hcju/isabelleps huggingface dataset:
Basis_list_vec3: shows "Basis_list = [axis 0 1, axis 1 1, axis 2 1]"
map_le_refl: fixes f :: "'f \<Rightarrow> 'g option" shows "f \<subseteq>\<^sub>m f"
wrap_def: shows "wrap \<equiv> \<Lambda> f (x :@ xs). f\<cdot>x\<cdot>xs"
prefs_wf: fixes i :: "'agent" assumes "i \<in> agents" shows "finite_total_preorder_on alts (permute_profile \<sigma> R i)"
f_accessible_def: fixes \<delta> :: "('d, 'e) ta_rule set"   and Q0 :: "'d set" shows "f_accessible \<delta> Q0 \<equiv> (f_succ \<delta>)\<^sup>* `` Q0"
hn_refine_guessI: fixes P :: "assn"   and f :: "'a Heap"   and P' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and f' :: "'b nres"   and f_conc :: "'a Heap" assumes "hn_refine P f P' R f'"   and "f = f_conc" shows "hn_refine P f_conc P' R f'"
real_quad_ge: fixes an :: "real"   and bn :: "real" shows "2 * (an * bn) \<le> an * an + bn * bn"
ir_disj: fixes P\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare P\<^sub>1 c c' Q\<^sub>1"   and "ir_hoare P\<^sub>2 c c' Q\<^sub>2" shows "ir_hoare (\<lambda>s s'. P\<^sub>1 s s' \<or> P\<^sub>2 s s') c c' (\<lambda>t t'. Q\<^sub>1 t t' \<or> Q\<^sub>2 t t')"
sigma_sets_X: fixes i :: "'i" assumes "i \<in> I" shows "sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)} = sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> E i}"
sub_less_eq_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.subset_extended minus uminus inf sup bot less_eq less top" shows "less_eq (uminus x) (uminus y) = (sup (uminus x) (uminus y) = uminus y)"
ide_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VxVxV.ide f = (B.ide (fst f) \<and> B.VxV.ide (snd f))"
full\<^sub>i_ins: fixes n :: "nat"   and t :: "'b tree23"   and a :: "'b" assumes "full n t" shows "full\<^sub>i n (ins a t)"
set_concat: fixes xs :: "'d list list" shows "set (concat xs) = \<Union> (set ` set xs)"
simps_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat"   and w :: "nat"   and ws :: "nat list" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.len plus m u v (w # ws) = plus (m u w) (linordered_ab_monoid_add.len plus m w v ws)"
adjunct_ord_large_a: fixes a :: "'a" assumes "Order D"   and "a \<notin> carrier D" shows "\<forall>x\<in>carrier D. x \<prec>\<^bsub>adjunct_ord D a\<^esub> a"
abs_cpayload_rep: fixes x :: "msg" shows "(x \<in> Ab ` cpayload) = (Re x \<in> cpayload)"
6: shows "lincomb (\<lambda>a. v $ find_first a (cols A)) (set (cols A)) = 0\<^sub>v n"
le_floor_add: fixes x :: "'a"   and y :: "'a" shows "\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> \<le> \<lfloor>x + y\<rfloor>"
step_2: shows "w\<^sub>2 =\<^sub>S w\<^sub>3"
mangoldt_def: fixes n :: "nat" shows "mangoldt n = (if primepow n then of_real (ln (real (aprimedivisor n))) else (0::'b))"
f0_is_0: shows "f \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>N\<^esub>"
real_cond_exp_gr_c: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable MA f"   and "AE x in MA. c < f x" shows "AE x in MA. c < real_cond_exp MA A.Invariants f x"
measurable_fun_upd: fixes I :: "'b set"   and J :: "'b set"   and i :: "'b"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd"   and N :: "'c measure"   and M :: "'b \<Rightarrow> 'd measure"   and h :: "'c \<Rightarrow> 'd" assumes "I = J \<union> {i}"   and "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M J M"   and "h \<in> N \<rightarrow>\<^sub>M M i" shows "(\<lambda>x. (f x)(i := h x)) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
DD: shows "DD ars r (\<gamma>1'', \<gamma>2, \<sigma>', \<gamma>1''')"
inv_comp_left_2: fixes g :: "'b"   and f :: "'b" assumes "B.iso (g \<cdot>\<^sub>B f)"   and "B.iso g" shows "B.iso f"
rep_eq: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and xb :: "('c, 'd) RBT.rbt" shows "RBT.foldi x xa xb = foldi x xa (rbt.impl_of xb)"
