hcju/isabelleps huggingface dataset:
par_comp_prot_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and \<P> :: "('d strand_label \<times> ('a, 'b) strand_step) list set"   and Sec :: "('a, 'b) Term.term set" assumes "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2" shows "labeled_typed_model.par_comp_prot arity public Ana \<Gamma> \<P> Sec \<equiv> (\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> labeled_typed_model.GSMP_disjoint arity public Ana \<Gamma> (\<Union>\<A>\<in>\<P>. trms\<^sub>l\<^sub>s\<^sub>t (proj l1 \<A>)) (\<Union>\<A>\<in>\<P>. trms\<^sub>l\<^sub>s\<^sub>t (proj l2 \<A>)) Sec) \<and> ground Sec \<and> (\<forall>s\<in>Sec. \<forall>s'\<in>subterms s. intruder_model.intruder_synth arity public {} s' \<or> s' \<in> Sec) \<and> labeled_typed_model.typing_cond_prot arity Ana \<Gamma> \<P>"
param_concat: fixes R :: "('a \<times> 'b) set" shows "(concat, concat) \<in> \<langle>\<langle>R\<rangle>list_rel\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
fractrel_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.idom times one plus zero minus uminus" shows "idom.fractrel times zero = (\<lambda>x y. snd x \<noteq> zero \<and> snd y \<noteq> zero \<and> times (fst x) (snd y) = times (fst y) (snd x))"
untransfer_start: fixes Q :: "bool"   and P :: "bool" assumes "Q"   and "Transfer.Rel (=) P Q" shows "P"
set: fixes x :: "'a" shows "set_discr (Discr x) = {x}"
igSubstBinp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'b, 'j) Iteration.model_scheme"   and ys :: "'e"   and Y :: "'i"   and y :: "'h"   and binp :: "'a \<Rightarrow> 'b option" shows "igSubstBinp MOD ys Y y binp \<equiv> lift (igSubstAbs MOD ys Y y) binp"
intros_2: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x"   and ln :: "'a \<Rightarrow>f nat" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "ln \<noteq> no_wait_locks" shows "not_final_thread s t"
abs_content_int: fixes p :: "int poly" shows "\<bar>content p\<bar> = content p"
split_max_inorder: fixes t :: "'a btree"   and k :: "nat" assumes "nonempty_lasttreebal t"   and "t \<noteq> Leaf" shows "inorder_pair (split_max k t) = inorder t"
set_rewrite1: fixes m :: "nat"   and k :: "nat" shows "{p. p partitions Suc m \<and> sum p {..Suc m} = Suc k \<and> p 1 \<noteq> 0} = (\<lambda>p. p(1 := p 1 + 1)) ` {p. p partitions m \<and> sum p {..m} = k}"
residuated_conjugate_ineq: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "(conjugate f x \<le> y) = (x \<le> - f (- y))"
IH: fixes e3 :: "exp" assumes "e3a__ \<longrightarrow>* e3" shows "e2 \<longrightarrow>* e3"
R'_intros_3: fixes x :: "nat"   and y' :: "nat"   and n :: "nat" assumes "(x, y') \<in> R"   and "x \<le> n"   and "n < y'" shows "(x, n + 1) \<in> R' n"
allk: shows "\<forall>i\<in>K. \<forall>j\<in>K. i \<noteq> j \<longrightarrow> transition_idx s T {i, j} = k"
simps_2: fixes m :: "nat" shows "acklist [m] = m"
inst_ok_code2: fixes \<Theta> :: "exetheory"   and insts :: "(('b \<times> String.literal set) \<times> typ) list" assumes "exe_wf_theory' \<Theta>" shows "list_all (\<lambda>((idn, S), T). has_sort (osig (sig (translate_theory \<Theta>))) T S) insts = list_all (\<lambda>((idn, S), T). exehas_sort (exesorts (exesig \<Theta>)) T S) insts"
L_def: fixes n :: "nat" shows "L n = (LEAST i. i \<in> I \<and> n \<le> i)"
pdom_pran_res_finite: fixes f :: "('a, 'b) pfun"   and A :: "'b set" assumes "finite (pdom f)" shows "finite (pdom (f \<rhd>\<^sub>p A))"
insert_Collect: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
