hcju/isabelleps huggingface dataset:
source_determination: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and \<rho> :: "'e"   and v :: "'d" assumes "vwb_lens x"   and "get\<^bsub>x\<^esub> \<sigma> = get\<^bsub>x\<^esub> \<rho>"   and "put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v" shows "\<sigma> = \<rho>"
assms_4: shows "io \<in> L A \<inter> L B"
sig_red_top_tail_cases: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_red sing_reg (\<preceq>) F p q = (sig_red sing_reg (=) F p q \<or> sig_red sing_reg (\<prec>) F p q)"
simps_2: fixes n :: "String.literal"   and uu :: "int"   and S :: "variable set" shows "variant_variable (Var (n, uu)) S = (let s' = fresh_name' (name ` S) in (Var (s', 0), insert (Var (s', 0)) S))"
list_of_set_def: fixes s :: "'b set" shows "list_of_set s = set_choose (list_of_set_set s)"
eta_norm_collapse: fixes t :: "Core.term" shows "eta_norm (eta_norm t) = eta_norm t"
assms: shows "\<Gamma> (Fun f X) = \<Gamma> (Fun g Y)"
similar_mat_def: fixes A :: "'a mat"   and B :: "'a mat" shows "similar_mat A B = (\<exists>P Q. similar_mat_wit A B P Q)"
ia_less_length: shows "ia < length [mod_type_class.to_nat i..<IArray.length (vec_to_iarray A)]"
infsetsum'_add: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "infsetsum'_converges f A"   and "infsetsum'_converges g A" shows "infsetsum' (\<lambda>x. f x + g x) A = infsetsum' f A + infsetsum' g A"
rational_root_test_main_def: fixes df :: "int \<Rightarrow> int list"   and dp :: "int \<Rightarrow> int list"   and p :: "rat poly" shows "rational_root_test_main df dp p \<equiv> let ip = snd (rat_to_normalized_int_poly p); a0 = coeff ip 0; an = lead_coeff ip in if a0 = 0 then Some 0 else let d0 = df a0; dn = dp an in map_option fst (find_map_filter (\<lambda>x. (x, poly p x)) (\<lambda>(uu_, res). res = 0) (concat (map (\<lambda>b0. concat (map (\<lambda>bn. if coprime b0 bn then [rat_of_int b0 / rat_of_int bn] else []) dn)) d0)))"
primrec_dummyfac: shows "\<forall>i<Suc (Suc 0). primerec ([recf.id (Suc 0) 0, Cn (Suc 0) s [Cn (Suc 0) rec_dummyfac [recf.id (Suc 0) 0, recf.id (Suc 0) 0]]] ! i) (Suc 0)"
fixp_above_Kleene_iter: fixes k :: "nat" assumes "\<forall>M\<in>Chains leq. finite M"   and "(f ^^ Suc k) a = (f ^^ k) a" shows "fixp_above a = (f ^^ k) a"
ANNOT_def: fixes x :: "'a"   and a :: "annot" shows "Autoref_Tagging.ANNOT x a \<equiv> x"
floor_add_int: fixes x :: "'b"   and z :: "int" shows "\<lfloor>x\<rfloor> + z = \<lfloor>x + of_int z\<rfloor>"
Bernstein_coeffs_eq_rescale: fixes c :: "real"   and d :: "real"   and p :: "nat"   and P :: "real poly" assumes "c \<noteq> d" shows "Bernstein_coeffs p c d P = Bernstein_coeffs_01 p (P \<circ>\<^sub>p [:c, 1:] \<circ>\<^sub>p [:0, d - c:])"
init_def: fixes h :: "'grp \<times> 'grp"   and w :: "nat" shows "init h w = (let (h, h') = h in sample_uniform (order \<G>) \<bind> (\<lambda>r. return_spmf (r, \<^bold>g [^] r, g' [^] r)))"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<tau>s :: "'b list" shows "merges f [] \<tau>s = \<tau>s"
fold_insert: fixes A :: "'a set"   and x :: "'a"   and z :: "'c" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) z (insert x A) = upd x (map_default d g x) (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) z A)"
