hcju/isabelleps huggingface dataset:
of_int_minus: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (- z) = uminus (ring_1.of_int one plus zero minus z)"
IH: fixes e :: "exp" assumes "set T \<subseteq> range Arg" shows "(\<Gamma>, eta_expand (length T) e, T @ S) \<Rightarrow>\<^sup>* (\<Gamma>, e, T @ S)"
prod_pOp_commute: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and a :: "'d \<Rightarrow> 'e"   and b :: "'d \<Rightarrow> 'e" assumes "\<forall>k\<in>I. aGroup (A k)"   and "a \<in> carr_prodag I A"   and "b \<in> carr_prodag I A" shows "prod_pOp I A a b = prod_pOp I A b a"
labels_psimps_3: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp"   and p1 :: "('c, 'd, 'b, 'e) seqp"   and p2 :: "('c, 'd, 'b, 'e) seqp" assumes "wellformed \<Gamma>" shows "labels \<Gamma> (p1 \<oplus> p2) = labels \<Gamma> p1 \<union> labels \<Gamma> p2"
diff_funI: fixes f :: "'a \<Rightarrow> 'd" assumes "\<And>x. x \<in> carrier \<Longrightarrow> \<exists>c1\<in>atlas. x \<in> domain c1 \<and> k-smooth_on (codomain c1) (f \<circ> inv_chart c1)" shows "diff_fun k charts f"
det_col_add: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and a :: "('c, 'd) vec"   and b :: "('c, 'd) vec" shows "Square_Matrix.det (upd_col M i (a + b)) = Square_Matrix.det (upd_col M i a) + Square_Matrix.det (upd_col M i b)"
Int_insert_left: fixes a :: "'k"   and B :: "'k set"   and C :: "'k set" shows "insert a B \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
fconverse_ftimes: fixes A :: "V"   and B :: "V" shows "(A \<times>\<^sub>\<bullet> B)\<inverse>\<^sub>\<bullet> = B \<times>\<^sub>\<bullet> A"
eq_iff: fixes x :: "'c"   and y :: "'c" shows "(x = y) = (y \<le> x \<and> x \<le> y)"
minVertex_in: fixes f :: "face" assumes "vertices f \<noteq> []" shows "minVertex f \<in> \<V> f"
cindex_poly_0_1: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "cindex_poly a b 0 p = 0"
get_mode_pure: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "pure (get_mode shadow_root_ptr) h"
feq: fixes x :: "'c" assumes "x \<in> U" shows "(h \<circ> (k \<circ> f)) x = f x"
sorted_list_of_set_lessThan_Suc: fixes k :: "nat" shows "sorted_list_of_set {..<Suc k} = sorted_list_of_set {..<k} @ [k]"
ids_imply_messages_same: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and ir :: "'a"   and e2 :: "'b" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Add i e1) \<in> set (node_deliver_messages xs)"   and "(ir, Rem is e2) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
finite_measure_Diff': fixes A :: "real set"   and B :: "real set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M (A \<inter> B)"
incr_bounds_eq: fixes k :: "nat"   and n :: "nat"   and t :: "term"   and i :: "int" assumes "k \<le> n"   and "wellformed' k t" shows "incr_bounds i n t = t"
mono_closed_real: fixes S :: "real set" assumes "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"   and "closed S" shows "S = {} \<or> S = UNIV \<or> (\<exists>a. S = {a..})"
fps_binomial_of_nat: fixes n :: "nat" shows "fps_binomial (of_nat n) = (1 + fps_X) ^ n"
image_image: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "\<P> f (\<P> g A) = \<P> (\<lambda>x. f (g x)) A"