set_iterator_rev_linord_I: fixes l0 :: "'a list"   and S :: "'a set"   and iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" assumes "distinct l0"   and "S = set l0"   and "sorted (rev l0)"   and "iti = foldli l0" shows "set_iterator_rev_linord iti S"
pathstart_def: fixes g :: "real \<Rightarrow> 'a" shows "pathstart g = g 0"
a: shows "Cut <a>.M x.N \<longrightarrow>\<^sub>a R"
prems: shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> A"
FGModIso_composite_left: fixes smult'' :: "('f, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'w \<Rightarrow> 'b"   and X :: "'b set" assumes "FGModuleIso G (\<star>) W smult'' S X" shows "FGModuleIso G (\<cdot>) V smult'' (S \<circ> T) X"
cycle_root_end_empty_var: fixes r :: "'a"   and x :: "'a"   and e :: "'a" assumes "terminating_path_root_end r x e"   and "x \<noteq> 0" shows "\<not> many_strongly_connected x"
prefs_undefined': fixes i :: "'agent" assumes "i \<notin> agents" shows "R i = (\<lambda>_ _. False)"
embedding_lemma: fixes \<Pi> :: "'c strips_problem"   and \<pi> :: "'c strips_operator list" assumes "is_valid_problem_strips \<Pi>"   and "STRIPS_Semantics.is_serial_solution_for_problem \<Pi> \<pi>" shows "STRIPS_Semantics.is_parallel_solution_for_problem \<Pi> (List_Supplement.embed \<pi>)"
intro: fixes P :: "('a \<times> bool) list set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>ps. ps \<in> P \<Longrightarrow> freeword_funlift f (Abs_freeword ps) = (0::'b)" shows "GroupByPresentationInducedFun_axioms P f"
TermP_Var: fixes x :: "name" shows "{} \<turnstile> TermP \<guillemotleft>Var x\<guillemotright>"
map_pair_cong: fixes xs :: "('a \<times> 'b) list"   and f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c"   and g :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c" assumes "\<And>kv. kv \<in> set xs \<Longrightarrow> f kv = g kv" shows "map_pair f xs = map_pair g xs"
pnet_never_newpkt': fixes s :: "'c net_state"   and a :: "'d node_action"   and s' :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and n :: "net_tree" assumes "(s, a, s') \<in> automaton.trans (pnet np n)" shows "\<forall>i d di. a \<noteq> i:newpkt(d, di)"
irrefl: fixes a :: "'b" shows "\<not> codomain.greater a a"
pmf_of_mapping: fixes m :: "('b, real) mapping"   and x :: "'b" assumes "finite (Mapping.keys m)"   and "All_mapping m (\<lambda>_. (\<le>) 0)"   and "sum (Mapping.lookup_default 0 m) (Mapping.keys m) = 1" shows "pmf (pmf_of_mapping m) x = Mapping.lookup_default 0 m x"
lookup_reorder: fixes ko :: "'c nat_term_order"   and xs :: "('c, 'b) oalist_ntm"   and k :: "'c" shows "OAlist_lookup_ntm (OAlist_reorder_ntm ko xs) k = OAlist_lookup_ntm xs k"
WHILE_def: shows "WHILE \<equiv> iWHILE (\<bind>) RETURN"
induct: fixes g :: "('b, 'a) generator"   and P :: "'a \<Rightarrow> bool"   and a0 :: "'a" assumes "\<And>s. (\<And>x xa y. \<lbrakk>list.has_next g s; x = list.next g s; (xa, y) = x\<rbrakk> \<Longrightarrow> P y) \<Longrightarrow> P s" shows "P a0"
countable_insert_eq: fixes x :: "'f"   and A :: "'f set" shows "countable (insert x A) = countable A"
auto_def: fixes G :: "('c, 'd) monoid_scheme" shows "auto G = hom G G \<inter> Bij (carrier G)"
tmf_c_correct: fixes a :: "real list"   and A :: "float interval list"   and prec :: "nat"   and f :: "floatarith"   and i :: "nat"   and I :: "float interval" assumes "a all_in A"   and "tmf_c prec A f i = Some I" shows "interpret_floatarith (deriv_rec f i) a / fact i \<in>\<^sub>r I"
lex_comp_prod_def: shows "lex_comp = (\<lambda>u v. case lex_comp' (fst u) (fst v) of Eq \<Rightarrow> comparator_of (snd u) (snd v) | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
