hcju/isabelleps huggingface dataset:
emeasure_pmf_of_set: fixes S :: "'c set"   and A :: "'c set" assumes "S \<noteq> {}"   and "finite S" shows "emeasure (measure_pmf (local.pmf_of_set S)) A = ennreal (real (card (S \<inter> A)) / real (card S))"
\<R>\<^sub>1_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" where "c\<^sub>2 = c\<^sub>1" and "mds\<^sub>2 = mds" and "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<Gamma>'__,\<S>'__,Pa__" and "tyenv_wellformed mds \<Gamma> \<S> P" and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" and "pred P mem\<^sub>1" and "pred P mem\<^sub>2" and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
Equivalence_def: shows "{(x, y). stk1.rel x y} = {(a, b). \<exists>A\<in>e1.Partition. a \<in> A \<and> b \<in> A}"
simps_2: fixes x3 :: "'b formula"   and y3 :: "'b formula" shows "(\<^bold>\<not> x3 = \<^bold>\<not> y3) = (x3 = y3)"
F: shows "\<forall>i<length ns. offs ns 0 ! i \<noteq> k"
wf_prog_def: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" shows "wf_prog wf_md P \<equiv> wf_syscls P \<and> (\<forall>c\<in>set P. wf_cdecl wf_md P c) \<and> distinct_fst P"
partition_on_mset_sum_size_eq: fixes A :: "'b multiset"   and P :: "'b multiset multiset" assumes "partition_on_mset A P" shows "\<Sum>\<^sub># (image_mset size P) = size A"
inter_mset_empty_distrib_right: fixes A :: "'k multiset"   and B :: "'k multiset"   and C :: "'k multiset" shows "(A \<inter># (B + C) = {#}) = (A \<inter># B = {#} \<and> A \<inter># C = {#})"
type_less_x_Collection_intro_4: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Bag \<rho>"   and "\<rho> \<le> \<sigma>" shows "\<tau> < Collection \<sigma>"
dec: fixes p :: "'b list"   and q :: "'b list"   and r :: "'b list"   and s :: "'b list" assumes "awalk u p v"   and "awalk_cyc_decomp p = (q, r, s)"   and "\<not> distinct (awalk_verts u p)"   and "P (q @ s)" shows "P p"
emeasure_std_simplex_aux: fixes A :: "'a set"   and t :: "real" assumes "finite A"   and "0 \<le> t" shows "emeasure (Pi\<^sub>M A (\<lambda>_. lborel)) ({x. (\<forall>i\<in>A. 0 \<le> x i) \<and> sum x A \<le> t} \<inter> space (Pi\<^sub>M A (\<lambda>_. lborel))) = ennreal (t ^ card A / fact (card A))"
fold_graph_determ: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and z :: "'e"   and B :: "'e set"   and x :: "'e"   and y :: "'e" assumes "comp_fun_commute_on f A"   and "fold_graph f z B x"   and "fold_graph f z B y"   and "B \<subseteq> A"   and "z \<in> A" shows "y = x"
simps_7: fixes \<mu> :: "'a ltln"   and \<mu>' :: "'a ltln" shows "pure_eventual (\<mu> W\<^sub>n \<mu>') = (pure_eventual \<mu> \<and> pure_eventual \<mu>')"
root_int_ceiling: fixes p :: "nat"   and x :: "int" shows "root_int_ceiling p x = \<lceil>root p (real_of_int x)\<rceil>"
span_explicit: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d set" assumes "module scale" shows "module.span scale b = {\<Sum>a\<in>t. scale (r a) a |t r. finite t \<and> t \<subseteq> b}"
respection: shows "rel_respects_binary_pred Rel (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
geometric_sum_aux: fixes x :: "int" shows "(x - 1) * sum ((^) x) {..<a} = x ^ a - 1"
prm_unit_involution: fixes a :: "'b"   and b :: "'b" shows "[a \<leftrightarrow> b] \<diamondop> [a \<leftrightarrow> b] = \<epsilon>"
infsetsum'_converges_Un_disjoint: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "infsetsum'_converges f A"   and "infsetsum'_converges f B"   and "A \<inter> B = {}" shows "infsetsum'_converges f (A \<union> B)"