rbt_union_with_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "ord.rbt_union_with less f = ord.rbt_union_with_key less (\<lambda>_. f)"
children_rank_less: fixes t :: "('c, 'd) SkewBinomialTree" assumes "tree_invar t" shows "\<forall>t'\<in>set (children t). rank t' < rank t"
eqOn_def: fixes P :: "'c set"   and env :: "'c \<Rightarrow> 'd"   and env1 :: "'c \<Rightarrow> 'd" shows "eqOn P env env1 \<equiv> \<forall>p. p \<in> P \<longrightarrow> env p = env1 p"
assigned_def: fixes V :: "'a"   and e :: "'a exp" shows "assigned V e \<equiv> \<exists>v e'. e = V:=Val v;; e'"
Grp_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "BNF_Def.Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
notIn_Un_bij_betw: fixes b :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and A' :: "'c set" assumes "b \<notin> A"   and "f b \<notin> A'"   and "bij_betw f A A'" shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
terminates_impl_no_infinite_computation: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<not> \<Gamma>\<turnstile> (c, s) \<rightarrow> \<dots>(\<infinity>)"
set_trieP_def: shows "set_trieP = set_trie \<circ> abs_trieP"
subadditive_def: fixes u :: "nat \<Rightarrow> real" shows "Fekete.subadditive u = (\<forall>m n. u (m + n) \<le> u m + u n)"
simps_2: shows "extended_to_set \<turnstile> = {}"
starlike_imp_simply_connected: fixes S :: "'a set" assumes "starlike S" shows "simply_connected S"
bit_double_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 a :: "'b"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (times (numeral.numeral one plus (num.Bit0 num.One)) a) n = (bit a (n - 1) \<and> n \<noteq> 0 \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
wtA_cOfFax: fixes f :: "'fsym" assumes "wtFsym f"   and "prot (resOf f) \<or> protFw (resOf f)" shows "GE.wtA (cOfFax f)"
inner_complex_def: fixes x :: "complex"   and y :: "complex" shows "inner x y = Re x * Re y + Im x * Im y"
sorted_list_of_set_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq = folding.F (linorder.insort_key less_eq (\<lambda>x. x)) []"
HEndPhase2_HInv4b_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase2 s s' p" shows "HInv4b s' p"
Int_left_absorb: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> (A \<inter> B) = A \<inter> B"
with_proj_simps_5: fixes G :: "'b pair_pre_digraph" shows "head (with_proj G) = snd"
llist_of_tllist_TNil: fixes x :: "'b" shows "llist_of_tllist (TNil x) = LNil"
representsD_1: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "p \<noteq> 0"
induct: fixes x1 :: "nat \<Rightarrow> classDef option"   and x2 :: "nat"   and x3 :: "nat"   and x4 :: "nat list"   and x5 :: "nat"   and P :: "(nat \<Rightarrow> classDef option) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> bool" assumes "mtype(x1,x2,x3) = x4 \<rightarrow> x5"   and "\<And>CT C CDef m mDef Bs B. \<lbrakk>CT C = Some CDef; lookup (cMethods CDef) (\<lambda>md. mName md = m) = Some mDef; varDefs_types (mParams mDef) = Bs; mReturn mDef = B\<rbrakk> \<Longrightarrow> P CT m C Bs B"   and "\<And>CT C CDef m D Bs B. \<lbrakk>CT C = Some CDef; lookup (cMethods CDef) (\<lambda>md. mName md = m) = None; cSuper CDef = D; mtype(CT,m,D) = Bs \<rightarrow> B; P CT m D Bs B\<rbrakk> \<Longrightarrow> P CT m C Bs B" shows "P x1 x2 x3 x4 x5"
min: shows "\<forall>g. (m, g) \<in> gseq \<longrightarrow> good (emb P) g"