closure_open_segment: fixes a :: "'a"   and b :: "'a" shows "closure (open_segment a b) = (if a = b then {} else closed_segment a b)"
sum_length_partition_by_median: fixes l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a"   and ps :: "(real, 'a) vec list" assumes "(l, m, r) = partition_by_median k ps" shows "length ps = length l + length r"
card_bit1: shows "CARD('b bit1) = Suc (2 * CARD('b))"
mk_disjoint_insert: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "\<exists>B. A = a \<triangleright> B \<and> a \<notin> B"
fconst_on_vsingleton: fixes a :: "V"   and c :: "V" shows "fconst_on (set {a}) c = set {[a, c]\<^sub>\<circ>}"
image_std_dual: assumes "subspace S" shows "local.std_dual ` S \<subseteq> dual_space S"
HPhase1or2ReadThen_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4d s q" shows "HInv4d s' q"
bdd_UN: shows "small (\<Union>x\<in>A. f ` B x)"
2: shows "orientation (x \<sqinter> - (1::'a)) y"
ereal_less_eq_5: fixes r :: "real" shows "(0 \<le> ereal r) = (0 \<le> r)"
id_hom: fixes A :: "'o" assumes "A \<in> Ob" shows "Id A \<in> Hom A A"
wellFoundedSatFlagLessState: shows "wf satFlagLessState"
l10_6_uniqueness: fixes P :: "'p"   and P1 :: "'p"   and A :: "'p"   and B :: "'p"   and P2 :: "'p" assumes "P P1 Reflect A B"   and "P P2 Reflect A B" shows "P1 = P2"
llength_LCons: fixes x :: "'b"   and xs :: "'b llist" shows "llength (x % xs) = eSuc (llength xs)"
hyp: fixes A :: "(('a, 'n) vec, 'm) vec" assumes "n < nrows A"   and "n < mod_type_class.to_nat a" shows "row_add_iterate A n i j $ a $ b = A $ a $ b"
induct: fixes P :: "('a, 'b) Combinators list \<Rightarrow> bool"   and a0 :: "('a, 'b) Combinators list" assumes "\<And>x. P [x]"   and "\<And>x v va. P (v # va) \<Longrightarrow> P (x # v # va)"   and "P []" shows "P a0"
bex_simps_7: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>x\<in>f ` A. P x) = (\<exists>x\<in>A. P (f x))"
hyps_free_Cons: fixes e :: "('d \<times> 'c) \<times> 'e \<times> 'f"   and pth :: "(('d \<times> 'c) \<times> 'e \<times> 'f) list" shows "hyps_free (e # pth) = (hyps_free pth \<and> hyps (nodeOf (fst (fst e))) (snd (fst e)) = None)"
oalist_ntm_eqI: fixes xs :: "('b, 'c) oalist_ntm"   and ys :: "('b, 'c) oalist_ntm" assumes "list_of_oalist_ntm xs = list_of_oalist_ntm ys" shows "xs = ys"
IH_2: fixes t' :: "'a" assumes "no_abs t'"   and "convert_term t\<^sub>2 = convert_term t'" shows "t\<^sub>2 = t'"
exists_base: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "ring R"   and "subfield K R"   and "ring.dimension R n K E" shows "\<exists>Vs. set Vs \<subseteq> carrier R \<and> ring.independent R K Vs \<and> length Vs = n \<and> ring.Span R K Vs = E"
nn_intergal_power_times_exp_Icc: fixes a :: "real"   and k :: "nat" assumes "0 \<le> a" shows "\<integral>\<^sup>+x\<in>{0..a}. ennreal (x ^ k * exp (- x))\<partial>lborel = ennreal ((1 - (\<Sum>n\<le>k. a ^ n * exp (- a) / fact n)) * fact k)"
lfilter_LCons: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a llist" shows "lfilter P (x #\<^sub>l xs) = (if P x then x #\<^sub>l lfilter P xs else lfilter P xs)"
normal_subgroup_quotientfun_classrep_equality: fixes H :: "'g set"   and a :: "'g" assumes "Group H \<and> order.greater_eq G H"   and "order.greater_eq Ker H"   and "a \<in> G" shows "quotientfun T (LCoset_rel H `` {a}) = T a"
fact: shows "\<b> * \<k> = \<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1))"
