hcju/isabelleps huggingface dataset:
isom_gch_units_transpTr3: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat" assumes "Ugp E"   and "Gchain n g"   and "i \<le> n" shows "finite ({i. i \<le> n \<and> g i \<cong> E} - {i})"
det_minor_row: fixes B :: "('c, 'd) sq_matrix"   and i :: "'d"   and j :: "'d" shows "Square_Matrix.det (minor B i j) = Square_Matrix.det (upd_row B i (axis j (1::'c)))"
vimageE: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "a \<in> f -` B" obtains x :: "'c" where "f a = x" and "x \<in> B"
invariantE: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "invariant rel I"   and "rel x y"   and "I x" shows "I y"
simps_2: fixes lam :: "'a"   and db :: "nat \<Rightarrow> 'a"   and i :: "nat"   and x :: "'b"   and s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" shows "lambda_encoding.subst_db lam db i x (App s t) = App (lambda_encoding.subst_db lam db i x s) (lambda_encoding.subst_db lam db (if head s = Sym lam then i + 1 else i) x t)"
tab_in_vhom': fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c" assumes "tabulation_data V H \<a> \<i> src trg r \<rho> f g" shows "partial_magma.in_hom V \<rho> g (H r f)"
boundedE: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}"   and "Max A \<le> x" obtains "\<And>a. a \<in> A \<Longrightarrow> a \<le> x"
var_imp_symvar_var2: fixes v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "'c \<Rightarrow> nat" assumes "v \<in> Bexp.vars e" shows "symvar v s \<in> Bexp.vars (adapt_bexp e s)"
not_ex: shows "\<not> extensible R (graph_of (X y')) f"
that: fixes \<alpha>s :: "V list"   and ms :: "nat list" assumes "list.set \<alpha>s \<subseteq> ON"   and "list.set ms \<subseteq> {0<..}"   and "length \<alpha>s = length ms"   and "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"   and "\<alpha> = Cantor_sum \<alpha>s ms" shows "thesis"
Dom_cod: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "Dom (cod f) = Cod f"
of_real_def: fixes r :: "real" shows "of_real r = r *\<^sub>R (1::'d)"
exhaust_sel: fixes language :: "'a language" obtains "language = Lang (\<oo> language) (\<dd> language)"
a1: shows "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"
Smith_mxn_pinduct_carrier: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "Smith_mxn_dom A"   and "A \<in> carrier_mat m n" shows "fst (Smith_mxn A) \<in> carrier_mat m m \<and> fst (snd (Smith_mxn A)) \<in> carrier_mat m n \<and> snd (snd (Smith_mxn A)) \<in> carrier_mat n n"
OV_inv3D: fixes x :: "'a obsv_state_scheme"   and r :: "nat"   and p :: "process"   and v :: "val" assumes "x \<in> OV_inv3"   and "obsv_state.obs x r p = Some v" shows "obs_safe r x v"
v_0_AE: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "(v cfg = 0) = (AE x in T cfg. \<not> (HLD S1 suntil HLD S2) (state cfg ## x))"
k_frontier_is_node: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and aOps :: "('k, 'j, 'd list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps"   and a :: "'b" assumes "AlgorithmForAgent jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps" shows "list_all (AlgorithmForAgent.k_isNode jkbp envInit envAction envTrans envVal jview simf simAbs a) (Algorithm.k_frontier envInit envObs simInit a)"
last_message_exists_nth': fixes xs :: "'b message_af list" assumes "last_message xs \<noteq> NoMsg" shows "\<exists>i<length xs. last_message xs = xs ! i"
t1_def: shows "t1 \<equiv> punit.lt (rep_list p)"
TermSetInSet: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T" shows "Expr2ZF ` TermEquivClGen T A e B \<in> range explode"
implies_curry_flip: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leadsto> (y \<leadsto> z) = y \<leadsto> (x \<leadsto> z)"
