hcju/isabelleps huggingface dataset:
representation_sum: fixes basis :: "'a set"   and I :: "'b set"   and v :: "'b \<Rightarrow> 'a" assumes "independent basis"   and "\<And>i. i \<in> I \<Longrightarrow> v i \<in> span basis" shows "representation basis (sum v I) = (\<lambda>b. \<Sum>i\<in>I. representation basis (v i) b)"
correct_transition_relation_def: fixes SES :: "('c, 'd) SES_rec" shows "correct_transition_relation SES \<equiv> \<forall>x y z. x y\<longrightarrow>\<^bsub>SES\<^esub> z \<longrightarrow> x \<in> S\<^bsub>SES\<^esub> \<and> y \<in> E\<^bsub>SES\<^esub> \<and> z \<in> S\<^bsub>SES\<^esub>"
det_Gauss_Jordan_upt_k_det_P: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "fst (Gauss_Jordan_upt_k_det_P A k) * det A = det (snd (Gauss_Jordan_upt_k_det_P A k))"
rbl_word_slice: fixes n :: "nat"   and w :: "'b word" shows "rev (to_bl (slice n w)) = takefill False LENGTH('a) (drop n (rev (to_bl w)))"
ovalid_wp_comb1: fixes P' :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q' :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "ovalid P' f Q"   and "ovalid P f Q'"   and "\<And>s. P s \<Longrightarrow> P' s" shows "ovalid P f (\<lambda>r s. Q r s \<and> Q' r s)"
less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y < z" shows "x < z"
rep_eq: fixes x :: "'b iso"   and xa :: "'b iso" shows "Rep_iso (x \<squnion> xa) = Rep_iso x \<sqinter> Rep_iso xa"
hom_delta: fixes x :: "'b"   and br :: "bool \<times> (bool \<times> 'b) rexp" shows "hom_ab (delta_a x br) = delta_b x (hom_ab br)"
Sup_upper: fixes x :: "'a \<times> 'b"   and A :: "('a \<times> 'b) set" assumes "x \<in> A" shows "less_eq_prod x (Sup_prod A)"
solve_depressed_cubic_complex_code: fixes e :: "complex"   and f :: "complex" shows "solve_depressed_cubic_complex e f = Let (if e = 0 then all_croots 3 (- f) else let f2 = f / 2; u = - f2 + csqrt (f2\<^sup>2 + e ^ 3 / 27) in Let (all_croots 3 u) (map (\<lambda>z. z - e / (3 * z)))) remdups"
simps_5: fixes \<C> :: "t_context"   and t :: "t"   and ux :: "binop_f"   and ts :: "checker_type" shows "check_single \<C> (Binop_f t ux) ts = (if is_float_t t then type_update ts [TSome t, TSome t] (Type [t]) else Bot)"
empty_interior_lowdim_gen: fixes S :: "'b set"   and T :: "'a set" assumes "DIM('a) < DIM('b)"   and "S homeomorphic T" shows "interior S = {}"
lucas_lehmer_to_real_inj: shows "inj lucas_lehmer_to_real"
par_comp_constr_stateful: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> Sec"   and "typing_cond\<^sub>s\<^sub>s\<^sub>t (unlabel \<A>)"   and "\<lbrakk>{}; {}; unlabel \<A>\<rbrakk>\<^sub>s \<I>"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<exists>\<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>) \<and> \<lbrakk>{}; {}; unlabel \<A>\<rbrakk>\<^sub>s \<I>\<^sub>\<tau> \<and> ((\<forall>n. \<lbrakk>{}; {}; proj_unl n \<A>\<rbrakk>\<^sub>s \<I>\<^sub>\<tau>) \<or> (\<exists>\<A>'. prefix \<A>' \<A> \<and> \<A>' leaks Sec under \<I>\<^sub>\<tau>))"
inf_le1: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> x"
assms_4: shows "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"
args_subst: fixes \<rho> :: "'d \<Rightarrow> ('c, 'd) tm"   and s :: "('c, 'd) tm" shows "args (subst \<rho> s) = (case head s of Var x \<Rightarrow> args (\<rho> x) | Sym f \<Rightarrow> []) @ map (subst \<rho>) (args s)"
