hcju/isabelleps huggingface dataset:
unsigned_less: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and w :: "'b word" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less (semiring_1.unsigned one plus zero w) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) LENGTH('b))"
assms_1: shows "p\<^sub>0 \<in> set ps"
rep_eq: fixes x :: "ennreal"   and xa :: "ennreal" shows "enn2ereal (x + xa) = enn2ereal x + enn2ereal xa"
pr_loop: fixes code :: "abc_inst list"   and n :: "nat"   and fft :: "nat"   and gft :: "nat"   and gap :: "abc_inst list"   and xs :: "nat list"   and x :: "nat"   and f :: "recf"   and g :: "recf"   and gar :: "nat"   and ft :: "nat"   and y :: "nat"   and anything :: "nat list" assumes "code = [Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]"   and "length xs = n"   and "\<forall>y<x. \<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything}"   and "rec_ci g = (gap, gar, gft)"   and "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"   and "ft = max (n + 3) (max fft gft)"   and "Suc y \<le> x" shows "\<exists>stp. abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything) code stp = (0, xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything)"
fset_cases: fixes y :: "'b set" assumes "y \<in> {A. finite A}" obtains x :: "'b fset" where "y = fset x"
degree_def: fixes g :: "graph"   and v :: "nat" shows "degree g v \<equiv> |facesAt g v|"
the_lub_equality: fixes A :: "'b set"   and x :: "'b" assumes "lub A x" shows "\<Squnion>A = x"
ntrs_mon_e_no_ctx: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), w, s', c') \<in> ntrs fg" shows "mon_w fg w \<inter> mon_c fg c = {}"
dom_proj_snapshot: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" shows "prob_dom (prob_proj PROB (prob_dom (snapshot PROB s))) = prob_dom (snapshot PROB s)"
syc_cong_ax_def: shows "syc_cong_ax = Var X1 EQ Var X2 IMP suc (Var X1) EQ suc (Var X2)"
ccw_axioms_2: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and p :: "'b"   and q :: "'b"   and r :: "'b" assumes "ccw_system1235 ccw S"   and "distinct3 p q r"   and "in3 S p q r" shows "ccw p q r \<or> ccw p r q"
unfold: fixes s :: "'b"   and h :: "'b \<rightarrow> ('c, 'b) Step" assumes "s \<noteq> \<bottom>" shows "unfold\<cdot>h\<cdot>s = unfold2\<cdot>(unfold\<cdot>h)\<cdot>(h\<cdot>s)"
rel_pred_pos_distr_iff: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_pred_pos_distr_cond A B = True"
primitive_dchar_iff: shows "primitive_dchar d \<Phi> = (\<not> (\<exists>da<d. \<Phi>.induced_modulus da))"
local_help: fixes vs :: "'a list"   and a :: "'a list"   and r :: "'a"   and b :: "'a list"   and c :: "'a list"   and d :: "'a list" assumes "vs = a @ r # b"   and "vs = c @ r # d"   and "r \<notin> set a"   and "r \<notin> set b" shows "a = c \<and> b = d"
put_nondet_def: fixes put :: "'i \<Rightarrow> 'h \<Rightarrow> 'h"   and s :: "'i"   and m :: "('g, 'h) nondetT" shows "nondetM_base.put_nondet put s m = NondetT (put s (run_nondet m))"
assms_1: shows "has_parametrized_subprob_density M N R f"
is_certified_associated_string_unique: fixes M :: "('a, 'b) message set"   and s :: "'c + ('a, 'b option) insert_message list"   and t :: "'c + ('a, 'b option) insert_message list" assumes "is_certified_associated_string M s"   and "is_certified_associated_string M t" shows "s = t"
ih_2: fixes p :: "nat" assumes "n \<noteq> 0"   and "n div base < p" shows "local.encode (e + 1) (n div base) < local.encode (e + 1) p"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Oldf x1 of Oldf x \<Rightarrow> f1 x | Wit x \<Rightarrow> f2 x) = f1 x1"