equivp_vimage2p: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "equivp R" shows "equivp (vimage2p f f R)"
Inf_dual_var: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and dual :: "'b \<Rightarrow> 'b"   and X :: "'b set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "dual (Inf X) = Sup (dual ` X)"
ms: fixes i :: "nat" shows "X (Suc i) = (case selector (X i) of None \<Rightarrow> X i | Some (R, f) \<Rightarrow> let g = extend (fst (X i)) R f in (max (fst (X i)) (nextMax (Range g)), on_triple g `` edges (snd R) \<union> snd (X i)))"
pred_subset_eq: fixes R :: "'b set"   and S :: "'b set" shows "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) = (R \<subseteq> S)"
left_assoc: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ac_operator f" shows "f a (f b c) = f (f a b) c"
Image_def: fixes r :: "('h \<times> 'g) set"   and s :: "'h set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
homeomorphic_space_sym: fixes X :: "'b topology"   and Y :: "'c topology" shows "(X homeomorphic_space Y) = (Y homeomorphic_space X)"
ereal_abs_mult: fixes x :: "ereal"   and y :: "ereal" shows "\<bar>x * y\<bar> = \<bar>x\<bar> * \<bar>y\<bar>"
get_subsets_R_def: fixes data :: "rat mat \<times> (nat list \<times> nat list) list \<times> rat list list" shows "get_subsets_R data = fst (snd data)"
deriv_monom: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "deriv (monom P a n) b = [n] \<cdot> a \<otimes> b [^] (n - 1)"
simps_1: fixes uu :: "iface \<Rightarrow> ('b word \<times> nat) list option" shows "oiface_rewrite uu MatchAny = MatchAny"
sets_PiF_singleton: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "sets (Pi\<^sub>F {I} M) = sigma_sets (\<Pi>' j\<in>I. space (M j)) {Pi' I X |X. X \<in> (\<Pi> j\<in>I. sets (M j))}"
assms: shows "Di \<in> set (subseqs (dbproj k D))"
inM: shows "0 < zcount M (results_in t'' (sum_weights (map (\<lambda>(s, l, t). l) xs)))"
extend_evals: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es'' :: "char list exp list"   and s'' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es'',s''\<rangle>"   and "P \<turnstile> \<langle>es'',s''\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
Act: fixes f :: "'effect"   and F :: "'effect fs_set"   and \<alpha> :: "'act"   and x :: "('b, 'pred, 'act, 'effect) formula" assumes "f \<in>\<^sub>f\<^sub>s F"   and "bn \<alpha> \<sharp>* (F, f)"   and "x \<in> \<A>[L (\<alpha>, F, f)]" shows "Act f \<alpha> x \<in> \<A>[F]"
bdd_lookup_append: fixes n :: "nat"   and B :: "'a bdd"   and bs :: "bool list"   and w :: "bool list" assumes "bddh n B"   and "n \<le> length bs" shows "bdd_lookup B (bs @ w) = bdd_lookup B bs"
vdisjnt_vempty2: fixes A :: "V" shows "vdisjnt A 0"
prefixes_at_Suc: fixes t :: "nat"   and j :: "nat" assumes "local.prefixes t j \<down>" shows "local.prefixes (Suc t) j = nxt (the (local.prefixes t j))"
continuous_on_snd: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. snd (f x))"
Gamma_Weierstrass_complex: fixes z :: "complex" shows "Gamma_series_Weierstrass z \<longlonglongrightarrow> Gamma z"
numeral_pow: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "num"   and l :: "num" assumes "class.semiring_numeral one times plus" shows "numeral.numeral one plus (Num.pow k l) = power.power one times (numeral.numeral one plus k) (numeral l)"
sim_cfunc_usr: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and iu :: "'iusr_core"   and s1 :: "'s_core"   and s2 :: "'s_core'"   and ou :: "'ousr_core"   and s1' :: "'s_core"   and s2' :: "'s_core'" assumes "S p q"   and "iu \<in> IU"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(ou, s1') \<in> set_spmf (cfunc_usr core1 s1 iu)"   and "(ou, s2') \<in> set_spmf (cfunc_usr core2 s2 iu)" shows "S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_usr core2 s2 iu)) ou)"
