hcju/isabelleps huggingface dataset:
of_nat_mono: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and i :: "nat"   and j :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "i \<le> j" shows "less_eq (semiring_1.of_nat one plus zero i) (semiring_1.of_nat one plus zero j)"
strict_mono_comp: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "strict_mono g"   and "strict_mono f" shows "strict_mono (f \<circ> g)"
fimage_ident: fixes Y :: "'a fset" shows "(\<lambda>x. x) |`| Y = Y"
return_bind_spmf: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c spmf" shows "return_spmf x \<bind> f = f x"
bheight_rbtreeify_f_eq_plog2I: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list"   and t :: "('a, 'b) rbt"   and kvs' :: "('a \<times> 'b) list" assumes "rbtreeify_f n kvs = (t, kvs')"   and "n \<le> length kvs" shows "bheight t = RBT_Impl.plog2 n"
awalk_verts_non_Nil: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list" shows "pre_digraph.awalk_verts G u p \<noteq> []"
annos_ne: fixes C :: "'b acom" shows "annos C \<noteq> []"
mr_first: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators"   and c :: "((int \<times> int) set set, int) Combinators list"   and s :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C b)"   and "\<forall>r. r \<in> set c \<longrightarrow> x \<notin> dom (C r)"   and "s = b # c" shows "applied_rule_rev C x s = \<lfloor>b\<rfloor>"
union_disjoint: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum_0 g (A \<union> B) = sum_0 g A + sum_0 g B"
paval'dom: fixes a :: "aexp"   and ps :: "char list \<Rightarrow> int option"   and v :: "int" assumes "paval' a ps = Some v" shows "vars a \<subseteq> dom ps"
weak_put_eq: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and k :: "'e"   and u :: "'e"   and \<rho> :: "'f"   and v :: "'e" assumes "mwb_lens x"   and "\<sigma> \<in> \<S>\<^bsub>x\<^esub>"   and "get\<^bsub>x\<^esub> \<sigma> = k"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "put\<^bsub>x\<^esub> \<rho> k = \<sigma>"
set_eq: shows "{Inl vid1, Inr vid1} = {Inr vid1, Inl vid1}"
assms: fixes x :: "'a" assumes "x \<notin> A" shows "f x = unity"
adm_all: fixes P :: "'e \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>y. adm (\<lambda>x. P x y)" shows "adm (\<lambda>x. \<forall>y. P x y)"
fps_div_rep_coeffs: fixes p :: "nat"   and n :: "nat"   and k :: "nat" assumes "prime p" shows "((1 + fps_X ^ p) ^ (n div p) * (1 + fps_X) ^ (n mod p)) $ k = int ((n div p choose k div p) * (n mod p choose k mod p))"
whilenc: shows "denote_while_n M0 M1 DS n \<rho> \<in> carrier_mat d d"
a_l_coset_def: fixes G :: "('a, 'b) ring_scheme" shows "(<+\<^bsub>G\<^esub>) = (<#\<^bsub>add_monoid G\<^esub>)"
D_in: shows "D \<in> \<G>_Fset N"
WhileBodyThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and c :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
Inf_many_def: fixes P :: "'c \<Rightarrow> bool" shows "Inf_many P = infinite {x. P x}"
scaleC: fixes fa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and r :: "complex"   and b :: "'a" shows "cblinfun_apply fa (r *\<^sub>C b) = r *\<^sub>C cblinfun_apply fa b"
r_dummy_def: fixes n :: "nat"   and f :: "recf" shows "r_dummy n f \<equiv> Cn (arity f + n) f (map (recf.Id (arity f + n)) [0..<arity f])"
simps_2: fixes V :: "Vars.var set"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "restr_stack V (Alts e1 e2 # S) = Alts e1 e2 # restr_stack V S"
e1: shows "set (remdups (map (signs_at qs) [0])) \<subseteq> consistent_sign_vectors_R qs UNIV"
discr_cont: fixes c :: "('test, 'atom, 'choice) cmd"   and i :: "nat"   and s :: "'state" assumes "discr c"   and "i < brn c" shows "discr (local.cont c s i)"
sel_5: fixes x11 :: "'a"   and x12 :: "'a"   and x13 :: "'a"   and x14 :: "'b" shows "new_id (Add x11 x12 x13 x14) = x13"
simps_1: fixes t :: "nat"   and v :: "nat" shows "valid_sops_expr t (Const v) = True"
