hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "com"   and x2 :: "nat"   and P :: "com \<Rightarrow> nat \<Rightarrow> bool" assumes "\<turnstile> x1 : x2"   and "\<And>l. P SKIP l"   and "\<And>x a. sec a \<le> sec x \<Longrightarrow> P (x ::= a) (sec x)"   and "\<And>c\<^sub>1 l\<^sub>1 c\<^sub>2 l\<^sub>2. \<lbrakk>\<turnstile> c\<^sub>1 : l\<^sub>1; P c\<^sub>1 l\<^sub>1; \<turnstile> c\<^sub>2 : l\<^sub>2; P c\<^sub>2 l\<^sub>2\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) (min l\<^sub>1 l\<^sub>2)"   and "\<And>b l\<^sub>1 l\<^sub>2 c\<^sub>1 c\<^sub>2. \<lbrakk>sec b \<le> min l\<^sub>1 l\<^sub>2; \<turnstile> c\<^sub>1 : l\<^sub>1; P c\<^sub>1 l\<^sub>1; \<turnstile> c\<^sub>2 : l\<^sub>2; P c\<^sub>2 l\<^sub>2\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (min l\<^sub>1 l\<^sub>2)"   and "\<And>b l c. \<lbrakk>sec b \<le> l; \<turnstile> c : l; P c l\<rbrakk> \<Longrightarrow> P (WHILE b DO c) l" shows "P x1 x2"
pa_minus_def: fixes x :: "'a"   and y :: "'a" shows "- x - - y = - (- - x \<squnion> - y)"
t_1: shows "t \<in> existence_ivl t0 x0"
axioms_2: fixes \<Delta> :: "('a, 'b) network_scheme" assumes "flow_attainability \<Delta>" shows "flow_attainability_axioms \<Delta>"
elims_2: fixes P :: "ast_problem"   and x :: "plan_action" assumes "ast_problem.wf_plan_action P x" obtains n :: "char list" and args :: "object list" where "x = PAction n args" and "case ast_domain.resolve_action_schema (domain P) n of None \<Rightarrow> False | Some a \<Rightarrow> ast_problem.action_params_match P a args \<and> ast_problem.wf_effect_inst P (ground_action.effect (instantiate_action_schema a args))"
rep_eq: fixes x :: "\<kappa>" shows "d\<^sub>\<kappa> x = eval\<kappa> x"
Labels_WhileBody: fixes c' :: "cmd"   and l :: "nat"   and c :: "cmd"   and b :: "expr" assumes "labels c' l c" shows "labels (while (b) c') (l + 2) (c;; while (b) c')"
match_pres_correct: fixes pres :: "(nat \<times> nat) list"   and s :: "nat \<Rightarrow> nat option" assumes "distinct (map fst pres)"   and "s \<in> valid_states" shows "match_pres pres s = (s \<in> subsuming_states (map_of pres))"
divisors_def: fixes m :: "nat" shows "divisors m \<equiv> {n. n dvd m}"
wf_Restrict: fixes k :: "'k"   and i :: "nat"   and idx :: "'n" assumes "LESS k i idx" shows "local.wf idx (Restrict k i)"
interior_unique: fixes T :: "'b set"   and S :: "'b set" assumes "T \<subseteq> S"   and "open T"   and "\<And>T'. \<lbrakk>T' \<subseteq> S; open T'\<rbrakk> \<Longrightarrow> T' \<subseteq> T" shows "interior S = T"
exhaust: fixes y :: "'a el_step" obtains x1 :: "'a" where "y = LOC x1"   | x2 :: "'a" where "y = ENV x2"
3: fixes k :: "nat"   and n :: "nat" assumes "f k < n"   and "n < f (Suc k)" shows "w n = w (f k)"
prop_equiv_card: fixes P :: "'a ltln set" assumes "finite P" shows "card {abs_ltln\<^sub>P \<psi> |\<psi>. prop_atoms \<psi> \<subseteq> P} \<le> 2 ^ 2 ^ card P"
ineq_on_Og: fixes l :: "nat"   and n :: "nat"   and x :: "'a" assumes "l \<in> {1..n}"   and "n \<notin> bad_times x"   and "x \<in> Og" shows "- delta l * real l < u n x - u (n - l) x"
h11: shows "(\<exists>(a', b', c')\<in>set d. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
rel_simps_47: shows "(1::'b) \<le> (1::'b)"
A: shows "\<forall>x y. ((x, y) \<in> r) = (\<exists>u. p x y u = 0)"
lubH_is_fixp: fixes H :: "'a set" assumes "H = {x \<in> A. (x, f x) \<in> r}" shows "lub H cl \<in> fix f A"
mult_powr_eq_mult_powr_iff_asym: fixes m1 :: "int"   and e1 :: "int"   and e2 :: "int"   and m2 :: "int" assumes "odd m1"   and "e1 \<le> e2" shows "(real_of_int m1 * 2 powr real_of_int e1 = real_of_int m2 * 2 powr real_of_int e2) = (m1 = m2 \<and> e1 = e2)"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'a" assumes "A.ide x" shows "FG.\<epsilon> x = \<psi> x (G x)"
interval_cbox: fixes a :: "'c"   and b :: "'c" shows "{a..b} = cbox a b"
