hcju/isabelleps huggingface dataset:
fv\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('b, 'a) stateful_strand_step"   and A :: "('b, 'a) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t (a # A) = fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> fv\<^sub>s\<^sub>s\<^sub>t A"
imp_map_contains_key_axioms: shows "imp_map_contains_key hassoc_map_assn hassoc.contains_key"
Float_num_7: shows "real_of_float (Float (- 1) 0) = - 1"
fmrel_code: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'f \<rightharpoonup>\<^sub>f 'd"   and n :: "'f \<rightharpoonup>\<^sub>f 'e" shows "fmrel R m n = (fBall (fmdom m) (\<lambda>x. rel_option R (fmlookup m x) (fmlookup n x)) \<and> subtuple_fun R m n)"
assms: shows "\<forall>t\<in>M. M' \<turnstile> t"
int_simps_7: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "\<turnstile> P \<noteq> Q = (P = \<not> Q)"
terminal_char2: fixes t :: "'s" shows "S.terminal t = (S.ide t \<and> S.set t = {t})"
cont: fixes f :: "nat \<Rightarrow> ordinal" shows "F (oLimit f) = oLimit (\<lambda>n. F (f n))"
permute_minus_cancel_1: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "perm"   and x :: "'a" assumes "class.pt permute" shows "permute p (permute (- p) x) = x"
setdist_eq_infdist: fixes A :: "'b set"   and B :: "'b set" shows "setdist A B = (if A = {} then 0 else INF a\<in>A. infdist a B)"
hom_mult: fixes x :: "'c"   and y :: "'c" shows "[:x * y:] = [:x:] * [:y:]"
llast_sswSts: shows "llast sswSts = llast wSts"
fst_of_sp_is_vert: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "SubRel.subpath g v1 es v2 subs" shows "v1 \<in> Graph.vertices g"
restrict_complement_singleton_eq: fixes f :: "'g \<Rightarrow> 'h option"   and x :: "'g" shows "f |` (- {x}) = f(x := None)"
simple_cipher: fixes \<eta> :: "nat" shows "simple_cipher (prf_key_gen \<eta>) (prf_fun \<eta>) (prf_domain \<eta>) (prf_dlen \<eta>) (prf_clen \<eta>) (upf_key_gen \<eta>)"
jn: shows "(\<pi> @\<^bsup>l\<^esup> \<pi>') j = \<pi> n"
flow_eq_rev0: fixes y :: "'a"   and t :: "real" shows "flow0 y t = rev.flow0 y (- t)"
card_injective_functions_domain_and_range_permutation: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) = iverson (card A \<le> card B)"
test_leq_mult_def_var: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "(p \<le> q) = (p \<cdot> q = p)"
closed_subterm_split: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e :: "exp"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile> e : C"   and "\<Gamma> = Map.empty" shows "(\<exists>C0 es fi. FieldProj (New C0 es) fi \<in> subexprs(e)) \<or> (\<exists>C0 es m ds. MethodInvk (New C0 es) m ds \<in> subexprs(e)) \<or> (\<exists>C0 D es. Cast D (New C0 es) \<in> subexprs(e)) \<or> val(e)"
assms_2: shows "D r \<tau> \<sigma> \<sigma>' \<tau>'"
B: shows "\<forall>\<^sub>F n in sequentially. Sigma_Algebra.measure (distr P borel (\<lambda>x. (g x + birkhoff_sum f n x) / B n)) {x. C0 \<le> \<bar>x\<bar>} < epsilon"
hyps: shows "x\<^bsup>n\<^esup> + x\<^bsub>0\<^esub>\<^bsup>n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>n\<^esup>"
strict_trans2: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "b \<sqsubset> a"   and "c \<sqsubseteq> b" shows "c \<sqsubset> a"
of_to_name: fixes of_name :: "name \<Rightarrow> 'c"   and a :: "'c" assumes "class.is_name of_name" shows "of_name (is_name.to_name of_name a) = a"
simps_2: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) list"   and vs :: "'d set" shows "as_proj (a # as) vs = (if fmdom' (fmrestrict_set vs (snd a)) \<noteq> {} then action_proj a vs # as_proj as vs else as_proj as vs)"
