hcju/isabelleps huggingface dataset:
endofunctor_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'd" shows "endofunctor A F \<equiv> functor A A F"
that: shows "ASSUMPTION (map_spmf f p = map_spmf g q)"
exists_trace_for_any_i: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "distributed_system channel"   and "\<exists>c'. distributed_system.trace channel trans send recv c t c'" shows "distributed_system.trace channel trans send recv c (take i t) (distributed_system.s channel trans send recv c t i)"
assms: fixes b :: "'a set" assumes "b \<in># B" shows "\<not> ps \<subseteq> b"
smallD: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and c :: "real" assumes "f \<in> \<omega>[F](g)"   and "0 < c" shows "\<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)"
atMost_iff: fixes i :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and k :: "'d" shows "(i \<in> ord.atMost less_eq k) = less_eq i k"
path_snoc: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and ns :: "'b list"   and n :: "'b"   and m :: "'b" assumes "graph_path \<alpha>e \<alpha>n invar inEdges'"   and "graph_path_base.path \<alpha>n invar inEdges' g (ns @ [n])"   and "n \<in> set (graph_path_base.predecessors inEdges' g m)" shows "graph_path_base.path \<alpha>n invar inEdges' g (ns @ [n, m])"
wt_start_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and Ts :: "ty list"   and mxl\<^sub>0 :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "wt_start P C Ts mxl\<^sub>0 \<tau>s \<equiv> P \<turnstile> \<lfloor>([], OK (Class C) # map OK Ts @ replicate mxl\<^sub>0 Err)\<rfloor> \<le>' \<tau>s ! 0"
le_add_diff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c \<le> b + c - a"
fpr_target_is_head': fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and path' :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list" shows "find_path_rec_f P ctx cl path' = Some path \<longrightarrow> (\<forall>fqn ctxcld. cl = cl_fqn fqn \<and> find_cld_f P ctx fqn = Some ctxcld \<longrightarrow> (\<exists>path''. path = path' @ ctxcld # path''))"
cancel_comm_semigroup_axioms: shows "cancel_comm_semigroup f\<^sub>b fi\<^sub>b"
primitive_dchar_axioms_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" shows "primitive_dchar_axioms n \<chi> \<equiv> \<not> (\<exists>d<n. dcharacter.induced_modulus n \<chi> d)"
cond_abs: fixes s :: "'C" assumes "s \<in> invar_addc"   and "\<alpha> s \<in> wa_invar WAA"   and "s \<in> condc" shows "\<alpha> s \<in> wa_cond WAA"
prv_eqv_sym: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (Syntax_with_Connectives.eqv cnj imp \<phi> \<chi>)" shows "prv (Syntax_with_Connectives.eqv cnj imp \<chi> \<phi>)"
assms_1: shows "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2"
M_cone_car: fixes x :: "'a vec"   and t :: "'a" assumes "x \<in> carrier_vec n" shows "(x @\<^sub>v vec_of_scal t \<in> next_dim.polyhedral_cone M) = (A *\<^sub>v x - t \<cdot>\<^sub>v b \<le> 0\<^sub>v nr \<and> (0::'a) \<le> t)"
app_vfieldI2: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r"
decr_grading_p_dgrad_p_setI: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "hom_grading d" shows "local.punit.decr_grading_p d m p \<in> punit.dgrad_p_set d m"
spmf_try_spmf: fixes p :: "'c spmf"   and q :: "'c spmf"   and x :: "'c" shows "spmf (TRY p ELSE q) x = spmf p x + pmf p None * spmf q x"
i: fixes a :: "real" assumes "a \<le> 0"   and "0 \<le> y"   and "y \<le> 1"   and "- 1 / 2 < a" shows "a * (1 - 2 * y) \<le> 1 / 2"
hlinear: shows "\<forall>f\<in>set (map (\<lambda>a. fm.Atom (linear_substitution var (- isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) a)) L @ map (linear_substitution_fm var (- isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0))) F). freeIn var f"
