hcju/isabelleps huggingface dataset:
dickson_lessI: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and s :: "'a"   and m :: "nat"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "d s \<le> m"   and "d t \<le> m"   and "ord_strict s t" shows "gd_powerprod.dickson_less ord_strict d m s t"
P2_security: fixes D :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf"   and n :: "nat"   and m2 :: "bool"   and HCP_ad :: "real"   and m1 :: "bool \<times> bool" assumes "\<forall>a. lossless_spmf (D a)"   and "\<forall>b\<^sub>\<sigma>. etp_rsa.HCP_adv n etp.\<A> m2 b\<^sub>\<sigma> D \<le> HCP_ad" shows "etp.OT_12.adv_P2 n m1 m2 D \<le> 2 * HCP_ad"
that: fixes D :: "'a set" assumes "countable D"   and "\<And>X. \<lbrakk>p X; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>d\<in>D. d \<in> X" shows "thesis"
comp_fun_idem_axioms_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "comp_fun_idem_axioms f \<equiv> \<forall>x. f x \<circ> f x = f x"
diamond_y_valid_1: shows "diamond_y piecewise_C1_differentiable_on {- d / 2..d / 2}"
rel_mset_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'a multiset"   and Y :: "'b multiset" shows "rel_mset R X Y = (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
finprod_mono_neutral_cong_left: fixes B :: "'b set"   and A :: "'b set"   and h :: "'b \<Rightarrow> 'a vec"   and g :: "'b \<Rightarrow> 'a vec" assumes "finite B"   and "A \<subseteq> B"   and "\<And>i. i \<in> B - A \<Longrightarrow> h i = 0\<^sub>v n"   and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"   and "h \<in> B \<rightarrow> carrier_vec n" shows "finsum V g A = finsum V h B"
continuous_on_const: fixes s :: "'b set"   and c :: "'c" shows "continuous_on s (\<lambda>x. c)"
map_ident: fixes t :: "'a \<Rightarrow> 'b" shows "(\<lambda>x. x) \<cdot> t = t"
fcomp_comp: fixes f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "fcomp f g = g \<circ> f"
mult_left_dist_sup: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<odot> (y \<oplus> z) = x \<odot> y \<oplus> x \<odot> z"
assms_2: fixes i :: "'a" assumes "i \<in> S" shows "finite (T i)"
overrides_eq_sigD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old" shows "msig old = msig new"
elims: fixes x :: "'c state"   and xa :: "'a state \<times> 'b"   and y :: "'a state \<times> 'b" assumes "fix_clock x xa = y" obtains s :: "'c state" and s' :: "'a state" and res :: "'b" where "x = s" and "xa = (s', res)" and "y = (update_clock (\<lambda>_. if clock s' \<le> clock s then clock s' else clock s) s', res)"
confluentD: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and y' :: "'b" assumes "confluent R"   and "(x, y) \<in> R"   and "(x, y') \<in> R" shows "(y, y') \<in> joinable R"
simps_2: fixes t1 :: "trm"   and t2 :: "trm"   and Y :: "tyvrs"   and T2 :: "ty" shows "(t1 \<cdot> t2)[Y \<mapsto>\<^sub>\<tau> T2] = t1[Y \<mapsto>\<^sub>\<tau> T2] \<cdot> t2[Y \<mapsto>\<^sub>\<tau> T2]"
upcast_min: shows "0xFFFFFFFF80000001 = SCAST(32 \<rightarrow> 64) (- 0x7FFFFFFF)"
assms_2: shows "set ss \<subseteq> {..<length ls}"
filterlim_cong: fixes F1 :: "'b filter"   and F1' :: "'b filter"   and F2 :: "'c filter"   and F2' :: "'c filter"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "F1 = F1'"   and "F2 = F2'"   and "\<forall>\<^sub>F x in F2. f x = g x" shows "filterlim f F1 F2 = filterlim g F1' F2'"