vp_mem_val_poss: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(x \<in> vp K v) = (0 < v x)"
monotone_dual: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b" assumes "PO cl"   and "Tarski.monotone f (pset cl) (order cl)" shows "Tarski.monotone f (pset (dual cl)) (order (dual cl))"
in_face_cycle_sets: fixes a :: "'b" assumes "a \<in> arcs G" shows "G.face_cycle_set a \<in> G.face_cycle_sets"
compute_nr_of_jordan_blocks: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and k :: "nat"   and ev :: "'a" assumes "jordan_nf A n_as"   and "k \<noteq> 0" shows "local.compute_nr_of_jordan_blocks A ev k = length (filter ((=) (k, ev)) n_as)"
simps_94: fixes x51 :: "'b formula"   and x52 :: "'b formula" shows "atoms (x51 \<^bold>\<or> x52) = atoms x51 \<union> atoms x52"
pred_mapping_empty: fixes Q :: "'a \<Rightarrow> bool" shows "pred_mapping Q Map.empty"
RedT_def: shows "h.mthr.if.RedT \<equiv> h.mthr.if.mthr.Trsys"
mset_filter: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "mset (filter P xs) = filter_mset P (mset xs)"
modified_variables_are_not_assumed_not_written: fixes cms\<^sub>A :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list"   and mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and \<R>n :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and n :: "nat"   and cms\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and \<R>i :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and i :: "nat"   and cm\<^sub>Cn' :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)"   and mem\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val"   and m\<^sub>A :: "nat"   and cm\<^sub>An' :: "'Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)"   and mem\<^sub>A' :: "'Var\<^sub>A \<Rightarrow> 'Val"   and x\<^sub>C :: "'Var\<^sub>C" assumes "abs.sound_mode_use (cms\<^sub>A, mem\<^sub>A)"   and "preserves_modes_mem \<R>n"   and "((cms\<^sub>A ! n, mem\<^sub>A), cms\<^sub>C ! n, mem\<^sub>C) \<in> \<R>n"   and "preserves_modes_mem \<R>i"   and "((cms\<^sub>A ! i, mem\<^sub>A), cms\<^sub>C ! i, mem\<^sub>C) \<in> \<R>i"   and "n < length cms"   and "length cms\<^sub>A = length cms"   and "length cms\<^sub>C = length cms"   and "\<And>i. i < length cms \<Longrightarrow> priv_mem\<^sub>C ! i \<subseteq> snd (cms\<^sub>C ! i) AsmNoReadOrWrite"   and "\<And>i j. \<lbrakk>i < length cms; j < length cms; i \<noteq> j\<rbrakk> \<Longrightarrow> priv_mem\<^sub>C ! i \<subseteq> snd (cms\<^sub>C ! j) GuarNoReadOrWrite"   and "\<And>i. i < length cms \<Longrightarrow> (snd (cms\<^sub>C ! i) AsmNoReadOrWrite \<union> snd (cms\<^sub>C ! i) AsmNoWrite) \<inter> - range var\<^sub>C_of \<subseteq> priv_mem\<^sub>C ! i"   and "(cms\<^sub>C ! n, mem\<^sub>C) \<leadsto>\<^sub>C (cm\<^sub>Cn', mem\<^sub>C')"   and "abs.neval (cms\<^sub>A ! n, mem\<^sub>A) m\<^sub>A (cm\<^sub>An', mem\<^sub>A')"   and "((cm\<^sub>An', mem\<^sub>A'), cm\<^sub>Cn', mem\<^sub>C') \<in> \<R>n"   and "mem\<^sub>C x\<^sub>C \<noteq> mem\<^sub>C' x\<^sub>C \<or> dma\<^sub>C mem\<^sub>C x\<^sub>C \<noteq> dma\<^sub>C mem\<^sub>C' x\<^sub>C"   and "i \<noteq> n"   and "i < length cms"   and "preserves_local_guarantee_compliance \<R>n" shows "\<not> var_asm_not_written (snd (cms\<^sub>C ! i)) x\<^sub>C"
assms_2: shows "(\<lambda>x. t x (g x)) \<in> L F h"