intros_3: fixes c :: "nat" shows "Regions.valid_intv c (Regions.intv.Greater c)"
mult_eq_0_iff: fixes x :: "V"   and y :: "V" shows "(x * y = 0) = (x = 0 \<or> y = 0)"
nonsat_push_\<Delta>: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "push_effect_locale c s t f l u v"   and "Network.nonsat_push_precond c f l (u, v)" shows "push_effect_locale.\<Delta> c f u v = Network.excess c f u"
ipv4addr_of_nat_suc: fixes x :: "nat" shows "ipv4addr_of_nat (Suc x) = word_succ (ipv4addr_of_nat x)"
icd_pd_intermediate': fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and j :: "nat" assumes "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "k < j"   and "j < i" shows "\<pi> i pd\<rightarrow> \<pi> j"
injectivity_1: fixes B :: "'b set"   and C :: "'b set"   and P :: "'b set set"   and j :: "nat"   and P' :: "'b set set"   and j' :: "nat"   and B' :: "'b set"   and k :: "nat"   and B'' :: "'b set"   and k' :: "nat"   and Q :: "'b set set"   and Q' :: "'b set set"   and f :: "'b \<Rightarrow> 'b set"   and g :: "'b \<Rightarrow> 'b set"   and P'' :: "'b set set"   and P''' :: "'b set set" assumes "B \<inter> C = {}"   and "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "f = g"
exhaust_sel: fixes interval :: "'a interval" obtains "interval = Interval (low interval) (high interval)"
lm009: fixes G :: "'c set"   and n :: "'d"   and N :: "'d set" assumes "G \<noteq> {}"   and "n \<in> N" shows "{(G, n)} \<in> possible_allocations_rel G N"
perm_append1_eq: fixes zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" shows "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
upd_lock_Some_thread_idD: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and L :: "lock_action"   and t' :: "'b"   and n :: "nat" assumes "upd_lock l t L = \<lfloor>(t', n)\<rfloor>"   and "t \<noteq> t'" shows "\<exists>n. l = \<lfloor>(t', n)\<rfloor>"
phiUses_exI: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and m :: "'b"   and g :: "'a"   and n :: "'b"   and v :: "'d"   and vs :: "'d list" assumes "CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"   and "m \<in> set (graph_path_base.predecessors inEdges' g n)"   and "phis g (n, v) = Some vs"   and "n \<in> set (\<alpha>n g)" obtains v' :: "'d" where "v' \<in> CFG_SSA_base.phiUses \<alpha>n inEdges' phis g m" and "v' \<in> set vs"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "??.Quickcheck_Narrowing.narrowing_type \<Rightarrow> (??.Quickcheck_Narrowing.narrowing_term list \<Rightarrow> 'b) list \<Rightarrow> 'a"   and narrowing_cons :: "'b ??.Quickcheck_Narrowing.narrowing_cons" shows "P (case narrowing_cons of Narrowing_cons x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. narrowing_cons = Narrowing_cons x1 x2 \<longrightarrow> P (f x1 x2))"
sectionI: fixes g :: "'a"   and f :: "'a" assumes "ide (g \<cdot> f)" shows "section f"
f_repr: shows "list_repr (i - 1) (dec_i (update_im1 (update_i f (fs'' ! i)) (fs'' ! (i - 1)))) (map ((!) fs'') [0..<m])"
LTL_output_vend: fixes d :: "value"   and t :: "(String.literal \<times> value list) stream" shows "alw (\<lambda>xs. label_eq ''vend'' xs \<and> nxt (output_eq [Some d]) xs \<longrightarrow> check_exp (Ge (V (Rg 2)) (L (Num 100))) xs) (watch drinks t)"
nt_mult_pp: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A"   and "0 \<le> m"   and "0 \<le> n" shows "m\<triangleright>n\<triangleright>a\<^bsub>A\<^esub>\<^bsub>A\<^esub> = (m * n)\<triangleright>a\<^bsub>A\<^esub>"
