hcju/isabelleps huggingface dataset:
induct: fixes P :: "'c rexp \<times> ('b val \<Rightarrow> 'b val) \<Rightarrow> 'c rexp \<times> ('b val \<Rightarrow> 'b val) \<Rightarrow> bool"   and a0 :: "'c rexp \<times> ('b val \<Rightarrow> 'b val)"   and a1 :: "'c rexp \<times> ('b val \<Rightarrow> 'b val)" assumes "\<And>f\<^sub>1 r\<^sub>2 f\<^sub>2. P (One, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"   and "\<And>f\<^sub>1 f\<^sub>2. P (Zero, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Atom v, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Plus v va, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Times v va, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Star v, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>f\<^sub>1 f\<^sub>2. P (Zero, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>f\<^sub>1 v f\<^sub>2. P (Zero, f\<^sub>1) (Atom v, f\<^sub>2)"   and "\<And>f\<^sub>1 v va f\<^sub>2. P (Zero, f\<^sub>1) (Plus v va, f\<^sub>2)"   and "\<And>f\<^sub>1 v va f\<^sub>2. P (Zero, f\<^sub>1) (Times v va, f\<^sub>2)"   and "\<And>f\<^sub>1 v f\<^sub>2. P (Zero, f\<^sub>1) (Star v, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Atom v, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Atom v, f\<^sub>1) (Atom va, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Atom v, f\<^sub>1) (Plus va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Atom v, f\<^sub>1) (Times va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Atom v, f\<^sub>1) (Star va, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Plus v va, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Plus v va, f\<^sub>1) (Atom vb, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Plus v va, f\<^sub>1) (Plus vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Plus v va, f\<^sub>1) (Times vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Plus v va, f\<^sub>1) (Star vb, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Times v va, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Times v va, f\<^sub>1) (Atom vb, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Times v va, f\<^sub>1) (Plus vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Times v va, f\<^sub>1) (Times vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Times v va, f\<^sub>1) (Star vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Star v, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Star v, f\<^sub>1) (Atom va, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Star v, f\<^sub>1) (Plus va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Star v, f\<^sub>1) (Times va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Star v, f\<^sub>1) (Star va, f\<^sub>2)" shows "P a0 a1"
has_field_derivative_subset: fixes f :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c"   and s :: "'c set"   and t :: "'c set" assumes "(f has_field_derivative y) (at x within s)"   and "t \<subseteq> s" shows "(f has_field_derivative y) (at x within t)"
inf_prefix_length_le_min: fixes xs :: "'b list"   and ys :: "'b list" shows "inf_prefix_length xs ys \<le> min (length xs) (length ys)"
Realfract_def: fixes p :: "int"   and q :: "int" shows "Realfract p q = Real.real_of_int p / Real.real_of_int q"
hlp1: assumes "a \<noteq> b" shows "a # word_upto (a + 1) b = word_upto a b"
list_of_vec_map: fixes xs :: "'b vec" shows "list_of_vec xs = map (($) xs) [0..<dim_vec xs]"