d_def: fixes t :: "real" shows "delayed_safe_distance.d t = (if t \<le> 0 then other.s \<delta> - ego.q \<delta> else if t \<le> ego2.t_stop \<and> t \<le> delayed_safe_distance.other.t_stop then delayed_safe_distance.ego_other.p t else if ego2.t_stop \<le> t \<and> t \<le> delayed_safe_distance.other.t_stop then delayed_safe_distance.other.p t - ego2.s_stop else if delayed_safe_distance.other.t_stop \<le> t \<and> t \<le> ego2.t_stop then delayed_safe_distance.other.s_stop - ego2.p t else delayed_safe_distance.other.s_stop - ego2.s_stop)"
sel_1: fixes x1 :: "'b"   and x2 :: "'d" shows "fst (x1, x2) = x1"
rep_eq: fixes x :: "complex"   and xa :: "complex"   and xb :: "complex"   and xc :: "complex" shows "Rep_moebius_mat (mk_moebius_mat x xa xb xc) = mk_moebius_cmat x xa xb xc"
assoc\<^sub>F\<^sub>M\<^sub>C_def: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" shows "\<a>[a, b, c] = mkarr \<^bold>\<a>\<^bold>[rep a, rep b, rep c\<^bold>]"
induct: fixes P :: "'a \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'b list"   and a2 :: "'a" assumes "\<And>u v. P u [] v"   and "\<And>u e es v. P (head G e) es v \<Longrightarrow> P u (e # es) v" shows "P a0 a1 a2"
blocksRead_allBlocksRead: fixes rb :: "BlockProc"   and s :: "state"   and p :: "Proc"   and d :: "Disk" assumes "rb \<in> blocksRead s p d" shows "block rb \<in> allBlocksRead s p"
suminf_ereal_eq_SUP: fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" shows "(\<Sum>x. f x) = (\<Squnion>n. sum f {..<n})"
sup_bot_left: fixes x :: "'a \<times> 'b" shows "sup_prod bot_prod x = x"
fold_set_fold: fixes y :: "'a"   and xs :: "'a list" shows "Finite_Set.fold (\<squnion>) y (set xs) = fold (\<squnion>) xs y"
mkarr_Assoc': fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Arr v" shows "\<F>C.mkarr \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = \<F>C.\<alpha>' (\<F>C.mkarr t, \<F>C.mkarr u, \<F>C.mkarr v)"
coefficients_Nil: fixes p :: "'b poly" shows "coefficients p \<noteq> []"
objI_src: fixes a :: "'a" assumes "arr a"   and "src a = a" shows "obj a"
conv: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "SN_on r {a} = (\<forall>b. (a, b) \<in> r \<longrightarrow> SN_on r {b})"
ineq_dfa_accepts: fixes n :: "nat"   and bss :: "bool list list"   and ks :: "int list"   and l :: "int" assumes "list_all (is_alph n) bss" shows "dfa_accepts (ineq_dfa n ks l) bss = (eval_dioph ks (nats_of_boolss n bss) \<le> l)"
is_ground_atm_def: fixes A :: "('a, 'b) Term.term" shows "is_ground_atm A = (\<forall>\<sigma>. A = subst_atm_abbrev A \<sigma>)"
lookup_unit_vec: fixes j :: "nat"   and n :: "nat"   and i :: "nat" assumes "j < n" shows "lookup (unit_vec n i) [j] = (if i = j then 1::'a else (0::'a))"
cindex_const: fixes a :: "real"   and b :: "real"   and c :: "real" shows "cindex a b (\<lambda>_. c) = 0"
intro: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice" assumes "seq_distrib (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"
in_ap_setI: fixes f' :: "'c \<Rightarrow> 'd"   and f :: "('c \<Rightarrow> 'd) set"   and x' :: "'c"   and x :: "'c set" assumes "f' \<in> f"   and "x' \<in> x" shows "f' x' \<in> f \<diamondop> x"
add_minus: fixes x :: "'a" assumes "x \<in> E" shows "x + - x = (0::'a)"
diff_delta_ln_upper_11_def: shows "diff_delta_ln_upper_11 \<equiv> \<lambda>x. (x - 1) ^ 11 / ((6 * x ^ 5 + 75 * x ^ 4 + 200 * x ^ 3 + 150 * x\<^sup>2 + 30 * x + 1)\<^sup>2 * x)"
assms_5: shows "h \<turnstile> to_tree_order ptr' \<rightarrow>\<^sub>r to"
rep_eq: fixes x :: "integer" shows "nat_of_integer x = nat (int_of_integer x)"