Fract_of_int_eq: fixes k :: "int" shows "Fract k 1 = rat_of_int k"
enat_defs_3: fixes i :: "enat" shows "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
MOST_conjI: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<forall>\<^sub>\<infinity>x. P x"   and "\<forall>\<^sub>\<infinity>x. Q x" shows "\<forall>\<^sub>\<infinity>x. P x \<and> Q x"
simps_2: fixes val :: "'b"   and l :: "nat"   and i :: "nat"   and k :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_3_c_inner_loop val l i (Suc k) A = ??.Jordan_Normal_Form_Existence.step_3_c_inner_loop val (l - 1) (i - 1) k (add_col_sub_row val i l A)"
trancl_unfold_right: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>+ = r\<^sup>* O r"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> yahalom"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> yahalom; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> yahalom; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> yahalom; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Agent A, Nonce NA\<rbrace> # evs1)"   and "\<And>evs2 NB B A NA. \<lbrakk>evs2 \<in> yahalom; P evs2; Nonce NB \<notin> used evs2; Gets B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Agent B, Nonce NB, Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> # evs2)"   and "\<And>evs3 KAB B NB A NA. \<lbrakk>evs3 \<in> yahalom; P evs3; Key KAB \<notin> used evs3; KAB \<in> symKeys; Gets Server \<lbrace>Agent B, Nonce NB, Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, Nonce NA, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key KAB\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 A K B NA NB X. \<lbrakk>evs4 \<in> yahalom; P evs4; A \<noteq> Server; K \<in> symKeys; Gets A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, X\<rbrace> \<in> set evs4; Says A B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> # evs4)" shows "P x"
polar_inj: fixes p :: "proj2"   and q :: "proj2" assumes "polar p = polar q" shows "p = q"
simps: fixes a :: "(name \<times> ty) list" shows "valid a = (a = [] \<or> (\<exists>\<Gamma> aa T. a = (aa, T) # \<Gamma> \<and> valid \<Gamma> \<and> aa \<sharp> \<Gamma>))"
assms_4: shows "run\<^sub>t \<delta> q\<^sub>0 w = f \<circ> run\<^sub>t \<delta>' q\<^sub>0' w"
mem_frontier_diff: fixes t :: "'t"   and M :: "'t zmultiset"   and N :: "'t zmultiset" assumes "t \<in>\<^sub>A frontier M - frontier N" shows "zcount (frontier_changes M N) t = 1"
nths_all: fixes xs :: "'g list"   and I :: "nat set" assumes "\<forall>i<length xs. i \<in> I" shows "nths xs I = xs"
enc_Inl: fixes x :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list"   and m :: "nat"   and p :: "nat" assumes "x \<in> enc (w, I)"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "m < length I"   and "I ! m = Inl p" shows "p < length x \<and> snd (x ! p) ! m"
DomInterAllowsMT: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set"   and p :: "int"   and po :: "int" assumes "twoNetsDistinct a b c d" shows "dom (C (AllowPortFromTo a b p)) \<inter> dom (C (AllowPortFromTo c d po)) = {}"
ord: fixes x :: "enat" shows "ord bot (f x (eSuc x))"
pwd_spy_con_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_con n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_con_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
selberg_aux: shows "(\<chi>x. of_real ((ln (real x))\<^sup>2)) * fds moebius_mu = (fds mangoldt)\<^sup>2 - fds_deriv (fds mangoldt)"
\<eta>_eq: fixes x :: "'a" assumes "x \<in> R" shows "\<eta>' x = \<eta> x"