subst_dnf'_product: fixes A :: "'b ltln list set"   and B :: "'b ltln list set"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_dnf' (A \<otimes>\<^sub>l B) m = subst_dnf' A m \<otimes>\<^sub>l subst_dnf' B m"
that_1: assumes "i < length l"   and "x = y" shows "thesis"
augment_mono_map: fixes A :: "nat \<Rightarrow> bool option"   and B :: "nat \<Rightarrow> bool option"   and x :: "nat set"   and C :: "nat set" assumes "A \<subseteq>\<^sub>s B" shows "A \<oplus>\<^bsub>x\<^esub> C \<subseteq>\<^sub>s B \<oplus>\<^bsub>x\<^esub> C"
i_Exec_Stream_Acc_Output__State_Idle_nth: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "nat"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. n \<le> i \<Longrightarrow> State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState l localState trans_fun input c n)" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c i = i_Exec_Comp_Stream_Acc_Output l output_fun trans_fun input c i"
d: shows "ball w d \<subseteq> S - path_image \<gamma>"
bounded_iff: fixes b :: "'e"   and c :: "'e"   and a :: "'e" shows "(max b c \<le> a) = (b \<le> a \<and> c \<le> a)"
diff_add_eq_diff_diff_swap: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.group_add minus plus zero uminus" shows "minus a (plus b c) = minus (minus a c) b"
struct_map_comp: fixes g :: "'g \<Rightarrow> 'h"   and T :: "('g, 'i) partial_object_scheme"   and S :: "('h, 'j) partial_object_scheme"   and f :: "'h \<Rightarrow> 'k"   and U :: "('k, 'l) partial_object_scheme" assumes "g \<in> struct_maps T S"   and "f \<in> struct_maps S U" shows "compose (carrier T) f g \<in> struct_maps T U"
sup_continuous_enn2ereal: fixes f :: "'a \<Rightarrow> ennreal" assumes "sup_continuous f" shows "sup_continuous (\<lambda>x. enn2ereal (f x))"
closedin_subtopology: fixes U :: "'b topology"   and V :: "'b set"   and S :: "'b set" shows "closedin (subtopology U V) S = (\<exists>T. closedin U T \<and> S = T \<inter> V)"
hyps_1: fixes xvec :: "name list"   and yvec :: "name list"   and y :: "name"   and B' :: "('a, 'b, 'c) boundOutput" assumes "n = length xvec"   and "\<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>x\<rparr>B = \<lparr>\<nu>*yvec\<rparr>\<lparr>\<nu>y\<rparr>B'"   and "length xvec = length yvec"   and "x \<sharp> B'"   and "x \<sharp> xvec"   and "x \<sharp> yvec" shows "\<lparr>\<nu>*xvec\<rparr>B = \<lparr>\<nu>*yvec\<rparr>[(x, y)] \<bullet> B'"
inside_of_triangle: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "DIM('a) = 2" shows "inside (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a) = interior (convex hull {a, b, c})"
assms_3: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<in># roots p"   and "[\<one>, \<ominus> a] \<in> carrier (poly_ring R)"   and "[\<one>, \<ominus> a] pdivides p" shows "P"
closure_act_2: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<forall>\<alpha>. \<^bold>\<A>(\<^bold>\<A>\<phi> \<alpha> \<^bold>\<equiv> \<phi> \<alpha>) in v]"
span_subspace: fixes A :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a"   and B :: "(('a \<Rightarrow> real) \<Rightarrow> real) set" assumes "A \<subseteq> sub.tangent_space p"   and "B \<subseteq> sub.tangent_space p"   and "A \<subseteq> B"   and "B \<subseteq> span A"   and "subspace B" shows "span A = B"
small_to_csmall_nstep_pres: fixes Q :: "'prog \<Rightarrow> 'state \<Rightarrow> bool"   and P :: "'prog"   and \<sigma> :: "'state"   and \<sigma>' :: "'state"   and coll :: "'coll"   and n :: "nat" assumes "\<And>P \<sigma> \<sigma>'. \<lbrakk>Q P \<sigma>; \<sigma>' \<in> small P \<sigma>\<rbrakk> \<Longrightarrow> Q P \<sigma>'"   and "Q P \<sigma>"   and "(\<sigma>', coll) \<in> csmall_nstep P \<sigma> n" shows "Q P \<sigma>'"