vars2'_def: shows "c.vars2' = ind_in_set c.vars0 ` vars3"
cases: fixes x :: "nat \<times> nat" obtains n :: "nat" and a :: "nat" where "n__ \<le> 2" and "x = (a__, n__)"   | n :: "nat" and a :: "nat" where "2 < n" and "x = (a, n)"
step_Normal_elim_cases_3: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Basic f, Normal s) \<rightarrow> u" obtains "u = (Skip, Normal (f s))"
finfun_Diag_apply: fixes f :: "'c \<Rightarrow>f 'd"   and g :: "'c \<Rightarrow>f 'e" shows "($) ($f, g$) = (\<lambda>x. (f $ x, g $ x))"
quot_of_fract_def: shows "quot_of_fract \<equiv> map_fun rep_fract id normalize_quot"
da: shows "Env\<turnstile> B \<guillemotright>t\<guillemotright> A"
lrev_wwfusion: fixes xs :: "'b llist" shows "list2H\<cdot>(wrapH\<cdot>lrev_work2\<cdot>xs) = lrev_work2\<cdot>xs"
Y_ide_arr_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a"   and g :: "'a"   and b' :: "'a"   and b :: "'a" assumes "yoneda_functor C S \<phi>"   and "partial_magma.ide C a"   and "partial_magma.in_hom C g b' b" shows "yoneda_functor.Y C S \<phi> a g = set_category_given_img.mkArr S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set C \<phi> (b, a)) (hom_functor.set C \<phi> (b', a)) (\<phi> (b', a) \<circ> dual_category.comp C g \<circ> hom_functor.\<psi> C \<phi> (b, a))"
complex_exp_exists: fixes z :: "complex" shows "\<exists>a r. z = complex_of_real r * exp a"
index_distinct_alt_ss: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "\<And>G. G \<in> \<G> \<Longrightarrow> \<not> ps \<subseteq> G" shows "int (\<B> index ps) = 1"
IH: shows "changes (replicate (p - length xs) (0::'a) @ rev xs) = changes xs"
poly_div_minus_left: fixes x :: "'b poly"   and y :: "'b poly" shows "- x div y = - (x div y)"
IH: shows "nths (nths xs A) (ind_in_set A ` B) = nths xs B"
g: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "(cms\<^sub>2 ! k, subst \<sigma> (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), subst \<sigma> mem\<^sub>h\<rangle> \<and> (cms\<^sub>1' ! k, subst \<sigma> mem\<^sub>1') \<approx> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), subst \<sigma> mem\<^sub>h\<rangle>"
hext_objD: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option" assumes "h \<unlhd> h'"   and "h a = \<lfloor>(C, fs)\<rfloor>" shows "\<exists>fs'. h' a = \<lfloor>(C, fs')\<rfloor>"
eigenvector_hom: fixes A :: "'a mat"   and n :: "nat"   and v :: "'a vec"   and ev :: "'a" assumes "A \<in> carrier_mat n n"   and "eigenvector A v ev" shows "eigenvector (mat\<^sub>h A) (vec\<^sub>h v) (hom ev)"
elims: fixes x :: "('c, 'd) result"   and xa :: "('c, 'd) result"   and y :: "('c, 'd) result" assumes "x <|> xa = y" obtains xb :: "'d" and uu_ :: "('c, 'd) result" where "x = Ok xb__" and "xa = uu_" and "y = Ok xb__"   | v :: "'c" and xb :: "'d" where "x = Error v__" and "xa = Ok xb" and "y = Ok xb"   | e :: "'c" and v :: "'c" where "x = Error e" and "xa = Error v" and "y = Error e"
elims: fixes x :: "'b tree"   and y :: "'b multiset" assumes "mset_tree x = y" obtains uu_ :: "nat" and a :: "'b" and ts :: "'b tree list" where "x = Node uu_ a ts" and "y = {#a#} + \<Sum>\<^sub># (image_mset mset_tree (mset ts))"
between_vs: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "pre_between vs ram1 ram2" shows "vs = fst (splitAt ram1 vs) @ ram1 # between vs ram1 ram2 @ ram2 # snd (splitAt ram2 vs)"
l6_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> P"   and "B \<noteq> P"   and "C \<noteq> P"   and "Bet A P C" shows "Bet B P C = Tarski_neutral_dimensionless.Out Bet P A B"
bg_impl_rel_eext_internal_def: fixes Rm :: "('b \<times> 'd) set"   and Rv :: "('a \<times> 'c) set" shows "bg_impl_rel_eext Rm Rv \<equiv> \<langle>Rm, \<langle>Rv\<rangle>fun_set_rel\<rangle>gen_bg_impl_rel_eext"
