hcju/isabelleps huggingface dataset:
mult_le_cancel_right2: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (times a c) c = ((less zero c \<longrightarrow> less_eq a one) \<and> (less c zero \<longrightarrow> less_eq one a))"
bisimE_3: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<otimes> \<Psi>' \<rhd> P \<sim> Q"
elims: fixes x :: "val"   and y :: "pdf_type" assumes "val_type x = y" obtains b :: "bool" where "x = BoolVal b" and "y = BOOL"   | i :: "int" where "x = IntVal i" and "y = INTEG"   | "x = UnitVal" and "y = UNIT"   | r :: "real" where "x = RealVal r" and "y = REAL"   | v1 :: "val" and v2 :: "val" where "x = <|v1, v2|>" and "y = PRODUCT (val_type v1) (val_type v2)"
goodEnv_def: fixes rho :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option" shows "goodEnv rho \<equiv> (\<forall>ys. liftAll good (rho ys)) \<and> (\<forall>ys. |{y. rho ys y \<noteq> None}| <o |UNIV|)"
coalg_T: shows "coalg carT1 carT2 strT1 strT2"
norm_assertion_simps_15: fixes x :: "assn"   and y :: "assn"   and z :: "assn" shows "(x \<or>\<^sub>A y) \<or>\<^sub>A z = x \<or>\<^sub>A y \<or>\<^sub>A z"
moebius_similarity_preserve_orientation: fixes a :: "complex"   and H :: "ocircline"   and b :: "complex" assumes "a \<noteq> 0"   and "\<infinity>\<^sub>h \<notin> ocircline_set H" shows "pos_oriented H = pos_oriented (moebius_ocircline (moebius_similarity a b) H)"
deny_dom: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set" assumes "twoNetsDistinct a b c d" shows "dom (C (DenyAllFromTo a b)) \<inter> dom (C (DenyAllFromTo c d)) = {}"
recurrent_acc_4: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" assumes "MC_syntax.recurrent K x"   and "(x, y) \<in> (SIGMA x:UNIV. set_pmf (K x))\<^sup>*" shows "(x, y) \<in> MC_syntax.communicating K"
dtor_corec: fixes s :: "'d \<Rightarrow> ('c, 'c coF + 'd) coF0"   and x :: "'d" shows "dtor_coF (dtor_corec_coF s x) = map_pre_coF id (case_sum id (dtor_corec_coF s)) (s x)"
language_def: shows "M.language \<equiv> {xs. M.nextl (init M) xs \<in> final M}"
bot_acom: fixes c' :: "'b option acom"   and c :: "com" assumes "strip c' = c" shows "\<bottom>\<^sub>c c \<sqsubseteq> c'"
sigma_finite_embed_measure: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "sigma_finite_measure M"   and "inj f" shows "sigma_finite_measure (embed_measure M f)"
dlist_eq_iff: fixes dxs :: "'d dlist"   and dys :: "'d dlist" shows "(dxs = dys) = (list_of_dlist dxs = list_of_dlist dys)"
dX_singular: fixes X :: "'a set"   and x :: "'a"   and d :: "'b" assumes "inj_on Xd X"   and "x \<in> X"   and "d = Xd x" shows "dX X d = {x}"
inject_2: fixes x2 :: "'m"   and y2 :: "'m" shows "(Ok x2 = Ok y2) = (x2 = y2)"
l9_18_R1: fixes X :: "'p"   and Y :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" assumes "Col X Y P"   and "Col A B P"   and "X Y TS A B" shows "Bet A P B \<and> \<not> Col X Y A \<and> \<not> Col X Y B"
diff_Suc_numeral: fixes n :: "nat"   and k :: "num" shows "Suc n - numeral k = n - pred_numeral k"
equivalence_of_monoidal_categories_axioms: shows "equivalence_of_monoidal_categories (\<cdot>\<^sub>S) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C \<F>C.D D.\<phi> local.map \<F>C.\<nu> \<F>C.\<mu>"
prebicategory_with_homs_and_units_axioms: shows "prebicategory_with_homs_and_units (\<cdot>) (\<star>) \<a> \<i> src trg"
last_map: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
inconsistent_weaken: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set"   and S' :: "(('c, 'b) fm list \<times> 'b) set" assumes "\<not> consistent A S"   and "S \<subseteq> S'" shows "\<not> consistent A S'"
finite_tree_edge_tree: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" shows "tfinite (tree v p pth) = tfinite (edge_tree v p)"
eq_0_height: fixes t :: "'a tree" shows "(0 = height t) = (t = \<langle>\<rangle>)"