ideal_remap_incl: fixes A' :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "finite A'"   and "\<forall>a'\<in>A'. \<exists>a\<in>A. a - a' \<in> B" shows "More_Modules.ideal (A' \<union> B) \<subseteq> More_Modules.ideal (A \<union> B)"
assms_1: shows "e \<in> UNIV \<rightarrow> {0<..<1}"
rel_fun_conversep: fixes A :: "'c \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun A\<inverse>\<inverse> B\<inverse>\<inverse> = (rel_fun A B)\<inverse>\<inverse>"
beukers_nn_integral3_le: shows "local.beukers_nn_integral3 \<le> ennreal (2 * (1 / 27) ^ n * Re (zeta 3))"
simps_2: fixes p :: "nat"   and a :: "int"   and n :: "nat" shows "gen_lucas_lehmer_sequence'' p a (Suc n) = gen_lucas_lehmer_sequence'' p (mersenne_mod2 (a\<^sup>2 - 2) p) n"
c_secure_implies_secure: fixes I :: "('a \<times> 'a) set"   and step :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and out :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and s\<^sub>0 :: "'b"   and D :: "'c \<Rightarrow> 'a" assumes "refl I"   and "c_secure step out s\<^sub>0 I D" shows "secure (c_process step out s\<^sub>0) I (c_dom D)"
OstrowskiTr3: fixes v :: "'a \<Rightarrow> ant"   and s :: "'a"   and t :: "'a" assumes "valuation K v"   and "s \<in> carrier K"   and "t \<in> carrier K"   and "0 \<le> v t"   and "v s < 0" shows "0 < v (t \<cdot>\<^sub>r (s \<plusminus> t)\<^bsup>\<hyphen> K\<^esup>)"
assms: fixes X'' :: "'x set"   and h :: "'h" assumes "X'' = Ch h (X \<union> X'')"   and "X'' \<noteq> Ch h X"   and "X'' \<subseteq> CD_on ds (X \<union> X'')" shows "False"
distinct4_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" shows "distinct4 A B C D \<equiv> A \<noteq> B \<and> A \<noteq> C \<and> A \<noteq> D \<and> B \<noteq> C \<and> B \<noteq> D \<and> C \<noteq> D"
fold_congs_2: fixes r :: "('b, 'c) node_scheme"   and r' :: "('b, 'c) node_scheme"   and v' :: "nat set"   and f :: "nat set \<Rightarrow> nat set"   and f' :: "nat set \<Rightarrow> nat set" assumes "r = r'"   and "incoming r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "incoming_update f r = incoming_update f' r'"
aux: shows "\<forall>m. y - ((1 - m) *\<^sub>R x + m *\<^sub>R y) = (1 - m) *\<^sub>R (y - x)"
base_simps_2: shows "hseq (f \<star> g) h"
preserves_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "src\<^sub>D (F \<mu>) = F.map\<^sub>0 (src\<^sub>C \<mu>)"
g_real_def: fixes x :: "real" shows "g_real x = g (nat \<lfloor>x\<rfloor>)"
arrE: fixes f :: "('o, 'a) arr" assumes "arr f" obtains "f \<noteq> Null" and "Dom f \<in> Obj" and "Cod f \<in> Obj" and "Map f \<in> Hom (Dom f) (Cod f)"
hd_tl: fixes l :: "'b list" assumes "1 < length l" shows "hd (tl l) = l ! 1"
max_in_G: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> G"   and "g \<in> G" shows "(\<lambda>x. max (g x) (f x)) \<in> G"
dgrad_set_def: fixes d :: "'c \<Rightarrow> nat"   and m :: "nat" shows "dgrad_set d m = {t. d t \<le> m}"
code: fixes m :: "('d, 'e, 'f) resumption"   and f :: "'f \<Rightarrow> ('d, 'e, 'f) resumption" shows "bind_resumption m f = (if is_Done m then f (result m) else Pause (output m) (\<lambda>i. bind_resumption (resume m i) f))"
llast_def: fixes xs :: "'c llist" shows "llast xs = (case llength xs of enat (Suc n') \<Rightarrow> xs $ n')"