get_def: fixes h :: "heap" shows "Ref.get h = from_nat \<circ> refs h TYPEREP('a) \<circ> addr_of_ref"
Suc: fixes j :: "nat" shows "(htuple i = htuple j) = (i = j)"
MSF_def: fixes B :: "(nat \<times> int \<times> nat) set" shows "s.MSF B \<equiv> s.SpanningForest B \<and> (\<forall>B'. s.SpanningForest B' \<longrightarrow> (\<Sum>(u, w, v)\<in>B. w) \<le> (\<Sum>(u, w, v)\<in>B'. w))"
induct: fixes P :: "('c \<Rightarrow> 'c list) \<Rightarrow> ('c, 'd) Term.term \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'c list"   and a1 :: "('c, 'd) Term.term" assumes "\<And>P x. P P (Var x)"   and "\<And>P f T. (\<And>x. x \<in> set T \<Longrightarrow> P P x) \<Longrightarrow> P P (Fun f T)" shows "P a0 a1"
fmrestrict_fset_drop: fixes S :: "'a fset"   and b :: "'a"   and m :: "('a, 'b) fmap" shows "fmrestrict_fset S (fmdrop b m) = fmrestrict_fset (S |-| {|b|}) m"
constr_def: fixes F :: "'d upred \<Rightarrow> 'd upred"   and E :: "nat \<Rightarrow> 'd upred" shows "constr F E = (chain E \<and> (\<forall>X n. (F X \<and> E (n + 1)) = (F (X \<and> E n) \<and> E (n + 1))))"
intro: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" assumes "Prim2.prim_invar2_ctd w g r Q \<pi>" shows "Prim_Invar2_ctd_loc w g r Q \<pi>"
np: fixes r :: "sequent list \<times> sequent"   and R :: "(sequent list \<times> sequent) set"   and S :: "sequent" assumes "r \<in> R"   and "case r of (x, xa) \<Rightarrow> (x, xa) \<in> nprovRules" shows "extendRule S r \<in>  R*"
consequence_graph_def: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and G :: "('d, 'f) labeled_graph" shows "consequence_graph Rs G \<equiv> graph G \<and> (\<forall>R\<in>Rs. subgraph (fst R) (snd R) \<and> maintained R G)"
open_contains_cball_eq: fixes S :: "'b set" assumes "open S" shows "\<forall>x. (x \<in> S) = (\<exists>e>0. cball x e \<subseteq> S)"
connected_iff5: fixes x :: "'a" shows "connected x = (x\<^sup>+ ; 1 ; x\<^sup>+ \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>)"
Restrict_def: fixes ord :: "WS1S_Prelim.order"   and i :: "'a" shows "Restrict ord i = (case ord of FO \<Rightarrow> FBase (Fo i) | SO \<Rightarrow> FBool True)"
aform_val_degree_cong: fixes b :: "'a \<times> 'a pdevs"   and d :: "'a \<times> 'a pdevs"   and a :: "nat \<Rightarrow> real"   and c :: "nat \<Rightarrow> real" assumes "b = d"   and "\<And>i. i < degree_aform d \<Longrightarrow> a i = c i" shows "aform_val a b = aform_val c d"
wf_app: fixes r :: "('b \<times> 'b) set"   and f :: "'c \<Rightarrow> 'b" assumes "wf r" shows "wf {(x, y). (f x, f y) \<in> r}"
min_maxsimpchainD_min_betw: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set"   and ys :: "'a set list" assumes "min_gallery (x # xs @ [y])"   and "folding.gallery (x # ys @ [y])" shows "order.greater_eq (length ys) (length xs)"
word_length_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b" assumes "class.monoid_add plus zero" shows "monoid_add.word_length plus zero A a \<equiv> length (ARG_MIN length as. as \<in> lists A \<and> monoid_add.sum_list plus zero as = a)"
weak_greatest_unique: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and A :: "'a set"   and y :: "'a" assumes "weak_partial_order L"   and "greatest L x A"   and "greatest L y A" shows "x .=\<^bsub>L\<^esub> y"
subset_image_lepoll: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "B \<subseteq> f ` A" shows "B \<lesssim> A"