finite_proots: fixes p :: "'a poly"   and s :: "'a set" assumes "p \<noteq> 0" shows "finite (proots_within p s)"
as_proj_valid_in_prob_proj: fixes as :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) list"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" assumes "as \<in> valid_plans PROB" shows "as_proj as vs \<in> valid_plans (prob_proj PROB vs)"
SD_is_preorder': fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "Order_Predicates.preorder_on carrier le" shows "Order_Predicates.preorder_on (lotteries_on carrier) (SD le)"
add: fixes b1 :: "'d"   and b2 :: "'d" shows "g (b1 + b2) = g b1 + g b2"
indecomposable_ordertype_finite_ge: fixes \<alpha> :: "V"   and \<A> :: "V set set"   and A :: "V set" assumes "indecomposable \<alpha>"   and "finite \<A>"   and "A \<subseteq> \<Union> \<A>"   and "\<A> \<noteq> {}"   and "\<alpha> \<le> tp A"   and "small (\<Union> \<A>)" shows "\<exists>X\<in>\<A>. \<alpha> \<le> tp X"
ntrunc_def: fixes k :: "nat"   and N :: "('c, 'd) Old_Datatype.node set" shows "ntrunc k N \<equiv> {n \<in> N. ndepth n < k}"
order_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
simps: fixes cfl :: "(('test, 'atom) com \<times> 'state) list"   and s :: "'state" shows "finTrace (cfl, s) = (parTrace cfl \<and> last cfl \<rightarrow>t s)"
if_False: fixes x :: "'d"   and y :: "'d" shows "(if False then x else y) = y"
writes_union_left_I: fixes S :: "('a, 'b, 'c) prog set"   and f :: "('a, 'b, 'd) prog"   and h :: "'a"   and h' :: "'a"   and S' :: "('a, 'b, 'c) prog set" assumes "writes S f h h'" shows "writes (S \<union> S') f h h'"
simps: 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 a1 :: "char list"   and a2 :: "char list list"   and a3 :: "char list"   and a4 :: "ty list \<times> ty \<times> char list list \<times> expr"   and a5 :: "char list list" shows "(P \<turnstile> (a1,a2) selects a3 = a4 via a5) = ((\<exists>C M mthd Cs' Cs. a1 = C \<and> a2 = Cs \<and> a3 = M \<and> a4 = mthd \<and> a5 = Cs' \<and> P \<turnstile> C has least M = mthd via Cs') \<or> (\<exists>C M Cs mthd Cs'. a1 = C \<and> a2 = Cs \<and> a3 = M \<and> a4 = mthd \<and> a5 = Cs' \<and> (\<forall>mthd Cs'. \<not> P \<turnstile> C has least M = mthd via Cs') \<and> P \<turnstile> (C, Cs) has overrider M = mthd via Cs'))"
ts_tuple_rel_auxlist': shows "ts_tuple_rel (set auxlist') = {tas \<in> ts_tuple_rel (set (linearize data_prev) \<union> set (linearize data_in) \<union> {(nt, X)}). valid_tuple tuple_since' tas}"
cpu_reg_val_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and cr :: "CPU_register" assumes "low_equal s1 s2" shows "cpu_reg_val cr s1 = cpu_reg_val cr s2"
InitBlockRed: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and v :: "val"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and 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 v' :: "val"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V \<mapsto> v))\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l' V = \<lfloor>v'\<rfloor>" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow> \<langle>{V:T; V:=Val v';; e'},(h', l'(V := l V))\<rangle>"
finite_basis: shows "\<exists>basis. finite basis \<and> cindependent basis \<and> cspan basis = UNIV"
arr_dom_iff_arr: fixes f :: "'a" shows "C.arr (C.dom f) = C.arr f"
domain_wall_compose: fixes xs :: "wall"   and ys :: "wall" shows "domain_wall (xs \<circ> ys) = domain_wall xs"
hom_integer_group_pow: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.group G"   and "x \<in> carrier G" shows "([^]\<^bsub>G\<^esub>) x \<in> hom integer_group G"
sup_state_conv: fixes P :: "'b prog"   and ST :: "ty list"   and LT :: "ty err list"   and ST' :: "ty list"   and LT' :: "ty err list" shows "P \<turnstile> (ST, LT) \<le>\<^sub>i (ST', LT') = (P \<turnstile> ST [\<le>] ST' \<and> P \<turnstile> LT [\<le>\<^sub>\<top>] LT')"