l2_refines_l1: shows "refines (R12s \<inter> reach skl1 \<times> (l2_inv1 \<inter> l2_inv2 \<inter> l2_inv3 \<inter> l2_inv4 \<inter> l2_inv5 \<inter> l2_inv6 \<inter> l2_inv6' \<inter> l2_inv7)) med12s skl1 l2"
hyps_correct: fixes n :: "('form, 'rule) graph_node"   and p1 :: "('form, 'var) out_port"   and p2 :: "('form, 'var) antecedent" assumes "hyps n p1 = Some p2" shows "p1 |\<in>| outPorts n \<and> p2 |\<in>| inPorts n"
tau_mem: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and m :: "'f"   and n :: "'h"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "n \<in> carrier N"   and "FM\<^bsub>R\<^esub> P M N" shows "tau R M N P (m, n) \<in> carrier P"
eq_equiv_class_iff: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "equiv A r"   and "x \<in> A"   and "y \<in> A" shows "(r `` {x} = r `` {y}) = ((x, y) \<in> r)"
g_1_Ex1: shows "g_1 \<in> CofGroups.Ex1"
nz_prod_eq: fixes c :: "'b"   and x :: "'b" assumes "c \<noteq> (0::'b)" shows "c * x = (0::'b) \<equiv> x = (0::'b)"
simps_1: shows "pass\<^sub>2 [] = heap.Empty"
kE_satA: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and at :: "('a, 'b) atm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"   and "Signature.wtA wtFsym wtPsym arOf resOf parOf at"   and "Struct.wtE intT (InfModel.kE intT \<xi>)" shows "Struct.satA intF intP (InfModel.kE intT \<xi>) at = Struct.satA (InfModel.intFF arOf resOf intT intF) (InfModel.intPF parOf intT intP) \<xi> at"
f_nz: shows "\<forall>\<^sub>F x in at_top. f x \<noteq> 0"
2: shows "\<Psi> (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x)) (inC \<circ> \<psi> x \<circ> HomD.\<psi> (y, G x))"
Rep_ty_cases: fixes y :: "ty_Rep" assumes "y \<in> Collect ty_Rep_set" obtains x :: "ty" where "y = Rep_ty x"
unsat_mono: fixes cs :: "constraint set"   and ds :: "constraint set" assumes "\<nexists>v. v \<Turnstile>\<^sub>c\<^sub>s cs"   and "cs \<subseteq> ds" shows "\<nexists>v. v \<Turnstile>\<^sub>c\<^sub>s ds"
b: fixes x :: "'b" assumes "x \<in> S" shows "b \<le> norm (l x)"
topspace_euclidean_subtopology: fixes S :: "'b set" shows "topspace (top_of_set S) = S"
f4_red_aux_irredudible: fixes h :: "'t \<Rightarrow>\<^sub>0 'e"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and b :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f" assumes "h \<in> set (f4_red_aux bs ps)"   and "b \<in> set bs"   and "fst b \<noteq> 0" shows "\<not> lt (fst b) adds\<^sub>t lt h"
antisym: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "linorder_list le S"   and "a \<in> S"   and "b \<in> S"   and "le a b"   and "le b a" shows "a = b"
simps_15: fixes x31 :: "jump"   and x32 :: "stmt" shows "x31\<bullet> x32 \<noteq> Skip"
me_K: shows "measurable_envelope M (K \<inter> {x \<in> space M. f x \<le> a}) K"
open_imp_subset: fixes U :: "'d set" assumes "U = {}" shows "U \<subseteq> {}"
simps: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and a1 :: "'a language"   and a2 :: "'a language" shows "Plus_Times_pre_cong R a1 a2 = ((\<exists>x y. a1 = x \<and> a2 = y \<and> x \<le> y) \<or> (\<exists>x y z. a1 = x \<and> a2 = z \<and> Plus_Times_pre_cong R x y \<and> Plus_Times_pre_cong R y z) \<or> (\<exists>x y. a1 = x \<and> a2 = y \<and> R x y) \<or> (\<exists>x y x' y'. a1 = Plus x x' \<and> a2 = Plus y y' \<and> Plus_Times_pre_cong R x y \<and> Plus_Times_pre_cong R x' y') \<or> (\<exists>x y x' y'. a1 = Times x x' \<and> a2 = Times y y' \<and> Plus_Times_pre_cong R x y \<and> Plus_Times_pre_cong R x' y'))"
