hcju/isabelleps huggingface dataset:
bind_assoc: fixes x :: "'m"   and f :: "'a list \<Rightarrow> 'm"   and g :: "'a list \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
\<phi>_def: shows "\<phi> \<equiv> \<lambda>n x. \<Sum>k = Suc 0..\<sigma> n. if real k / real (\<sigma> n) \<le> g x then f x /\<^sub>R real (\<sigma> n) else 0"
DiffD2: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set"   and P :: "bool" assumes "c \<in> A - B"   and "c \<in> B" shows "P"
tauPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel" shows "\<tau>.P \<leadsto><Rel> \<tau>.Q"
acute_trivial: fixes A :: "'p"   and B :: "'p" assumes "A \<noteq> B" shows "Acute A B A"
A_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Simulation_Invariant A B sim PA PB"   and "PA a"   and "A a b" shows "PA b"
Tableau_def: fixes istate :: "'a istate" shows "Tableau istate = (case istate of IState x1 x2 x3 x4 x5 \<Rightarrow> x2)"
powr_less_cancel_iff: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "1 < x" shows "(x powr a < x powr b) = (a < b)"
in_hom_char: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> = (arr a \<and> arr b \<and> arr f \<and> \<F>C.in_hom f a b)"
one_closed: shows "\<one>\<^bsub>DirProds G I\<^esub> \<in> carrier (DirProds G I)"
less_setsD: fixes A :: "'a set"   and B :: "'a set"   and a :: "'a"   and b :: "'a" assumes "A \<lless> B"   and "a \<in> A"   and "b \<in> B" shows "a < b"
tllist_ord_TNil_TCons: fixes b1 :: "'b"   and y :: "'d"   and ys :: "('d, 'b) tllist" shows "local.tllist_ord (TNil b1) (TCons y ys) = (b1 = b)"
d_def: shows "d = A\<^sup>T *\<^sub>v vec (length fs) c"
simps: fixes a1 :: "'c"   and a2 :: "'c"   and succs :: "'c \<Rightarrow> 'c list" shows "(a1 [succs]\<rightarrow>* a2) = ((\<exists>g. a1 = g \<and> a2 = g) \<or> (\<exists>g g' g''. a1 = g \<and> a2 = g'' \<and> g' \<in> set (succs g) \<and> g' [succs]\<rightarrow>* g''))"
apfst_conv: fixes f :: "'d \<Rightarrow> 'a"   and x :: "'d"   and y :: "'c" shows "apfst f (x, y) = (f x, y)"
th0: shows "\<forall>xa\<in>set (remdups (uset (simpfm p))). \<forall>y\<in>set (remdups (uset (simpfm p))). Ifm vs (x # bs) (msubst (simpfm p) (xa, y)) = Ifm vs (x # bs) (msubst (simpfm p) (y, xa))"
iFROM_plus: fixes x :: "nat"   and n :: "nat"   and k :: "nat" assumes "x \<in> [n\<dots>]" shows "x + k \<in> [n\<dots>]"
connected_rel_iff: fixes a :: "nat"   and b :: "nat"   and n :: "nat"   and S :: "(nat \<times> nat) set" shows "((a, b) \<in> connected_rel n S) = has_path n S a b"
nz: fixes x :: "complex" assumes "x \<in> X" shows "\<beta> x \<noteq> 0"
ennreal_sum_less_top: fixes I :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "finite I" shows "(sum f I < top) = (\<forall>i\<in>I. f i < top)"
ltl_simps_2: fixes x21 :: "'c"   and x22 :: "'c llist" shows "ltl (LCons x21 x22) = x22"
funpow_shift1_head: fixes p :: "poly"   and n0 :: "nat"   and k :: "nat" assumes "isnpolyh p n0"   and "p \<noteq> 0\<^sub>p" shows "head ((shift1 ^^ k) p) = head p"
bbox_def: fixes x :: "'a"   and y :: "'a" shows "[x| y = ar (ar y \<cdot> x)"
Cons_in_lists_iff: fixes x :: "'c"   and xs :: "'c list"   and A :: "'c set" shows "(x # xs \<in> lists A) = (x \<in> A \<and> xs \<in> lists A)"
wVarU: fixes x :: "fin_string"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([Var x]<>\<nu>) = (\<nu> (Inl x), \<nu> (Inr x))"
card_Compl: fixes A :: "'a set" assumes "finite A" shows "card (- A) = CARD('a) - card A"