UP_subring_pderiv_equal: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "pderiv g = UP_cring.pderiv (R\<lparr>carrier := S\<rparr>) g"
empty_valuation_def: shows "\<A>\<^sub>0 \<equiv> \<lambda>_. False"
right_idem: fixes a :: "'b word"   and b :: "'b word" shows "signed.min (signed.min a b) b = signed.min a b"
abs_square_less_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (power.power one times x 2) one = less (abs x) one"
vimage2p_mono: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'g"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'d"   and y :: "'f"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool" assumes "BNF_Def.vimage2p f g R x y"   and "R \<le> S" shows "BNF_Def.vimage2p f g S x y"
1: shows "top * e \<le> v\<^sup>T * w\<^sup>\<star>"
traverse_tree_pure_tree: fixes path :: "dir list"   and x :: "'a" shows "traverse_tree path (pure x) = pure x"
map_of_from_function_graph_is_some_if: fixes xs :: "'a list"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "set xs \<noteq> {}"   and "x \<in> set xs" shows "map_of (map (\<lambda>x. (x, f x)) xs) x = Some (f x)"
sorted_wrt_hd_drop_less_drop: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and n :: "nat" assumes "sorted_wrt f xs"   and "\<And>x. f x x" shows "\<forall>x\<in>set (drop n xs). f (hd (drop n xs)) x"
map_poly_hom_add: fixes p :: "int poly"   and q :: "int poly" shows "of_int_poly (p + q) = of_int_poly p + of_int_poly q"
LAssRed: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and V :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow> \<langle>V:=e',s'\<rangle>"
hmem_shift_iff: fixes m :: "hf"   and y :: "hf"   and f :: "hf"   and delta :: "hf" shows "(\<langle>m, y\<rangle> \<^bold>\<in> shift f delta) = (\<exists>n. m = delta @+ n \<and> \<langle>n, y\<rangle> \<^bold>\<in> f)"
wset_actions_ok_def: fixes ws :: "'a \<Rightarrow> 'b wait_set_status option"   and t :: "'a"   and was :: "('a, 'b) wait_set_action list" shows "wset_actions_ok ws t was = (if Notified \<in> set was then ws t = \<lfloor>PostWS WSNotified\<rfloor> else if WokenUp \<in> set was then ws t = \<lfloor>PostWS WSWokenUp\<rfloor> else ws t = None)"
unit_prod: fixes a :: "'a"   and b :: "'a" assumes "is_unit a"   and "is_unit b" shows "is_unit (a * b)"
foundedIncLevel: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool"   and A :: "(nat \<times> 'b) set" shows "founded subs P (incLevel ` A) = founded subs P A"
assms_2: shows "X0 \<subseteq> {l..u}"
const_expansion_ms_def: fixes c :: "real" shows "const_expansion c = MS (const_ms_aux c) (\<lambda>_. c)"
strict_mono_mono: fixes f :: "'a \<Rightarrow> 'b" assumes "strict_mono f" shows "mono f"
not_lnull_conv: fixes xs :: "'b llist" shows "(\<not> lnull xs) = (\<exists>x xs'. xs = x % xs')"
regular_closed_star: fixes x :: "'a" assumes "regular x" shows "regular top"
not_choice_def: fixes p :: "('ap, 'aq, 'ar, 'as) seqp" shows "not_choice p \<equiv> \<forall>p1 p2. p \<noteq> p1 \<oplus> p2"
simps_2: fixes x :: "char list"   and e :: "aexp"   and S :: "'a"   and x' :: "char list"   and e' :: "aexp"   and S' :: "'a" shows "(x ::= e {S} \<le> x' ::= e' {S'}) = (x = x' \<and> e = e' \<and> S \<le> S')"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and index :: "int" assumes "incomplete_design point_set block_collection u_block_size"   and "t_wise_balance point_set block_collection grouping index"   and "t_design_axioms u_block_size grouping" shows "t_design point_set block_collection u_block_size grouping index"
