs \<noteq> ys \<or> y \<notin> {z1, z2}"
assms_2: shows "\<alpha> \<in>\<^sub>\<circ> \<beta>"
diag: shows "(0::'a) \<notin> set (diag_mat A)"
hyps: fixes C :: "char list"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>RI (C,throw e) ; Cs' \<leftarrow> e\<^sub>0,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "\<exists>a s\<^sub>t. e' = throw a \<and> P \<turnstile> \<langle>throw e,s\<rangle> \<Rightarrow> \<langle>throw a,s\<^sub>t\<rangle>"
mem_lclss_of_state_1: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "((C, New) \<in> lclss_of_state (N, P, Q)) = (C \<in> N)"
vnat_le: fixes n :: "nat" shows "VNat n \<sqsubseteq> VNat n"
better_deriv: shows "((\<lambda>v. v $ x) has_derivative (\<lambda>h. \<nu> \<bullet> 0 + h \<bullet> axis x 1)) (at \<nu>)"
finite_UN_I: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (\<Union> (B ` A))"
IH: assumes "j < length xs"   and "j \<in> J" shows "nths xs J ! card {j0. j0 < j \<and> j0 \<in> J} = xs ! j"
elem_set_implies_set_eq_singleton: fixes a :: "'s"   and b :: "'s" assumes "a \<in> local.set b" shows "local.set a = {a}"
fsets: shows "\<forall>x. f -` {f x} \<in> sets lebesgue"
ord_spmf_eq_pmf_None_eq: fixes p :: "'b spmf"   and q :: "'b spmf" assumes "ord_spmf (=) p q"   and "pmf p None = pmf q None" shows "p = q"
rtrancl_is_final: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "r\<^sup>*\<^sup>* a b"   and "relation.is_final r a" shows "a = b"
PossWorldAux: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists>p. p \<^bold>& F \<^bold>= (\<^bold>\<lambda>x. p))) \<^bold>\<rightarrow> PossibleWorld (x\<^sup>P) in v]"
R_def: fixes Rel :: "('a \<times> 'b) set"   and valid_pub :: "'a set"   and init :: "'a \<Rightarrow> 'b \<Rightarrow> ('f \<times> 'c) spmf"   and response :: "'f \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> 'e spmf"   and h :: "'a"   and w :: "'b"   and e :: "'d" assumes "\<Sigma>_protocols_base Rel valid_pub" shows "\<Sigma>_protocols_base.R init response h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
subst_empty: fixes t :: "'a" shows "subst t fmempty = t"
project: fixes a :: "'a"   and n :: "nat" assumes "a \<in> \<Sigma> (Suc n)" shows "project a \<in> \<Sigma> n"
intros_2: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wary s"   and "wary t"   and "enat (num_args s) < arity_hd (head s)" shows "wary (App s t)"
shift_simps_2: fixes xs :: "'b list"   and s :: "'b stream" shows "stl (xs @- s) = (if xs = [] then stl s else tl xs @- s)"
bounded_linearI': fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x" shows "bounded_linear f"
ee2: shows "essentially_equal G (ah # take i as' @ drop (Suc i) as') (as' ! i # take i as' @ drop (Suc i) as')"
ex1_imp_Collect_singleton: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "((\<exists>!x. P x) \<and> P x) = (Collect P = {x})"
height_l_bal: fixes l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0"   and n :: "'a" assumes "height l = height r + 2" shows "height (l_bal\<^sub>0 n l r) = height r + 2 \<or> height (l_bal\<^sub>0 n l r) = height r + 3"
odd_card_imp_not_empty: fixes A :: "'a set" assumes "odd (card A)" shows "A \<noteq> {}"
rel_simps_76: fixes n :: "num" shows "(0::'c) \<noteq> numeral n"
inv_comp_right_1: fixes g :: "'b"   and f :: "'b" assumes "\<phi>.B.iso (g \<cdot>\<^sub>B f)"   and "\<phi>.B.iso f" shows "\<phi>.B.inv (g \<cdot>\<^sub>B f) = \<phi>.B.inv f \<cdot>\<^sub>B \<phi>.B.inv g"