Step1_4_6b: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MClkRetry memCh crCh cst p \<and> $S6 rmhist p \<and> S3 rmhist p$ \<and> unchanged (e p, r p, m p, rmhist!p) \<longrightarrow> MemFail memCh (resbar rmhist) p"
add_mult_distrib2: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> V" shows "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
onlI': fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'h, 'f, 'e) seqp"   and p :: "('g, 'h, 'f, 'e) seqp"   and P :: "'i \<times> 'e \<Rightarrow> bool"   and \<xi> :: "'i" assumes "\<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)" shows "onl \<Gamma> P (\<xi>, p)"
SubsetSmall: fixes A' :: "ZF"   and A :: "ZF"   and B :: "ZF" assumes "HOLZF.subset A' A"   and "HOLZF.subset A B" shows "HOLZF.subset A' B"
simps_10: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and \<psi> :: "('sf, 'sc, 'sz) formula"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "step_result (SG, C) (i, CE \<phi> \<psi> \<sigma>) = (closeI SG i, C)"
in_components_subset: fixes c :: "'c set"   and s :: "'c set" assumes "c \<in> components s" shows "c \<subseteq> s"
Inr_def_alt: shows "Inr \<equiv> \<lambda>a. Basic_BNF_LFPs.xtor (BNF_Composition.id_bnf (Inr a))"
take_bit_eq_mod: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n a = modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
disc_eq_case_2: fixes option :: "'a option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | Some uu_ \<Rightarrow> True)"
assms_1: shows "tiTable_SampleT req a1 stop lose st_in b ack i1 vc st_out"
poly_nate_polypoly': fixes bs :: "'b list"   and p :: "poly" shows "poly_nate bs p = polypoly' bs (polynate p)"
is_2sat_def: fixes cnf :: "lit uprod set" shows "is_2sat cnf = (\<forall>cl\<in>cnf. proper_uprod cl)"
phis_wf: fixes g :: "'a"   and n :: "'b"   and v :: "'e"   and args :: "'e list" assumes "Mapping.lookup (phis g) (n, v) = Some args" shows "length (old.predecessors g n) = length args"
simps_1: fixes p :: "real mpoly"   and L :: "real list" shows "aEval (Eq p) L = (insertion (nth_default 0 L) p = 0)"
hyp: fixes A :: "(('a, 'n) vec, 'm) vec" assumes "n < nrows A"   and "mod_type_class.to_nat i \<le> n" shows "row_add_iterate A n i j $ i $ b = A $ i $ b"
cases: fixes P :: "'a prog"   and a1 :: "char list"   and a2 :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> a1 has_fields a2" obtains C :: "char list" and D :: "char list" and fs :: "fdecl list" and ms :: "'a mdecl list" and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" and FDTs' :: "((char list \<times> char list) \<times> staticb \<times> ty) list" where "a1 = C" and "a2 = FDTs'" and "class P C = \<lfloor>(D__, fs__, ms__)\<rfloor>" and "C \<noteq> Object" and "P \<turnstile> D__ has_fields FDTs__" and "FDTs' = map (\<lambda>(F, b, T). ((F, C), b, T)) fs__ @ FDTs__"   | D :: "char list" and fs :: "fdecl list" and ms :: "'a mdecl list" and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" where "a1 = Object" and "a2 = FDTs" and "class P Object = \<lfloor>(D, fs, ms)\<rfloor>" and "FDTs = map (\<lambda>(F, b, T). ((F, Object), b, T)) fs"
taut_conj_elim: fixes A :: "'a upred"   and B :: "'a upred" assumes "`A \<and> B`" obtains "`A`" and "`B`"