transforms_to_unique_keys_def: fixes m :: "'a \<Rightarrow> 'b option"   and f :: "'a \<times> 'b \<Rightarrow> ('c \<times> 'd) option" shows "transforms_to_unique_keys m f \<equiv> \<forall>k1 k2 v1 v2 k' v1' v2'. m k1 = Some v1 \<and> m k2 = Some v2 \<and> f (k1, v1) = Some (k', v1') \<and> f (k2, v2) = Some (k', v2') \<longrightarrow> k1 = k2"
proj_instance_ex: fixes B :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and P :: "('c \<Rightarrow> ('b, 'c) Term.term) \<Rightarrow> bool"   and b :: "'a strand_label \<times> ('b, 'c) stateful_strand_step"   and l :: "'a" assumes "\<forall>b\<in>set B. \<exists>a\<in>set A. \<exists>\<delta>. b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<and> P \<delta>"   and "b \<in> set (proj l B)" shows "\<exists>a\<in>set (proj l A). \<exists>\<delta>. b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<and> P \<delta>"
Cd_domain: fixes d :: "'d"   and X :: "'x set" shows "Cd d (X \<inter> Field (Pd d)) = Cd d X"
eq\<^sub>Q: shows "equal_class.equal \<equiv> local.ltln\<^sub>Q_eq_test"
suffix_plus: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" shows "s |\<^sub>s n |\<^sub>s m = s |\<^sub>s m + n"
finite_conj: fixes E :: "(('d \<Rightarrow> 'e) \<Rightarrow> bool) set" assumes "finite E"   and "\<forall>e\<in>E. finite (Bexp.vars e)" shows "finite (Bexp.vars (conjunct E))"
refusals_def: fixes P :: "'c process"   and xs :: "'c list" shows "refusals P xs \<equiv> failures P `` {xs}"
path_empty_conv: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and v :: "'b" shows "path E u [] v = (u = v)"
sort_key_arg_min_on: fixes zs :: "'c list"   and p :: "'c \<Rightarrow> 'd" assumes "zs \<noteq> []"   and "inj_on p (set zs)" shows "sort_key p zs = (let z = arg_min_on p (set zs) in z # sort_key p (remove1 z zs))"
unsigned_less: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and w :: "'d word" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less (semiring_1.unsigned one plus zero w) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) LENGTH('d))"
simps: fixes H :: "('a, 'b, 'c) hashedTa_scheme"   and f :: "'b"   and ts :: "'b tree list" shows "faccs' H (NODE f ts) = (let Qs = map (faccs' H) ts in ll_set_xy.g_image_filter (\<lambda>r. case r of q \<rightarrow> f' qs \<Rightarrow> if list_all_zip (\<lambda>Q q. ls.memb q Q) Qs qs then Some (lhs r) else None) (hta_lookup_f f H))"
transpos_def: fixes i :: "nat"   and j :: "nat" shows "\<tau>\<^bsub>i j\<^esub> = (\<lambda>k. if k = i then j else if k = j then i else k)"
zer_left_mult: fixes x :: "'a" shows "zer * x = zer"
linorder_neq_iff: fixes x :: "'b"   and y :: "'b" shows "(x \<noteq> y) = (x < y \<or> y < x)"
generate_valid_stateful_policy_IFSACS_2_all_security_requirements_fulfilled_IFS: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and edgesList :: "('b \<times> 'b) list" assumes "valid_reqs M"   and "wf_graph G"   and "all_security_requirements_fulfilled M G"   and "set edgesList \<subseteq> edges G" shows "all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph (generate_valid_stateful_policy_IFSACS_2 G M edgesList))"
eval_monom: fixes r :: "'a"   and s :: "'a"   and n :: "nat" assumes "r \<in> carrier R"   and "s \<in> carrier R" shows "eval R R id s (monom P r n) = id r \<otimes> s [^] n"
minimal_def: fixes A :: "'b set"   and P :: "'b set \<Rightarrow> bool" shows "minimal A P = (P A \<and> (\<forall>B. B \<subseteq> A \<and> P B \<longrightarrow> B = A))"
sum_upto_dirichlet_prod: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and x :: "real" shows "sum_upto (dirichlet_prod f g) x = sum_upto (\<lambda>d. f d * sum_upto g (x / real d)) x"