simps_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and N :: "'b" assumes "infinitely_many2 P N" shows "infinitely_many2.enumchain P N 0 = N"
rot_circle_div_circle: assumes "0 < d" shows "gen_division (cubeImage circle_cube) (cubeImage ` {rot_circle_cube})"
inject_2: fixes x21 :: "'b list set"   and x22 :: "'b rexp"   and y21 :: "'b list set"   and y22 :: "'b rexp" shows "(Trn x21 x22 = Trn y21 y22) = (x21 = y21 \<and> x22 = y22)"
mset_isort_key: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "mset (isort_key f xs) = mset xs"
cball_def: fixes x :: "'c"   and e :: "real" shows "cball x e = {y. dist x y \<le> e}"
complex_eq: fixes a :: "complex" shows "a = complex_of_real (Re a) + \<i> * complex_of_real (Im a)"
P_verifies_AuthReq: fixes AuthReqData :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and HOIData :: "msg"   and HOD :: "msg"   and M :: "agent"   and P_I :: "msg"   and evs :: "event list" assumes "AuthReqData = \<lbrace>Number LID_M, Number XID, HOIData, HOD\<rbrace>"   and "Crypt (priSK M) (Hash \<lbrace>AuthReqData, Hash P_I\<rbrace>) \<in> parts (knows Spy evs)"   and "evs \<in> set_pur"   and "M \<notin> bad" shows "\<exists>j trans KM OIData HPIData. Notes M \<lbrace>Number LID_M, Agent (PG j), trans\<rbrace> \<in> set evs \<and> Gets M \<lbrace>P_I, OIData, HPIData\<rbrace> \<in> set evs \<and> Says M (PG j) (EncB (priSK M) KM (pubEK (PG j)) AuthReqData P_I) \<in> set evs"
alwD: fixes P :: "'c stream \<Rightarrow> bool"   and x :: "'c stream" assumes "alw P x" shows "P x"
Cond: fixes pre :: "'c set"   and rely :: "('c \<times> 'c) set"   and P1 :: "'c com"   and b :: "'c set"   and guar :: "('c \<times> 'c) set"   and post :: "'c set"   and P2 :: "'c com" assumes "stable pre rely"   and "\<turnstile> P1 sat [pre \<inter> b, rely, guar, post]"   and "\<turnstile> P2 sat [pre \<inter> - b, rely, guar, post]"   and "\<forall>s. (s, s) \<in> guar" shows "\<turnstile> Cond b P1 P2 sat [pre, rely, guar, post]"
hyps_1: fixes B :: "nat set" assumes "n = card B"   and "finite A"   and "B \<subseteq> A"   and "\<forall>b\<in>B. f b \<noteq> 0" shows "prod f (A - B) = prod f A / prod f B"
differentiable_chain_at: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and g :: "'b \<Rightarrow> 'c" assumes "f differentiable at x"   and "g differentiable at (f x)" shows "g \<circ> f differentiable at x"
append_eq_append_conv: fixes xs :: "'e list"   and ys :: "'e list"   and us :: "'e list"   and vs :: "'e list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
assms: shows "\<Gamma> \<turnstile> s \<equiv> t : T"
set_cat_def: fixes U :: "'b set" shows "set_cat U = \<lparr>ob = Pow U, ar = {f. set_arrow U f}, dom = set_dom, cod = set_cod, id = set_id U, comp = (\<odot>)\<rparr>"
d_rec: fixes n :: "nat" assumes "2 \<le> n" shows "d n = 1 + p * d (nat \<lfloor>p * real n + \<delta>\<rfloor>) + q * d (nat \<lfloor>q * real n - \<delta>\<rfloor>)"
constrains_Int: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A co A'"   and "F \<in> B co B'" shows "F \<in> A \<inter> B co A' \<inter> B'"
zero_le_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a" shows "less_eq zero (power.power one times a n)"
ptr_kinds_M_ok: fixes h :: "(_) heap" shows "h \<turnstile> ok element_ptr_kinds_M"
ideal_inverse: fixes a :: "'a"   and A :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "a \<in> A"   and "ideal A R (+) mult \<zero> \<one>" shows "comm.additive.inverse a \<in> A"