isOK_case_prod: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> 'a + 'b"   and lr :: "'c \<times> 'd" shows "isOK (case lr of (l, r) \<Rightarrow> P l r) = (case lr of (l, r) \<Rightarrow> isOK (P l r))"
honest_verifier_ZK: shows "Schnorr_\<Sigma>.HVZK"
Norm_Distr_2: fixes r :: "'b \<times> 'c \<Rightarrow> 'a"   and P1 :: "'e \<Rightarrow> 'b decision option"   and P2 :: "('f \<Rightarrow> 'c decision option) list"   and d :: "'d \<Rightarrow> 'e \<times> 'f"   and x :: "'d" shows "(r o\<^sub>f (P1 \<Otimes>\<^sub>2 list2policy P2 \<circ> d)) x = list2policy ((P1 \<Otimes>\<^sub>L P2) (\<Otimes>\<^sub>2) r d) x"
hconf_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "P \<turnstile> h \<surd> \<equiv> (\<forall>a obj. h a = \<lfloor>obj\<rfloor> \<longrightarrow> P,h \<turnstile> obj \<surd>) \<and> preallocated h"
good_time_def: shows "good_time = (\<lambda>K. \<Sqinter> {i. 0 < i \<and> 0 < emeasure M ((T ^^ i) -` K \<inter> A)})"
cong: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e" shows "abel_semigroup_set.F f z \<equiv> abel_semigroup_set.F f z"
float_defs_1: fixes a :: "('a, 'b) IEEE.float" shows "is_finite a = (is_normal a \<or> is_denormal a \<or> is_zero a)"
A_proj_wf_trms': fixes l :: "'lbl" shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) (remdups (trms_list\<^sub>s\<^sub>s\<^sub>t (proj_unl l A) @ map (pair' Pair) (setops_list\<^sub>s\<^sub>s\<^sub>t (proj_unl l A))))"
g_def: shows "g \<equiv> \<lambda>x. sum_list (map h (freeword x))"
globs_def2: fixes g :: "oref \<Rightarrow> obj option"   and l :: "lname \<Rightarrow> val option" shows "globs (st g l) = g"
dghm_comp_assoc: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and \<HH> :: "V"   and \<BB> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<HH> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"   and "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> = \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M (\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)"
simps_4: fixes \<sigma> :: "'a basic_type" shows "Boolean \<squnion> \<sigma> = (if Boolean \<le> \<sigma> then \<sigma> else if \<sigma> \<le> Boolean then Boolean else OclAny)"
parent_child_rel_finite: fixes h :: "(_) heap" shows "finite (parent_child_rel h)"
surj_f_inv_f: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "surj f" shows "f (inv f y) = y"
poly_funI: fixes f :: "int \<Rightarrow> int"   and p :: "rat poly" assumes "\<And>a. rat_of_int (f a) = poly p (rat_of_int a)" shows "poly_fun f"
intro: fixes m :: "int" assumes "m = int CARD('b)" shows "poly_mod_type TYPE('b) m"
assms_6: shows "\<D>\<^sub>\<circ> b = I"
fmrel_on_fset_alt_def: fixes S :: "'e fset"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "'e \<rightharpoonup>\<^sub>f 'f"   and n :: "'e \<rightharpoonup>\<^sub>f 'g" shows "fmrel_on_fset S P m n = fBall S (\<lambda>x. rel_option P (fmlookup m x) (fmlookup n x))"
set_less_eq_aux_rec: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "A \<noteq> {}"   and "B \<noteq> {}" shows "(A \<sqsubseteq>' B) = (Min B < Min A \<or> Min A = Min B \<and> A - {Min A} \<sqsubseteq>' B - {Min A})"
gcd_mult_distrib_int: fixes k :: "int"   and m :: "int"   and n :: "int" shows "\<bar>k\<bar> * gcd m n = gcd (k * m) (k * n)"
start_value_def: fixes n :: "int"   and p :: "nat" shows "start_value n p = 2 ^ nat \<lceil>rat_of_nat (log_ceiling 2 n) / rat_of_nat p\<rceil>"