annotate_rlen_len: fixes r :: "'b list" shows "fst (annotate_rlen_code r) = length r"
power_one: fixes n :: "nat" shows "(0::'a) +^ n = (0::'a)"
sign_group_hom: fixes n :: "nat" shows "group_hom (sym_group n) sign_img sign"
dg_Par_is_arrD_1: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B" shows "arr_Par \<alpha> S"
assms: shows "integrable (lebesgue_on {a..b}) f"
eq_nextl_def: shows "eq_nextl \<equiv> {(u, v). nextl (dfa.init M) u = nextl (dfa.init M) v}"
sets_completionI_sets: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "A \<in> sets (completion M)"
CUD: shows "C \<union> D \<subseteq> {x + y |x y. x \<in> S \<and> y \<in> T}"
removed_cond: shows "all_acquired (x # sb) \<inter> dom \<S> - dom \<S>' = {}"
elims: fixes x :: "'v \<times> 'v \<Rightarrow> ennreal"   and xa :: "'v vertex \<times> 'v vertex"   and y :: "ennreal" assumes "split x xa = y" obtains f :: "'v \<times> 'v \<Rightarrow> ennreal" and xb :: "'v" and x' :: "'v" and ya :: "'v" where "x = f__" and "xa = (Vertex xb__, Edge x' ya__)" and "y = (if x' = xb__ then f__ (xb__, ya__) else 0)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and xb :: "'v" and y' :: "'v" and ya :: "'v" where "x = fa__" and "xa = (Edge xb y', Vertex ya)" and "y = (if y' = ya then fa__ (xb, ya) else 0)"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and vb :: "'v" and vc :: "'v" and v :: "'v" and vd :: "'v" where "x = fb__" and "xa = (Edge vb__ vc, Edge v vd)" and "y = 0"   | f :: "'v \<times> 'v \<Rightarrow> ennreal" and va :: "'v" and vb :: "'v" where "x = f" and "xa = (Vertex va, Vertex vb)" and "y = 0"
word_test_bit_def: fixes a :: "'b word" shows "(!!) a = bit (uint a)"
xfer_reachable_aux: fixes as\<^sub>0 :: "'b"   and alstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and cs\<^sub>0 :: "'e"   and clstep :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e" assumes "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R"   and "system.reachable cs\<^sub>0 clstep c" shows "\<exists>a. R a c \<and> system.reachable as\<^sub>0 alstep a"
Rep1: fixes x :: "'b" shows "T1 (Rep1 x) x"
simps_2: fixes s :: "nat \<Rightarrow> 'd list"   and i :: "nat" shows "inf_last_ti s (Suc i) = (if s (Suc i) \<noteq> [] then s (Suc i) else inf_last_ti s i)"
nth_Cons': fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" shows "(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))"
root: shows "h \<turnstile> get_root_node (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr) \<rightarrow>\<^sub>r cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr"
preserves_iso: fixes f :: "'b \<times> 'b \<times> 'b" assumes "VVV.iso f" shows "local.iso \<a>[fst f, fst (snd f), snd (snd f)]"
that: fixes g :: "real \<Rightarrow> real" assumes "continuous_on UNIV g"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> g x = f x"   and "\<And>x. g (x + (b - a)) = g x" shows "thesis"
rawpsubst_compose_freshVar2: fixes \<phi> :: "'fmla"   and ts :: "'trm list"   and xs :: "'var list"   and us :: "'var list"   and vs :: "'var list" assumes "\<phi> \<in> fmla"   and "set ts \<subseteq> trm"   and "set xs \<subseteq> var"   and "distinct xs"   and "set us \<subseteq> var"   and "distinct us"   and "set us \<inter> Fvars \<phi> = {}"   and "set us \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set us \<inter> set xs = {}"   and "set vs \<subseteq> var"   and "distinct vs"   and "set vs \<inter> Fvars \<phi> = {}"   and "set vs \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set vs \<inter> set xs = {}"   and "length us = length xs"   and "length vs = length xs"   and "length ts = length xs" shows "rawpsubst (rawpsubst \<phi> (zip (map Var us) xs)) (zip ts us) = rawpsubst (rawpsubst \<phi> (zip (map Var vs) xs)) (zip ts vs)"