connected_open_delete_finite: fixes S :: "'a set"   and T :: "'a set" assumes "open S"   and "connected S"   and "2 \<le> DIM('a)"   and "finite T" shows "connected (S - T)"
sum: fixes g :: "'c \<Rightarrow> 'a"   and S :: "'c set" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
of_nat_of_natural: fixes n :: "natural" shows "of_nat (nat_of_natural n) = n"
approx_subset_monad: fixes x :: "'b star"   and y :: "'b star" assumes "x \<approx> y" shows "{x, y} \<subseteq> monad x"
IH_2: shows "size1 r \<le> 2 ^ height r"
simps_2: fixes n :: "int"   and m :: "int" shows "(Const n \<sqsubseteq> Const m) = (n = m)"
Worder_def: fixes D :: "('c, 'd) Order_scheme" shows "Worder D \<equiv> Torder D \<and> Worder_axioms D"
intros: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>c (c', s') \<and> mayDiverge c' s'" shows "mayDiverge c s"
leD: fixes y :: "complex"   and x :: "complex" assumes "y \<le>\<^sub>\<complex> x" shows "\<not> x <\<^sub>\<complex> y"
map2_val_compatD1: fixes f :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'f) list"   and zs :: "('a \<times> 'e) list" assumes "map2_val_compat f"   and "oalist_inv_raw zs" shows "oalist_inv_raw (f zs)"
list_all2_Nil_iff: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" assumes "list_all2 R xs ys" shows "(xs = []) = (ys = [])"
comp_converter_assoc: fixes conv1 :: "('a, 'b, 'g, 'h) converter"   and conv2 :: "('g, 'h, 'e, 'f) converter"   and conv3 :: "('e, 'f, 'c, 'd) converter" shows "comp_converter (comp_converter conv1 conv2) conv3 = comp_converter conv1 (comp_converter conv2 conv3)"
prob_le_1: fixes M :: "'h pmf"   and A :: "'h set" shows "measure_pmf.prob M A \<le> 1"
not_none_2: shows "obs \<sigma>' k \<noteq> None"
bind_pure_returns_result_I: fixes f :: "('ap, 'aq, 'ar) prog"   and h :: "'ap"   and x :: "'ar"   and g :: "'ar \<Rightarrow> ('ap, 'aq, 'as) prog"   and y :: "'as" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
random_serial_dictatorship_def: fixes agents :: "'b set"   and alts :: "'a set"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "random_serial_dictatorship agents alts R = fold_bind_random_permutation (\<lambda>i. Max_wrt_among (R i)) pmf_of_set alts agents"
satisfies_eqvt: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and \<phi> :: "'b"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "satisfies P \<phi>" shows "satisfies (p \<bullet> P) (p \<bullet> \<phi>)"
leg1_commutes: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "C (Leg1 (Cod \<mu>)) (Chn \<mu>) = Leg1 (Dom \<mu>)"
that: fixes j :: "nat"   and r :: "'a list" assumes "j < \<^bold>|ws\<^bold>|"   and "r <p ws ! j"   and "concat (take j ws) \<cdot> r = u" shows "thesis"
dom_dom: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.dom (CC.dom f) = CC.dom f"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b list" assumes "inter_sorted_rev x xa = y" obtains l2 :: "'b list" where "x = []" and "xa = l2__" and "y = []"   | v :: "'b" and va :: "'b list" where "x = v # va" and "xa = []" and "y = []"   | x1 :: "'b" and l1 :: "'b list" and x2 :: "'b" and l2 :: "'b list" where "x = x1 # l1" and "xa = x2 # l2" and "y = (if x2 < x1 then inter_sorted_rev l1 (x2 # l2) else if x1 = x2 then x1 # inter_sorted_rev l1 l2 else inter_sorted_rev (x1 # l1) l2)"
is_acc_ifaceD: fixes G :: "prog"   and P :: "pname"   and I :: "qtname" assumes "is_acc_iface G P I" shows "is_iface G I \<and> G \<turnstile> Iface I accessible_in P"
atrm_trm: shows "atrm \<subseteq> trm"