exhaust: fixes y :: "'b environment" obtains x1 :: "char list \<Rightarrow> 'b option" where "y = Env x1"   | "y = Malformed"
exhaust: fixes y :: "ev_signal_stage_t" obtains "y = EV_SIGNAL_PREP"   | "y = EV_SIGNAL_FINISH"
that: fixes k :: "nat \<Rightarrow> nat" assumes "strict_mono k"   and "\<And>x. convergent (\<lambda>n. f (k n) x)" shows "thesis"
wf_tuple_def: fixes n :: "nat"   and V :: "nat set"   and x :: "'b option list" shows "wf_tuple n V x = (length x = n \<and> (\<forall>i<n. (x ! i = None) = (i \<notin> V)))"
magicl_applied_def: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and F :: "'a \<Rightarrow> bool" shows "magicl_applied h1 h2 h0 F \<equiv> (h1,h2\<triangleright>h0) \<and> F h2"
rt_fresher_trans_rt_strictly_fresher: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt3"
termMOD_igSubstAllIPresIGWlsAll: shows "igSubstAllIPresIGWlsAll termMOD"
big_step_pres_approx_restrict: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int option" assumes "(c, s) \<Rightarrow> s'"   and "approx (t |` (- lvars c)) s" shows "approx (t |` (- lvars c)) s'"
bounded_linear_right: fixes a :: "'a \<Rightarrow>\<^sub>L 'b" shows "bounded_linear (blinfun_apply a)"
leq_squared: fixes z :: "int" shows "z \<le> z\<^sup>2"
Collect_case_prodD: fixes x :: "'c \<times> 'd"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
S_separates: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and S :: "'c set"   and xs :: "'c list" assumes "Separation G v0 v1 S"   and "v0 \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> v1" shows "set xs \<inter> S \<noteq> {}"
a_in: shows "a \<in> read_only (share sb \<S>)"
cClose2: fixes a :: "name"   and y :: "name"   and P' :: "pi"   and x :: "name"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>y> \<prec> P'"   and "!P \<longmapsto> a<x> \<prec> P''"   and "y \<sharp> P"   and "x \<sharp> C"   and "y \<sharp> C"   and "\<And>C. F C (!P) (a<x> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>y>(P' \<parallel> P''[x::=y]))"
min_le1: fixes w :: "'a"   and y :: "'a" assumes "w \<in> S"   and "y \<in> S" shows "le (min_for w y) y"
effective_row_formula: fixes i :: "nat"   and M1 :: "'a list list"   and M2 :: "'a list list" assumes "i < row_length M1 * row_length M2"   and "mat (row_length M1) (length M1) M1"   and "mat (row_length M2) (length M2) M2" shows "row (M1 \<otimes> M2) i = vec_vec_Tensor (row M1 (i div row_length M2)) (row M2 (i mod row_length M2))"
simps_3: fixes e :: "name list"   and t :: "tm"   and u :: "tm" shows "trans_tm e (Eats t u) = DBEats (trans_tm e t) (trans_tm e u)"
absorb_iff2: fixes a :: "'d upred"   and b :: "'d upred" shows "(b \<sqsubseteq> a) = ((a \<or> b) = b)"
cases: fixes x :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr" obtains \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option" and \<tau> :: "'d \<Rightarrow> ('e, 'd, 'f) val option" and e :: "('e, 'd, 'f) expr" where "x = (\<sigma>, \<tau>, e)"
seq_cons: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and t :: "state"   and rs :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, t\<rangle> \<Rightarrow> t'"   and "no_matching_Goto \<gamma> p [r]" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs, Undecided\<rangle> \<Rightarrow> t'"
inj_on_imp_bij_betw: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "bij_betw f A (f ` A)"
intros_2: 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 Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n'' :: "'b SDG_node"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.call_cdep_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None None n'" shows "SDG.call_cdep_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ [n'']) n'"
empty_not_insert: fixes a :: "'d"   and A :: "'d set" shows "{} \<noteq> insert a A"