S_def: fixes x :: "real"   and y :: "real" shows "local.S x y = (ln (G y) - ln (G x)) / (y - x)"
content_smult: fixes c :: "'a"   and p :: "'a poly" shows "content (Polynomial.smult c p) = normalize c * content p"
D_imp_front_tickFree: fixes s :: "'c event list"   and P :: "'c process" assumes "s \<in> D P" shows "front_tickFree s"
Color_Target: shows "\<turnstile> Color_Target  \<lbrace>\<acute>Mut_init \<and> \<acute>z\<rbrace>"
simps_23: fixes g1 :: "'a gexp"   and g2 :: "'a gexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_gexp_aux (Nor g1 g2) (Eq v va) = False"
M_FullModel_axioms: shows "M_FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
map_prod_def: fixes f :: "'p \<Rightarrow> 'r"   and g :: "'q \<Rightarrow> 's" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
almost_all_suffixes_model_\<F>: fixes \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "closed \<G> w"   and "G \<phi> \<in> \<G>" shows "\<forall>\<^sub>\<infinity>j. suffix j w \<Turnstile> eval\<^sub>G \<G> (\<F> \<phi> w \<G> j)"
length_dims0: shows "length d.dims0 = card ({0..<length dims} \<inter> d.vars0)"
dominate_def: fixes G :: "nat graph_rec"   and n1 :: "nat"   and n2 :: "nat" assumes "cfg_doms G" shows "cfg_doms.dominate G n1 n2 \<equiv> \<forall>pa. path_entry (g_E G) pa n2 \<longrightarrow> n1 \<in> set pa \<or> n1 = n2"
sub_monom: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "f = monom P a n"   and "g \<in> carrier P" shows "deg R (Cring_Poly.compose R f g) = n * deg R g"
prj_chine_assoc_2: shows "Prj\<^sub>1\<^sub>0 \<cdot> chine_assoc = Prj\<^sub>0\<^sub>1"
Send_ev: shows "ev = Send i' (occurs_on ev) r u u' m"
take_eq: shows "take (length A_pos) (L_pos @ L_neg) = L_pos"
sigma_finite_measure_axioms: shows "sigma_finite_measure (density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x)))"
frees_free: fixes name :: "name" shows "frees (free name) = {|name|}"
sigma_algebra_axioms: fixes M :: "'c measure" shows "sigma_algebra (space M) (sets M)"
simps_10: fixes f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'e"   and x21 :: "'f"   and x22 :: "('f, 'g) Term.term list" shows "map_term f1 f2 (Fun x21 x22) = Fun (f1 x21) (map (map_term f1 f2) x22)"
inj_g: shows "inj_on ((\<circ>) (Fun.swap j k id)) {p \<in> {p. p permutes UNIV}. evenperm p}"
inv_baliL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "invc2 l"   and "invc r"   and "bheight l = bheight r" shows "invc (baliL l a r) \<and> invh (baliL l a r) \<and> bheight (baliL l a r) = Suc (bheight l)"
inf_sup_aci_1: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<otimes> y = y \<otimes> x"
filter_mono': fixes F1 :: "'b filter"   and F2 :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "F1 \<le> F2" shows "O[F2](f) \<subseteq> O[F1](f)"
kernel_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" shows "kernel G H h = {x \<in> carrier G. h x = \<one>\<^bsub>H\<^esub>}"
properfactor_of_zero_2: fixes b :: "'a list" assumes "b \<in> carrier (poly_ring R)" shows "properfactor (poly_ring R) b \<zero>\<^bsub>poly_ring R\<^esub> = (b \<noteq> \<zero>\<^bsub>poly_ring R\<^esub>)"
subsumes_list_filterD: fixes Ls :: "('a, 'b) Term.term literal list"   and P :: "('a, 'c) Term.term literal \<Rightarrow> bool"   and Ks :: "('a, 'c) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option" assumes "subsumes_list Ls (filter P Ks) \<sigma>" shows "subsumes_list Ls Ks \<sigma>"
ide_is_iso: fixes a :: "'a1" assumes "C1.ide a" shows "C1.iso a"
nsqn_sqn_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat" shows "nsqn rt dip = (if flag rt dip = Some val \<or> sqn rt dip = 0 then sqn rt dip else sqn rt dip - 1)"
min_list_ex: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "\<exists>x\<in>set xs. Missing_List.min_list xs = x"