is_generic_def: fixes p :: "'b point"   and q :: "'b point" shows "is_generic p q = (p \<noteq> Infinity \<and> q \<noteq> Infinity \<and> p \<noteq> q \<and> p \<noteq> opp q)"
assms_2: shows "\<forall>x\<in>S. finite (cl_ecl x)"
cases: fixes x :: "exp" obtains xa :: "Vars.var" where "x = Var xa__"   | xa :: "Vars.var" and e :: "exp" where "x = Lam [xaa__]. ea__"   | e :: "exp" and xa :: "Vars.var" where "x = App eb__ xa"   | as :: "(Vars.var \<times> exp) list" and e :: "exp" where "x = Let as e"   | b :: "bool" where "x = Bool b"   | scrut :: "exp" and e1 :: "exp" and e2 :: "exp" where "x = (scrut ? e1 : e2)"
prefixes_Suc: fixes j :: "nat"   and t :: "nat" shows "local.prefixes j (Suc t) = e_snoc (e_append_zeros (local.prefixes j t) (the (eval r_find_k [local.prefixes j t]))) 1"
D_subset: shows "D \<subseteq> {x. 0 < x \<and> x \<le> int ((p - 1) div 2)}"
homeomorphic_maps_sym: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "homeomorphic_maps X Y f g = homeomorphic_maps Y X g f"
least_def: fixes uu :: "'b itself"   and Rs :: "(('c, 'd) labeled_graph \<times> ('c, 'd) labeled_graph) set"   and S :: "('c, 'e) labeled_graph"   and G :: "('c, 'e) labeled_graph" shows "least uu Rs S G \<equiv> subgraph S G \<and> (\<forall>C. consequence_graph Rs C \<longrightarrow> maintained (S, G) C)"
assms_2: shows "(b2 - B2)\<^sup>2 + (b1 - B1)\<^sup>2 \<noteq> 0"
alt_matrix_A_R_def: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" shows "alt_matrix_A_R signs subsets = mat (length subsets) (length signs) (\<lambda>(i, j). z_R (subsets ! i) (signs ! j))"
imonad1: fixes u :: "unit"   and f :: "unit \<Rightarrow> 'maa" shows "abind (areturn u) f = f u"
comp_assoc_assoc'_2: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] \<cdot>\<^sub>D \<a>\<^sub>D[f, g, h] = (f \<star>\<^sub>D g) \<star>\<^sub>D h"
S'_telescope_trapezium: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "S' n x = (\<Sum>r<n. T (real r + x))"
SUP_least: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
chop_emb_step_at: fixes t :: "('c, 'd) tm" assumes "is_App t" shows "chop t = emb_step_at (replicate (num_args (fun t)) dir.Left) dir.Right t"
quant_le_iff_magn_le: fixes x :: "'a['b, 'c]"   and y :: "'a['b, 'c]" shows "(x \<le> y) = (\<lbrakk>x\<rbrakk>\<^sub>Q \<le> \<lbrakk>y\<rbrakk>\<^sub>Q)"
Vagree_or: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set"   and W :: "variable set" shows "Vagree \<nu> \<nu>' V \<or> Vagree \<nu> \<nu>' W \<longrightarrow> Vagree \<nu> \<nu>' (V \<inter> W)"
pre_3: shows "\<forall>type\<in>dom (tcsigs (translate_osig sa)). \<forall>ars\<in>ran (the (tcsigs (translate_osig sa) type)). the (map_of tao type) = length ars"
le_rexp_antisym: fixes r :: "'a rexp"   and s :: "'a rexp" assumes "local.le_rexp r s"   and "local.le_rexp s r" shows "r = s"
assms_3: fixes w :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a" assumes "w \<in> SW"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "P h h'"
setFinal_notin_finals: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "\<not> final f"   and "minGraphProps g" shows "setFinal f \<notin> set (finals g)"
takeWhile_dropWhile: fixes f :: "'b \<Rightarrow> bool"   and l :: "'b list"   and l' :: "'b list" assumes "takeWhile f l @ dropWhile f l = l'" shows "l' = l"