rel_cases: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "('d, 'f, 'h) generat"   and b :: "('e, 'g, 'i) generat" assumes "rel_generat R1 R2 R3 a b" obtains x :: "'d" and y :: "'e" where "a = Pure x" and "b = Pure y" and "R1 x y"   | x1a :: "'f" and x2 :: "'h" and y1a :: "'g" and y2 :: "'i" where "a = IO x1a x2" and "b = IO y1a y2" and "R2 x1a y1a" and "R3 x2 y2"
fs_by_gso_def: fixes i :: "nat" assumes "i < length fs" shows "fs ! i = gso i + M.sumlist (map (\<lambda>ja. \<mu> i ja \<cdot>\<^sub>v gso ja) [0..<i])"
elims: fixes x :: "'a option \<times> 'a option"   and y :: "'a option" assumes "merge_option x = y" obtains "x = (None, None)" and "y = None"   | xa :: "'a" where "x = (Some xa__, None)" and "y = Some xa__"   | xa :: "'a" where "x = (None, Some xa)" and "y = Some xa"   | a :: "'a" and b :: "'a" where "x = (Some a, Some b)" and "y = Some a"
rep_eq: fixes x :: "'a covector" shows "Rep_covector (- x) = - Rep_covector x"
LFP_weak_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "LFP f .= f (LFP f)"
diff_less_0_iff_less: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (minus a b) zero = less a b"
compose: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b"   and G :: "'d filter" assumes "f \<in> \<Omega>[F](g)"   and "filterlim h' F G" shows "(\<lambda>x. f (h' x)) \<in> \<Omega>[G](\<lambda>x. g (h' x))"
DataSpace_select: fixes L :: "'b dataspace" shows "DataSpace (Rep_dataspace L)"
cone_subset_PolysD_2: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and X :: "'x set" assumes "cone (h, U) \<subseteq> P[X]"   and "h \<noteq> 0" shows "U \<subseteq> X"
cod_dom: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.cod (VV.dom f) = VV.dom f"
in_components_connected: fixes c :: "'a set"   and s :: "'a set" assumes "c \<in> components s" shows "connected c"
get_env_def: fixes get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> ('a, 'b) envT" shows "get_env get f = EnvT (\<lambda>r. get (\<lambda>s. run_env (f s) r))"
assms_2: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> y" shows "f y \<le> f x"
fmran'_alt_def: fixes m :: "'f \<rightharpoonup>\<^sub>f 'e" shows "fmran' m = fset (fmran m)"
in_set_conv_decomp: fixes x :: "'e"   and xs :: "'e list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
span_subset_spanI: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> span B" shows "span A \<subseteq> span B"
nth_zip: fixes i :: "nat"   and xs :: "'c list"   and ys :: "'d list" assumes "i < length xs"   and "i < length ys" shows "(xs || ys) ! i = (xs ! i, ys ! i)"
length_negGorbit_list: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> 'w"   and as :: "'a list" shows "length (HRep.negGorbit_list gs T as) = length gs"
monic_prod_list: fixes as :: "'c poly list" assumes "\<And>a. a \<in> set as \<Longrightarrow> monic a" shows "monic (prod_list as)"
right_minus_eq: fixes a :: "'a"   and b :: "'a" shows "(a - b = (0::'a)) = (a = b)"
iAll_cong: fixes I :: "nat set"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> nat \<Rightarrow> bool" assumes "\<box> t I. f t = g t" shows "(\<box> t I. P (f t) t) = (\<box> t I. P (g t) t)"