H2: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)"   and "P b (a mod b)" shows "P a b"
dom_cod: fixes f :: "'a2" assumes "C2.arr f" shows "C2.dom (C2.cod f) = C2.cod f"
star_rtc_least: fixes x :: "'a"   and y :: "'a" assumes "1' + x + y ; y \<le> y" shows "x\<^sup>\<star> \<le> y"
ext: fixes g :: "'a"   and f :: "'a" assumes "Right.comp g f \<noteq> Right.null" shows "Right.seq g f"
dom_update_with_aux: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and ps :: "('a \<times> 'b) list" shows "fst ` list.set (update_with_aux v k f ps) = {k} \<union> fst ` list.set ps"
cd'cb: shows "c \<otimes> somegcd G a b divides c \<otimes> b"
Abs_one_or_two_cases: fixes x :: "'a one_or_two" obtains y :: "'a" where "x = Abs_one_or_two y" and "y \<in> one_or_two"
in_homE: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "Dop.in_hom f a b" obtains "Dop.arr f" and "Dop.dom f = a" and "Dop.cod f = b"
bin_to_bl_trunc: fixes n :: "nat"   and m :: "nat"   and w :: "int" assumes "n \<le> m" shows "bin_to_bl n (take_bit m w) = bin_to_bl n w"
Id_onE: fixes c :: "'a \<times> 'a"   and A :: "'a set" assumes "c \<in> Id_on A" obtains x :: "'a" where "x \<in> A" and "c = (x, x)"
wf_cdecl_def: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "wf_cdecl wf_md P \<equiv> \<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> (D, C) \<notin> (subcls1 P)\<^sup>* \<and> (\<forall>(M, Ts, T, m)\<in>set ms. \<forall>D' Ts' T' m'. P \<turnstile> D sees M: Ts'\<rightarrow>T' = m' in D' \<longrightarrow> P \<turnstile> Ts' [\<le>] Ts \<and> P \<turnstile> T \<le> T'))"
prts: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "a = plus (lattice_ab_group_add.pprt zero sup a) (lattice_ab_group_add.nprt zero inf a)"
restrict_def: fixes R :: "('d \<times> 'e) set"   and X :: "'d set" shows "R || X = X \<times> Range R \<inter> R"
lucas_lehmer_add_commute: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "lucas_lehmer_add m x y = lucas_lehmer_add m y x"
image_constant_conv: fixes c :: "'f"   and A :: "'g set" shows "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"
merge2: fixes x :: "'b"   and lx :: "'b heap list"   and h :: "'b heap" shows "merge (Hp x lx) h = (case h of heap.Empty \<Rightarrow> Hp x lx | Hp y ly \<Rightarrow> if x < y then Hp x (Hp y ly # lx) else Hp y (Hp x lx # ly))"
Rep_regular_cases: fixes y :: "'c" assumes "y \<in> regular_elements" obtains x :: "'c regular" where "y = Rep_regular x"
Inf_greatest: fixes A :: "'b set"   and z :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "\<Squnion> A \<le> z"
length_subsets_smash_R: fixes n :: "nat"   and subs1 :: "(nat list \<times> nat list) list"   and subs2 :: "(nat list \<times> nat list) list" shows "length (subsets_smash_R n subs1 subs2) = length subs1 * length subs2"
l_inv: fixes x :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x = \<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub>"
preorder_on_restrict_subset: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "Order_Predicates.preorder_on carrier le"   and "A \<subseteq> carrier" shows "Order_Predicates.preorder_on A (restrict_relation A le)"
delete_eq: fixes k :: "'c" shows "delete k = filter (\<lambda>(k', uu_). k \<noteq> k')"
find_base_vectors_3: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "vectorspace.basis class_ring (module_vec TYPE('a) nc\<lparr>carrier := mat_kernel A\<rparr>) (set (find_base_vectors A))"
simps_8: fixes t :: "'c local.term" shows "Arr \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Arr t"