all_term: shows "\<forall>i. terminate f (args @ [i]) \<and> 0 < rec_exec f (args @ [i])"
reflexive_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_idempotent_semiring.reflexive f = (\<forall>i. f (i, i) = top)"
un_empty_absorb2: fixes i :: "nat_int" shows "bot \<squnion> i = i"
sups_def: fixes x :: "'a"   and y :: "'a" shows "sups x y = \<Squnion> {x, y}"
mult: fixes a :: "'a"   and b :: "'a" shows "a \<otimes>\<^bsub>law_restrict A\<^esub> b = a \<otimes>\<^bsub>law_restrict B\<^esub> b"
mult_strict_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less c d"   and "less zero b"   and "less_eq zero c" shows "less (times a c) (times b d)"
borel_measurable_halfspacesI: fixes F :: "real \<times> 'd \<Rightarrow> 'c set"   and S :: "real \<Rightarrow> 'd \<Rightarrow> 'e set"   and f :: "'e \<Rightarrow> 'c"   and M :: "'e measure" assumes "borel = sigma UNIV (F ` (UNIV \<times> Basis))"   and "\<And>a i. S a i = f -` F (a, i) \<inter> space M" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M)"
blocks_nempty: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and bl :: "'b set" assumes "design point_set block_collection"   and "bl \<in># block_collection" shows "bl \<noteq> {}"
decelerate_other: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o" shows "a\<^sub>o < 0"
isin_set: fixes t :: "'a tree"   and x :: "'a" assumes "Sorted_Less.sorted (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
fp_cop_F_allocation: fixes ds :: "'b set" shows "inj_on Xd (ManiPref'.cop ds)"
upset_set_ext: fixes dual :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder_with_dual dual less_eq less" shows "id \<le> ord.upset_set less_eq"
iso_integer_zeroth_homology_group: fixes X :: "'b topology"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" assumes "path_connected_space X"   and "singular_simplex 0 X f" shows "([^]\<^bsub>homology_group 0 X\<^esub>) (homologous_rel_set 0 X {} (frag_of f)) \<in> Group.iso integer_group (homology_group 0 X)"
derivation_eq_localI: fixes U :: "'a set"   and p :: "'a"   and X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "open U"   and "p \<in> U"   and "U \<subseteq> carrier"   and "X \<in> tangent_space p"   and "f \<in> diff_fun_space"   and "g \<in> diff_fun_space"   and "\<And>x. x \<in> U \<Longrightarrow> f x = g x" shows "X f = X g"
steps: shows "(x, y) \<in> (NS \<union> S)\<^sup>*"
pp_increasing: fixes x :: "'a" shows "- - x \<le> x"
simps_1: fixes uu :: "clock"   and uv :: "'b tag_const"   and uw :: "clock" shows "positive_atom (uu sporadic uv on uw) = True"
ord_ge_1: fixes a :: "'a" assumes "finite (carrier G)"   and "a \<in> carrier G" shows "1 \<le> ord a"
components_are_equivalences: fixes a :: "'b" assumes "C.obj a" shows "D.equivalence_map (\<tau>\<^sub>0 a)"
PrefixLists: shows "prefixList exec exec'' \<and> prefixList trace trace''"
prems_1: shows "(E(x' # xs' [\<mapsto>] ys)) x = \<lfloor>y\<rfloor>"
WT_restD_3: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and rest :: "('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme" assumes "WT_rest \<I>_adv \<I>_usr I rest" shows "I (rinit rest)"
prime_elem_associated: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "comm_semiring_1.prime_elem times one zero q"   and "dvd.dvd times q p" shows "normalize q = normalize p"