le_imp_head_\<omega>_le: fixes m :: "hmultiset"   and n :: "hmultiset" assumes "m \<le> n" shows "head_\<omega> m \<le> head_\<omega> n"
p: shows "p = (\<Sum>f\<in>F''. q' f * f)"
elims_1: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and x :: "('b \<times> nat) tree"   and y :: "bool" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "WBT.wbt \<Delta>1 \<Delta>2 x = y" obtains "x = \<langle>\<rangle>" and "y"   | l :: "('b \<times> nat) tree" and uu_ :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (uu_, n), r\<rangle>" and "y = (n = size l + size r + 1 \<and> WBT.balanced1 \<Delta>1 \<Delta>2 l r \<and> WBT.balanced1 \<Delta>1 \<Delta>2 r l \<and> WBT.wbt \<Delta>1 \<Delta>2 l \<and> WBT.wbt \<Delta>1 \<Delta>2 r)"
assms_2: shows "B\<^sub>1 \<noteq> {}"
simps_10: fixes b :: "bool"   and v :: "Term.var"   and e :: "expr" shows "assigns_if b (v:=e) = assignsE (v:=e)"
negligible_imp_measure0: fixes S :: "'a set" assumes "negligible S" shows "Sigma_Algebra.measure lebesgue S = 0"
and_and_not: fixes a :: "'b word"   and b :: "'b word" shows "(a AND b) AND NOT b = 0"
top_filter_def: shows "top \<equiv> Filters.filter.Abs_filter UNIV"
subgraph_forest: fixes H :: "('v, 'w) graph" assumes "subgraph H G" shows "forest H"
simps_29: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x4 :: "'b" shows "(case Snapshot x4 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f4 x4"
Le_LCons: fixes xs :: "'c llist"   and ys :: "'c llist"   and x :: "'c" assumes "lprefix xs ys" shows "lprefix (LCons x xs) (LCons x ys)"
I_subst\<^sub>0: fixes a :: "atom"   and c :: "real"   and r :: "real"   and cs :: "real list"   and xs :: "real list" assumes "depends\<^sub>R a"   and "c \<noteq> 0" shows "I\<^sub>R (subst\<^sub>0 (Eq r (c # cs)) a) xs = I\<^sub>R a ((r - \<langle>cs,xs\<rangle>) / c # xs)"
wfi: fixes relr :: "((('c \<times> ('d \<times> 'c) list) \<times> 'c \<times> ('d \<times> 'c) list) \<times> ('c \<times> ('d \<times> 'c) list) \<times> 'c \<times> ('d \<times> 'c) list) set"   and r :: "('d \<times> 'd) set" assumes "relr = pex r"   and "\<not> wf relr" shows "\<not> wf (mul r)"
arr_inv: fixes f :: "'a" assumes "Right.iso f" shows "Right.arr (Right.inv f)"
hom_one: fixes p :: "'a poly" shows "1 \<circ>\<^sub>p p = 1"
delete_index_def: fixes i :: "nat" shows "delete_index i \<equiv> \<lambda>i'. if i' < i then i' else i' - Suc 0"
Sup_fold_sup: fixes A :: "'a set" assumes "finite A" shows "Sup A = Finite_Set.fold sup bot A"
lgc_complete_Liminf: fixes Ns :: "('f inference set \<times> ('f \<times> 'l) set) llist"   and B :: "'f" assumes "chain (\<leadsto>LGC) Ns"   and "active_subset (snd (lhd Ns)) = {}"   and "passive_subset (Liminf_llist (lmap snd Ns)) = {}"   and "\<forall>\<iota>\<in>Inf_F. prems_of \<iota> = [] \<longrightarrow> \<iota> \<in> fst (lhd Ns)"   and "Liminf_llist (lmap fst Ns) = {}"   and "B \<in> Bot_F"   and "fst ` snd (lhd Ns) \<Turnstile>\<inter>\<G> {B}" shows "\<exists>BL\<in>Bot_FL. BL \<in> Liminf_llist (lmap snd Ns)"
exhaust: fixes y :: "['b]" obtains "y = \<bottom>"   | "y = []"   | a :: "'b" and list :: "['b]" where "y = a : list"
assms_3: shows "poly (sturm p ! (i + 1)) x = 0"
fmdom_notI: fixes m :: "('d, 'c) fmap"   and x :: "'d" assumes "fmlookup m x = None" shows "x |\<notin>| fmdom m"