exec_bind_SE_success: fixes A :: "'b \<Rightarrow> ('a \<times> 'b) option"   and \<sigma> :: "'b"   and b :: "'a"   and \<sigma>' :: "'b"   and M :: "'a \<Rightarrow> 'b \<Rightarrow> (bool \<times> 'b) option" assumes "A \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M) = (\<sigma>' \<Turnstile> M b)"
poly_map_poly_eval_poly: fixes p :: "real poly" shows "poly (map_poly of_real p) = eval_poly of_real p"
InvariantReasonClauses_def: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list" shows "InvariantReasonClauses F M \<equiv> \<forall>literal. literal el elements M \<and> \<not> literal el decisions M \<longrightarrow> (\<exists>clause. formulaEntailsClause F clause \<and> isReason clause literal (elements M))"
topological_space_ow_def: fixes U :: "'b set"   and "open" :: "'b set \<Rightarrow> \<bool>" shows "topological_space_ow U open \<equiv> open U \<and> (\<forall>K\<in>Pow (Pow U). Ball K open \<longrightarrow> open (\<Union> K)) \<and> (\<forall>S\<in>Pow U. \<forall>T\<in>Pow U. open S \<longrightarrow> open T \<longrightarrow> open (S \<inter> T))"
inverse_le_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse x) one = (less_eq x zero \<or> less_eq one x)"
assms_1: shows "landau_function_family_pair F (powr_closure g) H (\<lambda>x. g x powr 1)"
cinner_add_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "cinner x (y + z) = cinner x y + cinner x z"
right_neutral: fixes a :: "'b" shows "a \<sqinter> \<top> = a"
natural_extension_1: shows "pmpt J.natural_extension_measure J.natural_extension_map"
inner_diff_left: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (minus x y) z = inner x z - inner y z"
attach_compose: fixes conv1 :: "('p, 'q, 't, 'u) converter"   and conv2 :: "('t, 'u, 'r, 's) converter"   and res :: "('r, 's) resource" shows "conv1 \<odot> conv2 \<rhd> res = conv1 \<rhd> conv2 \<rhd> res"
uminus_ms_aux_MSLNil: shows "uminus_ms_aux MSLNil = MSLNil"
assms: fixes m :: "'a" assumes "invar m" shows "map_iterator_linord (itoi m) (\<alpha> m)"
bind_commute: fixes m :: "'m"   and m' :: "'m"   and f :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m' (f x)) = bind m' (\<lambda>y. bind m (\<lambda>x. f x y))"
Sigma_mono: fixes A :: "'d set"   and C :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and D :: "'d \<Rightarrow> 'e set" assumes "A \<subseteq> C"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> D x" shows "Sigma A B \<subseteq> Sigma C D"
head_if: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {m..n} = (if n < m then 0::'a else sum g {m..<n} + g n)"
p_gt_2: fixes p :: "nat" assumes "noninert_gauss_int_prime p" shows "2 < p"
well_def2: shows "well_def_signs (length qs2) signs2"
elims_3: fixes x :: "('a \<times> constraint) list"   and xa :: "'a set"   and xb :: "'a simplex_state" assumes "\<not> checked_simplex x xa xb" obtains cs :: "('a \<times> constraint) list" and J :: "'a set" and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" where "x = cs" and "xa = J" and "xb = Simplex_State s" and "\<not> checked_simplex' cs J s"
update: shows "GEN_OP upd op_map_update (A \<rightarrow> B \<rightarrow> \<langle>A, B\<rangle>Rm \<rightarrow> \<langle>A, B\<rangle>Rm)"
distinct_filterI: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "\<And>i j. \<lbrakk>i < j; i < length xs; j < length xs; P (xs ! i); P (xs ! j)\<rbrakk> \<Longrightarrow> xs ! i \<noteq> xs ! j" shows "distinct (filter P xs)"
is_mult_sum4sq_int: fixes x :: "int"   and y :: "int" assumes "is_sum4sq_int x"   and "is_sum4sq_int y" shows "is_sum4sq_int (x * y)"
lt_eqI_keys: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p"   and "\<And>u. u \<in> keys p \<Longrightarrow> le_of_nat_term_order cmp_term u v" shows "lt p = v"
iter_narrow_termination: fixes P :: "'a acom \<Rightarrow> bool"   and f :: "'a acom \<Rightarrow> 'a acom"   and c0 :: "'a acom" assumes "\<And>c. P c \<Longrightarrow> P (c \<triangle>\<^sub>c f c)"   and "wf ({uu_. \<exists>c c'. uu_ = (c, c \<triangle>\<^sub>c f c) \<and> P c \<and> \<not> c \<sqsubseteq> c \<triangle>\<^sub>c f c}\<inverse>)"   and "P c0" shows "\<exists>c. iter_narrow f c0 = Some c"