degree_exists_monom: fixes p :: "'a mpoly"   and var :: "nat" assumes "p \<noteq> 0" shows "\<exists>m\<in>monomials p. lookup m var = MPoly_Type.degree p var"
unitary11_sgn_det_orientation: fixes k :: "complex"   and a :: "complex"   and b :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "k \<noteq> 0"   and "mat_det (a, b, cnj b, cnj a) \<noteq> 0"   and "M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a)" shows "\<exists>k'. sgn k' = sgn (Re (mat_det (a, b, cnj b, cnj a))) \<and> congruence M (1, 0, 0, - 1) = cor k' *\<^sub>s\<^sub>m (1, 0, 0, - 1)"
prim_list_impl_def: fixes l :: "(('b \<times> 'b) \<times> nat) list"   and r :: "'b" shows "prim_list_impl l r \<equiv> if G_valid_wgraph_repr l then Some (prim_impl (G_from_list l) r) else None"
conversep_iff: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'a" shows "r\<inverse>\<inverse> a b = r b a"
prod_le_def: fixes P1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "prod_le P1 P2 = (\<lambda>(p1, p2) (q1, q2). P1 p1 q1 \<and> P2 p2 q2)"
comparator_option_simps_4: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c" shows "comparator_option comp\<^sub>'\<^sub>a (Some x) (Some y) = comp\<^sub>'\<^sub>a x y"
indiff_trans: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "preference carrier relation"   and "x \<approx>[relation] y"   and "y \<approx>[relation] z" shows "x \<approx>[relation] z"
add_right_mono: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "a + c \<subseteq># b + c"
map_sel: fixes a :: "'e option"   and f :: "'e \<Rightarrow> 'f" assumes "a \<noteq> None" shows "the (map_option f a) = f (the a)"
lins_is_lins2_with_empty_S: fixes G :: "diagram" shows "lins G = lins2 {||} G"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<times> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat list \<times> nat" assumes "\<And>a x xs s. (\<And>xa. \<lbrakk>xa = s + a * x; C (x # xs) xa\<rbrakk> \<Longrightarrow> P a (Suc x) (xs, s)) \<Longrightarrow> P a x (xs, s)" shows "P a0 a1 a2"
r_inconsist_recfn: shows "recfn 2 r_inconsist"
chop_eq_Nil_iff: fixes n :: "nat"   and xs :: "'a list" shows "(chop n xs = []) = (n = 0 \<or> xs = [])"
HMA_M3_def: fixes A :: "'j mat \<times> 'j mat \<times> 'j mat"   and B :: "(('j, 'k) Finite_Cartesian_Product.vec, 'k) Finite_Cartesian_Product.vec \<times> (('j, 'l) Finite_Cartesian_Product.vec, 'k) Finite_Cartesian_Product.vec \<times> (('j, 'l) Finite_Cartesian_Product.vec, 'l) Finite_Cartesian_Product.vec" shows "HMA_M3 A B = (Mod_Type_Connect.HMA_M (fst A) (fst B) \<and> Mod_Type_Connect.HMA_M (fst (snd A)) (fst (snd B)) \<and> Mod_Type_Connect.HMA_M (snd (snd A)) (snd (snd B)))"
neg_le_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq a (divide b c) = less_eq b (times a c)"
rel_stateT_def: fixes S :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m :: "('e, 'g) stateT"   and m' :: "('f, 'h) stateT" shows "rel_stateT S M m m' = (S ===> M) (run_state m) (run_state m')"
is_sum4sq_int_def: fixes n :: "int" shows "is_sum4sq_int n = (\<exists>a b c d. n = sum4sq_int (a, b, c, d))"
antisym_conv: fixes y :: "'b word"   and x :: "'b word" assumes "y \<le>s x" shows "(x \<le>s y) = (x = y)"
symmetric_reachable': fixes G :: "('c, 'd) pre_digraph"   and v :: "'c"   and w :: "'c" assumes "sym_digraph G"   and "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w" shows "w \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