cod_char: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.cod f = (if A1.arr (fst f) \<and> A2.arr (snd f) then (A1.cod (fst f), A2.cod (snd f)) else (A1.null, A2.null))"
Fence: fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Fence # is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is, \<theta>, x, m, False, \<O>, Map.empty, \<S>)"
in_per_union: fixes u :: "'b"   and x :: "'b"   and y :: "'b"   and u' :: "'b"   and V :: "'b set"   and R :: "('b \<times> 'b) set" assumes "u \<in> {x, y}"   and "u' \<in> {x, y}"   and "x \<in> V"   and "y \<in> V"   and "refl_on V R"   and "part_equiv R" shows "(u, u') \<in> per_union R x y"
take_cols_def: fixes A :: "'a mat"   and inds :: "nat list" shows "take_cols A inds = mat_of_cols (dim_row A) (map ((!) (cols A)) (filter (\<lambda>y. y < dim_col A) inds))"
simps: fixes "is" :: "int list"   and i :: "int"   and off :: "int" shows "nullable0 (presb.Eq is i off) = (i = off)"
winding_number_split_linepath: fixes c :: "complex"   and a :: "complex"   and b :: "complex"   and z :: "complex" assumes "c \<in> closed_segment a b"   and "z \<notin> closed_segment a b" shows "winding_number (linepath a b) z = winding_number (linepath a c) z + winding_number (linepath c b) z"
verit_minus_simplify_4: fixes b :: "'q" shows "- (- b) = b"
free_tv_Fun: fixes t1 :: "typ"   and t2 :: "typ" shows "free_tv (t1 -> t2) = free_tv t1 \<union> free_tv t2"
LEX_trans: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and h :: "nat \<Rightarrow> 'b" assumes "transp_on P A"   and "f \<in> SEQ A"   and "g \<in> SEQ A"   and "h \<in> SEQ A"   and "LEX P f g"   and "LEX P g h" shows "LEX P f h"
less_nmultiset_induct: fixes P :: "'b nmultiset \<Rightarrow> 'b nmultiset \<Rightarrow> bool"   and a0 :: "'b nmultiset"   and a1 :: "'b nmultiset" assumes "\<And>a b. P (Elem a) (Elem b)"   and "\<And>a M. P (Elem a) (MSet M)"   and "\<And>M a. P (MSet M) (Elem a)"   and "\<And>M N. (\<And>X Y k a. \<lbrakk>X \<noteq> {#}; X \<subseteq># N; M = N - X + Y; k \<in># Y\<rbrakk> \<Longrightarrow> P k a) \<Longrightarrow> P (MSet M) (MSet N)" shows "P a0 a1"
approximation_preproc_nat_13: fixes n :: "nat"   and m :: "nat" shows "(n \<le> m) = (real n \<le> real m)"
adh_imp_closure: fixes x :: "'a llist"   and A :: "'a llist set" assumes "x \<in> carrier"   and "x adh A" shows "x \<in> closure A"
1: shows "\<forall>j. (f \<odot> mtop) (i, j) = top \<and> (\<forall>k. i \<noteq> k \<longrightarrow> (f \<odot> mtop) (k, j) = bot)"
pdevs_apply: fixes x :: "'b pdevs" shows "pdevs_apply x \<in> {x. finite {i. x i \<noteq> (0::'b)}}"
le_plus_2: fixes s' :: "'lbl"   and s :: "'lbl" shows "s' \<le> s + s'"
gf: fixes x :: "'a" assumes "x \<in> S" shows "g (f x) = x"
rel_interior_convex_nonempty_aux: fixes S :: "'b set" assumes "convex S"   and "(0::'b) \<in> S" shows "rel_interior S \<noteq> {}"
Sum_any_when_equal': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.Sum_any plus zero (\<lambda>a. zero.when zero (f a) (b = a)) = f b"
eq: fixes a :: "'a" assumes "a \<in> A" shows "emeasure M {a} = emeasure N {a}"
arg_le_Vset_rank: fixes a :: "V" shows "a \<le> Vset (rank a)"
return_edges_only: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<forall>a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<longrightarrow> (\<exists>Qx fx. kind a' = Qx\<hookleftarrow>\<^bsub>p\<^esub>fx)"
of_nat_Suc: fixes m :: "nat" shows "of_nat (Suc m) = (1::'a) + of_nat m"
seqI: fixes f :: "(bool, bool list) concrete_category.arr"   and g :: "(bool, bool list) concrete_category.arr" assumes "J.arr f"   and "J.arr g"   and "J.dom g = J.cod f" shows "J.seq g f"