to_fun_nat_pow: fixes h :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "h \<in> carrier (UP R)"   and "a \<in> carrier R" shows "to_fun (h [^]\<^bsub>UP R\<^esub> n) a = to_fun h a [^] n"
p': fixes s :: "'a" assumes "guard s" shows "p' \<le> Sigma_Algebra.measure (measure_spmf (iter (N s) s)) {s. \<not> guard s}"
good_item_simps_68: fixes Aa :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rhob :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodAbs Aa"   and "goodEnv rhob"   and "\<And>zs z. freshEnv zs z rhob \<or> freshAbs zs z Aa" shows "Aa $[rhob] = Aa"
rbt_bulkload_def: fixes xs :: "('a \<times> 'c) list" shows "rbt_bulkload xs = foldr (\<lambda>(k, v). rbt_insert k v) xs Empty"
fw_upd_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "fw_upd m k i j \<equiv> upd m i j (min (m i j) (m i k + m k j))"
geodesic_segment_param_8: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "geodesic_segment_between G x y"   and "z \<in> G" shows "z = geodesic_segment_param G x (dist x z)"
CA_Says_2_lemma: fixes i :: "nat"   and C :: "agent"   and NC1 :: "nat"   and evs :: "event list" assumes "Crypt (priSK (CA i)) (Hash \<lbrace>Agent C, Nonce NC1\<rbrace>) \<in> parts (knows Spy evs)"   and "evs \<in> set_cr"   and "CA i \<notin> bad" shows "\<exists>Y. Says (CA i) C \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce NC1\<rbrace>, Y\<rbrace> \<in> set evs"
wf_tr_def: shows "wf_tr = (\<lambda>M T. length T = length (fst M) - 1 \<and> (\<forall>i<length T. length (T ! i) = i + 1))"
gen_contr_graph_def: fixes G :: "('b, 'c) pre_digraph"   and V :: "'b set" shows "gen_contr_graph G V \<equiv> \<lparr>pverts = V, parcs = {(u, v). \<exists>p. pre_digraph.gen_iapath G V u p v}\<rparr>"
apply_icrit_def: fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "apply_icrit crit data gs bs hs ps = (let c = crit data gs bs hs in map (\<lambda>(q_in_bs, p, q). (c p q, q_in_bs, p, q)) ps)"
ncols_eq_card_columns: fixes A :: "(('d, 'e) vec, 'f) vec" shows "ncols_iarray (matrix_to_iarray A) = CARD('e)"
valid_cfg_cont: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and cfg :: "'a cfg"   and s :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "s \<in> set_pmf (action cfg)" shows "cont cfg s \<in> Finite_Markov_Decision_Process.valid_cfg K S"
targets_char: fixes a :: "'a" assumes "weak_unit a" shows "targets a = {x. x \<cong> a}"
twoCisTypeI_6: shows "g2 piecewise_C1_differentiable_on {a..b}"
simps_1: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'b"   and xs :: "'b list" shows "rm_MT_rules C (x # xs) = (if dom (C x) = {} then rm_MT_rules C xs else x # rm_MT_rules C xs)"
subcls1p_subcls1_eq: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "subcls1p P = subcls1' P"
sys_block_sizes_def: fixes n :: "nat" shows "mult_sys.sys_block_sizes n \<equiv> {int (card bl) |bl. bl \<in># multiple_blocks n}"
predicate1I: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "P \<le> Q"
euler_liar_1_p: fixes p :: "nat" assumes "p \<noteq> 0" shows "euler_liar 1 p"
src_dom: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.src (S.dom \<mu>) = S.src \<mu>"
hyps_2: shows "bn \<alpha> \<sharp>* (P \<parallel> Q \<parallel> R)"
trunc_degree: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)"   and "0 < deg R f" shows "deg R (Cring_Poly.truncate R f) < deg R f"
assms_3: assumes "\<bar>b\<bar> < \<bar>a\<bar>" shows "thesis"