\<delta>\<^sub>p_def_1: shows "p\<^sub>1 = find_closest_bf p\<^sub>0 (p\<^sub>2 # p\<^sub>3 # ps)"
simps_9: fixes uv :: "typ"   and v :: "Core.term"   and vb :: "typ"   and vc :: "Core.term" shows "eta_reducible (Abs uv (v $ Abs vb vc)) = eta_reducible (v $ Abs vb vc)"
exp_diff: fixes x :: "'b"   and y :: "'b" shows "exp (x - y) = exp x / exp y"
Amicable_pair_equiv_def: fixes m :: "nat"   and n :: "nat" assumes "m Amic n"   and "1 \<le> m"   and "1 \<le> n" shows "Esigma m = Esigma n \<and> Esigma m = m + n"
arl_empty_def: shows "arl_empty \<equiv> Array.new initial_capacity default \<bind> (\<lambda>a. return (a, 0::'b))"
12377: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (x \<sqinter> y) = - x"
simps_7: fixes p :: "'b" shows "redexes (Call p) = {Call p}"
has_derivativeI: fixes f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "bounded_linear f'"   and "((\<lambda>y. (f y - f x - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> (0::'b)) (at x within s)" shows "(f has_derivative f') (at x within s)"
compliant_stateful_IFS: shows "all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<T>)"
pair_not_snd: fixes A :: "'b trm"   and B :: "'b trm"   and P :: "'b trm" shows "SimplyTyped.Pair A B \<noteq> Snd P"
Z: shows "OpposedThinChamberComplexFoldings X f g Z"
ir_Seq1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and d :: "com"   and R :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare P c c' Q"   and "ir_hoare Q d SKIP R" shows "ir_hoare P (c;; d) c' R"
annir: fixes x :: "'a" shows "x \<odot> (0::'a) = (0::'a)"
homotopy_equivalent_space_def: fixes X :: "'b topology"   and Y :: "'c topology" shows "X homotopy_equivalent_space Y \<equiv> \<exists>f g. continuous_map X Y f \<and> continuous_map Y X g \<and> homotopic_with (\<lambda>x. True) X X (g \<circ> f) id \<and> homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id"
ticks_sub: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and a :: "clock" assumes "dilating f sub r" shows "hamlet (Rep_run sub n a) = hamlet (Rep_run r (f n) a)"
bottom_neq_one_Integer: shows "\<bottom> \<noteq> 1"
dg_SemiCAT_components_4: fixes \<alpha> :: "V" shows "dg_SemiCAT \<alpha>\<lparr>Cod\<rparr> = (\<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomCod\<rparr>)"
ex_is_poincare_line_points': fixes i1 :: "complex_homo"   and H :: "circline"   and i2 :: "complex_homo"   and a :: "complex_homo" assumes "i1 \<in> circline_set H \<inter> unit_circle_set"   and "i2 \<in> circline_set H \<inter> unit_circle_set"   and "i1 \<noteq> i2"   and "a \<in> circline_set H"   and "a \<notin> unit_circle_set" shows "\<exists>b. b \<noteq> i1 \<and> b \<noteq> i2 \<and> b \<noteq> a \<and> b \<noteq> inversion a \<and> b \<in> circline_set H"
the_f: shows "(THE f. \<forall>i. v $ i = (\<Sum>x\<in>UNIV. f x * X $ x $ i)) = f"
OUTfromChCorrect_data24: shows "OUTfromChCorrect data24"
mset_neg_as_pos: fixes M :: "'b zmultiset" shows "zmset_of (mset_neg M) = zmset_of (mset_pos M) - M"
Pyz_nn: fixes x :: "'d \<times> 'e" assumes "x \<in> space (T \<Otimes>\<^sub>M P)" shows "0 \<le> Pyz x"
left_commute: fixes b :: "'d upred"   and a :: "'d upred"   and c :: "'d upred" shows "(b \<and> a \<and> c) = (a \<and> b \<and> c)"
absolutely_integrable_continuous: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on (cbox a b) f" shows "f absolutely_integrable_on cbox a b"
not_Cons_self: fixes xs :: "'c list"   and x :: "'c" shows "xs \<noteq> x # xs"
aux0_5: fixes x :: "('b, 'c) Combinators list" assumes "\<nexists>a b. a \<oplus> b \<in> set x" shows "singleCombinators x"