equivalence_def: fixes S :: "('d, 'e) eq_object_scheme" shows "equivalence S \<equiv> (\<forall>x. x \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y. x .=\<^bsub>S\<^esub> y \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> y .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y z. x .=\<^bsub>S\<^esub> y \<longrightarrow> y .=\<^bsub>S\<^esub> z \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> z \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> z)"
elims_3: fixes x :: "iface"   and xa :: "iface" assumes "\<not> x \<le> xa" obtains v :: "char" and va :: "char list" where "x = Iface (v # va)" and "xa = Iface []"   | a :: "char" and as :: "char list" and b :: "char" and bs :: "char list" where "x = Iface (a # as)" and "xa = Iface (b # bs)" and "\<not> (if a = b then Iface as \<le> Iface bs else a \<le> b)"
absorb1: fixes a :: "nat"   and b :: "nat" assumes "a dvd b" shows "gcd a b = a"
is_moebius_similarity: fixes a :: "complex_homo"   and b :: "complex_homo" assumes "a \<noteq> 0\<^sub>h"   and "a \<noteq> \<infinity>\<^sub>h"   and "b \<noteq> \<infinity>\<^sub>h" shows "is_moebius (\<lambda>z. a *\<^sub>h z +\<^sub>h b)"
simps_2: fixes st :: "'b state"   and env :: "v sem_env"   and v2 :: "v"   and p :: "pat"   and e :: "exp0"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" shows "fun_evaluate_match st env v2 ((p, e) # pes) err_v = (if allDistinct (pat_bindings p []) then case pmatch (c env) (refs st) p v2 [] of No_match \<Rightarrow> fun_evaluate_match st env v2 pes err_v | Match_type_error \<Rightarrow> (st, Rerr (Rabort Rtype_error)) | Match env_v' \<Rightarrow> fun_evaluate st (update_v (\<lambda>_. nsAppend (alist_to_ns env_v') (sem_env.v env)) env) [e] else (st, Rerr (Rabort Rtype_error)))"
has_densityD_3: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f" shows "space N \<noteq> {}"
intros: fixes n :: "'a"   and m :: "'a" assumes "n \<le> m" shows "\<tau>\<^sub>c\<^sub>s\<^sub>t n \<le> \<tau>\<^sub>c\<^sub>s\<^sub>t m"
enc_dec: fixes n :: "nat"   and w :: "('a \<times> bool list) list"   and FO :: "nat set" assumes "wf_word n w"   and "\<forall>i\<in>FO. i < n \<longrightarrow> (\<exists>!p. p < length w \<and> snd (w ! p) ! i)" shows "enc (dec_word w, dec_interp n FO w) = w"
CatIdCompId: fixes X :: "'a" assumes "X \<in> obj" shows "id X ;; id X = id X"
distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b \<squnion> c) = a * b \<squnion> a * c"
simps_2: fixes a :: "'b" shows "literal_formula_to_literal (\<^bold>\<not> (Atom a)) = a\<inverse>"
weak_reduction_bisimulation_and_closures_5: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_bisimulation (Rel\<^sup>*) Cal"
unity_root_eqD: fixes k :: "nat"   and i :: "int"   and j :: "int" assumes "0 < k"   and "unity_root k i = unity_root k j" shows "i mod int k = j mod int k"
Cv: shows "C *\<^sub>v v = 0\<^sub>v n"
trancl_induct: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (a, y) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>+; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
norm_sum_Pythagorean: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite I"   and "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I" shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
strict_lunit: fixes a :: "'d" assumes "D.ide a" shows "\<l>\<^sub>D[a] = a"
trancl_into_rtrancl: fixes a :: "'g"   and b :: "'g"   and r :: "('g \<times> 'g) set" assumes "(a, b) \<in> r\<^sup>+" shows "(a, b) \<in> r\<^sup>*"
poly_split: fixes m :: "'d monom"   and p :: "('d monom \<times> 'c) list"   and c :: "'c"   and q :: "('d monom \<times> 'c) list" assumes "poly_split m p = (c, q)" shows "p =p (m, c) # q"