column_iarray_def: fixes k :: "nat"   and A :: "'a iarray iarray" shows "column_iarray k A = IArray.of_fun (\<lambda>m. A !! m !! k) (IArray.length A)"
subtensor_eqI: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "dims A \<noteq> []"   and "dims A = dims B"   and "\<And>i. i < hd (dims A) \<Longrightarrow> subtensor A i = subtensor B i" shows "A = B"
INT_lower: fixes a :: "'c"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "a \<in> A" shows "\<Inter> (B ` A) \<subseteq> B a"
indep_subset_carrier: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "indep_system carrier indep"   and "indep X" shows "X \<subseteq> carrier"
120: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> x \<sqinter> - z) = y \<squnion> x"
that: fixes x :: "nat" assumes "xa = Suc x"   and "ya_even x" shows "thesis"
t: fixes f :: "'a \<Rightarrow> 'b stream" assumes "F f" shows "F (\<lambda>x. stl (f x))"
Im_Arccos: fixes z :: "complex" shows "Im (Arccos z) = - ln (cmod (z + \<i> * csqrt (1 - z\<^sup>2)))"
in_alphabet_imp_sibling_in_alphabet: fixes a :: "'a"   and t :: "'a tree" assumes "a \<in> alphabet t" shows "sibling t a \<in> alphabet t"
zero: fixes f :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f" shows "f 0 = (0::'b)"
x: shows "\<not> lookup (lpp f) x \<le> lookup (lpp q) x"
weakCongSimWeakSim: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" assumes "P \<leadsto><Rel> Q" shows "P \<leadsto>\<^sup>^<Rel> Q"
ucast_ucast_len: fixes x :: "'b word" assumes "x < 2 ^ LENGTH('c)" shows "UCAST('c \<rightarrow> 'b) (UCAST('b \<rightarrow> 'c) x) = x"
measures_less: fixes f :: "'g \<Rightarrow> nat"   and x :: "'g"   and y :: "'g"   and fs :: "('g \<Rightarrow> nat) list" assumes "f x < f y" shows "(x, y) \<in> measures (f # fs)"
Cons_1: fixes S :: "nat set" assumes "S \<subseteq> {..<length as}" shows "map_pmf (\<lambda>l. nths l S) (bv (length as)) = bv (length (nths as S))"
C: shows "class P C = \<lfloor>(S, fs, ms)\<rfloor>"
assms_1: shows "A \<prec>\<^sub>\<circ> B"
vdisjnt_z\<omega>: shows "z \<inter>\<^sub>\<circ> \<omega> = []\<^sub>\<circ>"
ord_of_nat_zero: shows "0\<^sub>\<nat> = []\<^sub>\<circ>"
Ring1_RG: fixes G :: "'a set" assumes "Group G" shows "Ring1 (aezfun_setspan G)"
quicksort_ref_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b" shows "quicksort_ref R h = (\<lambda>(lo, hi, xs0). REC\<^sub>T (\<lambda>f (lo, hi, xs). ASSERT (lo \<le> hi \<and> hi < length xs0 \<and> mset xs = mset xs0) \<bind> (\<lambda>_. partition_between_ref R h lo hi xs \<bind> (\<lambda>(xs, p). ASSERT (mset xs = mset xs0 \<and> lo \<le> p \<and> p < length xs0) \<bind> (\<lambda>_. (if p - 1 \<le> lo then RETURN xs else f (lo, p - 1, xs)) \<bind> (\<lambda>xs. ASSERT (mset xs = mset xs0) \<bind> (\<lambda>_. if hi \<le> p + 1 then RETURN xs else f (p + 1, hi, xs))))))) (lo, hi, xs0))"
mult_isol: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.pre_dioid plus times less_eq less"   and "less_eq x y" shows "less_eq (times z x) (times z y)"
conjugate_dist_mul: fixes a :: "'b"   and b :: "'b" shows "conjugate (a * b) = conjugate a * conjugate b"
termMOD_igFreshIGAbs1: shows "igFreshIGAbs1 termMOD"
take_append: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" shows "take n (xs @ ys) = take n xs @ take (n - length xs) ys"
gt_irrefl: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "arity.wary arity_sym arity_var s" shows "\<not> tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym s s"
product_swap_mset: fixes A :: "'b multiset"   and B :: "'a multiset" shows "image_mset prod.swap (A \<times># B) = B \<times># A"
A2: shows "A2 = \<Union> (B2 ` N)"
count_image_mset_Pair: fixes a :: "'b"   and B :: "'c multiset"   and x :: "'b"   and b :: "'c" shows "count (image_mset (Pair a) B) (x, b) = (if x = a then count B b else 0)"