matrix_mul_assoc: fixes A :: "(('b, 'e) vec, 'd) vec"   and B :: "(('b, 'f) vec, 'e) vec"   and C :: "(('b, 'c) vec, 'f) vec" shows "A ** (B ** C) = A ** B ** C"
wt_App_fun_\<delta>\<^sub>h: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A"   and "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym (App s t))) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t))" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s)) = zhmset_of (of_nat \<delta>)"
old_next_eq: shows "old nd = old n \<and> next nd = next n"
fac_acc_body3_body2: shows "fac_acc_body3 oo unwrapA oo wrapA = fac_acc_body2"
CONS_not_NIL: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" shows "CONS M N \<noteq> NIL"
map_dirvec_polychain_of: fixes x :: "'a"   and xs :: "'a list" shows "map dirvec (polychain_of x xs) = xs"
chop_induct': fixes P :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and n :: "nat"   and xs :: "'a list" assumes "\<And>n xs. n = 0 \<or> xs = [] \<Longrightarrow> P n xs"   and "\<And>n xs. \<lbrakk>0 < n; xs \<noteq> []; P n (drop n xs)\<rbrakk> \<Longrightarrow> P n xs" shows "P n xs"
fg_free1: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and i :: "'e \<Rightarrow> 'e"   and s :: "'a \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and M :: "('e, 'a, 'f) Module_scheme" assumes "A \<noteq> {}"   and "fgmodule_condition R f i s A z"   and "free_generator R (fgmodule R A z i f s) A"   and "R module M"   and "free_generator R M A" shows "M \<cong>\<^bsub>R\<^esub> fgmodule R A z i f s"
the_ext_prop_2: fixes y' :: "'c"   and g :: "'d" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y y' g" shows "g = G (y\<eta>.the_ext y' g) \<cdot>\<^sub>D \<eta>o y"
c: fixes sh :: "('a, 'b, 'c, 'd, 'e) system_state"   and p1 :: "'c"   and \<alpha> :: "'d"   and \<beta> :: "'a"   and ls1' :: "('a, 'b, 'd, 'e) com list \<times> 'b option \<times> 'e"   and p2 :: "'c"   and ls2' :: "('a, 'b, 'd, 'e) com list \<times> 'b option \<times> 'e" assumes "reachable_state sys sh"   and "P sh"   and "GST sh p1 \<rightarrow>\<^bsub>\<guillemotleft>\<alpha>, \<beta>\<guillemotright>\<^esub> ls1'"   and "GST sh p2 \<rightarrow>\<^bsub>\<guillemotright>\<alpha>, \<beta>\<guillemotleft>\<^esub> ls2'"   and "p1 \<noteq> p2" shows "P \<lparr>GST = (GST sh)(p1 := ls1', p2 := ls2'), HST = HST sh @ [(\<alpha>, \<beta>)]\<rparr>"
changes_all_nonneg: fixes xs :: "'b list" assumes "\<And>i. (0::'b) \<le> nth_default (0::'b) xs i" shows "changes xs = 0"
simps_1: fixes k :: "nat" shows "from_WW k [] = []"
add_eq_0_iff_both_eq_0: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
NDComm: fixes a :: "'a set set"   and b :: "'a set set" shows "netsDistinct a b = netsDistinct b a"
prob_space_restrict_space: fixes S :: "'a set"   and M :: "'a measure" assumes "S \<in> sets M"   and "emeasure M S = 1" shows "prob_space (restrict_space M S)"
get_and_nonempty: fixes \<phi> :: "Formula.formula" assumes "safe_formula \<phi>" shows "get_and_list \<phi> \<noteq> []"
col_trivial_2: fixes A :: "'p"   and B :: "'p" shows "Col A B B"
simps_2: fixes f :: "'f"   and g :: "'d \<Rightarrow> 'd" shows "mark_guards f (Basic g) = Basic g"
simps_6: fixes xT :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and T :: "Unary_PCF.type"   and e :: "expr" shows "subst xT t (\<Lambda>\<langle>T\<rangle> e) = \<Lambda>\<langle>T\<rangle> subst xT t e"