prj0_in_hom: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.cospan f g"   and "a = f \<down>\<down> g"   and "b = C.dom g" shows "\<guillemotleft>\<p>\<^sub>0[f, g] : a \<rightarrow>\<^sub>C b\<guillemotright>"
sup_None_2: fixes x :: "'a option" shows "sup x None = x"
Bd_ab: shows "Brouwer_degree2 (n - Suc 0) f = a + b"
meet_cont1: fixes y :: "'c" shows "cont (\<lambda>x. x \<sqinter> y)"
vlrestriction_mono: fixes A :: "V"   and B :: "V"   and r :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "r \<restriction>\<^sup>l\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> B"
globally_consistent_def: fixes A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "globally_consistent A mds mem\<^sub>1 mem\<^sub>2 \<equiv> (\<forall>x. case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1 x \<noteq> v \<or> mem\<^sub>2 x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x \<noteq> dma mem\<^sub>1 x \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x = Low \<and> (x \<notin> mds AsmNoReadOrWrite \<or> x \<in> \<C>) \<longrightarrow> mem\<^sub>1 [\<parallel>\<^sub>1 A] x = mem\<^sub>2 [\<parallel>\<^sub>2 A] x)"
split_conv: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and a :: "'h"   and b :: "'i" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
ldropn_lappend2: fixes xs :: "'a llist"   and n :: "nat"   and ys :: "'a llist" assumes "llength xs \<le> enat n" shows "ldropn n (lappend xs ys) = ldropn (n - the_enat (llength xs)) ys"
nn_integral_indicator': fixes A :: "'c set"   and M :: "'c measure" assumes "A \<inter> space M \<in> sets M" shows "integral\<^sup>N M (indicator A) = emeasure M (A \<inter> space M)"
module_with_overloaded: shows "module = module_with (+) (0::'e) (-) uminus"
p3: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All k' (SeqSubstFormP v a x y s (Var k) IMP SeqSubstFormP v a x (Var j') s' (Var k') IMP Var j' EQ y))(j'::=y')"
ctr_1: fixes xyax :: "x_y_ax" assumes "xyax = ??.Process.x_y_ax.x" shows "F xyax = Choice (F ??.Process.x_y_ax.ax) (F ??.Process.x_y_ax.y)"
assms_1: shows "\<guillemotleft>f : a \<otimes> \<I> \<rightarrow> a\<guillemotright>"
singletons_dotP: fixes v :: "'a"   and w :: "'a" shows "[v]\<^sub>v \<bullet> [w]\<^sub>v = v * w"
MGFn_Body: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and D :: "qtname"   and c :: "stmt" assumes "wf_prog G"   and "G,A\<turnstile>{=:n} In1r (Init D)\<succ> {G\<rightarrow>}"   and "G,A\<turnstile>{=:n} In1r c\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{=:n} In1l (Body D c)\<succ> {G\<rightarrow>}"
assms_2: fixes \<sigma> :: "'a"   and p :: "'b" assumes "(\<sigma>, p) \<in> init A" shows "P \<sigma> p"
SkipE: fixes \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (SKIP, s) \<Rightarrow> t" obtains "t = s"
data_prev'_def: shows "data_prev' \<equiv> append_queue (nt, X) data_prev"
the_pmf_return: fixes x :: "'d" shows "option_to_spmf.the_pmf (return_pmf x) = x"
below_trans: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.po below"   and "below x y"   and "below y z" shows "below x z"
abs_le_iff: fixes abs :: "'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a" 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)"
hd_exp_basis: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist" shows "ms_aux_hd_exp (scale_shift_ms_aux x xs) = map_option ((+) (snd x)) (ms_aux_hd_exp xs)"
r_limr_total: shows "Partial_Recursive.total r_limr"
h7: shows "\<forall>k<n. nSend k t = []"
scalar_right_mono: fixes u :: "'b vec"   and n :: "nat"   and v :: "'b vec"   and w :: "'b vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n"   and "\<And>i. i < n \<Longrightarrow> w $ i \<le> v $ i"   and "\<And>i. i < n \<Longrightarrow> (0::'b) \<le> u $ i" shows "u \<bullet> w \<le> u \<bullet> v"
induced_induce: fixes G :: "('c, 'd) pre_digraph"   and vs :: "'c set" assumes "wf_digraph G"   and "vs \<subseteq> verts G" shows "induced_subgraph (G \<restriction> vs) G"