orthogonal_nullspace_rowspace: fixes A :: "((real, 'b) vec, 'a) vec"   and x :: "(real, 'b) vec"   and y :: "(real, 'b) vec" assumes "A *v x = 0"   and "y \<in> span (rows A)" shows "orthogonal x y"
M_inv_M_id: fixes x :: "int" shows "M (inv_M x) = M x"
comp_Abs_inverse: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "f \<circ> Abs = g" shows "g \<circ> Rep = f"
ipp_cond2_minus: fixes a :: "'e"   and i :: "'e \<Rightarrow> 'e"   and z :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f" shows "ipp_cond2 z {\<^sub>i- a} i f"
ent_disjI2_direct: fixes B :: "assn"   and A :: "assn" shows "B \<Longrightarrow>\<^sub>A A \<or>\<^sub>A B"
redT_updW_None_implies_None: fixes t' :: "'d"   and ws :: "'d \<Rightarrow> 'e wait_set_status option"   and wa :: "('d, 'e) wait_set_action"   and ws' :: "'d \<Rightarrow> 'e wait_set_status option"   and t :: "'d" assumes "redT_updW t' ws wa ws'"   and "ws t = None"   and "t \<noteq> t'" shows "ws' t = None"
ad1: shows "adjoint (tensor_P (1\<^sub>m N) M1) = tensor_P (1\<^sub>m N) M1"
sum_list_replicate: fixes n :: "nat"   and c :: "'c" shows "sum_list (replicate n c) = of_nat n * c"
rp: shows "(\<sigma>, p) \<in> oreachable A S U"
4: shows "w \<le> f \<squnion> f\<^sup>T"
comm_mult_left_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "(0::'b) \<le> c" shows "c * a \<le> c * b"
rsd_winners_subset: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a list"   and alts' :: "'b set" assumes "pref_profile_wf agents alts R"   and "set agents' \<subseteq> agents" shows "rsd_winners R alts' agents' \<subseteq> alts'"
NextPropagate'_causes_safe_2: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" assumes "NextPropagate' c0 c1 p"   and "inv_imps_work_sum (prop_config c1 p)"   and "inv_implications_nonneg (prop_config c1 p)" shows "impl_safe (prop_config c1 p)"
is_sol: shows "b \<bullet> ys = a \<bullet> xs"
transfer: shows "rel_fun pcr_float (rel_fun pcr_float pcr_float) (+) (+)"
interiorI: fixes T :: "'c set"   and x :: "'c"   and S :: "'c set" assumes "open T"   and "x \<in> T"   and "T \<subseteq> S" shows "x \<in> interior S"
m_lcancel: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" assumes "a \<noteq> \<zero>"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp" shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
Set_def: fixes t :: "('a, unit) RBT.rbt" shows "RBT_Set.Set t = {x. RBT.lookup t x = Some ()}"
embed_bothWays_bij_betw: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "Well_order r"   and "Well_order r'"   and "embed r r' f"   and "embed r' r g" shows "bij_betw f (Field r) (Field r')"
is_dghmD_13: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
word32_bounds_2: fixes x :: "32 Word.word" shows "2 ^ (size x - 1) - 1 = 2147483647"
power_poly_f_mod_def: fixes modulus :: "'a poly" shows "power_poly_f_mod modulus = (\<lambda>a n. a ^ n mod modulus)"
prj_simps_15: shows "C.dom Prj\<^sub>0\<^sub>0 = \<mu>\<nu>\<pi>.chine"
kat_1_equiv: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<cdot> y \<le> y \<cdot> d z) = (y \<cdot> ad z \<le> ad x \<cdot> y)"
n2m_ctor_rec_G_F_def: fixes s1 :: "('e, 'e G F \<times> 'f) G0 \<Rightarrow> 'g"   and s2 :: "('e G \<times> 'g, 'e G F \<times> 'f) F0 \<Rightarrow> 'f" shows "n2m_ctor_rec_G_F s1 s2 = ctor_rec_F (s2 \<circ> map_pre_GF\<^sub>M id <id, n2m_ctor_rec_G s1 s2> id \<circ> BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf)"
wmin_lemma: fixes w1 :: "32 Word.word"   and r1 :: "real"   and w2 :: "32 Word.word"   and r2 :: "real" assumes "w1 \<equiv>\<^sub>E r1"   and "w2 \<equiv>\<^sub>E r2" shows "wmin w1 w2 \<equiv>\<^sub>E min r1 r2"
length_Cons: fixes x :: "'c"   and xs :: "'c list" shows "|x # xs| = Suc |xs|"
Trans: shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> \<alpha> \<prec> P'"