purgeIdle_simps_2: fixes st :: "'St"   and sl :: "('St, 'U, 'C) state list" shows "purgeIdle (Idle st # sl) = purgeIdle sl"
less_trans: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.preorder less_eq less"   and "less x y"   and "less y z" shows "less x z"
intro: fixes S :: "'a set"   and P :: "('a \<times> bool) list set"   and f :: "'a \<Rightarrow> 'b" assumes "GroupByPresentation S P"   and "GroupByPresentationInducedFun_axioms P f" shows "GroupByPresentationInducedFun S P f"
smallo_abs_powr_iff: fixes p :: "real"   and f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "0 < p" shows "((\<lambda>x. \<bar>f x\<bar> powr p) \<in> o[F](\<lambda>x. \<bar>g x\<bar> powr p)) = (f \<in> o[F](g))"
unit_factor_self: fixes a :: "'a" shows "unit_factor a dvd a"
axioms_2: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and g :: "'ah" assumes "constant_transformation A B g" shows "category B"
gr: shows "Lxy xs S = Lxy a S @ [x] @ Lxy b S @ [y] @ Lxy c S"
is_iso_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
eq0: shows "app_pair vec_of \<circ> Inr = Inr"
\<sigma>_def: shows "\<sigma> = (Re s - 2 * real n) / 2"
non_consistency: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and A :: "'b set"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "not_PL p \<in> Cn A"   and "p \<in> Cn A" shows "q \<in> Cn A"
null_setsI: fixes M :: "'b measure"   and A :: "'b set" assumes "emeasure M A = 0"   and "A \<in> sets M" shows "A \<in> null_sets M"
path_image_circlepath_minus_subset: fixes z :: "complex"   and r :: "real" shows "path_image (circlepath z (- r)) \<subseteq> path_image (circlepath z r)"
quadratic2: shows "\<forall>f\<in>set (map (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) (- 1) ((isolate_variable_sparse p var (Suc 0))\<^sup>2 - 4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) (2 * isolate_variable_sparse p var 2)) L @ map (quadratic_sub_fm var (- isolate_variable_sparse p var (Suc 0)) (- 1) ((isolate_variable_sparse p var (Suc 0))\<^sup>2 - 4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) (2 * isolate_variable_sparse p var 2)) F). freeIn var f"
assms: shows "xs = xs' #\<^sub>\<circ> x"
trans_strong: fixes zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "lexext gt zs ys"   and "lexext gt ys xs" shows "lexext gt zs xs"
assms: shows "b\<^sup>2 - 4 * c \<noteq> 0"
is_acc_typeD: fixes G :: "prog"   and P :: "pname"   and T :: "ty" assumes "is_acc_type G P T" shows "is_type G T \<and> G \<turnstile> T accessible_in P"
otherassms_7: shows "(\<xi>', q) \<in> reachable A I"
C_def: shows "C = (norm (integral {a..x0} (\<lambda>t. pbernpoly n t *\<^sub>R f t)) - D * g x0) / fact n"
reduce_column: fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "H \<in> carrier_mat m n"   and "(P, K) = reduce_column div_op H"   and "0 < m" shows "P \<in> carrier_mat m m \<and> K \<in> carrier_mat m n \<and> P * H = K \<and> invertible_mat P"
cond_abs: fixes s :: "'a" assumes "s \<in> wa_invar WC"   and "s \<in> wa_cond WC" shows "\<alpha>1 s \<in> wa_cond WB"
unlabel_subst: fixes S :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "unlabel S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta> = unlabel (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
traces_prefixclosed_def: fixes ES :: "'c ES_rec" shows "traces_prefixclosed ES \<equiv> prefixclosed Tr\<^bsub>ES\<^esub>"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and op :: "op" shows "P (case op of Empty \<Rightarrow> f1 | Ins \<Rightarrow> f2) = ((op = Empty \<longrightarrow> P f1) \<and> (op = Ins \<longrightarrow> P f2))"