assoc: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" shows "linordered_monoid.min (linordered_monoid.min a b) c = linordered_monoid.min a (linordered_monoid.min b c)"
simps_1: fixes var :: "nat"   and p :: "real mpoly"   and xs :: "real list" shows "convert_atom var (Less p) xs = map_option LessUni (convert_poly var p xs)"
r_def: shows "r \<equiv> Cn 2 (r_lifz r_nth (Cn 2 r_tau [Cn 2 r_hd [recf.Id 2 0], recf.Id 2 1])) [Cn 2 r_less [recf.Id 2 1, Cn 2 r_length [recf.Id 2 0]], recf.Id 2 0, recf.Id 2 1]"
that: fixes f :: "nat \<times> 'b \<Rightarrow> nat" assumes "ranking A w f" shows "thesis"
eq_\<I>_generat_relcompp: fixes A :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and D :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and D' :: "'h \<Rightarrow> 'f \<Rightarrow> bool" shows "eq_\<I>_generat A \<I> D OO eq_\<I>_generat A' \<I> D' = eq_\<I>_generat (A OO A') \<I> (D OO D')"
I_1: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t I"
that: shows "low_equal s1 s2 \<and> t1 = snd (fst (mul_instr instr s1)) \<and> t2 = snd (fst (mul_instr instr s2))"
H_consl: fixes p :: "'a"   and p' :: "'a"   and x :: "'a"   and q :: "'a" assumes "\<tt>\<tt> p \<le> \<tt>\<tt> p'"   and "H p' x q" shows "H p x q"
deterministic_if: fixes e :: "((nat \<times> nat) \<times> transition) fset" assumes "\<nexists>s l i r d1 d2 t1 t2. (d1 \<noteq> d2 \<or> t1 \<noteq> t2) \<and> ((s, d1), t1) |\<in>| e \<and> ((s, d2), t2) |\<in>| e \<and> Label t1 = Label t2 \<and> can_take_transition t1 i r \<and> can_take_transition t2 i r" shows "deterministic e"
open_map_def: fixes X1 :: "'b topology"   and X2 :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "open_map X1 X2 f \<equiv> \<forall>U. openin X1 U \<longrightarrow> openin X2 (f ` U)"
intros_1: fixes G :: "'b list set" shows "\<epsilon> \<in> \<langle>G\<rangle>\<^sub>F"
Pbij_insert: fixes \<beta> :: "(Location \<times> Location) set"   and l :: "Location"   and ll :: "Location" assumes "\<beta> \<in> Pbij"   and "l \<notin> Pbij_Rng \<beta>"   and "ll \<notin> Pbij_Dom \<beta>" shows "insert (ll, l) \<beta> \<in> Pbij"
dist_step_def: fixes l :: "'b"   and i :: "nat" shows "dist_step l i \<equiv> next_step_of l i - i"
is_zero_def: fixes a :: "('a, 'b) IEEE.float" shows "IEEE.is_zero a = (IEEE.exponent a = 0 \<and> fraction a = 0)"
aux: fixes a :: "nat" assumes "Suc n dvd Suc (Suc a)"   and "a < n + n"   and "n \<le> a" shows "False"
cut_ge_all_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>\<ge> t = I) = (\<forall>x\<in>I. t \<le> x)"
tag_name_eq_h: fixes ptr' :: "(_) element_ptr"   and disc_nodes :: "char list" shows "h \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes = h' \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b" assumes "monad_fail return bind fail" shows "monad return bind"
ccProd_comm: fixes S :: "Vars.var set"   and S' :: "Vars.var set" shows "S G\<times> S' = S' G\<times> S"
inheritedIncLevelEq: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and A :: "(nat \<times> 'b) set" assumes "inherited subs P" shows "P (incLevel ` A) = P A"
analz_Un_implSecureSet: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({sym.implSecure A B M |A B M. (A, B) \<in> Ag \<and> M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
induct_step: fixes st :: "'c"   and f :: "('a, 'b) urel" assumes "\<And>st'. (st', st) \<in> R \<Longrightarrow> (Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st'\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> f"   and "fp B = f" shows "(Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B f"
LIMSEQ_offset: fixes f :: "nat \<Rightarrow> 'c"   and k :: "nat"   and a :: "'c" assumes "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a" shows "f \<longlonglongrightarrow> a"
gr0I: fixes n :: "nat" assumes "n = 0 \<Longrightarrow> False" shows "0 < n"