pos_less_divide_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less a (divide b c) = less (times a c) b"
EndPhase2_HInv3_nL: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase2 s s' t"   and "\<not> HInv3_L s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "\<not> HInv3_L s' p q d"
check'_code_def: shows "check'_code = PivotUpdateMinVars.check' eq_idx_for_lvar min_rvar_incdec_eq pivot_and_update_code"
isAbr_simps_1: fixes s :: "'a" shows "isAbr (Normal s) = False"
all_lt_Max_imp_lt_mset: fixes N :: "'a multiset"   and M :: "'a multiset" assumes "N \<noteq> {#}"   and "\<forall>a\<in>#M. a < Max_mset N" shows "M < N"
p': shows "path \<Delta> x (y # p) (sink \<Delta>)"
ChiRel_ChiStar_trans: fixes S :: "'d"   and U :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and T :: "'d" assumes "(S, U) \<in> ChiStar A"   and "(U, T) \<in> ChiRel A" shows "(S, T) \<in> ChiStar A"
has_densityI: fixes f :: "'c \<Rightarrow> ennreal"   and N :: "'c measure"   and M :: "'c measure" assumes "f \<in> borel_measurable N"   and "M = density N f"   and "space N \<noteq> {}" shows "has_density M N f"
ws_cdecl_Ext: shows "ws_cdecl tprg Ext Base"
f_1: fixes s :: "'a set" assumes "s \<in> f" shows "\<exists>a b. s = cbox a b"
suc: fixes x :: "ordinal" assumes "P x" shows "P (oSuc x)"
comp_ide_self: fixes a :: "'a" assumes "B.ide a" shows "a \<cdot>\<^sub>B a = a"
Asm: fixes P :: "'d set"   and p :: "'e"   and Q :: "'d set"   and A :: "'d set"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and F :: "'f set" assumes "(P, p, Q, A) \<in> \<Theta>" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Call p Q,A"
Von_Neumann_Morgenstern_Utility_Theorem: fixes outcomes :: "'b set"   and \<R> :: "('b pmf \<times> 'b pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}" shows "(rational_preference (\<P> outcomes) \<R> \<and> independent_vnm (\<P> outcomes) \<R> \<and> continuous_vnm (\<P> outcomes) \<R>) = (\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u))"
a1: shows " t \<longrightarrow>\<^sub>\<beta>\<^sup>* t1"
u_gt_s_if_ui: assumes "is_App u"   and "fun u \<ge>\<^sub>t t \<or> arg u \<ge>\<^sub>t t" shows "u >\<^sub>t s"
ord_p_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.ord_p ord_term_strict p q \<equiv> ordered_term.ord_strict_p ord_term_strict p q \<or> p = q"
mset_contains_eq: fixes m :: "'a"   and M :: "'a multiset" shows "(m \<in># M) = ({#m#} + (M - {#m#}) = M)"
sum_mono: fixes K :: "'e set"   and f :: "'e \<Rightarrow> 'f multiset"   and g :: "'e \<Rightarrow> 'f multiset" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<subseteq># g i" shows "comm_monoid_add.sum (+) {#} f K \<subseteq># comm_monoid_add.sum (+) {#} g K"
assms_2: assumes "s = r'"   and "c = c'" shows "P"
zero_le_divide_abs_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide a (abs b)) = (less_eq zero a \<or> b = zero)"
mu_no_change: fixes i' :: "nat"   and j' :: "nat" assumes "i' < m"   and "j' < m"   and "i' \<noteq> i \<or> j < j'" shows "gs2.\<mu> i' j' = gs1.\<mu> i' j'"
disc_1: fixes x1 :: "'f option" shows "Resumption.resumption.is_Done (Resumption.resumption.Done x1)"
components_are_iso: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide a" shows "Span.iso (Span.src a)"
cf_sol_def: fixes R :: "('d, 'e) Ring_scheme"   and S :: "('d, 'f) Ring_scheme"   and X :: "'d"   and p :: "'d"   and c :: "nat \<times> (nat \<Rightarrow> 'd)" shows "cf_sol R S X p c = (pol_coeff S c \<and> p = polyn_expr R X (fst c) c)"