game\<^sub>3_bad: shows "map_spmf snd (sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> game\<^sub>3 (\<lambda>b m\<^sub>1 m\<^sub>2. spmf_of_set (nlists UNIV len_plain)) x)) = game (elgamal_reduction \<A>)"
pdevs_apply_inject: fixes x :: "'a pdevs"   and y :: "'a pdevs" shows "(pdevs_apply x = pdevs_apply y) = (x = y)"
\<C>_def: shows "\<C> \<equiv> \<Union> (range \<C>_vars)"
ordinal_oSuc_minus: fixes y :: "ordinal"   and x :: "ordinal" assumes "y \<le> x" shows "oSuc x - y = oSuc (x - y)"
bc: shows "(b, c) \<in> S\<^sup>* O R O S\<^sup>*"
mult_sign_intros_1: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) \<le> a"   and "(0::'c) \<le> b" shows "(0::'c) \<le> a * b"
not_in_iff: fixes x :: "'k"   and M :: "'k multiset" shows "(x \<notin># M) = (count M x = 0)"
Max_in: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "linorder.Max (\<sqsubseteq>) A \<in> A"
hyps: assumes "sorted_wrt r (insort_wrt s x xs)" shows "sorted_wrt r xs"
delete_edges_correct: fixes G :: "'c list_graph"   and E :: "('c \<times> 'c) list" shows "FiniteGraph.delete_edges (list_graph_to_graph G) (set E) = list_graph_to_graph (FiniteListGraph.delete_edges G E)"
linorder_linorder_of_list: fixes xs :: "'a list" assumes "distinct xs" shows "linorder_on (set xs) (linorder_of_list xs)"
i_drop_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Up> n \<equiv> \<lambda>x. f (n + x)"
suff_mono2: fixes s :: "'b llist"   and t :: "'b llist"   and A :: "'b set" assumes "s \<le> t" shows "suff A t \<subseteq> suff A s"
codomains_def: fixes f :: "'c" shows "C.codomains f \<equiv> {b. C.ide b \<and> C b f \<noteq> C.null}"
funas_term_ctxt_apply: fixes C :: "('c, 'd) ctxt"   and t :: "('c, 'd) Term.term" shows "funas_term C\<langle>t\<rangle> = funas_ctxt C \<union> funas_term t"
irbd_def: shows "bd\<^sup>-\<^sub>\<R> = \<R> \<circ> bd\<^sup>-\<^sub>\<F>"
exp_times_self_strict_mono: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x < y" shows "x * exp x < y * exp y"
corthogonal_sort: fixes us' :: "'b vec list"   and us :: "'b vec list" assumes "distinct us'"   and "set us = set us'"   and "corthogonal us" shows "corthogonal us'"
norm_bound_max: fixes A :: "'a mat" shows "local.norm_bound A (Max {norm (A $$ (i, j)) |i j. i < dim_row A \<and> j < dim_col A})"
filter_min_aux_distinct: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "reflp rel"   and "distinct ys" shows "distinct (General.filter_min_aux rel xs ys)"
test_inf_bot: fixes x :: "'a" assumes "test x" shows "x \<sqinter> bot = bot"
wordinterval_subset_set_eq: fixes r1 :: "'a wordinterval"   and r2 :: "'a wordinterval" shows "wordinterval_subset r1 r2 = (wordinterval_to_set r1 \<subseteq> wordinterval_to_set r2)"
csin_eq_minus1: fixes z :: "complex" shows "(sin z = - 1) = (\<exists>x. z = complex_of_real (of_real (2 * real_of_int x * pi) + 3 / 2 * pi))"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "comm_semiring_ow U plus times"   and "comm_monoid_add_ow U plus zero"   and "mult_zero_ow U times zero" shows "comm_semiring_0_ow U plus zero times"
AlphaAnd: fixes A :: "('a, 'b) form"   and B :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<stileturn> A # B # G" shows "\<stileturn> And A B # G"
bisims_Val_loc_eq_xcp_None: fixes P :: "'a prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and vs :: "'addr val list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,es,h \<turnstile> (map Val vs, xs) [\<leftrightarrow>] (stk, loc, pc, xcp)" shows "xs = loc \<and> xcp = None"