eq_is_equal: fixes L :: "('d, 'e) gorder_scheme" assumes "partial_order L" shows "(.=\<^bsub>L\<^esub>) = (=)"
mset_union_size_inter: fixes A :: "'a multiset"   and B :: "'a multiset" shows "size (A \<union># B) = size A + size B - size (A \<inter># B)"
can_mkarr_Can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Can C t" shows "free_monoidal_category.can C (free_monoidal_category.mkarr C t)"
rel_coinduct: fixes P :: "('c, 'd) resource \<Rightarrow> ('c, 'e) resource \<Rightarrow> bool"   and x :: "('c, 'd) resource"   and y :: "('c, 'e) resource"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "P x y"   and "\<And>resource resource'. P resource resource' \<Longrightarrow> rel_fun (=) (rel_spmf (rel_prod R2 P)) (run_resource resource) (run_resource resource')" shows "rel_resource' R2 x y"
hyps_2: fixes x :: "int" shows "M_poly (Mp p) x = M_poly p x"
inv_begin_step_E: fixes i :: "nat"   and j :: "nat" assumes "0 < i"   and "0 < j" shows "\<exists>ia>0. ia + j - Suc 0 = i + j \<and> Oc # Oc \<up> i = Oc \<up> ia"
lcs_adds_plus: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "class.lcs_powerprod plus minus zero lcs" shows "plus.adds plus (lcs s t) (plus s t)"
simps: fixes \<A>1 :: "'a \<Rightarrow> ((nat \<times> nat) \<times> 'c) spmf"   and \<A>2 :: "'a \<Rightarrow> 'c \<Rightarrow> bool spmf" shows "ped_commit.hiding_game_ind_cpa (\<A>1, \<A>2) = TRY key_gen \<bind> (\<lambda>(ck, vk). \<A>1 vk \<bind> (\<lambda>((m0, m1), \<sigma>). assert_spmf (valid_msg m0 \<and> valid_msg m1) \<bind> (\<lambda>_. coin_spmf \<bind> (\<lambda>b. commit ck (if b then m0 else m1) \<bind> (\<lambda>(c, d). \<A>2 c \<sigma> \<bind> (\<lambda>b'. return_spmf (b' = b))))))) ELSE coin_spmf"
ordermap_bij: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "wf r"   and "total_on A r"   and "small A" shows "bij_betw (ordermap A r) A (elts (ordertype A r))"
simps: fixes a :: "nat"   and b :: "nat" shows "balanced2_arith a b = (balanced1_arith a b \<and> balanced1_arith b a)"
r_comp_rtrancl_eq: fixes r :: "('b \<times> 'b) set" shows "r O r\<^sup>* = r\<^sup>* O r"
simps_4: shows "eval_ternary_Or TernaryFalse TernaryFalse = TernaryFalse"
a_2: fixes f :: "'a"   and P :: "'a \<Rightarrow> 'a list"   and T :: "('a, 'b) Term.term list" shows "set (map (Fun f) (U P T)) \<subseteq> set (term_variants P (Fun f T))"
Up_To_Technique: fixes d :: "'d"   and PP :: "nat set"   and R' :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set" assumes "disj_dlHPP_Bisimulation_Up_To_R' d PP R' R"   and "Arefl R \<inter> Arefl R' \<subseteq> {[]}" shows "SdlHPPB d PP (R \<union> R')"
functor_axioms: shows "functor D.CC.comp (\<cdot>\<^sub>D) D'.T"
wf_syscls_is_class_Object: fixes P :: "'b prog" assumes "wf_syscls P" shows "is_class P Object"
induct: fixes x1 :: "'var"   and x2 :: "'edge list"   and x3 :: "'edge list"   and n' :: "'node"   and P :: "'var \<Rightarrow> 'edge list \<Rightarrow> 'edge list \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> dependent_live_vars n'"   and "\<And>V. V \<in> Use n' \<Longrightarrow> P V [] []"   and "\<And>V a as' n'' as''. \<lbrakk>V \<in> Use (sourcenode a); sourcenode a -a # as'\<rightarrow>\<^sub>c\<^sub>d n''; n'' -as''\<rightarrow>\<^sub>d* n'\<rbrakk> \<Longrightarrow> P V [] (a # as' @ as'')"   and "\<And>V as' as V' a. \<lbrakk>(V, as', as) \<in> dependent_live_vars n'; P V as' as; V' \<in> Use (sourcenode a); n' = last (targetnodes (a # as)); sourcenode a -{V}a # as'\<rightarrow>\<^sub>d\<^sub>d last (targetnodes (a # as'))\<rbrakk> \<Longrightarrow> P V' [] (a # as)"   and "\<And>V as' as a. \<lbrakk>(V, as', as) \<in> dependent_live_vars n'; P V as' as; n' = last (targetnodes (a # as)); \<not> sourcenode a -{V}a # as'\<rightarrow>\<^sub>d\<^sub>d last (targetnodes (a # as'))\<rbrakk> \<Longrightarrow> P V (a # as') (a # as)" shows "P x1 x2 x3"