optimum_def: fixes t :: "'a tree" shows "optimum t = (\<forall>u. consistent u \<longrightarrow> alphabet t = alphabet u \<longrightarrow> freq t = freq u \<longrightarrow> cost t \<le> cost u)"
ground_instance_exists: fixes C :: "'b literal set" assumes "finite C" shows "\<exists>\<sigma>. ground_clause (subst_cl C \<sigma>)"
some_trg_comp: fixes \<tau> :: "'a"   and \<sigma> :: "'a" assumes "\<tau> \<star> \<sigma> \<noteq> null" shows "some_trg (\<tau> \<star> \<sigma>) = some_trg \<tau>"
pcr_hmultiset_def: shows "pcr_hmultiset \<equiv> rel_nmultiset (=) OO cr_hmultiset"
assms_2: shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l A))"
excised_relcycle_exists: fixes X :: "'b topology"   and U :: "'b set"   and T :: "'b set"   and S :: "'b set"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "X closure_of U \<subseteq> X interior_of T"   and "T \<subseteq> S"   and "singular_relcycle p (subtopology X S) T c" obtains c' :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" where "singular_relcycle p (subtopology X (S - U)) (T - U) c'" and "homologous_rel p (subtopology X S) T c c'"
lex_sym_eqI: fixes a :: "real \<times> real"   and b :: "real \<times> real" assumes "Counterclockwise_2D_Arbitrary.lex a b"   and "Counterclockwise_2D_Arbitrary.lex b a" shows "a = b"
equivalence_in_bicategory_axioms: shows "internal_equivalence (src \<mu>) (src \<mu>) (local.inv \<i>[src \<mu>]) \<i>[src \<mu>]"
FC: fixes \<sigma> :: "'b" assumes "I l0 [] \<sigma>"   and "c \<sigma>" shows "P \<sigma>"
vars_of_subterm: fixes x :: "'b"   and s :: "'b trm"   and t :: "'b trm"   and p :: "indices list" assumes "x \<in> vars_of s"   and "subterm t p s" shows "x \<in> vars_of t"
atLeast_Suc_lessThan: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m < n" shows "sum g {m..<n} = g m + sum g {Suc m..<n}"
minimal: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "minimal_element P A"   and "x \<in> A"   and "Q x" shows "\<exists>y\<in>A. P\<^sup>=\<^sup>= y x \<and> Q y \<and> (\<forall>z\<in>A. P z y \<longrightarrow> \<not> Q z)"
inf_bot_left: fixes x :: "'e multiset" shows "{#} \<inter># x = {#}"
order_iff: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a = min a b)"
mode_update_add_anno: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and upd :: "'Var ModeUpd" assumes "mds_consistent mds \<Gamma> \<S> P" shows "mds_consistent (update_modes upd mds) \<Gamma> \<oplus>\<^sub>\<S> upd (add_anno_stable \<S> upd) P |` {v. stable (add_anno_stable \<S> upd) v}"
unitary11_gen_mat_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary11_gen M"   and "mat_det M \<noteq> 0" shows "unitary11_gen (mat_inv M)"
rbt_greater_simps_2: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and c :: "color"   and lt :: "('b, 'c) RBT_Impl.rbt"   and kt :: "'b"   and v :: "'c"   and rt :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_greater less k (Branch c lt kt v rt) = (less k kt \<and> ord.rbt_greater less k lt \<and> ord.rbt_greater less k rt)"
pderivs_lang_def: fixes A :: "'a list set"   and r :: "'a rexp" shows "pderivs_lang A r \<equiv> \<Union>x\<in>A. pderivs x r"
simps: fixes x :: "'b extT"   and y :: "'b extT" shows "x * y = x \<sqinter> y"
A'0: fixes i :: "nat" assumes "i < n - 1" shows "A' $$ (Suc i, 0) = 0"
"4_1": assumes "length ys \<le> length xs"   and "\<forall>i<length ys. xs ! i = ys ! i" shows "take (length ys) xs = ys"
acceptI: fixes G :: "('h, 'i, 'j) sa_rec_scheme"   and r :: "nat \<Rightarrow> 'h"   and w :: "nat \<Rightarrow> 'i" assumes "sa G"   and "graph_defs.is_run G r"   and "w = sa_L G \<circ> r" shows "sa.accept G w"
elims: fixes x :: "('a \<times> 'a) list"   and y :: "('a \<times> 'a) list" assumes "rembiflowdups x = y" obtains "x = []" and "y = []"   | s :: "'a" and r :: "'a" and as :: "('a \<times> 'a) list" where "x = (s, r) # as" and "y = (if (s, r) \<in> set as \<or> (r, s) \<in> set as then rembiflowdups as else (s, r) # rembiflowdups as)"