assms: shows "bal (Node (ls @ (a, b) # rs) t)"
scaleC_left_mono_neg: fixes b :: "'b"   and a :: "'b"   and c :: "complex" assumes "b \<le> a"   and "c \<le> 0" shows "c *\<^sub>C a \<le> c *\<^sub>C b"
psiFreshVec_4: fixes xvec :: "name list"   and P :: "('e, 'f, 'g) psi"   and Q :: "('e, 'f, 'g) psi" shows "xvec \<sharp>* (P \<parallel> Q) = (xvec \<sharp>* P \<and> xvec \<sharp>* Q)"
path_nodes_edges: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and p :: "'b list"   and v :: "'b" assumes "path E u p v" shows "set p \<subseteq> fst ` E"
is_unit_unit_factor: fixes a :: "'a" assumes "is_unit a" shows "unit_factor a = a"
rep_eq: fixes x :: "linear_poly" shows "coeff x = fun_coeff (Rep_linear_poly x)"
adm_subset: fixes f :: "'a \<Rightarrow> 'b set"   and S :: "'b set" assumes "cont f" shows "adm (\<lambda>x. f x \<subseteq> S)"
elims_2: fixes x :: "'b"   and xa :: "'b \<times> 'b" assumes "elem x xa" obtains xb :: "'b" and l :: "'b" and u :: "'b" where "x = xb" and "xa = (l, u)" and "l < xb \<and> xb < u"
getRel_def: fixes l :: "'c"   and G :: "('c, 'b) labeled_graph" shows "getRel l G = {(x, y). (l, x, y) \<in> edges G}"
birkhoff_sum_meas: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "qmpt M T"   and "f \<in> borel_measurable M" shows "qmpt.birkhoff_sum T f n \<in> borel_measurable M"
conversep_conversep: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
REFSTEP: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a"   and x' :: "'Sa"   and x :: "'S" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi> it \<sigma>"   and "\<Phi>' it' \<sigma>'"   and "\<forall>x\<in>S - it. \<forall>y\<in>it. RR x y"   and "\<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y"   and "x' = \<alpha> x"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "x \<in> it"   and "\<forall>y\<in>it - {x}. RR x y"   and "x' \<in> it'"   and "\<forall>y'\<in>it' - {x'}. RR' x' y'" shows "f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"
abs_mult_sgn: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'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 a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "times (abs a) (sgn a) = a"
frontier_def: fixes S :: "'b set" shows "frontier S = closure S - interior S"
no_state_change_if_no_event: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and p :: "'d" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "occurs_on ev \<noteq> p" shows "states c p = states c' p \<and> ps c p = ps c' p"
all_red_crit: shows "\<forall>q\<in>Q. calculus Bot Inf (entails_q q) (Red_I_q q) (Red_F_q q)"
cong_minus_minus_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "[- b = - c] (mod a) = [b = c] (mod a)"
test_dvd_exec_def: fixes p :: "int"   and n :: "nat"   and sl_impl :: "(int poly, int \<times> int poly list, 'a) subseqs_foldr_impl"   and sli :: "int \<times> int poly list \<Rightarrow> int poly list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> bool"   and lu :: "int"   and u :: "int poly"   and ws :: "int poly multiset" assumes "prime p"   and "p = p ^ n"   and "n \<noteq> 0"   and "correct_subseqs_foldr_impl (\<lambda>x. map_prod (p.mul_const x) ((#) x)) sl_impl sli" shows "p.test_dvd_exec lu u ws = (\<not> p.inv_Mp (p.Mp (Polynomial.smult lu (\<Prod>\<^sub># ws))) dvd Polynomial.smult lu u)"