separating_hyperplane_set_point_inaff: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "S \<noteq> {}"   and "z \<notin> S" obtains a :: "'a" and b :: "real" where "z + a \<in> affine hull insert z S" and "a \<noteq> (0::'a)" and "a \<bullet> z \<le> b" and "\<And>x. x \<in> S \<Longrightarrow> b \<le> a \<bullet> x"
brc_pref_br': fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist" assumes "rqrm_invar rqrm"   and "rqrm_prop (ls.\<alpha> \<delta>) rqrm"   and "ls.invar \<delta>" shows "wa_precise_refine (det_wa_wa (brc_det_algo rqrm \<delta>)) (br'_algo (ls.\<alpha> \<delta>)) brc_\<alpha>"
exhaust: fixes y :: "'c decision" obtains x1 :: "'c" where "y = allow x1"   | x2 :: "'c" where "y = deny x2"
tauLaw4StatImpLeft: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q, \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q) \<in> Rel" shows "\<Psi> \<rhd> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<lessapprox><Rel> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
continuous_at_ennreal: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> real" assumes "continuous F f" shows "continuous F (\<lambda>x. ennreal (f x))"
update_convs_1: fixes getF' :: "Literal list list \<Rightarrow> Literal list list"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getF_update getF' \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = \<lparr>getF = getF' getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr>"
favorites_permute: fixes i :: "'agent"   and \<sigma> :: "'alt \<Rightarrow> 'alt" assumes "i \<in> agents"   and "\<sigma> permutes alts" shows "favorites (permute_profile \<sigma> R) i = \<sigma> ` favorites R i"
FactGroup_iso_set: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h"   and "h ` carrier G = carrier H" shows "(\<lambda>X. the_elem (h ` X)) \<in> Group.iso (G Mod kernel G H h) H"
no_collision_with_Unit: fixes t :: "Syntax.term" assumes "hash Unit = hash t" shows "t = Unit"
indRelLTPO_impl_TRel_weakly_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_respects_barbs (indRelLTPO TRel) (STCalWB SWB TWB)" shows "rel_weakly_respects_barbs TRel TWB"
iso_unit: fixes a :: "'c" assumes "C.obj a" shows "C.iso \<i>\<^sub>C[a]"
assms_1: shows "(\<^bold>\<not> \<^bold>\<not> p) at a in (ps, a) # branch"
keys_minus: fixes a :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (- a) = keys a"
sum_upto_cong: fixes f :: "nat \<Rightarrow> 'a"   and f' :: "nat \<Rightarrow> 'a"   and n :: "real"   and n' :: "real" assumes "\<And>n. 0 < n \<Longrightarrow> f n = f' n"   and "n = n'" shows "sum_upto f n = sum_upto f' n'"
nop_bij_vdomain: shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
Graph11: fixes M :: "node list"   and j :: "nat" assumes "M ! j \<noteq> Black"   and "j < length M" shows "Blacks M \<subset> Blacks (M[j := Black])"
assms: assumes "c \<noteq> 0" shows "f abs_summable_on A"
elims_2: fixes x :: "value" assumes "is_Num x" obtains uu_ :: "int" where "x = Num uu_"
subgraph_def: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" shows "subgraph G\<^sub>1 G\<^sub>2 = (graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2)"