subalbegra_incl_imp_finite_dimension: fixes K :: "'a list set set"   and E :: "'a list set set"   and V :: "'a list set set" assumes "subfield K (Rupt A P)"   and "Rupt.finite_dimension K E"   and "subalgebra K V (Rupt A P)"   and "V \<subseteq> E" shows "Rupt.finite_dimension K V"
assms: shows "finite {a. f a \<noteq> (0::'a)}"
simps_3: fixes x :: "name"   and y :: "name"   and t' :: "lam"   and t :: "lam" assumes "x \<sharp> y"   and "x \<sharp> t'" shows "Lam [x].t[y::=t'] = Lam [x].t[y::=t']"
fst: shows "pmf (mix_pmf (\<beta> / \<alpha>) (mix_pmf \<alpha> p q) q) e = \<beta> / \<alpha> * pmf (mix_pmf \<alpha> p q) e + (1 - \<beta> / \<alpha>) * pmf q e"
bit_word_log2: fixes w :: "'b word" assumes "w \<noteq> 0" shows "bit w (word_log2 w)"
P3: shows "D Out G0 G \<and> Cong D G A B"
lhd_llist_inf: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "xs \<noteq> LNil"   and "ys \<noteq> LNil"   and "lhd xs = lhd ys" shows "lhd (xs \<sqinter> ys) = lhd ys"
exec_to_exec_strip_guards_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and f :: "'f"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f"   and "f \<notin> F" shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> \<Rightarrow> Fault f"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "hd (drop n xs) = xs ! n"
concat_hd_pref: fixes ws :: "'b list list" assumes "ws \<noteq> \<epsilon>" shows "hd ws \<le>p concat ws"
pmf_eqI: fixes M :: "'c pmf"   and N :: "'c pmf" assumes "\<And>i. pmf M i = pmf N i" shows "M = N"
inject_1: fixes x1 :: "agent"   and y1 :: "agent" shows "(sharK x1 = sharK y1) = (x1 = y1)"
tdev_def: fixes x :: "'d pdevs" shows "tdev x = (\<Sum>i<degree x. \<bar>pdevs_apply x i\<bar>)"
cond_unit_F: fixes P :: "('b, 'c) uexpr"   and Q :: "('b, 'c) uexpr" shows "P \<triangleleft> false \<triangleright> Q = Q"
space_null_measure: fixes M :: "'b measure" shows "space (null_measure M) = space M"
max_lin_independent_set_in_Span: fixes vs :: "'v list"   and us :: "'v list" assumes "set vs \<subseteq> V"   and "set us \<subseteq> RSpan vs"   and "R_lin_independent us" shows "length us \<le> length vs"
approx_sym: fixes x :: "'c star"   and y :: "'c star" assumes "x \<approx> y" shows "y \<approx> x"
exhaust: fixes y :: "Enum.finite_5" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"   | "y = a\<^sub>3"   | "y = a\<^sub>4"   | "y = a\<^sub>5"
nth_Cons': fixes x :: "'e"   and xs :: "'e list"   and n :: "nat" shows "(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))"
that: fixes n :: "nat" assumes "\<sigma> m = \<tau> n"   and "\<forall>j<n. \<exists>k<m. \<sigma> k = \<tau> j" shows "thesis"
is_binqueue_select: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list"   and t :: "('c, 'd) bintree" assumes "is_binqueue l xs"   and "Some t \<in> set xs" shows "\<exists>k. is_bintree_list k (children t) \<and> is_heap_list (priority t) (children t)"
evContainsEx_instance: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and body :: "formula"   and gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "EV (contains f (i, FAll Neg body))"   and "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "EV (contains f (0, instanceF (X i) body))"
untime_eq_filter2: fixes s :: "'b message_af list" shows "untime s = map the_af (filter (\<lambda>x. x \<noteq> NoMsg) s)"
rd: shows "\<forall>asset\<in>stocks Mkt. borel_adapt_stoch_proc F (prices Mkt asset)"
i_f_join_nth: fixes n :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "n < card I" shows "f \<Join>\<^sub> I ! n = f (I \<rightarrow> n)"
map_zip_map2: fixes f :: "'c \<times> 'd \<Rightarrow> 'b"   and xs :: "'c list"   and g :: "'e \<Rightarrow> 'd"   and ys :: "'e list" shows "map f (zip xs (map g ys)) = map2 (\<lambda>x y. f (x, g y)) xs ys"
sum_pos_ex_elem_pos: fixes f :: "'a \<Rightarrow> int"   and M :: "'a set" assumes "0 < sum f M" shows "\<exists>m\<in>M. 0 < f m"
o_def: fixes f :: "'r \<Rightarrow> 'q"   and g :: "'p \<Rightarrow> 'r" shows "f \<circ> g = (\<lambda>x. f (g x))"