elims: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "'edge list" assumes "upd_rev_cs x xa = y" obtains cs :: "'edge list" where "x = csb__" and "xa = []" and "y = csb__"   | cs :: "'edge list" and as :: "'edge list" and a :: "'edge" where "x = cs" and "xa = as @ [a]" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> upd_rev_cs cs as | c' # cs' \<Rightarrow> upd_rev_cs cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> upd_rev_cs (a # cs) as | _ \<Rightarrow> upd_rev_cs cs as)"
part: shows "partn_lst_VWF \<alpha> [ord_of_nat k, \<gamma>] 2"
mult_isol_equiv_subdistl: shows "(\<forall>x y z. x \<le> y \<longrightarrow> z \<cdot> x \<le> z \<cdot> y) = (\<forall>x y z. z \<cdot> x \<le> z \<cdot> (x + y))"
scast_eq: fixes w :: "'c word" shows "SCAST('c \<rightarrow> 'b) w = word_of_int (sint w)"
root_main'_sound: fixes x :: "int"   and n :: "int"   and y :: "int"   and b :: "bool" assumes "0 \<le> x"   and "0 \<le> n"   and "root_main' x n = (y, b)" shows "b = (y ^ p = n)"
show_lawD: fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and x :: "'a"   and p :: "nat"   and y :: "char list"   and z :: "char list" assumes "show_law s x" shows "s p x (y @ z) = s p x y @ z"
DisjointSchedules_def: fixes n :: "nat"   and nC :: "nat \<Rightarrow> Config" shows "DisjointSchedules n nC \<equiv> \<forall>i j. i < n \<and> j < n \<and> i \<noteq> j \<longrightarrow> disjoint (schedule (nC i)) (schedule (nC j))"
uncross_def: fixes x :: "wall"   and y :: "wall" shows "uncross x y \<equiv> uncross_positive_straighten x y \<or> uncross_positive_flip x y \<or> uncross_negative_straighten x y \<or> uncross_negative_flip x y"
zip_Cons_Cons: fixes x :: "'g"   and xs :: "'g list"   and y :: "'h"   and ys :: "'h list" shows "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
abs_of_nonpos: fixes abs :: "'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a zero" shows "abs a = uminus a"
fixp_induct: fixes P :: "(('a, 'out, 'ret) gpv \<Rightarrow> ennreal) \<Rightarrow> bool" assumes "lfp.admissible P"   and "P (\<lambda>expectation_gpv. \<Squnion> {})"   and "\<And>gpv. \<lbrakk>lfp.le_fun gpv local.expectation_gpv; lfp.le_fun gpv (\<lambda>gpva. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. gpv (c r)) \<partial>measure_spmf (the_gpv gpva) + fail * ennreal (pmf (the_gpv gpva) None)); P gpv\<rbrakk> \<Longrightarrow> P (\<lambda>gpva. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. gpv (c r)) \<partial>measure_spmf (the_gpv gpva) + fail * ennreal (pmf (the_gpv gpva) None))" shows "P local.expectation_gpv"
hurwitz_zeta_cnj: fixes a :: "real"   and z :: "complex" assumes "0 < a" shows "hurwitz_zeta a (cnj z) = cnj (hurwitz_zeta a z)"
pCons_2: shows "degree (pCons a2 p2) < degree (pCons a1 p1)"
O4: fixes x :: "'var"   and \<phi> :: "'fmla"   and n :: "nat" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<And>i. i \<le> n \<Longrightarrow> prv (subst \<phi> (Num i) x)" shows "prv (ball x (Num n) \<phi>)"
full_abstraction_wrt_preorders_impl_indRelSTEQ_to_SRel_and_TRel_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "trans TRel"   and "refl SRel"   and "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q" shows "\<forall>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<longrightarrow> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel"
gt2: shows "prob {x \<in> space M. a < Y x} = exp (- a * u)"
Err_def: shows "Err \<equiv> \<Lambda> e. error_abs\<cdot>(sinl\<cdot>(up\<cdot>e))"