terminal_def: fixes tllist :: "('b, 'a) tllist" shows "terminal tllist = (case tllist of TNil x1 \<Rightarrow> x1 | TCons _dummy_ xs \<Rightarrow> terminal0 xs)"
b_e_type_tee_local_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Tee_local i" shows "\<exists>ts'' t. ts = ts'' @ [t] \<and> ts' = ts'' @ [t] \<and> local \<C> ! i = t"
span_add: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "x \<in> pmdl S"   and "y \<in> pmdl S" shows "x + y \<in> pmdl S"
finite_SMP_representationD_2: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "finite_SMP_representation arity Ana \<Gamma> M" shows "has_all_wt_instances_of \<Gamma> (subterms\<^sub>s\<^sub>e\<^sub>t (set M)) (set M)"
set_binop_elemD1: fixes a :: "'d"   and b :: "'e"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and z1 :: "'f"   and z2 :: "'g"   and xs :: "('d \<times> 'f) list"   and ys :: "('d \<times> 'g) list" assumes "(a, b) \<in> set (binop f z1 z2 xs ys)" shows "a \<in> set (map fst xs) \<or> a \<in> set (map fst ys)"
prems_2: shows "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (r # sb) \<subseteq> A"
assms_2: fixes x :: "'a" assumes "x \<in> B" shows "\<exists>y\<in>A. dist x y \<le> D"
IH: assumes "var \<notin> vars p" shows "var \<notin> vars (p ^ i)"
no_unacquired_write_to_read_only'': fixes \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and \<O> :: "nat set" assumes "no_write_to_read_only_memory \<S> sb"   and "sharing_consistent \<S> \<O> sb" shows "read_only \<S> \<inter> outstanding_refs is_Write\<^sub>s\<^sub>b sb \<subseteq> \<O> \<union> all_acquired sb"
sum_distrib_right: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and r :: "'a" shows "sum f A * r = (\<Sum>n\<in>A. f n * r)"
lossless1: fixes s1 :: "'s1"   and x :: "'a" assumes "bad1 s1"   and "x \<in> outs_\<I> \<I>" shows "lossless_spmf (oracle1 s1 x)"
finite_cspan_complete: fixes B :: "'d set" assumes "finite B" shows "complete (cspan B)"
order_max: fixes a :: "'a"   and k :: "nat"   and p :: "'a poly" assumes "[:- a, 1::'a:] ^ k dvd p"   and "p \<noteq> 0" shows "k \<le> order a p"
transp_on_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "transp_on P A = (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. P x y \<and> P y z \<longrightarrow> P x z)"
basis_reduction_swap_1: fixes i :: "nat"   and fs :: "int vec list"   and upw' :: "bool"   and i' :: "nat"   and fs' :: "int vec list" assumes "LLL_invariant False i fs"   and "basis_reduction_swap i fs = (upw', i', fs')"   and "\<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 < \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2"   and "i < m"   and "i \<noteq> 0" shows "LLL_invariant upw' i' fs'"
rep_eq: fixes x :: "uint8" shows "lsb x = lsb (Rep_uint8 x)"
list_all2_shortcircuit_mono: fixes R :: "'a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool" assumes "R \<le> Q" shows "list_all2_shortcircuit R \<le> list_all2_shortcircuit Q"
induced_inj_on: shows "inj_on additive.induced kernel.additive.Partition = (additive.Ker = additive.Ker)"
enumerator_completeness_help: fixes ls :: "nat list"   and ks :: "nat"   and nmax :: "nat" assumes "increasing ls"   and "ls \<noteq> []"   and "|ls| = Suc ks"   and "list_all (\<lambda>x. x < Suc nmax) ls" shows "ls \<in> set ((enumAppend nmax ^^ ks) (enumBase nmax))"
C_to_CValue_reach: fixes x :: "C \<rightarrow> CValue" shows "(\<Squnion>n. \<psi>\<^sup>A\<^bsub>n\<^esub>\<cdot>x) = x"
stable_on_Xd: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set"   and x :: "'d" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_on Pd Ch ds X"   and "x \<in> X" shows "Xd x \<in> ds"
eq1: shows "\<Union> \<F> = \<Union> \<D>"
double_compl: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
qml_1: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>)]]"
qf3_oddprimedivisor: fixes P :: "int"   and a :: "int"   and b :: "int" assumes "prime P"   and "odd P"   and "coprime a b"   and "P dvd a\<^sup>2 + 3 * b\<^sup>2" shows "is_qfN P 3"