cindex_pathE_circlepath_upper_2: fixes r :: "real"   and z0 :: "complex"   and z :: "complex" assumes "r < Im (z0 - z)"   and "\<bar>Re (z0 - z)\<bar> < r" shows "cindex_pathE (part_circlepath z r 0 pi) z0 = 1"
l2_inv8_init: shows "init l2 \<subseteq> l2_inv8"
set_sorted_list_of_set: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
split_list_last_prop_iff: fixes xs :: "'m list"   and P :: "'m \<Rightarrow> bool" shows "(\<exists>x\<in>set xs. P x) = (\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z\<in>set zs. \<not> P z))"
IH: fixes P :: "nat fset" shows "WS1S.sat (Extend SO 0 \<AA> P) (\<epsilon> bs1 (X # bs2) \<phi>) = satisfies (dec_I1 (Extend SO 0 \<AA> P) bs1) (dec_I2 (Extend SO 0 \<AA> P) (X # bs2)) \<phi>"
HFinite_hcmod_iff: fixes x :: "complex star" shows "(hcmod x \<in> HFinite) = (x \<in> HFinite)"
gen_akiyama_tanigawa_row_Suc: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and l :: "nat"   and u :: "nat" shows "gen_akiyama_tanigawa_row f (Suc n) l u = akiyama_tanigawa_step_aux (Suc l) (gen_akiyama_tanigawa_row f n l (Suc u))"
0: shows "map (GE.int \<xi> \<circ> gT) Tl = map (Ik.int \<xi>) Tl"
preserves_ide: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide a" shows "Span.ide (Span.src a)"
wf_permute_agents: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "pref_profile_wf agents alts R"   and "\<sigma> permutes agents" shows "pref_profile_wf agents alts (R \<circ> \<sigma>)"
simps_3: fixes x :: "('fun, 'var) strand_step"   and S :: "('fun, 'var) extstrand_step list" shows "decomp_rm\<^sub>e\<^sub>s\<^sub>t (Step x # S) = Step x # decomp_rm\<^sub>e\<^sub>s\<^sub>t S"
inside_canonical_retraction_chamber_images_2: shows "canonical_retraction B C ` D = E"
outstanding_volatile_writes_unowned_by_others_store_buffer: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and sb' :: "'a memref list"   and p' :: "'a"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb' \<subseteq> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb"   and "all_acquired sb' \<subseteq> all_acquired sb" shows "outstanding_volatile_writes_unowned_by_others (ts\<^sub>s\<^sub>b[i := (p', is', \<theta>', sb', \<D>', \<O>, \<R>')])"
trg_src: fixes \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>" shows "S.trg (S.src \<mu>) = S.src \<mu>"
inf_assertion: fixes x :: "'a"   and y :: "'a" assumes "x \<in> assertion"   and "y \<in> assertion" shows "x \<sqinter> y \<in> assertion"
idem_D: fixes P :: "'a process" shows "P \<sqsubseteq>\<^sub>D P"
rs_1: fixes j :: "nat" assumes "j < n" shows "q j \<le> r j \<and> r j \<le> q j + 1"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in># X"   and "y \<in># Y"   and "(x, y) \<in> s" shows "(f x, g y) \<in> s'"
smult_l_null: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and x :: "'g" assumes "module R M"   and "x \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
cartesian_product: fixes C :: "('a \<times> 'b) set"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite C"   and "{a. \<exists>b. g a b \<noteq> (0::'c)} \<times> {b. \<exists>a. g a b \<noteq> (0::'c)} \<subseteq> C" shows "(\<Sum>a. Sum_any (g a)) = (\<Sum>(a, b). g a b)"