simps_2: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and wf0 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'b"   and a :: "'c" shows "Formula_Operations.wf SUC wf0 n (FBase a) = wf0 n a"
simps: fixes E :: "'b set set"   and G :: "'b set set" assumes "Graph_E E" shows "Graph_E.vertices G = \<Union> G"
mask_lower_twice: fixes n :: "nat"   and m :: "nat"   and x :: "'a word" assumes "n \<le> m" shows "(x && ~~ mask n) && ~~ mask m = x && ~~ mask m"
Inr_transfer: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "(T ===> rel_sum S T) Inr Inr"
rsp: shows "eq_onp ((\<le>) 0) 0 0"
wpo_s_mono: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term"   and f :: "'f"   and bef :: "('f, 'v) Term.term list"   and aft :: "('f, 'v) Term.term list" assumes "s \<succ> t" shows "Fun f (bef @ s # aft) \<succ> Fun f (bef @ t # aft)"
find_nu_measures_existence: fixes \<nu> :: "nat" assumes "mu \<le> \<nu>" shows "\<exists>i. seq (\<nu> + \<nu> + i) = seq \<nu>"
nlistToList_app: fixes xs :: "('a, 'b) Node list"   and ys :: "('a, 'b) Node list" shows "FingerTreeStruc.nlistToList (xs @ ys) = FingerTreeStruc.nlistToList xs @ FingerTreeStruc.nlistToList ys"
valid_plan_def: fixes P :: "ast_problem" shows "ast_problem.valid_plan P \<equiv> ast_problem.valid_plan_from P (ast_problem.I P)"
termMOD_igSwapIPresIGWls: shows "igSwapIPresIGWls termMOD"
stationary_distribution_iterate': fixes N :: "'s pmf"   and y :: "'s"   and n :: "nat" assumes "stationary_distribution N" shows "measure_pmf.prob N {y} = measure_pmf.expectation N (\<lambda>x. p x y n)"
coprime_add_one_left: fixes a :: "'a" shows "coprime (a + (1::'a)) a"
index_mat_swapcols_1: fixes i :: "nat"   and A :: "'d mat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "swapcols k l A $$ (i, j) = (if k = j then A $$ (i, l) else if l = j then A $$ (i, k) else A $$ (i, j))"
fls_X_def: shows "fls_X \<equiv> Abs_fls (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
assms_3: shows "(f \<longlongrightarrow> l) (at a within S)"
cast\<^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\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject: fixes x :: "(_) shadow_root_ptr"   and y :: "(_) shadow_root_ptr" shows "(cast x = cast y) = (x = y)"
truncate_down_switch_sign_mono: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "x \<le> 0"   and "0 \<le> y"   and "x \<le> y" shows "truncate_down prec x \<le> truncate_down prec y"
mem_idx_sound_output: fixes x :: "'c"   and as :: "'c list"   and i :: "nat" assumes "mem_idx x as = Some i" shows "i < length as \<and> as ! i = x"
diff_card_le_card_Diff: fixes B :: "'c set"   and A :: "'c set" assumes "finite B" shows "card A - card B \<le> card (A - B)"
writes_bind: fixes S :: "('x, 'y, 'z) prog set"   and f :: "('x, 'y, 'aa) prog"   and h :: "'x"   and g :: "'aa \<Rightarrow> ('x, 'y, 'ab) prog"   and h' :: "'x" assumes "\<And>h2. writes S f h h2"   and "\<And>x h2. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h2\<rbrakk> \<Longrightarrow> writes S (g x) h2 h'" shows "writes S (Heap_Error_Monad.bind f g) h h'"
left_add_mult_distrib_mset: fixes i :: "nat"   and u :: "'k multiset"   and j :: "nat"   and k :: "'k multiset" shows "repeat_mset i u + (repeat_mset j u + k) = repeat_mset (i + j) u + k"
cbox: fixes K :: "'M set" assumes "K \<in> \<D>" shows "interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
dom_in_domains: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "partial_magma C"   and "partial_magma.domains C f \<noteq> {}" shows "partial_magma.dom C f \<in> partial_magma.domains C f"