G2_bigo': shows "(\<lambda>x. G2 x - (x * (ln x)\<^sup>2 - x * ln x)) \<in> O(\<lambda>x. (ln x)\<^sup>2)"
sum_dcharacter_block': shows "sum \<chi>\<^sub>0 {Suc 0..n} = (if \<chi>\<^sub>0 = \<chi>\<^sub>0 then of_nat (totient n) else 0)"
lattice_latticeof: fixes r :: "('b \<times> 'b) set" shows "lattice (latticeof r)"
not_c_secure_counterexample: shows "\<not> c_secure g_step g_out g_s\<^sub>0 g_I g_D"
simpleDefs_phiDefs_disjoint_inv: fixes n :: "'node" assumes "n \<in> set (\<alpha>n g)" shows "defs g n \<inter> step.phiDefs g n = {}"
combine_options_simps_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a option" shows "combine_options f x None = x"
simps_2: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" shows "\<Phi> \<langle>l, a, r\<rangle> = \<phi> \<langle>l, a, r\<rangle> + \<Phi> l + \<Phi> r"
negate_embed: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c" shows "\<guillemotleft> \<N> P \<guillemotright> s = 1 - \<guillemotleft> P \<guillemotright> s"
subspace_tangent_space: fixes p :: "'a" shows "subspace (tangent_space p)"
combine_rep_number: fixes x :: "'a" shows "\<B>\<^sup>+ rep x = \<B> rep x + \<B>' rep x"
v2: shows "((\<lambda>x. sqrt (Re (cinner x x))) has_derivative (\<lambda>xa. (Re (cinner x xa) + Re (cinner xa x)) * (inverse (sqrt (Re (cinner x x))) / 2))) (at x)"
corthogonal_def: fixes vs :: "'c vec list" shows "corthogonal vs \<equiv> \<forall>i<length vs. \<forall>j<length vs. (inner_prod (vs ! j) (vs ! i) = (0::'c)) = (i \<noteq> j)"
Abs_freeword_inverse: fixes y :: "('b \<times> bool) list" assumes "y \<in> {as. proper_signed_list as}" shows "freeword (Abs_freeword y) = y"
card_deg_sect: fixes X :: "'b set"   and d :: "nat" assumes "finite X"   and "X \<noteq> {}" shows "card (deg_sect X d) = d + (card X - 1) choose (card X - 1)"
non_neg_vec_st_vec: shows "non_neg_vec (st_vec x)"
assms: shows "P \<sim>\<^sup>s\<^sub>l Q"
tagged_partial_division_ofD_4: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "s tagged_partial_division_of i"   and "(x, K) \<in> s" shows "\<exists>a b. K = cbox a b"
a: shows "put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> b\<^sub>1 (\<lbrakk>&y\<rbrakk>\<^sub>e b\<^sub>1)) (\<lbrakk>&x\<rbrakk>\<^sub>e b\<^sub>1) = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> b\<^sub>2 (\<lbrakk>&y\<rbrakk>\<^sub>e b\<^sub>2)) (\<lbrakk>&x\<rbrakk>\<^sub>e b\<^sub>2)"
induct: fixes P :: "'a rexp \<Rightarrow> 'a rexp \<Rightarrow> bool"   and a0 :: "'a rexp"   and a1 :: "'a rexp" assumes "\<And>r. P Zero r"   and "P One Zero"   and "\<And>v. P (Atom v) Zero"   and "\<And>v va. P (Plus v va) Zero"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v. P (Star v) Zero"   and "\<And>r s. \<lbrakk>P s One; P r (pnPlus s One)\<rbrakk> \<Longrightarrow> P (Plus r s) One"   and "\<And>r s v. \<lbrakk>P s (Atom v); P r (pnPlus s (Atom v))\<rbrakk> \<Longrightarrow> P (Plus r s) (Atom v)"   and "\<And>r s v va. \<lbrakk>P s (Plus v va); P r (pnPlus s (Plus v va))\<rbrakk> \<Longrightarrow> P (Plus r s) (Plus v va)"   and "\<And>r s v va. \<lbrakk>P s (Times v va); P r (pnPlus s (Times v va))\<rbrakk> \<Longrightarrow> P (Plus r s) (Times v va)"   and "\<And>r s v. \<lbrakk>P s (Star v); P r (pnPlus s (Star v))\<rbrakk> \<Longrightarrow> P (Plus r s) (Star v)"   and "\<And>s t. (\<lbrakk>One \<noteq> s; \<not> le_rexp One s\<rbrakk> \<Longrightarrow> P One t) \<Longrightarrow> P One (Plus s t)"   and "\<And>v s t. (\<lbrakk>Atom v \<noteq> s; \<not> le_rexp (Atom v) s\<rbrakk> \<Longrightarrow> P (Atom v) t) \<Longrightarrow> P (Atom v) (Plus s t)"   and "\<And>v va s t. (\<lbrakk>Times v va \<noteq> s; \<not> le_rexp (Times v va) s\<rbrakk> \<Longrightarrow> P (Times v va) t) \<Longrightarrow> P (Times v va) (Plus s t)"   and "\<And>v s t. (\<lbrakk>Star v \<noteq> s; \<not> le_rexp (Star v) s\<rbrakk> \<Longrightarrow> P (Star v) t) \<Longrightarrow> P (Star v) (Plus s t)"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Atom v) (Atom va)"   and "\<And>v va vb. P (Atom v) (Times va vb)"   and "\<And>v va. P (Atom v) (Star va)"   and "\<And>v va. P (Times v va) One"   and "\<And>v va vb. P (Times v va) (Atom vb)"   and "\<And>v va vb vc. P (Times v va) (Times vb vc)"   and "\<And>v va vb. P (Times v va) (Star vb)"   and "\<And>v. P (Star v) One"   and "\<And>v va. P (Star v) (Atom va)"   and "\<And>v va vb. P (Star v) (Times va vb)"   and "\<And>v va. P (Star v) (Star va)" shows "P a0 a1"
simps: fixes idx0 :: "'a bound \<Rightarrow> 'a \<Rightarrow> nat"   and idx1 :: "'b bound \<Rightarrow> 'b \<Rightarrow> nat"   and size1 :: "'b bound \<Rightarrow> nat"   and l0 :: "'a"   and r0 :: "'b"   and l1 :: "'a"   and r1 :: "'b"   and a :: "'a"   and b :: "'b" shows "prod_index_def.idx idx0 idx1 size1 (Bound (l0, r0) (l1, r1)) (a, b) = idx0 (Bound l0 l1) a * size1 (Bound r0 r1) + idx1 (Bound r0 r1) b"
leD: fixes ko :: "'a key_order"   and y :: "'a"   and x :: "'a" assumes "ko.le ko y x" shows "\<not> ko.lt ko x y"
nxt: shows "local.next_subseqs (head, tail, prevs) = (out, head', tail', prevs')"
\<Delta>'''_def: fixes \<Delta> :: "('a, 'b) network_scheme" assumes "countable_network \<Delta>" shows "countable_network.\<Delta>''' \<Delta> = \<lparr>edge = \<lambda>x y. edge \<Delta> x y \<and> 0 < capacity \<Delta> (x, y) \<and> y \<noteq> source \<Delta> \<and> x \<noteq> sink \<Delta>, capacity = \<lambda>(x, y). if x = sink \<Delta> \<or> y = source \<Delta> then 0 else capacity \<Delta> (x, y), source = source \<Delta>, sink = sink \<Delta>\<rparr>"
GOAL_INDICATION_def: fixes uu :: "'c" shows "\<paragraph>uu \<equiv> True"
IH: assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> A" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> (A @ D)"
h_in: shows "fst h' \<in> fst ` set hs"
ground_eval_ztpoly_wt_eq: fixes s :: "('s, 'v) tm"   and A :: "'v pvar \<Rightarrow> zhmultiset"   and B :: "'v pvar \<Rightarrow> zhmultiset" assumes "ground s" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
cr_CoCalls_def: shows "cr_CoCalls \<equiv> \<lambda>x y. x = Rep_CoCall y"
integrable_on_null: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "content (cbox a b) = 0" shows "f integrable_on cbox a b"
sorted_on_many: fixes x :: "'a"   and y :: "'a"   and zs :: "'a list" assumes "(x, y) \<in> r"   and "sorted_on r (y # zs)" shows "sorted_on r (x # y # zs)"
has_real_derivative_powr': fixes f :: "real \<Rightarrow> real"   and f' :: "real"   and x :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real" assumes "(f has_real_derivative f') (at x)"   and "(g has_real_derivative g') (at x)"   and "0 < f x" shows "((\<lambda>x. f x powr g x) has_real_derivative f x powr g x * (g' * ln (f x) + f' * g x / f x)) (at x)"