between_eq2: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2"   and "before vs ram2 ram1" shows "\<exists>as bs cs. between vs ram1 ram2 = cs @ as \<and> vs = as @ [ram2] @ bs @ [ram1] @ cs"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F \<tau>"
NextRecvUpd'_preserves_prop_invs: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('d, 'c, 'a) Combined.configuration"   and c1 :: "('d, 'c, 'a) Combined.configuration"   and p :: "'d"   and q :: "'d" assumes "dataflow_topology summary results_in"   and "order.InvMsgInGlob (dataflow_topology.cri_less_eq summary results_in) (exchange_config c0)"   and "dataflow_topology.inv_init_imp_prop_safe summary results_in c0"   and "\<forall>p. inv_implications_nonneg (prop_config c0 p) \<and> dataflow_topology.inv_imps_work_sum summary results_in (prop_config c0 p)"   and "dataflow_topology.InvGlobPointstampsEq c0"   and "dataflow_topology.NextRecvUpd' c0 c1 p q" shows "\<forall>p. inv_implications_nonneg (prop_config c1 p) \<and> dataflow_topology.inv_imps_work_sum summary results_in (prop_config c1 p)"
takefill_Suc_Cons: fixes fill :: "'b"   and n :: "nat"   and x21 :: "'b"   and x22 :: "'b list" shows "takefill fill (Suc n) (x21 # x22) = x21 # takefill fill n x22"
quot_cases: fixes A :: "'b quot" obtains a :: "'b" where "A = \<lfloor>a\<rfloor>"
right_neutral: fixes a :: "'q" shows "a * (1::'q) = a"
edge_or_swap: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> set p" shows "(u, v) \<in> E \<or> (v, u) \<in> E"
take_length_ib: fixes j :: "nat"   and s :: "'a list" assumes "0 < j"   and "j \<le> length s" shows "take (length (intrinsic_border (take j s))) s = intrinsic_border (take j s)"
axioms_1: fixes G :: "('a, 'b) graph" assumes "forest G" shows "valid_graph G"
mult_left_le_imp_le: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c * a \<le> c * b"   and "(0::'b) < c" shows "a \<le> b"
\<psi>: shows "natural_isomorphism A B G H \<psi>"
l_zero: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<zero>\<^bsub>K [X]\<^esub> \<oplus>\<^bsub>K [X]\<^esub> x = x"
Y1_carrier: shows "Y1 \<subseteq> carrier_vec (n + 1)"
fully_unsorted_poly_rel_extend_vars: fixes A :: "(char list list \<times> int) list"   and B :: "int mpoly"   and x1c :: "char list set"   and x1a :: "nat set" assumes "(A, B) \<in> fully_unsorted_poly_rel O mset_poly_rel"   and "(x1c, x1a) \<in> \<langle>var_rel\<rangle>set_rel" shows "RETURN (x1c \<union> vars_llist A) \<le> \<Down> (\<langle>var_rel\<rangle>set_rel) (SPEC ((\<subseteq>) (x1a \<union> vars B)))"
that: fixes ns' :: "'node SDG_node list"   and ns'' :: "'node SDG_node list" assumes "ns = ns' @ ns''"   and "n ics-ns'\<rightarrow>\<^sub>d* n''"   and "n'' ics-ns''\<rightarrow>\<^sub>d* n'" shows "thesis"
induct: fixes x1 :: "val list"   and x2 :: "val set list"   and x3 :: "(ty \<times> nat) list \<times> nat"   and P :: "val list \<Rightarrow> val set list \<Rightarrow> (ty \<times> nat) list \<times> nat \<Rightarrow> bool" assumes "\<turnstile> x1,x2 : x3"   and "P [] [] ([], 0)"   and "\<And>\<rho> \<eta> \<Gamma> v \<tau>. \<lbrakk>\<turnstile> \<rho>,\<eta> : \<Gamma>; P \<rho> \<eta> \<Gamma>; v \<in> T \<tau> \<eta>\<rbrakk> \<Longrightarrow> P (v # \<rho>) \<eta> (push_ty \<tau> \<Gamma>)"   and "\<And>\<rho> \<eta> \<Gamma> V. \<lbrakk>\<turnstile> \<rho>,\<eta> : \<Gamma>; P \<rho> \<eta> \<Gamma>\<rbrakk> \<Longrightarrow> P \<rho> (V # \<eta>) (push_tyvar \<Gamma>)" shows "P x1 x2 x3"