G_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.G problem \<equiv> ast_problem.subsuming_states problem (map_of (ast_problem.astG problem))"
\<epsilon>_def: shows "\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (F \<circ> G) \<epsilon>o"
assert_all_s_def: shows "assert_all_s = Incremental_Atom_Ops.assert_all_s assert_s"
that: fixes T' :: "ty" assumes "\<Gamma> \<turnstile> t1 : T'\<rightarrow>T"   and "\<Gamma> \<turnstile> t2 : T'" shows "thesis"
is_rreq_def: fixes \<xi> :: "'b state_scheme" shows "is_rreq \<xi> \<equiv> case msg \<xi> of Rreq hops' dip' dsn' dsk' oip' osn' sip' handled' \<Rightarrow> {\<xi>\<lparr>hops := hops', dip := dip', dsn := dsn', dsk := dsk', oip := oip', osn := osn', sip := sip', handled := handled'\<rparr>} | _ \<Rightarrow> {}"
Ch_range': fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and h :: "'f"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "x \<in> Ch h X" shows "x \<in> X \<and> Xh x = h"
Tree\<^sub>\<alpha>_abs_rep: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) FL_Formula.Tree\<^sub>\<alpha>" shows "FL_Formula.abs_Tree\<^sub>\<alpha> (FL_Formula.rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
minimum_elem_Ssub: fixes D :: "'c Order"   and T :: "'c set"   and X :: "'c set"   and a :: "'c" assumes "Order D"   and "T \<subseteq> carrier D"   and "X \<subseteq> T" shows "minimum_elem D X a = minimum_elem (SIod D T) X a"
fps_mult_assoc_lemma: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" shows "(\<Sum>j = 0..k. \<Sum>i = 0..j. f i (j - i) (n - j)) = (\<Sum>j = 0..k. \<Sum>i = 0..k - j. f j i (n - j - i))"
word_random: fixes p :: "'b word"   and x' :: "'b word"   and x :: "'b word" assumes "p \<le> p + x'"   and "x \<le> x'" shows "p \<le> p + x"
psiFreshVec_4: fixes xvec :: "name list"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* (P \<parallel> Q) = (xvec \<sharp>* P \<and> xvec \<sharp>* Q)"
continuous_vnmD: fixes C :: "'b pmf set"   and P :: "('b pmf \<times> 'b pmf) set" assumes "continuous_vnm C P" shows "\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. p \<succeq>[P] q \<and> q \<succeq>[P] r \<longrightarrow> (\<exists>\<alpha>\<in>{0..1}. mix_pmf \<alpha> p r \<approx>[P] q)"
coeffs_plus_eq_plus_coeffs: fixes p :: "'a poly"   and q :: "'a poly" shows "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
valid_def: fixes \<phi> :: "fml" shows "valid \<phi> \<equiv> \<forall>I \<omega>. \<omega> \<in> fml_sem I \<phi>"
sum_abs: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "\<bar>sum f A\<bar> \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
measure_preservingE_2: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and A :: "'c set" assumes "f \<in> measure_preserving M N"   and "A \<in> sets N" shows "emeasure M (f -` A \<inter> space M) = emeasure N A"
distinct_1: fixes x1 :: "('c, 'd) hd"   and x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm" shows "Hd x1 \<noteq> App x21 x22"
ltake_eSuc: fixes n :: "enat"   and xs :: "'c llist" shows "ltake (eSuc n) xs = (case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> LCons x (ltake n xs'))"
has_visible_field: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F:T in D" shows "P \<turnstile> C has F:T in D"