active_has_cf_outgoing: fixes u :: "nat" assumes "(0::'a) < excess f u" shows "cf.outgoing u \<noteq> {}"
mat_of_cols_carrier_3: fixes n :: "nat"   and vs :: "'b vec list" shows "dim_col (mat_of_cols n vs) = length vs"
zero_prefix_match_all: fixes m :: "'b prefix_match"   and ip :: "'b word" assumes "valid_prefix m"   and "pfxm_length m = 0" shows "prefix_match_semantics m ip"
Suc_0: shows "moebius_mu (Suc 0) = (1::'b)"
PARORD: fixes r :: "('c \<times> 'c) set" assumes "wo_rel r" shows "Partial_order r"
S_transform_formula_Pred: fixes \<phi> :: "'c" shows "S_transform_formula (Formula.Pred \<phi>) = Act\<^sub>\<alpha> (S_action.Pred \<phi>) (Conj\<^sub>\<alpha> bempty)"
injec_def: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" shows "injec\<^bsub>F,G\<^esub> f = (f \<in> aHom F G \<and> ker\<^bsub>F,G\<^esub> f = {\<zero>\<^bsub>F\<^esub>})"
ltake_Suc_conv_snoc_lnth: fixes m :: "nat"   and xs :: "'a llist" assumes "enat m < llength xs" shows "ltake (enat (Suc m)) xs = lappend (ltake (enat m) xs) (LCons (lnth xs m) LNil)"
eta_reducible_Abs: fixes T :: "typ"   and t :: "Core.term" assumes "\<not> eta_reducible (Abs T t)" shows "\<not> eta_reducible t"
rel_mset_induct: fixes x1 :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and x2 :: "'b multiset"   and x3 :: "'e multiset"   and P :: "('b \<Rightarrow> 'e \<Rightarrow> bool) \<Rightarrow> 'b multiset \<Rightarrow> 'e multiset \<Rightarrow> bool" assumes "rel_mset x1 x2 x3"   and "\<And>R. P R {#} {#}"   and "\<And>R a b M N. \<lbrakk>R a b; rel_mset R M N; P R M N\<rbrakk> \<Longrightarrow> P R (add_mset a M) (add_mset b N)" shows "P x1 x2 x3"
1: shows "reflexive t \<and> transitive t \<and> antisymmetric t \<and> regular t \<and> p \<le> t \<and> \<not> linear t"
simps_3: fixes y :: "'b"   and ys :: "'b list" shows "minus_poly_rev_list [] (y # ys) = []"
desargues_config_3D_coplanar_5_bis_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A', B', C', \<alpha>, \<gamma>} = 3"
tmod2_4: shows "[mn_] \<notin> defined_mods s \<and> no_dup_types ds_ \<and> evaluate_decs True [mn_] env s ds_ (s2_, Rerr err_)"
filterlim_at_bot_at_right: fixes Q :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "\<And>x y. \<lbrakk>Q x; Q y; x \<le> y\<rbrakk> \<Longrightarrow> f x \<le> f y"   and "\<And>x. P x \<Longrightarrow> f (g x) = x"   and "\<And>x. P x \<Longrightarrow> Q (g x)"   and "eventually Q (at_right a)"   and "\<And>b. Q b \<Longrightarrow> a < b"   and "eventually P at_bot" shows "filterlim f at_bot (at_right a)"
chine_assoc_def: shows "cods.chine_assoc \<equiv> \<langle>cods.Prj\<^sub>1\<^sub>1 \<lbrakk>cods.\<mu>.leg0, cods.\<nu>.leg1 \<cdot> cods.\<nu>\<pi>.prj\<^sub>1\<rbrakk> \<langle>cods.Prj\<^sub>0\<^sub>1 \<lbrakk>cods.\<nu>.leg0, cods.\<pi>.leg1\<rbrakk> cods.Prj\<^sub>0\<rangle>\<rangle>"
closure_of_def: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of S \<equiv> {x \<in> topspace X. \<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y\<in>S. y \<in> T)}"
root_in_redexes: fixes c :: "('a, 'b, 'c) com" shows "c \<in> redexes c"
simps_2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'b" shows "bisumTail f [a] = f a (0::'b)"
nat_le_iff_add: fixes m :: "nat"   and n :: "nat" shows "order.greater_eq n m = (\<exists>k. n = m + k)"
terminal_chain_Goto_decision: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and s :: "state"   and t :: "state" assumes "\<Gamma> chain = Some rs"   and "terminal_chain rs"   and "matches \<gamma> m p"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Goto chain)], s\<rangle> \<Rightarrow> t" shows "\<exists>X. t = Decision X"