call_FFI_def: fixes st :: "'b ffi_state"   and s :: "char list"   and conf :: "8 word list"   and bytes :: "8 word list" shows "call_FFI st s conf bytes = (if final_event st = None \<and> s \<noteq> [] then case oracle0 st s (ffi_state st) conf bytes of Oracle_return ffi' bytes' \<Rightarrow> if length bytes' = length bytes then (update_io_events (\<lambda>_. io_events st @ [IO_event s conf (zipSameLength bytes bytes')]) (update_ffi_state (\<lambda>_. ffi') st), bytes') else (update_final_event (\<lambda>_. Some (Final_event s conf bytes FFI_failed)) st, bytes) | Oracle_diverge \<Rightarrow> (update_final_event (\<lambda>_. Some (Final_event s conf bytes FFI_diverged)) st, bytes) | Oracle_fail \<Rightarrow> (update_final_event (\<lambda>_. Some (Final_event s conf bytes FFI_failed)) st, bytes) else (st, bytes))"
arrI: fixes f :: "'a" assumes "right a f" shows "Right_a.arr f"
Seq_insf: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "Seq s k" shows "Seq (insf s k y) (succ k)"
assms_3: shows "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"
finite_measure_mono: fixes A :: "real set"   and B :: "real set" assumes "A \<subseteq> B"   and "B \<in> events" shows "prob A \<le> prob B"
fST: fixes x :: "'a" assumes "x \<in> S" shows "f x \<subseteq> T"
prod_list_ge: fixes xs :: "nat list" assumes "\<forall>x\<in>set xs. 1 \<le> x" shows "1 \<le> prod_list xs"
assms_7: shows "quicksort_post R h (Suc p) hi xs' xs''"
Fvars_empty_ground: fixes t :: "tm" assumes "Fvars t = {}" shows "ground t"
rc_def: shows "rename_clause Sel C = rename_clause Sel P1 - {rename_literal Sel (Pos A)} \<union> (rename_clause Sel P2 - {rename_literal Sel (Neg A)})"
assms_5: shows "max_Var_floatariths fas \<le> DIM('a)"
R_lin_independent_imp_hd_n0: fixes m :: "'m"   and ms :: "'m list" assumes "R_lin_independent (m # ms)" shows "m \<noteq> (0::'m)"
lr: shows "(c::'a::refinement_lattice) ; ((d::'a::refinement_lattice) ; c)\<^sup>\<star> \<sqsubseteq> (c ; d)\<^sup>\<star> ; c"
distinct_5: fixes x2 :: "'b"   and x3 :: "'b" shows "Tag x2 \<noteq> Wit x3"
nat_cut_greater_ge_conv: fixes t :: "nat"   and I :: "nat set" assumes "0 < t" shows "I \<down>> (t - Suc 0) = I \<down>\<ge> t"
simps: fixes b :: "rat vec"   and A :: "rat mat" shows "x_mat_eq_vec b A = map (\<lambda>i. EQ (matrix_to_lpolies A ! i) (b $ i)) [0..<dim_vec b]"
mod_trap_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and new_traps :: "Trap set"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = s1\<lparr>traps := new_traps\<rparr> \<and> t2 = s2\<lparr>traps := new_traps\<rparr>" shows "low_equal t1 t2"
vars_def: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "Bexp.vars e = {v. \<exists>\<sigma> val. e (\<sigma>(v := val)) \<noteq> e \<sigma>}"
bit_or_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (or a b) n = (bit a n \<or> bit b n)"
linear_bounded: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
map_of_concat': fixes i :: "nat" assumes "k \<in> dom (map_of (l ! i)) \<and> i < length l" shows "k \<in> dom (map_of (concat l))"
instInp_def: fixes \<phi> :: "'fmla"   and t :: "'trm" shows "instInp \<phi> t \<equiv> subst \<phi> t inp"
ereal_of_real_of_ereal_iff_2: fixes x :: "ereal" shows "(x = ereal (real_of_ereal x)) = (x \<noteq> \<infinity> \<and> x \<noteq> - \<infinity>)"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data" shows "arr (\<rho>.map f) = arr f"
SI_insert_mod2: fixes inv0 :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and s :: "'s"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x. inv0 x \<Longrightarrow> I (f x)"   and "inv0 x"   and "SI s" shows "SI (insert_mod2 P f x s)"
alls_ne: fixes B :: "nat"   and as :: "nat list" shows "alls B as \<noteq> []"
cofactor_def: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "Determinant.cofactor A i j = (- (1::'a)) ^ (i + j) * Determinant.det (mat_delete A i j)"