mset_subset_eq_mono_add_right_cancel: fixes A :: "'b multiset"   and C :: "'b multiset"   and B :: "'b multiset" shows "(A + C \<subseteq># B + C) = (A \<subseteq># B)"
closed_limpt: fixes S :: "'a set" shows "closed S = (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
flop_adj: fixes C :: "'a set"   and D :: "'a set" assumes "folding.chamber C"   and "folding.chamber D"   and "C \<sim> D" shows "(if C \<in> f \<turnstile> folding.\<C> then THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C else f ` C) \<sim> (if D \<in> f \<turnstile> folding.\<C> then THE Da. Da \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` Da = D else f ` D)"
berlekamp_mat_closed_2: fixes u :: "'b mod_ring poly" shows "dim_row (berlekamp_mat u) = degree u"
p: shows "\<turnstile> p : T \<Rightarrow> \<Delta>"
pnsubstl_correct: fixes l :: "'a list"   and mps :: "(mon \<times> pol) list"   and P1 :: "pol"   and m :: "nat"   and n :: "nat" assumes "in_carrier l"   and "Imon_pol_list l mps" shows "Ipol l (pnsubstl P1 mps m n) = Ipol l P1"
emp_UNlift01_UNpart01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> {} \<notin> F n ` P n" shows "{} \<notin> UNlift01 cl dl P F ` (UNpart01 cl dl P - {BrnFT cl dl})"
is_scc_ex1: shows "\<exists>!scc. is_scc E scc \<and> v \<in> scc"
finite_subset: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
normalize_field: fixes a :: "'b" shows "normalize a = (if a = (0::'b) then 0::'b else (1::'b))"
hj_1: fixes x :: "'a" shows "j (h x) = x"
inv_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (homology_group p X)" shows "inv\<^bsub>homology_group p X\<^esub> x \<in> carrier (homology_group p X)"
f0_sum: shows "(\<Sum>v\<in>T. f0 v *\<^sub>R v) = (0::'a)"
small_these_tm_tdghms: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V" shows "small {\<NN>. \<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>}"
rab: shows "real (nat (a * b)) = real_of_int a * real_of_int b"
eq_card: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) = nrows A"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "finite_group (G i)"
q_def: shows "q = fcompose p [:a, b:] [:1, 1:]"
times_subset_iff: fixes A :: "'d set"   and C :: "'e set"   and B :: "'d set"   and D :: "'e set" shows "(A \<times> C \<subseteq> B \<times> D) = (A = {} \<or> C = {} \<or> A \<subseteq> B \<and> C \<subseteq> D)"
gcd_mod_ring_def: shows "gcd = normalization_euclidean_semiring_class.gcd"
inter_union_distrib_left: fixes A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset" shows "A \<inter># B + C = (A + C) \<inter># (B + C)"
simps_2: fixes v :: "'a" shows "Num_arc_delta v * MinInfty_delta = MinInfty_delta"
hyps_2: shows "P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'"
H\<^sub>L_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b" assumes "weak_composition V H" shows "weak_composition.H\<^sub>L H g \<equiv> H g"
"1_2": shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G \<^bold>\<rightarrow> F"
simps_1: fixes n :: "nat" shows "\<lbrakk>\<lbrakk> [] \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = {\<rho>. True}"
trans: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<preceq>\<^sub>t b"   and "b \<preceq>\<^sub>t c" shows "a \<preceq>\<^sub>t c"
d_def: shows "d = Fn\<cdot>(\<Lambda> e. e \<down>Fn \<bottom>)"
tt_tail: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.tail p \<noteq> 0" shows "punit.tt (punit.tail p) = punit.tt p"
bis_imp_1: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>s c2" shows "c1 \<approx>01T c2"
bit_mask_iff: fixes m :: "nat"   and n :: "nat" shows "bit (mask m) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> n < m)"
accepts_star: fixes A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "NAe.accepts (RegExp2NAe.star A) w = (\<exists>us. (\<forall>u\<in>set us. NAe.accepts A u) \<and> w = concat us)"
set_rec: fixes xs :: "'c list" shows "set xs = rec_list {} (\<lambda>x _. insert x) xs"