span_eq_iff: fixes s :: "('a \<Rightarrow> real) set" assumes "s \<subseteq> diff_fun_space" shows "(diff_fun_space.m1.span s = s) = diff_fun_space.m1.subspace s"
inner_add_left: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "inner (x + y) z = inner x z + inner y z"
WHILEIT_refine_genR: fixes I' :: "'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'d"   and R' :: "('d \<times> 'c) set"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres"   and R :: "('d \<times> 'c) set" assumes "I' x' \<Longrightarrow> (x, x') \<in> R'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; I' x'\<rbrakk> \<Longrightarrow> I x"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; I x; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; b x; b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R' (f' x')"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; \<not> b x; \<not> b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> (x, x') \<in> R" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f x \<le> \<Down> R (WHILE\<^sub>T\<^bsup>I'\<^esup> b' f' x')"
intro: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "group G composition unit" shows "left_translations_of_group G composition unit"
AllA_norm: shows "A\<^sub>p (id \<circ> Some) = (\<forall>Ax. \<lfloor>x\<rfloor>)"
l9_8_1: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and C :: "'p"   and B :: "'p" assumes "P Q TS A C"   and "P Q TS B C" shows "P Q OS A B"
sactionsI: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and P :: "'c prog" assumes "a \<in> actions E"   and "saction P (action_obs E a)" shows "a \<in> sactions P E"
gAnd_def: fixes v :: "'b gexp"   and va :: "'b gexp" shows "gAnd v va \<equiv> Nor (Nor v v) (Nor va va)"
arev_o_arev: shows "edge_rev M \<circ> edge_rev M = id"
i_ab_d': shows "ideal_generated {a, b} = ideal_generated {d'}"
l_set_shadow_root_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_shadow_root_get_shadow_root type_wf set_shadow_root set_shadow_root_locs get_shadow_root get_shadow_root_locs \<equiv> (\<forall>h ptr shadow_root_ptr_opt h'. type_wf h \<longrightarrow> h \<turnstile> set_shadow_root ptr shadow_root_ptr_opt \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_shadow_root ptr \<rightarrow>\<^sub>r shadow_root_ptr_opt) \<and> (\<forall>ptr ptr' w h h' r. ptr \<noteq> ptr' \<longrightarrow> w \<in> set_shadow_root_locs ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_shadow_root_locs ptr' \<longrightarrow> r h h')"
surj_imp_inj_inv: fixes f :: "'d \<Rightarrow> 'c" assumes "surj f" shows "inj (inv f)"
mk_poly2_pre: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_row A" shows "mk_poly2 A $ i = (\<Sum>j'<dim_col A. monom (A $$ (i, dim_col A - Suc j')) j')"
suc_ofilter_in: fixes A :: "'a set"   and b :: "'a" assumes "r.ofilter A"   and "r.AboveS A \<noteq> {}"   and "(b, r.suc A) \<in> r"   and "b \<noteq> r.suc A" shows "b \<in> A"
m: shows "matrix_to_lpolies A ! i \<lbrace> \<langle>X\<rangle> \<rbrace> \<le> b $ i"
tinstT_def: fixes T1 :: "typ"   and T2 :: "typ" shows "tinstT T1 T2 \<equiv> \<exists>\<rho>. tsubstT T2 \<rho> = T1"
esing_def: fixes x :: "'d" shows "esing x = (\<Lambda> a. (\<lambda>y. if x = y then a else \<bottom>))"
iffE: fixes P :: "bool"   and Q :: "bool" assumes "P = Q" obtains "P \<longrightarrow> Q" and "Q \<longrightarrow> P"
perm_supp_eq: fixes p :: "perm"   and x :: "'e" assumes "supp p \<sharp>* x" shows "p \<bullet> x = x"
fixp_def: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "fixp f a = (f a = a)"
of_int_le_iff: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w \<le> z)"
group_isomorphisms_sym: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and f :: "'a \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'a" assumes "group_isomorphisms G H f g" shows "group_isomorphisms H G g f"