wf_matchF_invar_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and st :: "(nat \<times> event_data option list set list \<times> event_data option list set) list \<times> (mregex, event_data option list set) mapping"   and i :: "nat" shows "wf_matchF_invar \<sigma> V n R I r st i = (case st of (aux, Y) \<Rightarrow> aux \<noteq> [] \<and> wf_matchF_aux \<sigma> V n R I r aux i 0 \<and> (case to_mregex r of (mr, \<phi>s) \<Rightarrow> \<forall>ms\<in>LPDs mr. qtable n (fv_regex r) (mem_restr R) (\<lambda>v. Regex.match (Formula.sat \<sigma> V (map the v)) (from_mregex ms \<phi>s) i (i + length aux - 1)) (lookup Y ms)))"
class_linorder_compare: shows "class.linorder (le_of_comp compare) (lt_of_comp compare)"
reflection: shows "rel_weakly_reflects_barbs (indRelLTPO TRel) (STCalWB SWB TWB)"
diagonal_step_PQ_JNF_def: fixes A :: "'b mat"   and i :: "nat"   and k :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "diagonal_step_PQ_JNF A i k bezout = (let m = dim_row A; n = dim_col A; (p, q, u, v, d) = bezout (A $$ (i, i)) (A $$ (k, k)); P = addrow (- v) k i (swaprows i k (addrow p k i (1\<^sub>m m))) in Let (multcol k (- (1::'b)) (addcol u k i (addcol q i k (1\<^sub>m n)))) (Pair P))"
equivalence_map_extends_to_adjoint_equivalence: fixes f :: "'b" assumes "D.equivalence_map f" shows "\<exists>g \<eta> \<epsilon>. adjoint_equivalence_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D f g \<eta> \<epsilon>"
filter_eq_nths: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "filter P xs = nths xs {i. i < length xs \<and> P (xs ! i)}"
ta_prod_rta: assumes "ranked_tree_automaton TA1 A"   and "ranked_tree_automaton TA2 A" shows "ranked_tree_automaton (ta_prod TA1 TA2) A"
induct: fixes P :: "('c, 'd) strand_step list \<Rightarrow> bool"   and a0 :: "('c, 'd) strand_step list" assumes "P []"   and "\<And>t S. P S \<Longrightarrow> P (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v S. P S \<Longrightarrow> P (send\<langle>v\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va vb S. P S \<Longrightarrow> P (\<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va S. P S \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0"
foldl_map: fixes g :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'e"   and f :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs"
AA_def: shows "AA \<equiv> {X. X \<subseteq> A \<and> f ` X \<subseteq> X \<and> (\<forall>Y s. Y \<subseteq> X \<longrightarrow> extreme_bound A (\<sqsubseteq>) Y s \<longrightarrow> s \<in> X)}"
pointwise_implies_pointbased: fixes f :: "'a set \<Rightarrow> 'b set" assumes "pointwise f" shows "pointbased f"
simps_69: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "nat set \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> 'a"   and f5 :: "'b \<Rightarrow> 'a"   and f6 :: "nat \<Rightarrow> 'a"   and f7 :: "'b \<Rightarrow> 'a"   and f8 :: "'a"   and x6 :: "nat" shows "(case deliver x6 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f6 x6"
n_def3: shows "(\<Sum>j. real_of_int (b (n + 1 + j)) / real_of_int (a (n + 1 + j))) \<le> real_of_int (2 * b (n + 1)) / real_of_int (a (n + 1))"
zero_le_constraint_def: shows "0 = Le_Constraint Leq_Rel 0 (0::'b)"
has_derivative_def: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f has_derivative f') F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (Lim F (\<lambda>x. x)) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> (0::'b)) F)"
absorb1: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "inf a b = a"
p: fixes i :: "nat" shows "\<integral>\<^sup>+ x. ennreal (U i x) \<partial>lborel = ennreal (p i)"
Rep_inverse: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'a" assumes "type_definition Rep Abs A" shows "Abs (Rep x) = x"
shunting: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (x \<le> z \<squnion> - y)"
of_real_iff: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" shows "((\<lambda>x. of_real (f x)) \<in> O[F](\<lambda>x. of_real (g x))) = (f \<in> O[F](g))"