VecHom_GMap_is_FGModuleHom: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'v \<Rightarrow> 'a"   and W :: "'a set" assumes "VectorSpaceHom (\<sharp>\<cdot>) V (aezfun_scalar_mult.fsmult smult') T"   and "FGModule G smult' W"   and "T ` V \<subseteq> W"   and "\<And>g v. \<lbrakk>g \<in> G; v \<in> V\<rbrakk> \<Longrightarrow> T (g *\<cdot> v) = aezfun_scalar_mult.Gmult smult' g (T v)" shows "FGModuleHom G (\<cdot>) V smult' T"
ladder_every_stage_2: fixes index :: "nat" assumes "index < length L" shows "ladder_j L' index = ladder_j L index"
E: shows "on_triple f `` E \<subseteq> edges (S (max i j))"
app_iso_eq_conv: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and H1 :: "('a, 'b) pre_digraph"   and H2 :: "('a, 'b) pre_digraph" assumes "digraph_isomorphism hom"   and "subgraph H1 G"   and "subgraph H2 G" shows "(app_iso hom H1 = app_iso hom H2) = (H1 = H2)"
that: assumes "2 \<le> p"   and "p \<le> n"   and "p dvd n"   and "prime p" shows "thesis"
moebius_circline_ocircline: fixes M :: "moebius"   and H :: "circline" shows "moebius_circline M H = of_ocircline (moebius_ocircline M (of_circline H))"
arr_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "partial_magma.arr (product_category.comp C1 C2) f = (partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f))"
e_psp_stable2: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and B :: "'d set" assumes "F \<in> A leadsTo[CC] A'"   and "F \<in> stable B"   and "\<forall>C\<in>CC. C \<inter> B \<in> CC" shows "F \<in> (B \<inter> A) leadsTo[CC] (B \<inter> A')"
additiveI: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>P Q s. \<lbrakk>sound P; sound Q\<rbrakk> \<Longrightarrow> t (\<lambda>s. P s + Q s) s = t P s + t Q s" shows "Transformers.additive t"
less_imp_le: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
solve_cubic_depressed_Cardano_complex: fixes e :: "complex"   and v :: "complex"   and u :: "complex"   and f :: "complex"   and y :: "complex" assumes "e \<noteq> 0"   and "v = - (e ^ 3 / 27)"   and "u\<^sup>2 + f * u + v = 0" shows "(y ^ 3 + e * y + f = 0) = (\<exists>z. z ^ 3 = u \<and> y = z - e / (3 * z))"
GuardSwap: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and R :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and g :: "'e set"   and f :: "'f" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c Q,A"   and "P \<subseteq> g \<inter> R" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g c Q,A"
relAPP_def: fixes f :: "('g \<times> 'h) set \<Rightarrow> 'f"   and x :: "('g \<times> 'h) set" shows "\<langle>x\<rangle>f \<equiv> f x"
ldropn_def: fixes n :: "nat"   and xs :: "'b llist" shows "ldropn n xs = (ltl ^^ n) xs"
nat_pow_Suc2: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "x [^] Suc n = x \<otimes> x [^] n"
map_def: fixes f :: "'a" shows "\<phi>.map f = (if hom.arr f then hom.comp ((d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, hom.cod f \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (hom.cod f \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[hom.cod f, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> hom.cod f \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[hom.cod f \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[hom.cod f]) (hom.map f) else hom.null)"
assms_2: fixes i :: "nat" assumes "0 < i" shows "A i \<noteq> {}"
Incr: assumes "lvl (update x y t) = lvl t + 1"   and "sngl (update x y t)" shows "thesis"
cod_tuple: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "cod \<langle>f, g\<rangle> = cod f \<otimes> cod g"
inj_minus: fixes x :: "nat"   and q :: "nat"   and ya :: "nat"   and y :: "nat" assumes "x < q"   and "ya < q"   and "(y + q - x) mod q = (y + q - ya) mod q" shows "x = ya"