sublist_def: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" shows "WB_Sort.sublist xs i j \<equiv> take (Suc j - i) (drop i xs)"
c_sources_mono: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and ys :: "'f list"   and zs :: "'f list"   and x :: "'f" assumes "c_sources I D u ys \<subseteq> c_sources I D u zs" shows "c_sources I D u (x # ys) \<subseteq> c_sources I D u (x # zs)"
fac_acc_work_final_def: shows "fac_acc_work_final \<equiv> \<Lambda> x. fix\<cdot>fac_acc_body_final\<cdot>x\<cdot>1"
usappconst_simp: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and f :: "char"   and r :: "trm"   and U :: "variable set" assumes "SConst \<sigma> f = Aterm r"   and "FVT r \<inter> U = {}" shows "usappconst \<sigma> U f = Aterm r"
differentiable_in_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c"   and s :: "'c set" assumes "f differentiable at (g x) within g ` s"   and "g differentiable at x within s" shows "(\<lambda>x. f (g x)) differentiable at x within s"
deadlock_eq_deadlocked': shows "r1.deadlock = r1.deadlocked'"
intros_2: fixes u :: "'a"   and p :: "'b"   and v :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and c :: "'a list multiset"   and r :: "'a list" assumes "(u, Call p, v) \<in> edges fg"   and "mon fg p \<inter> mon_c fg c = {}" shows "((u # r, c), LCall p, entry fg p # v # r, c) \<in> trss fg"
ex_productE: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 's" assumes "\<exists>a. has_as_product J D a" obtains a :: "'s" and \<pi> :: "'a \<Rightarrow> 's" where "product_cone J (\<cdot>) D a \<pi>"
Hermite_Hermite_of: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec" assumes "ass_function ass"   and "res_function res"   and "is_bezout_ext bezout" shows "Hermite (range ass) (\<lambda>c. range (res c)) (Hermite_of A ass res bezout)"
lift_simulation_plus: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and order :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and s2 :: "'c"   and s2' :: "'c"   and i1 :: "'d"   and s1 :: "'b" assumes "backward_simulation step1 step2 final1 final2 order match"   and "step2\<^sup>+\<^sup>+ s2 s2'"   and "match i1 s1 s2" shows "(\<exists>i2 s1'. step1\<^sup>+\<^sup>+ s1 s1' \<and> match i2 s1' s2') \<or> (\<exists>i2. match i2 s1 s2' \<and> order\<^sup>+\<^sup>+ i2 i1)"
decode_state_at_encoding_variables_equals_some_of_valuation_if: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k \<le> t"   and "v \<in> set (\<Pi>\<^sub>\<V>)" shows "(\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k) v = Some (\<A> (State k (index (\<Pi>\<^sub>\<V>) v)))"
simps_49: fixes \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "safe_formula (Until \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> (safe_formula \<phi> \<or> (case \<phi> of Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False)) \<and> safe_formula \<psi>)"
pow_mult_list: fixes u :: "'b list"   and m :: "nat"   and n :: "nat" shows "u \<^sup>@ (m * n) = (u \<^sup>@ m) \<^sup>@ n"
init_vmuaux_def: shows "init_vmuaux = (\<lambda>_. (0, []))"
mono_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
lens_override_plus: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and S\<^sub>1 :: "'f"   and S\<^sub>2 :: "'f" assumes "X \<bowtie> Y" shows "S\<^sub>1 \<triangleleft>\<^bsub>X +\<^sub>L Y\<^esub> S\<^sub>2 = S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> S\<^sub>2"
matrix_to_iarray_Gauss_Jordan: fixes A :: "(('a, 'b) vec, 'c) vec" shows "matrix_to_iarray (Gauss_Jordan A) = Gauss_Jordan_iarrays (matrix_to_iarray A)"
chain_transp_on_less: fixes f :: "nat \<Rightarrow> 'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. f i \<in> A \<and> P (f i) (f (Suc i))"   and "transp_on P A"   and "i < j" shows "P (f i) (f j)"