rel_G_conversep: fixes L1 :: "'q \<Rightarrow> 'j \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 'k \<Rightarrow> bool"   and Co1 :: "'s \<Rightarrow> 'l \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'm \<Rightarrow> bool"   and Contra1 :: "'u \<Rightarrow> 'n \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'o \<Rightarrow> bool" shows "rel_G L1\<inverse>\<inverse> L2\<inverse>\<inverse> Co1\<inverse>\<inverse> Co2\<inverse>\<inverse> Contra1\<inverse>\<inverse> Contra2\<inverse>\<inverse> = (rel_G L1 L2 Co1 Co2 Contra1 Contra2)\<inverse>\<inverse>"
Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2 # cs, css, Normal s) \<rightarrow> ([c\<^sub>1], (cs, c\<^sub>2 # cs) # css, Normal s)"
prj_simps_12: shows "C.seq \<nu>_\<pi>\<rho>.prj\<^sub>0 \<p>\<^sub>0[\<mu>.leg0, \<nu>.leg1 \<cdot> \<nu>_\<pi>\<rho>.prj\<^sub>1]"
bernoulli_fps_def: shows "bernoulli_fps = fps_X / (fps_exp (1::'b) - 1)"
lbI: fixes B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "\<And>y. y \<in> B \<Longrightarrow> P\<^sup>=\<^sup>= x y" shows "lb P B x"
inverse_contrasimulation_impl_reverse_pair_in_steps_closure: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus"   and P :: "'b"   and Q :: "'b" assumes "weak_reduction_contrasimulation (Rel\<inverse>) Cal"   and "(P, Q) \<in> Rel" shows "Q \<R>\<mapsto><Rel,Cal> P"
rsp: shows "eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b)) 1 1"
rep_eq: fixes x :: "integer" shows "int_of_integer (- x) = - int_of_integer x"
konig: fixes T :: "bool list set" assumes "infinite T"   and "wf_tree T" shows "\<exists>c. wf_infpath c \<and> (\<forall>n. c n \<in> T)"
arc_to_ends_def: fixes G :: "('a, 'b) pre_digraph"   and e :: "'b" shows "arc_to_ends G e \<equiv> (tail G e, head G e)"
sums_mult: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a" assumes "f sums a" shows "(\<lambda>n. c * f n) sums (c * a)"
termFSwMorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "termFSwMorph h hA MOD \<equiv> presWlsAll h hA MOD \<and> presCons h hA MOD \<and> presFreshAll h hA MOD \<and> presSwapAll h hA MOD"
right_idem: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.max (ord_term_lin.max a b) b = ord_term_lin.max a b"
set_intros_1: fixes x21 :: "'e"   and x22 :: "'e list" shows "x21 \<in> set (Cons x21 x22)"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'b option" assumes "\<forall>x\<in>A. g x = Some (0::'b)" shows "F g A = Some (0::'b)"
simps_4: fixes f :: "'f"   and c\<^sub>1 :: "('d, 'e, 'g) com"   and c\<^sub>2 :: "('d, 'e, 'g) com" shows "mark_guards f (Seq c\<^sub>1 c\<^sub>2) = Seq (mark_guards f c\<^sub>1) (mark_guards f c\<^sub>2)"
T_opt_def: fixes s :: "'state"   and rs :: "'request list" shows "T_opt s rs = \<Sqinter> {T s rs as |as. length as = length rs}"
igFreshClsSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igFreshClsSTR MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1STR MOD \<and> igFreshIGAbs2STR MOD \<and> igFreshIGOpSTR MOD"
tendsto_at_top_iff: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d" shows "(f \<longlongrightarrow> l) at_top = (\<forall>e>0. \<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e)"
finite_SigmaI: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (Sigma A B)"
powr_closure_def: fixes f :: "'a \<Rightarrow> real" shows "powr_closure f \<equiv> {\<lambda>x. f x powr p |p. True}"
select_convs_1: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and monom :: "'a \<Rightarrow> nat \<Rightarrow> 'b"   and coeff :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and more :: "'c" shows "monom \<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, smult = smult, monom = monom, coeff = coeff, \<dots> = more\<rparr> = monom"