indexed_const_def: fixes k :: "'c multiset \<Rightarrow> 'a" shows "L.indexed_const k = (\<lambda>m. if m = {#} then k else \<zero>\<^bsub>L\<^esub>)"
noetherian_ringI: assumes "\<And>I. ideal I R \<Longrightarrow> \<exists>A\<subseteq>carrier R. finite A \<and> I = Idl A" shows "noetherian_ring R"
simps: fixes prec :: "nat"   and x :: "float" shows "ub_arctan prec x = (let lb_horner = \<lambda>x. float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))); ub_horner = \<lambda>x. float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))) in if x < 0 then - lb_arctan prec (- x) else if x \<le> Float 1 (- 1) then ub_horner x else if x \<le> Float 1 1 then let y = float_divr prec x (float_plus_down (Suc prec) 1 (lb_sqrt prec (float_plus_down prec 1 (float_round_down prec (x * x))))) in if 1 < y then ub_pi prec * Float 1 (- 1) else Float 1 1 * ub_horner y else float_plus_up prec (ub_pi prec * Float 1 (- 1)) (- lb_horner (float_divl prec 1 x)))"
Rep_nat_int_cases: fixes y :: "nat set" assumes "y \<in> {S. \<exists>m n. {m..n} = S}" obtains x :: "nat_int" where "y = Rep_nat_int x"
Sup_llist_grounding_of_state_ground: fixes C :: "'a literal multiset" assumes "C \<in> Sup_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts)" shows "is_ground_cls C"
results'_gpv_map_gpv': fixes h :: "'b \<Rightarrow> 'a"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and gpv :: "('e, 'f, 'a) gpv" assumes "surj h" shows "results'_gpv (map_gpv' f g h gpv) = f ` results'_gpv gpv"
filter_mset_sum_list: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a multiset list" shows "filter_mset P (sum_list xs) = sum_list (map (filter_mset P) xs)"
hyps: assumes "P \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>k. P * f k) n = P * matrix_sum d f n"
logic_aux2: fixes j :: "'a"   and x :: "'b"   and f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and i :: "'c" shows "(\<exists>k. (k = j \<longrightarrow> x = f i j) \<and> (k \<noteq> j \<longrightarrow> x = (0::'b))) = ((\<exists>k. k = j \<and> x = f i j) \<or> (\<exists>k. k \<noteq> j \<and> x = (0::'b)))"
distrib_left: fixes a :: "'h"   and b :: "'h"   and c :: "'h" shows "a * (b + c) = a * b + a * c"
gfp_def: fixes f :: "'a \<Rightarrow> 'a" shows "complete_lattice.gfp Inf (\<lambda>x y. y \<le> x) f = \<Sqinter> {u. f u \<le> u}"
EX_RPG: shows "\<exists>f. NPreflow c s t f \<and> cf = cf_of f"
bind_spmf_of_pmf: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "spmf_of_pmf p \<bind> f = p \<bind> f"
index_zero_vec_1: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "0\<^sub>v n $v i = (0::'b)"
simps: fixes n :: "nat" shows "RBT_Impl.plog2 n = (if n \<le> 1 then 0 else RBT_Impl.plog2 (n div 2) + 1)"
is_empty_def: fixes q :: "('c, 'd) pq" shows "is_empty q = (alist_of q = [])"
set_tag_name_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and tag :: "char list" assumes "DocumentClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok set_tag_name element_ptr tag"
domains_subsume_def: fixes ls :: "('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr" shows "\<S> ls = (LID\<^sub>L ls \<subseteq> doms ls)"
ring_iso_def: fixes R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" shows "ring_iso R S = {h \<in> ring_hom R S. bij_betw h (carrier R) (carrier S)}"
yot_anything: fixes x :: "yxml_of_term"   and y :: "yxml_of_term" shows "x = y"
assms: fixes x :: "'a list" assumes "x \<in> set zs" shows "x \<noteq> []"