intros: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
lookup_except_min: fixes ko :: "'o"   and xs :: "'x"   and k :: "'a" shows "lookup (except_min ko xs) k = (if \<forall>k'\<in>fst ` set (fst (list_of_oalist xs)). le ko k k' then 0::'b else lookup xs k)"
d'_neg: shows "\<forall>y>c. y < d' \<longrightarrow> poly p y < 0"
vs: shows "lfinite us \<longrightarrow> vs = lzip (ldrop (llength us) xs) (ldrop (llength us) ys)"
d_IN_i: fixes i :: "nat"   and x :: "'v" shows "d_IN (f_i i) x \<le> value_flow \<Delta> (f_i i)"
finfun_All_All: fixes P :: "'a \<Rightarrow>f bool" shows "finfun_All P = All (($) P)"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "ide a"   and "seq f a" shows "local.dom f = a"
hby1: shows "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> y"
simps: fixes x :: "'b" shows "lsingleton\<cdot>x = x :@ lnil"
prems_2: fixes a :: "'c" assumes "a \<in> insert x F" shows "infsetsum'_converges f (B a)"
control_independence_def: fixes upd :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c control_state_scheme \<Rightarrow> 'c control_state_scheme" shows "\<sharp> upd \<equiv> \<forall>\<sigma> T b. break_status (upd T \<sigma>) = break_status \<sigma> \<and> return_status (upd T \<sigma>) = return_status \<sigma> \<and> upd T (\<sigma>\<lparr>return_status := b\<rparr>) = upd T \<sigma>\<lparr>return_status := b\<rparr> \<and> upd T (\<sigma>\<lparr>break_status := b\<rparr>) = upd T \<sigma>\<lparr>break_status := b\<rparr>"
eq_\<I>_gpv_conversep: fixes A :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and \<I> :: "('f, 'g) \<I>" shows "eq_\<I>_gpv A\<inverse>\<inverse> \<I> = (eq_\<I>_gpv A \<I>)\<inverse>\<inverse>"
var_asm_not_written_def: fixes mds :: "Mode \<Rightarrow> 'a set"   and x :: "'a" shows "var_asm_not_written mds x \<equiv> x \<in> mds AsmNoWrite \<or> x \<in> mds AsmNoReadOrWrite"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "min b (min a c) = min a (min b c)"
fst_y: shows "fst y = monom (1::'a) n * poly_shift n (fst y)"
sumZeroRight: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<leadsto>[Rel] P \<oplus> \<zero>"
simps_1: fixes prism_match :: "'c \<Rightarrow> 'd option"   and prism_build :: "'d \<Rightarrow> 'c"   and more :: "'e" shows "match\<^bsub>\<lparr>prism_match = prism_match, prism_build = prism_build, \<dots> = more\<rparr>\<^esub> = prism_match"
nat_pow_0: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
expands_to_mult: fixes basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and F :: "'a"   and g :: "real \<Rightarrow> real"   and G :: "'a" assumes "basis_wf basis"   and "(f expands_to F) basis"   and "(g expands_to G) basis" shows "((\<lambda>x. f x * g x) expands_to F * G) basis"
absorb_iff1: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (inf a b = a)"
finfun_left_compose: fixes y :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f" assumes "y \<in> finfun" shows "g \<circ> y \<in> finfun"
alt_matrix_A_def: fixes signs :: "rat list list"   and subsets :: "nat list list" shows "alt_matrix_A signs subsets = mat (length subsets) (length signs) (\<lambda>(i, j). z (subsets ! i) (signs ! j))"
sel_1: fixes x1 :: "(char list, char list, 'b) namespace"   and x2 :: "(char list, char list, nat \<times> tid_or_exn) namespace" shows "sem_env.v (make_sem_env x1 x2) = x1"