simps_13: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and e1' :: "'a aexp"   and e2' :: "'a aexp" shows "local.less_gexp_aux (Gt e1 e2) (Gt e1' e2') = (e1 < e1' \<or> e1 = e1' \<and> e2 < e2')"
rep_eq: fixes x :: "('a, 'b) pfun" shows "pfun_app x = (\<lambda>xa. if xa \<in> dom (Rep_pfun x) then the (Rep_pfun x xa) else undefined)"
valid: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "Labeling c s t f l"   and "(u, v) \<in> Graph.E (residualGraph c f)" shows "l u \<le> l v + 1"
hom_one: shows "(\<lambda>j. if j \<in> J then h (g j) (\<one>\<^bsub>DirProds Gs I\<^esub> (g j)) else undefined) = \<one>\<^bsub>DirProds Hs J\<^esub>"
ab_group_add_ow: shows "class.ab_group_add = ab_group_add_ow UNIV"
sorted_wrt_map: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "sorted_wrt R (map f xs) = sorted_wrt (\<lambda>x y. R (f x) (f y)) xs"
H: shows "H \<in> [\<beta>]\<^bsup>((n # \<alpha>) ! i)\<^esup>"
cupcake_single_preserve: fixes env :: "v sem_env"   and e :: "exp0"   and res :: "v" assumes "cupcake_evaluate_single env e (Rval res)"   and "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "is_cupcake_value res"
f_prime: fixes f :: "nat \<Rightarrow> 'b"   and f_prime :: "nat \<Rightarrow> 'b"   and p :: "nat" assumes "completely_multiplicative_function' f f_prime"   and "prime p" shows "f p = f_prime p"
neigh_shift: shows "\<forall>k. i + 1 < k \<and> k < j + 1 \<longrightarrow> \<not> regular_event (swap_events i j t ! k)"
small_big_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)"   and "g \<in> O[F](h)" shows "f \<in> o[F](h)"
WHATWHERE_Secure_def: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and E :: "'e \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and pp :: "'a \<Rightarrow> nat"   and DA :: "'b \<Rightarrow> 'f"   and lH :: "('f \<times> 'e \<times> nat) set"   and V :: "'a list" shows "WHATWHERE.WHATWHERE_Secure SR E pp DA lH V \<equiv> \<forall>d PP. \<exists>R. WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP R \<and> (V, V) \<in> R"
length_strict_mono: fixes f :: "nat \<Rightarrow> 'a list"   and i :: "nat" assumes "strict_mono f" shows "i + length (f 0) \<le> length (f i)"
ball_differentiable_atD: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>S. f differentiable at x" shows "f differentiable_on S"
guardedTau: fixes P :: "('a, 'b, 'c) psi" shows "guarded (\<tau>.P)"
red_singleton: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "red {f} p q = (\<exists>t. red_single p q f t)"
vinsert_set_insert_anyI: fixes B :: "V set"   and P :: "V \<Rightarrow> bool"   and a :: "V"   and b :: "V" assumes "small B"   and "P (vinsert a (set (insert b B)))" shows "P (set (insert a (insert b B)))"
execn_call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and n :: "nat"   and t :: "'d"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and u :: "('d, 'f) xstate" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Normal t"   and "\<Gamma>\<turnstile> \<langle>c s t,Normal (return s t)\<rangle> =Suc n\<Rightarrow> u" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> =Suc n\<Rightarrow> u"
induct: fixes P1 :: "pat \<Rightarrow> bool"   and P3 :: "(char list \<times> pat) list \<Rightarrow> bool"   and P2 :: "char list \<times> pat \<Rightarrow> bool"   and pat :: "pat" assumes "\<And>x. P1 (PVar x)"   and "\<And>x. P3 x \<Longrightarrow> P1 (PRcd x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P1 pat"
asm: shows "(\<forall>(d, e, f)\<in>set a. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