step_Stuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Stuck" shows "s' = Stuck"
Compl_subset_Compl_iff: fixes A :: "'e set"   and B :: "'e set" shows "(- A \<subseteq> - B) = (B \<subseteq> A)"
smult_smult: fixes a :: "'c"   and b :: "'c"   and p :: "'c poly" shows "smult a (smult b p) = smult (a * b) p"
arith_simps_51: shows "- (0::'a) = (0::'a)"
paths_second_disjoint: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<in> paths"   and "ys \<in> paths"   and "xs \<noteq> ys" shows "hd (tl xs) \<noteq> hd (tl ys)"
ununnecessaryPhis_disjoint_paths: fixes g :: "'g"   and r :: "'val"   and s :: "'val"   and V :: "'var" assumes "\<not> unnecessaryPhi g r"   and "\<not> unnecessaryPhi g s"   and "defNode g r \<noteq> defNode g s"   and "r \<in> allVars g"   and "s \<in> allVars g"   and "var g r = V"   and "var g s = V" obtains n :: "'node" and ns :: "'node list" and m :: "'node" and ms :: "'node list" where "V \<in> oldDefs g n" and "g \<turnstile> n-ns\<rightarrow>defNode g r" and "V \<in> oldDefs g m" and "g \<turnstile> m-ms\<rightarrow>defNode g s" and "set ns \<inter> set ms = {}"
assms_3: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p \<in> F"   and "q \<in> F"   and "p \<noteq> 0"   and "q \<noteq> 0"   and "p \<noteq> q"   and "component_of_term (lt p) = component_of_term (lt q)" shows "(red F)\<^sup>*\<^sup>* (spoly p q) 0"
is_red_monomial_iff: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and v :: "'t"   and c :: "'d" shows "is_red F (monomial c v) = (c \<noteq> (0::'d) \<and> (\<exists>f\<in>F. f \<noteq> 0 \<and> lt f adds\<^sub>t v))"
transfer': shows "state_val (transfer (kind a) s') = fst cfx'(ParamDefs (targetnode a) [:=] map (fst cf') outs)"
power_order_eq_one: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "comm_group G"   and "finite (carrier G)"   and "a \<in> carrier G" shows "a [^]\<^bsub>G\<^esub> card (carrier G) = \<one>\<^bsub>G\<^esub>"
LeadsETo_eq_leadsETo: fixes A :: "'d set"   and CC :: "'d set set"   and B :: "'d set" shows "A LeadsTo[CC] B = {F. F \<in> (reachable F \<inter> A) leadsTo[(\<inter>) (reachable F) ` CC] (reachable F \<inter> B)}"
simps_1: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream" shows "shd (sfilter P s) = shd (sdrop_while (Not \<circ> P) s)"
mult_scalar_pprod_def: shows "mult_scalar_pprod \<equiv> pprod.mult_scalar"
SN_SSN: fixes t :: "trm"   and k :: "stack" assumes "SN (t \<star> k)" shows "SSN k"
Unit_prod_eliminate_entries: fixes i :: "nat"   and nr :: "nat"   and "is" :: "('c \<times> nat) list"   and b :: "'d" assumes "i < nr"   and "\<And>a i'. (a, i') \<in> set is \<Longrightarrow> i' < nr \<and> i' \<noteq> i" shows "\<exists>P\<in>Units (ring_mat TYPE('c) nr b). \<forall>B nc. B \<in> carrier_mat nr nc \<longrightarrow> eliminate_entries_rec B i is = P * B"
Pfp: fixes y :: "'a" assumes "y \<in> P" shows "y \<sqsubset> f p"
Legendre_3_left: fixes p :: "nat" assumes "prime p"   and "3 < p" shows "Legendre 3 (int p) = (if p mod 12 \<in> {1, 11} then 1 else - 1)"
pnet_deliver_single_node: fixes p :: "net_tree"   and s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state" assumes "wf_net_tree p"   and "s \<in> reachable (pnet np p) TT"   and "(s, i:deliver(d), s') \<in> automaton.trans (pnet np p)" shows "(\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j) \<and> net_ip_action np (i:deliver(d)) i p s s'"