inject: fixes x :: "('e, 'f, ('e, 'f, 'g) rpv) generat spmf"   and ya :: "('e, 'f, ('e, 'f, 'g) rpv) generat spmf" shows "(GPV x = GPV ya) = (x = ya)"
th3_4: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec"   and c :: "(real, 'b) vec" assumes "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> b a c" shows "a = b"
assms_1: shows "(n, eD, m) \<in> \<alpha>e g"
i_in: shows "\<iota> \<in> Inf_from (Liminf_llist Ns)"
bind_lzero: fixes f :: "'b \<Rightarrow> 'a option" shows "Option.bind None f = None"
changes_itv_smods_0_2: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "changes_itv_smods a b p 0 = 0"
Tree_axioms_def: fixes G :: "('c, 'd) Graph_scheme" shows "Tree_axioms G \<equiv> (\<forall>v w. v \<in> V\<^bsub>G\<^esub> \<longrightarrow> w \<in> V\<^bsub>G\<^esub> \<longrightarrow> Graph.connected G v w) \<and> (\<forall>xs. \<not> Graph.cycle G xs)"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (trg f) = B.arr f"
case_4: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x4 :: "'b" shows "(case nprop\<^sub>n(x4) of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f4 x4"
wfactors_dividesI: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c"   and f :: "'c" assumes "Group.comm_monoid G"   and "wfactors G fs a"   and "set fs \<subseteq> carrier G"   and "a \<in> carrier G"   and "f \<in> set fs" shows "f divides\<^bsub>G\<^esub> a"
coerceQuantT_def: fixes t :: "'b itself"   and x :: "'c['a, 'd]" assumes "QD('a) = QD('b)" shows "coerceQuantT t x = toQ (fromQ x)"
isomorphisms_UNIV_UNIV: assumes "DIM('a) = DIM('b)" obtains f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'a" where "linear f" and "linear g" and "\<And>x. norm (f x) = norm x" and "\<And>y. norm (g y) = norm y" and "\<And>x. g (f x) = x" and "\<And>y. f (g y) = y"
iso_lunit': fixes f :: "'c" assumes "C.ide f" shows "C.iso \<l>\<^sub>C\<^sup>-\<^sup>1[f]"
stream_sel_1: fixes t :: "'a tree" shows "shd (stream t) = root t"
valid_drop_B: assumes "valid_point (B # xs) (i + 1)"   and "0 < i + 1" shows "valid_point xs (i + 1 - 1)"
C\<^sub>1: shows "\<^bold>\<forall>x xa xb. E (xa \<cdot> xb) \<^bold>\<leftarrow> E (x \<cdot> xa) \<^bold>\<and> E ((x \<cdot> xa) \<cdot> xb)"
lnths_lappend_lfinite: fixes xs :: "'b llist"   and k :: "nat"   and ys :: "'b llist"   and A :: "nat set" assumes "llength xs = enat k" shows "lselect A (xs $ ys) = lselect A xs $ lselect {n. n + k \<in> A} ys"
exp_count_up_def: shows "exp_count_up \<equiv> SCOPE (''n''[] ::= ''G_par_1'';; (''a'' ::= N 1;; ''c'' ::= N 0;; WHILE Cmpop (<) (V ''c'') (V ''n'') DO (''a'' ::= Binop (*) (N 2) (V ''a'');; ''c'' ::= Binop (+) (V ''c'') (N 1)));; ''G_ret_1''[] ::= ''a'')"
set_cases_1: fixes e :: "'e"   and a :: "('e, 'f) step" assumes "e \<in> set1_step a" obtains z2 :: "'f" where "a = Yield e z2"
secure_implies_failure_consistency_aux: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and X :: "'a set" assumes "secure P I D"   and "(xs @ ys @ zs, X) \<in> failures P"   and "ipurge_tr_rev_aux I D (D ` (X \<union> set zs)) ys = []" shows "(xs @ zs, X) \<in> failures P"