less_Suc_eq_le: fixes m :: "nat"   and n :: "nat" shows "order.greater (Suc n) m = order.greater_eq n m"
lcos_m_assoc: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and g :: "'c"   and h :: "'c" assumes "Group.group G"   and "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "g <#\<^bsub>G\<^esub> (h <#\<^bsub>G\<^esub> M) = g \<otimes>\<^bsub>G\<^esub> h <#\<^bsub>G\<^esub> M"
lm094: fixes pair :: "'c \<times> 'd set"   and N :: "'c set"   and G :: "'d set"   and bids :: "'c \<times> 'd set \<Rightarrow> 'e" assumes "pair \<in> N \<times> (Pow G - {{}})" shows "summedBidVector bids N G pair = summedBidSecond bids pair"
prj_tuple_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.commutative_square C f g h k" shows "C (prj1 f g) (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k) = h"
rel_inject_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" shows "rel_option R (pure x2) (pure y2) = R x2 y2"
complete_indiff_not: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "complete A r"   and "hasw [x, y] A"   and "\<not> x \<^bsub>r\<^esub>\<approx> y" shows "x \<^bsub>r\<^esub>\<prec> y \<or> y \<^bsub>r\<^esub>\<prec> x"
list_eq_iff_zip_eq: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). x = y))"
is_nth_power_conv_multiplicity: fixes n :: "nat"   and x :: "'a" assumes "0 < n" shows "is_nth_power n (normalize x) = (\<forall>p. prime p \<longrightarrow> n dvd multiplicity p x)"
indicator_eq_1_iff: fixes A :: "'f set"   and x :: "'f" shows "(indicator A x = (1::'e)) = (x \<in> A)"
balla: shows "\<forall>n\<ge>noa. \<bar>a n - x\<bar> < e"
fbox_one: fixes x :: "'a" shows "antidomain_semiringl.fbox a (\<cdot>) (1::'a) x = d x"
Pasch_fun_mono: fixes r1 :: "real"   and r2 :: "real" assumes "0 < r1"   and "r1 \<le> r2"   and "r2 < 1" shows "r2 + 1 / r2 \<le> r1 + 1 / r1"
divmod_numeral_simps_19: fixes m :: "num"   and n :: "num" shows "(num.Bit1 m \<le> num.Bit1 n) = (m \<le> n)"
is_regular_spair_nonzero: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "spair p q \<noteq> 0"
diff_right_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a c) b = minus (minus a b) c"
assms_2: fixes x1 :: "'a"   and x2 :: "'a" assumes "P x1"   and "P x2" shows "P (x1 \<nabla> x2)"
anno_annotate: fixes p :: "nat"   and c :: "com"   and f :: "nat \<Rightarrow> 'a" assumes "p < asize c" shows "anno (annotate f c) p = f p"
x: shows "x \<in> \<E> (TER f \<union> TER g)"
identity\<^sub>\<o>_def: shows "(\<^bold>=) \<equiv> \<lambda>F G. (\<^bold>\<lambda>y. F) \<^bold>= (\<^bold>\<lambda>y. G)"
dim_span_eq_card_independent: fixes B :: "'b set" assumes "local.independent B" shows "local.dim (local.span B) = card B"
subst_fmla_commute: fixes j :: "name"   and A :: "fmla"   and i :: "name"   and t :: "trm"   and u :: "trm" assumes "atom j \<sharp> A" shows "(A(i::=t))(j::=u) = A(i::=subst j u t)"
simps_3: fixes v :: "real" shows "preal_min preal.PInfty (preal v) = preal v"
ide_dom: fixes f :: "'b" assumes "C'.arr f" shows "C'.ide (C'.dom f)"
functor_axioms: shows "functor J (\<cdot>\<^sub>B) FoD.map"
extremum_uniqueI: fixes a :: "'f multiset" assumes "a \<subseteq># {#}" shows "a = {#}"
preim_subset_imp_subset: fixes I :: "'b set"   and J :: "'b set" assumes "f \<^sup>\<inverse> A I \<subseteq> f \<^sup>\<inverse> A J"   and "I \<subseteq> B" shows "I \<subseteq> J"