divideC_field_simps_2: fixes c :: "complex"   and b :: "'b"   and a :: "'b" assumes "c \<noteq> 0" shows "(b /\<^sub>C c = a) = (b = c *\<^sub>C a)"
IH: assumes "insert_ops xs"   and "head \<notin> fst ` set xs"   and "\<And>r. Some r \<in> snd ` set xs \<Longrightarrow> r \<noteq> head" shows "succ_rel (head # interp_ins xs) = interp_alt head xs"
implied_frontier_alt_least: fixes b :: "'t"   and c :: "('loc, 't, 'a) configuration_scheme"   and loc2 :: "'loc" assumes "b \<in>\<^sub>A implied_frontier_alt c loc2" shows "\<forall>loc a' s'. a' \<in>\<^sub>A frontier (c_pts c loc) \<longrightarrow> s' \<in>\<^sub>A path_summary loc loc2 \<longrightarrow> \<not> results_in a' s' < b"
simps_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # ys) = (Ball (ran\<^sub>l ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
neutral_map: fixes A :: "'c set" shows "(\<lambda>y. \<bottom>\<^sub>o\<^sub>w) ` A \<subseteq> U"
eq: fixes s :: "'a"   and c :: "'a \<Rightarrow>\<^sub>0 'b" shows "flatten (monomial c s * q) = flatten (monomial c s) * flatten q"
rtranclp_induct: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P a"   and "\<And>y z. \<lbrakk>r\<^sup>*\<^sup>* a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
insertion_zero: fixes f :: "nat \<Rightarrow> 'a" shows "insertion f 0 = (0::'a)"
hom_1_iff: fixes x :: "int poly" shows "(of_int_poly x = 1) = (x = 1)"
c: shows "\<not> (AE x in M. g x \<noteq> \<infinity>)"
below_ID: fixes d :: "'a \<rightarrow> 'a" assumes "deflation d" shows "d \<sqsubseteq> ID"
generate_subgroup_incl: fixes G :: "('f, 'g) monoid_scheme"   and K :: "'f set"   and H :: "'f set" assumes "Group.group G"   and "K \<subseteq> H"   and "subgroup H G" shows "generate G K \<subseteq> H"
disj2: shows "carrier (Fstack as S) \<inter> domA \<Delta> = {}"
fun_add_eval_assoc: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a"   and z :: "'c \<Rightarrow> 'a"   and a :: "'c" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "z \<in> carrier F"   and "a \<in> S" shows "(x \<oplus>\<^bsub>F\<^esub> y \<oplus>\<^bsub>F\<^esub> z) a = (x \<oplus>\<^bsub>F\<^esub> (y \<oplus>\<^bsub>F\<^esub> z)) a"
mem_ball_leI: fixes x :: "'c"   and y :: "'c"   and e :: "real"   and f :: "real" assumes "x \<in> ball y e"   and "e \<le> f" shows "x \<in> ball y f"
elims: fixes x :: "color"   and xa :: "('a, 'b) rbt"   and y :: "('a, 'b) rbt" assumes "paint x xa = y" obtains c :: "color" where "x = c__" and "xa = Leaf" and "y = Leaf"   | c :: "color" and l :: "('a, 'b) rbt" and c' :: "color" and xb :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = c" and "xa = Node l c' xb v r" and "y = Node l c xb v r"
rel_set_strong_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set" shows "rel_set_strong R A B = (\<forall>x y. R x y \<longrightarrow> (x \<in> A) = (y \<in> B))"
bot_edge_is_compos_of_scal_and_g2: shows "(\<lambda>x. twoC (0, x)) = (\<lambda>x. (g2 x, x)) \<circ> (\<lambda>x. a + (b - a) * x)"
axioms: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" assumes "eucl_ordinal_utility carrier relation u" shows "ordinal_utility carrier relation u"
r2f_comp_pres: fixes R :: "('a \<times> 'c) set"   and S :: "('c \<times> 'b) set" shows "\<F> (R ; S) = \<F> R \<circ>\<^sub>K \<F> S"
rel_mono: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R \<le> Ra" shows "list_all2 R \<le> list_all2 Ra"