topspace_final_topology: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Y :: "'c \<Rightarrow> 'd topology"   and X :: "'e set" assumes "\<And>i. f i \<in> topspace (Y i) \<rightarrow> X" shows "topspace (final_topology X Y f) = X"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>T<TRel> a2" obtains S :: "'procS" where "S__ \<in>S a1" and "\<lbrakk>S__\<rbrakk> \<in>T a2"   | S :: "'procS" where "\<lbrakk>S\<rbrakk> \<in>T a1" and "S \<in>S a2"   | T1 :: "'procT" and T2 :: "'procT" where "T1 \<in>T a1" and "T2 \<in>T a2" and "(T1, T2) \<in> TRel"
monotone2monotone: fixes ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ordc :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd" assumes "\<And>x. monotone ordb ordc (f x)"   and "monotone orda ordb t"   and "\<And>y. monotone orda ordc (\<lambda>x. f x y)"   and "transp ordc" shows "monotone orda ordc (\<lambda>x. f x (t x))"
subcls_antisym: fixes m :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "wf_prog m P"   and "P \<turnstile> C \<preceq>\<^sup>* D"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "C = D"
HomFtorContra': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "HomP\<^bsub>C\<^esub>[\<midarrow>,X] = HomP\<^bsub>Op C\<^esub>[X,\<midarrow>]"
zone_set_def: fixes Z :: "('a \<Rightarrow> 'b) set"   and r :: "'a list" shows "Z\<^bsub>r \<rightarrow> 0\<^esub> = {[r\<rightarrow>(0::'b)]u |u. u \<in> Z}"
pqr: shows "p ^ 4 + q ^ 4 = r\<^sup>2 \<and> p * q * r \<noteq> 0 \<and> odd p \<and> coprime p q \<and> r\<^sup>2 < x\<^sup>2"
simplify_correct: fixes w :: "nat \<Rightarrow> 'a set"   and m :: "mode"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n simplify m \<phi> = w \<Turnstile>\<^sub>n \<phi>"
abs_inverse: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b set \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b set"   and c :: "'b set" assumes "quot_type R Abs Rep"   and "\<exists>x. R x x \<and> c = Collect (R x)" shows "Rep (Abs c) = c"
upper: fixes t :: "real" assumes "t \<in> S"   and "t0 < t" shows "f t (v t) < v' t"
cartesian_product: fixes g :: "'e \<Rightarrow> 'f \<Rightarrow> 'a"   and B :: "'f set"   and A :: "'e set" shows "F.F (\<lambda>x. F.F (g x) B) A = F.F (\<lambda>(x, y). g x y) (A \<times> B)"
strand_substI_2: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" assumes "subst_domain \<theta> \<inter> vars\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>t \<theta> = S"
harry_step_True_start: fixes xs :: "bool list" shows "harry_step (True # xs) = (if True \<in> set xs then True # harry_step xs else False # xs)"
subgroup_def: shows "subgroup = {A. (0::'a) \<in> A \<and> (\<forall>a b. a \<in> A \<and> b \<in> A \<longrightarrow> a + b \<in> A \<and> - a \<in> A)}"
bdd_below_mono: fixes B :: "'a multiset set"   and A :: "'a multiset set" assumes "subset_mset.bdd_below B"   and "A \<subseteq> B" shows "subset_mset.bdd_below A"
mask_eq_sum_exp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one = comm_monoid_add.sum plus zero (power.power one times (numeral.numeral one plus (num.Bit0 num.One))) {q. q < n}"
eq_carr_eq_Vring: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "carrier (Vr K v) = carrier (Vr K v')" shows "Vr K v = Vr K v'"
vcg_conj_lift: fixes P :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and c :: "('answer, 'location, 'question, 'state) com"   and Q :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and P' :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and Q' :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool" assumes "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"   and "\<lbrace>P'\<rbrace> c \<lbrace>Q'\<rbrace>" shows "\<lbrace>\<lambda>s. P s \<and> P' s\<rbrace> c \<lbrace>\<lambda>s. Q s \<and> Q' s\<rbrace>"
Calling_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('c, 'd) chan"   and p :: "PrIds" shows "Calling ch p \<equiv> cbit<ch!p> \<noteq> rbit<ch!p>"
ack_less_mono1_aux: fixes i :: "nat"   and k :: "nat"   and i' :: "nat" shows "ack i k < ack (Suc (i + i')) k"