eventually_happens': fixes F :: "'c filter"   and P :: "'c \<Rightarrow> bool" assumes "F \<noteq> bot"   and "eventually P F" shows "\<exists>x. P x"
while_doesnt_modify: fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   and x :: "'Var" shows "doesnt_modify (While e c \<otimes> annos) x"
suminf_eq_SUP: fixes f :: "nat \<Rightarrow> 'b" shows "suminf f = (SUP n. sum f {..<n})"
getLastAssertedLiteralCharacterization: fixes clause :: "Literal list"   and valuation :: "Literal list" assumes "clauseFalse clause valuation"   and "clause \<noteq> []"   and "uniq valuation" shows "isLastAssertedLiteral (getLastAssertedLiteral (oppositeLiteralList clause) valuation) (oppositeLiteralList clause) valuation"
lprefix_refl: fixes xs :: "'f llist" shows "lprefix xs xs"
ih_14: shows "InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state'))"
\<alpha>2: shows "\<alpha>2 \<in> elts \<alpha>1"
to_fract_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(to_fract x = to_fract y) = (x = y)"
\<gamma>2_def: fixes x :: "real" shows "\<gamma>2 x = x + 2 * \<rho> * rmax"
univ_poly_carrier: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" shows "polynomial\<^bsub>R\<^esub> K p = (p \<in> carrier (K [X]\<^bsub>R\<^esub>))"
Some_image_these_eq: fixes A :: "'c option set" shows "Some ` Option.these A = {x \<in> A. x \<noteq> None}"
A2_def: shows "A2 = {x. jumpF f (at_right x) \<noteq> 0 \<and> b \<le> x \<and> x < c}"
cases: fixes x :: "'b\<^sub>\<bottom>" obtains "x = Ibottom"   | a :: "'b" where "x = Iup a"
eval_models: fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist"   and \<phi> :: "('a, 'p) Kform"   and s :: "'s" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t"   and "subjective a \<phi>"   and "s \<in> toSet aec" shows "eval envVal envObs (cec, aec) \<phi> = clock_repMC (toSet cec), s \<Turnstile> \<phi>"
wf_rcv_append: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term"   and S' :: "('d, 'c) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S')" shows "wf\<^sub>s\<^sub>t V (S @ S')"
finsum_cong': fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a list"   and f :: "'c \<Rightarrow> 'a list" assumes "A = B"   and "g \<in> B \<rightarrow> carrier (K [X])"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum (K [X]) f A = finsum (K [X]) g B"
old_iteratei_rule_P: fixes m :: "'s"   and I :: "'u set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'u \<times> 'v \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "invar m"   and "I (dom (\<alpha> m)) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; \<alpha> m k = Some v; it \<subseteq> dom (\<alpha> m); I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom (\<alpha> m); it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iteratei m c f \<sigma>0)"
that: assumes "refl_onP A r"   and "antisymp r"   and "transp r" shows "thesis"
coinduct: fixes X :: "'b stream \<Rightarrow> bool"   and x :: "'b stream" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>xa xs. x = xa ## xs \<and> xa \<notin> sset xs \<and> (X xs \<or> sdistinct xs)" shows "sdistinct x"
terminates_onp_terminates_on_eq: fixes g :: "'e \<Rightarrow> ('f, 'e) step" shows "terminates_onp g = (\<lambda>x. x \<in> terminates_on g)"
cart_basis_eq_set_of_vector_cart_basis': shows "cart_basis = set_of_vector cart_basis'"
coeff_sum: fixes p :: "'b \<Rightarrow> 'a poly"   and A :: "'b set"   and i :: "nat" shows "poly.coeff (sum p A) i = (\<Sum>x\<in>A. poly.coeff (p x) i)"
parSym: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<approx> Q \<parallel> P"