minimal_set: fixes S :: "'b set" shows "{} \<sqsubseteq> S"
rep_eq: fixes x :: "int poly" shows "map rep_real_alg_3 (roots_of_3 x) = roots_of_2 x"
connected_component_subset: fixes S :: "'a set"   and x :: "'a" shows "connected_component_set S x \<subseteq> S"
iptables_bigstep_deterministic: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t'" shows "t = t'"
modelAssigns_def: fixes M :: "model" shows "modelAssigns M = {phi. range phi \<subseteq> objects M}"
n_le_1: fixes s :: "'s" assumes "s \<in> S" shows "n s \<le> 1"
atU_union_cases: fixes U :: "'f set"   and c1 :: "'f list multiset"   and c2 :: "'f list multiset" assumes "atU U (c1 + c2)" obtains "atU U c1"   | "atU U c2"
quotient_composition_def: shows "([\<cdot>]) = (\<lambda>A\<in>Partition. \<lambda>B\<in>Partition. THE C. \<exists>a\<in>A. \<exists>b\<in>B. C = Class (a \<cdot> b))"
assms_2: shows "\<guillemotleft>\<phi> : g \<Rightarrow> g'\<guillemotright>"
class_rec: fixes C :: "cname"   and m :: "class"   and f :: "class \<Rightarrow> ('a \<times> 'b) list" assumes "class C = Some m"   and "ws_prog" shows "class_rec C f = (if C = Object then Map.empty else class_rec (super m) f) ++ map_of (f m)"
hom_mult: fixes x :: "'b"   and y :: "'b" shows "[:x * y:] = [:x:] * [:y:]"
subterms_dominated: fixes L :: "'a literal"   and C :: "'a literal set"   and t :: "'a trm"   and s :: "'a trm"   and p :: "sign"   and u :: "'a trm" assumes "maximal_literal L C"   and "orient_lit L t s p"   and "u \<in> subterms_of_cl C"   and "vars_of_lit L = {}"   and "vars_of_cl C = {}" shows "u = t \<or> (u, t) \<in> trm_ord"
insert_Diff_single: fixes a :: "'k"   and A :: "'k set" shows "insert a (A - {a}) = insert a A"
greaterThanLessThan_subseteq_greaterThanAtMost_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b \<subseteq> ord.greaterThanAtMost less_eq less c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
fun_union_def: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" shows "f + g \<equiv> \<lambda>p. f p \<union> g p"
indRelRTPO_to_TRel_3: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q" shows "\<forall>TP SQ. TP \<in>T P \<and> SQ \<in>S Q \<longrightarrow> False"
convex_bounds_lemma: fixes x :: "'b"   and e :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and y :: "'b" assumes "convex_on (cball x e) f"   and "\<forall>y\<in>cball x e. f y \<le> b"   and "y \<in> cball x e" shows "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
simps: fixes m :: "nat"   and q :: "nat"   and r :: "nat" shows "actn m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * scan r) else 4)"
subcringE_1: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subcring H R" shows "H \<subseteq> carrier R"
assms_3: shows "trg f \<star> \<mu> = (\<i>[trg f] \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[trg f, trg f, f]"
that: fixes k :: "nat" assumes "w = [a] \<^sup>@ k" shows "thesis"
DJ: shows "nodes g\<^sub>1 \<inter> nodes g\<^sub>2 = {}"
finite_fold_fold_keys: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c"   and t :: "('b, unit) RBT.rbt" assumes "comp_fun_commute f" shows "Finite_Set.fold f A (Set t) = fold_keys f t A"
verts_in_translation_finite_4: fixes X :: "'a allegorical_term" shows "Suc 0 \<in> vertices (translation X)"
pow_simps_2: fixes n :: "nat"   and P :: "pol" shows "Commutative_Ring.pow (2 * n) P = Commutative_Ring.pow n (sqr P)"
lift_CFG_wf: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and H :: "'i set"   and L :: "'i set" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "CFG_wf src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) state_val"
simps_2: fixes h :: "nat" shows "B (Suc h) = {N2 t1 a t2 |t1 a t2. t1 \<in> T h \<and> t2 \<in> B h \<or> t1 \<in> B h \<and> t2 \<in> T h}"
set_map: fixes f :: "'c \<Rightarrow> 'a"   and v :: "'b \<Rightarrow> 'c" shows "range (f \<circ> v) = f ` range v"