count_step: fixes k :: "nat"   and cs :: "('b \<Rightarrow> bool) list"   and a :: "'b" assumes "k < length cs"   and "(cs ! k) a" shows "count cs a k = Suc k mod length cs"
Contrapos1: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IMP B" shows "H \<turnstile> Robinson_Arithmetic.neg B IMP Robinson_Arithmetic.neg A"
Abstract_Task_axioms: shows "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"
gen_model_worldsD: fixes w' :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and w :: "'e" assumes "w' \<in> worlds (gen_model M w)" shows "w' \<in> worlds M"
lSup_finite_prefixes: fixes xs :: "'c llist" shows "lSup {ys. ys \<sqsubseteq> xs \<and> lfinite ys} = xs"
ring_def: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d" shows "ring R addition multiplication zero unit \<equiv> abelian_group R addition zero \<and> Group_Theory.monoid R multiplication unit \<and> ring_axioms R addition multiplication"
update_convs_2: fixes newAID' :: "(Arraytype \<Rightarrow> nat) \<Rightarrow> Arraytype \<Rightarrow> nat"   and newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a" shows "newAID_update newAID' \<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = \<lparr>newOID = newOID, newAID = newAID' newAID, vals = vals, \<dots> = more\<rparr>"
nth_zip: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'b list" assumes "i < length xs"   and "i < length ys" shows "(xs || ys) ! i = (xs ! i, ys ! i)"
inf_sup_aci_6: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup" shows "sup (sup x y) z = sup x (sup y z)"
implicit_finite_dimensional_vector_space_on: fixes U\<^sub>V\<^sub>S :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set" shows "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S (+) (0::'d) (-) uminus scale basis = finite_dimensional_vector_space_on U\<^sub>V\<^sub>S scale basis"
equiv_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "equiv A r = (refl_on A r \<and> sym r \<and> trans r)"
ivl_disj_un_one_3: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{..l} \<union> {l<..u} = {..u}"
that: fixes r :: "V" assumes "x = \<langle>r, r\<inverse>\<^sub>\<circ>\<rangle>"   and "r \<in>\<^sub>\<circ> A" shows "thesis"
abs_le_iff: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) b = (less_eq a b \<and> less_eq (uminus a) b)"
weaken_not_elem_A_minus_B: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "x \<notin> A \<or> x \<in> B" shows "x \<notin> A - B"
simps_2: fixes a :: "'k"   and x :: "'k list" shows "mset (a # x) = add_mset a (mset x)"
rel_map_2: fixes Sa :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c pmf"   and g :: "'e \<Rightarrow> 'd"   and y :: "'e pmf" shows "rel_pmf Sa x (map_pmf g y) = rel_pmf (\<lambda>x y. Sa x (g y)) x y"
execn_noSpec_no_Stuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "noSpec c"   and "\<forall>p\<in>dom \<Gamma>. noSpec (the (\<Gamma> p))"   and "procs c \<subseteq> dom \<Gamma>"   and "\<forall>p\<in>dom \<Gamma>. procs (the (\<Gamma> p)) \<subseteq> dom \<Gamma>"   and "s \<noteq> Stuck" shows "t \<noteq> Stuck"
1: shows "D'.in_hom (F.unit a) (map\<^sub>0 a) (local.map a)"
less_imp_neq: fixes y :: "'a list"   and x :: "'a list" assumes "y <lex x" shows "x \<noteq> y"
pmat_extension_def: fixes m :: "'b mat" shows "a.pmat_extension m = a.ptensor_mat m (1\<^sub>m a.d2)"
redT_updTs_finite_dom_inv: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and tas :: "('a, 'b, 'd) new_thread_action list" shows "finite (dom (redT_updTs ts tas)) = finite (dom ts)"