distinct_41: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "While x61 x62 \<noteq> Stop"
enforce_pred_id_option: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option" assumes "pred_option P x" shows "enforce_option P x = x"
unit_factor_mult_normalize: fixes a :: "'a" shows "unit_factor a * normalize a = a"
dbm_le_dbm_min: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<preceq> b" shows "a = dmin a b"
eval_wlp_DC: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" shows "wlp (a \<Sqinter> b) P = (\<lambda>s. min (wlp a P s) (wlp b P s))"
state_bound_def: fixes bp :: "instr array"   and s0 :: "integer" shows "state_bound bp s0 \<equiv> {s. bs_\<alpha> s - bprog_vars bp = bs_\<alpha> s0 - bprog_vars bp}"
prj1_in_hom': fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "\<guillemotleft>\<p>\<^sub>1[f, g] : C.dom \<p>\<^sub>0[f, g] \<rightarrow>\<^sub>C C.dom f\<guillemotright>"
RF_in_B: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" assumes "x \<in> B \<Gamma>" shows "(x \<in> RF\<^bsub>\<Gamma>\<^esub> S) = (x \<in> S)"
bij_betw_Pow: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "bij_betw ((`) f) (Pow A) (Pow B)"
hom_mult: fixes x :: "(bool \<times> unit) list"   and y :: "(bool \<times> unit) list" assumes "x \<in> carrier \<F>\<^bsub>{()}\<^esub>"   and "y \<in> carrier \<F>\<^bsub>{()}\<^esub>" shows "int.lift f (x \<otimes>\<^bsub>\<F>\<^bsub>{()}\<^esub>\<^esub> y) = int.lift f x \<otimes>\<^bsub>int_group\<^esub> int.lift f y"
mimic_single_step: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s\<^sub>1 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s\<^sub>1' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s\<^sub>2 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s\<^sub>1 \<approx> s\<^sub>1'"   and "(s\<^sub>1, s\<^sub>2) \<in> substitution.steps subst" shows "\<exists>s\<^sub>2'. s\<^sub>2 \<approx> s\<^sub>2' \<and> (s\<^sub>1', s\<^sub>2') \<in> substitution.steps subst"
infty_relE: fixes x :: "'a infty"   and x' :: "'b infty"   and R :: "('a \<times> 'b) set" assumes "(x, x') \<in> \<langle>R\<rangle>infty_rel" obtains "x = infty.Infty" and "x' = infty.Infty"   | a :: "'a" and a' :: "'b" where "x = Num a" and "x' = Num a'" and "(a, a') \<in> R"
fmdom'_restrict_set_precise: fixes A :: "'a set"   and m :: "('a, 'b) fmap" shows "fmdom' (fmrestrict_set A m) = fmdom' m \<inter> A"
gauss_int_norm_mod_less: fixes b :: "gauss_int"   and a :: "gauss_int" assumes "b \<noteq> 0" shows "gauss_int_norm (a mod b) < gauss_int_norm b"
dim_basis: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and A :: "'d set" assumes "vectorspace K V"   and "finite A"   and "vectorspace.basis K V A" shows "vectorspace.dim K V = card A"
fv_simps_1: fixes x1 :: "'c" shows "fv (Var x1) = {x1}"
bezout_matrix_works_transpose1: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and i :: "'d" assumes "is_bezout_ext bezout"   and "a \<noteq> b" shows "(A ** Finite_Cartesian_Product.transpose (bezout_matrix (Finite_Cartesian_Product.transpose A) a b i bezout)) $h i $h a = snd (snd (snd (snd (bezout (A $h i $h a) (A $h i $h b)))))"
ZF2mor_Op: fixes C :: "('e, 'd, 'f) LSCategory_scheme"   and f :: "ZF" shows "z2m\<^bsub>Op C\<^esub>f = z2m\<^bsub>C\<^esub>f"
nat_1_eq_mult_iff: fixes m :: "nat"   and n :: "nat" shows "(1 = m * n) = (m = 1 \<and> n = 1)"
assms: shows "run_one_step d i (s, vs, ves, $Call_indirect x13) = (s', vs', res)"