S2: assumes "c_len u = 1" shows "c_hd u = (if c_len u = 1 then c_snd (u - 1) else c_fst (c_snd (u - 1)))"
f20: shows "t' \<otimes> t \<otimes> (snd x \<otimes> snd x') \<otimes> (r \<otimes> r') = t' \<otimes> t \<otimes> (snd x \<otimes> r \<otimes> snd x' \<otimes> r')"
prod_vwb_lens: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'h" assumes "vwb_lens X"   and "vwb_lens Y" shows "vwb_lens (X \<times>\<^sub>L Y)"
assms: shows "v \<rightarrow>\<^bsub>del_arc u\<^esub> w"
intros_15: fixes \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" assumes "strict_subtuple (\<lambda>\<tau> \<sigma>. \<tau> \<sqsubset> \<sigma> \<or> \<tau> = \<sigma>) \<pi> \<xi>" shows "Tuple \<pi> \<sqsubset> Tuple \<xi>"
back_shift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set" shows "back cl n (local.shift cl n I) = I"
while_fixp_induct: fixes P :: "('a \<Rightarrow> 'a spmf) \<Rightarrow> bool" assumes "spmf.admissible P"   and "P (\<lambda>while. return_pmf None)"   and "\<And>while'. P while' \<Longrightarrow> P (\<lambda>s. if guard s then body s \<bind> while' else return_spmf s)" shows "P local.while"
intro: fixes z :: "char list \<Longrightarrow> 'a"   and x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a" assumes "vwb_lens z"   and "z \<bowtie> x"   and "x \<bowtie> z"   and "z \<bowtie> y"   and "y \<bowtie> z"   and "z \<bowtie> a"   and "a \<bowtie> z" shows "myss2_axioms x y a z"
simps_1: fixes xs :: "'a list" shows "repeat 0 xs = []"
assms: shows "\<S>\<bullet>\<C> \<turnstile> es @ es' @ es'' : t1s _> t2s"
map_of_Cons_code_1: fixes k :: "'b" shows "table_of [] k = None"
simps_5: fixes v :: "variable"   and va :: "typ"   and uu :: "nat"   and uv :: "Core.term list" shows "subst_bvs1' (Fv v va) uu uv = Fv v va"
liftE_handleE': fixes a :: "'d \<Rightarrow> ('f \<times> 'd) \<times> bool"   and b :: "'g \<Rightarrow> 'd \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool" shows "(liftE a <handle2> b) = liftE a"
base_more_bij_lens: shows "bij_lens (mylens.base\<^sub>L +\<^sub>L mylens.more\<^sub>L)"
Un: fixes a :: "'c set"   and M :: "'c measure"   and b :: "'c set" assumes "a \<in> null_sets M"   and "b \<in> null_sets M" shows "a \<union> b \<in> null_sets M"
assms_1: shows "valid ds (\<lambda>ds fp. R ds fp \<and> P ds fp \<and> Q ds fp) P"
wts'_iff_wts: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts :: "ty list" shows "(P,E,h \<turnstile> es [:'] Ts) = (P,E,h \<turnstile> es [:] Ts)"
cvalidt_def: 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 P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A \<equiv> (\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A) \<longrightarrow> \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
lprefix_cons_def: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and h :: "'e \<Rightarrow> ('d, 'e) step"   and sg :: "'c"   and sh :: "'e" shows "lprefix_cons g h sg sh = lprefix (lunstream g sg) (lunstream h sh)"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> less_eq x y \<or> less_eq y x" shows "connex A less_eq"
add_left_imp_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.cancel_semigroup_add plus"   and "plus a b = plus a c" shows "b = c"
of_int_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> '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 z :: "int"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (z ^ n) = power.power one times (ring_1.of_int one plus zero minus z) n"
R_seq: fixes p :: "'a"   and r :: "'a"   and q :: "'a" shows "\<lparr>Ref p r\<rparr>Ref r q\<lparr>Ref p q\<rparr>"
R'_walk_def: shows "R'_walk \<equiv> P_k_pre @ R_pre @ [z]"
id_is_mor_pr_rngs: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "morphism_presheaves_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF> \<rho> b add_str mult_str zero_str one_str (\<lambda>U. identity (\<FF> U))"