splits_wrtI: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "cone_decomp T (ps @ qs)"   and "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> cone (h, U) \<subseteq> P[X]"   and "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> h \<in> ideal F"   and "\<And>h U. (h, U) \<in> set qs \<Longrightarrow> cone (h, U) \<subseteq> P[X]"   and "\<And>h U a. \<lbrakk>(h, U) \<in> set qs; a \<in> cone (h, U); a \<in> ideal F\<rbrakk> \<Longrightarrow> a = 0" shows "splits_wrt (ps, qs) T F"
finfun_default_aux_update_const: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'b" assumes "f \<in> finfun" shows "finfun_default_aux (f(a := b)) = finfun_default_aux f"
trm_prm_apply_compose: fixes \<pi> :: "'b prm"   and \<sigma> :: "'b prm"   and A :: "'b trm" shows "\<pi> \<cdot> \<sigma> \<cdot> A = (\<pi> \<diamondop> \<sigma>) \<cdot> A"
normal: fixes g :: "'a"   and k :: "'a" assumes "g \<in> G"   and "k \<in> K" shows "inverse g \<cdot> k \<cdot> g \<in> K"
row_space_eq_phull: fixes ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and ts :: "'c list" assumes "Keys (set ps) \<subseteq> set ts" shows "row_to_poly ts ` row_space (polys_to_mat ts ps) = phull (set ps)"
dom_dom: fixes f :: "'s" assumes "S.arr f" shows "S.dom (S.dom f) = S.dom f"
plus_gt_right_mono: fixes x :: "'a"   and y :: "'a"   and a :: "'a" assumes "x \<succ> y" shows "a + x \<succ> a + y"
norm_add_rule_thm: fixes x1 :: "'b"   and b1 :: "real"   and x2 :: "'b"   and b2 :: "real" assumes "norm x1 \<le> b1"   and "norm x2 \<le> b2" shows "norm (x1 + x2) \<le> b1 + b2"
h4: shows "aff_dim (affine hull S) = aff_dim UNIV"
ldistinct_lfilter_ltake: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist"   and x :: "enat" assumes "ldistinct (lfilter P xs)" shows "ldistinct (lfilter P (ltake x xs))"
merge_length: fixes l1 :: "'a list"   and l2 :: "'a list" shows "length (merge l1 l2) \<le> length l1 + length l2"
gs: fixes x :: "'b" assumes "x \<in> R" shows "g x \<le> Sup (g ` R)"
remdups_adj_adjacent: fixes i :: "nat"   and xs :: "'a list" assumes "Suc i < length (remdups_adj xs)" shows "remdups_adj xs ! i \<noteq> remdups_adj xs ! Suc i"
compositionality: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and set :: "'c set" shows "f -` g -` set = (g \<circ> f) -` set"
Least_le: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and k :: "'d" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
fold_Option_bind_eq_Some_start_not_None': fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd \<Rightarrow> 'd option"   and list :: "('e \<times> 'f) list"   and start :: "'d option"   and res :: "'d" assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) list start = Some res" shows "start \<noteq> None"
add_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a"   and b :: "'b"   and b' :: "'b" assumes "bounded_bilinear prod" shows "prod a (b + b') = prod a b + prod a b'"
CMUD_axioms_def: shows "CMUD_axioms \<equiv> \<forall>x. \<midarrow> x = (THE z. \<forall>w. P w z = (\<not> O w x))"
2: fixes theta :: "('a \<times> 'a) set" assumes "sym theta \<and> trans theta" shows "sym (Retr theta) \<and> trans (Retr theta)"
sorted_list_lemma_var: fixes l :: "real list"   and x :: "real" assumes "1 < length l"   and "strict_sorted l"   and "\<not> List.member l x"   and "l ! 0 < x"   and "x < l ! (length l - 1)" shows "\<exists>n<length l - 1. l ! n < x \<and> x < l ! (n + 1)"
AB: shows "\<forall>a\<in>A. \<forall>b\<in>B. a \<sqsubseteq> b \<and> \<not> b \<sqsubseteq> a"
access_ibody: fixes i :: "iface" shows "access (ibody i) = access i"