Bernstein_reciprocal: fixes p :: "nat"   and i :: "nat" shows "reciprocal_poly p (Bernstein_Poly_01 i p) = smult (real (p choose i)) ([:- 1, 1:] ^ (p - i))"
lebesgue_integral_eq_integral: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "integrable (lebesgue_on S) f"   and "S \<in> sets lebesgue" shows "integral\<^sup>L (lebesgue_on S) f = integral S f"
substEnv_preserves_freshEnv_aux: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "freshEnv zs z rho"   and "fresh zs z Y"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "freshEnv zs z (rho &[Y / y]_ys)"
allocate_not_emptyI: fixes h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and hT :: "htype" assumes "(h', a) \<in> allocate h hT" shows "allocate h hT \<noteq> {}"
primepow_even_def: fixes q :: "nat" shows "primepow_even q = (\<exists>p k. 1 \<le> k \<and> prime p \<and> q = p ^ (2 * k))"
mult_def: fixes a :: "'a"   and b :: "'a" shows "mult a b = -r (a l\<rightarrow> -l b)"
next_execs_def: fixes s :: "'state_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "next_execs s execs = execs(current s := fst (snd (control s (current s) (execs (current s)))))"
mult_mat_vec_hom: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and v :: "'a Matrix.vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "vec\<^sub>h (A *\<^sub>v v) = mat\<^sub>h A *\<^sub>v vec\<^sub>h v"
correct_convergI: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "eval f xs \<down>"   and "reachable ((f, xs, ls) # rest, None) (rest, eval f xs)" shows "correct ((f, xs, ls) # rest, None)"
gt_imp_ge: fixes y :: "nat"   and x :: "nat" assumes "y < x" shows "y \<le> x"
EVAL_def: fixes x :: "'a" shows "EVAL x \<equiv> RETURN x"
tail_def: fixes s :: "nat \<Rightarrow> 'a" shows "tail s \<equiv> s |\<^sub>s 1"
inf_aci_1: fixes x :: "'a"   and y :: "'a" shows "inf x y = inf y x"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b \<preceq> a"
assms_3: fixes x :: "'a" shows "norm (f x) \<le> norm x * K"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "qSkel Y = qSkel qX" shows "\<forall>qX'. Y #= qX' \<longrightarrow> qInit MOD Y = qInit MOD qX'"
Equiv_Exec_def: fixes m :: "'d"   and equiv_states :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and localState1 :: "'g \<Rightarrow> 'e"   and localState2 :: "'h \<Rightarrow> 'f"   and input_fun1 :: "'d \<Rightarrow> 'i message_af"   and input_fun2 :: "'d \<Rightarrow> 'j message_af"   and output_fun1 :: "'g \<Rightarrow> 'k message_af"   and output_fun2 :: "'h \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and trans_fun2 :: "'j message_af \<Rightarrow> 'h \<Rightarrow> 'h"   and k1 :: "nat"   and k2 :: "nat"   and c1 :: "'g"   and c2 :: "'h" shows "Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 \<equiv> equiv_states (localState1 c1) (localState2 c2) \<longrightarrow> last_message (map output_fun1 (f_Exec_Comp_Stream trans_fun1 (input_fun1 m # NoMsg\<^bsup>k1 - Suc 0\<^esup>) c1)) = last_message (map output_fun2 (f_Exec_Comp_Stream trans_fun2 (input_fun2 m # NoMsg\<^bsup>k2 - Suc 0\<^esup>) c2)) \<and> equiv_states (localState1 (f_Exec_Comp trans_fun1 (input_fun1 m # NoMsg\<^bsup>k1 - Suc 0\<^esup>) c1)) (localState2 (f_Exec_Comp trans_fun2 (input_fun2 m # NoMsg\<^bsup>k2 - Suc 0\<^esup>) c2))"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_wt a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "vars_mset s \<subseteq># vars_mset t" and "wt s < wt t"