hcju/isabelleps huggingface dataset:
assms_1: shows "Complete_Partial_Order.chain local.tllist_ord A"
gcd_dvd_antisym: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times (gcd a b) (gcd c d)"   and "dvd.dvd times (gcd c d) (gcd a b)" shows "gcd a b = gcd c d"
KL_div_mul: fixes d :: "real"   and S :: "'b set"   and a :: "'b \<Rightarrow> real"   and e :: "'b \<Rightarrow> real" assumes "0 < d"   and "d \<le> 1"   and "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i"   and "\<And>i. i \<in> S \<Longrightarrow> 0 < e i" shows "KL_div S a (\<lambda>i. e i / d) \<le> KL_div S a e"
frameImpResFreshLeft: fixes x :: "name"   and F :: "'b frame" assumes "x \<sharp> F" shows "\<lparr>\<nu>x\<rparr>F \<hookrightarrow>\<^sub>F F"
intros: fixes xs :: "'c list"   and A :: "'c set"   and n :: "nat" assumes "set xs \<subseteq> A"   and "length xs = n" shows "xs \<in> nlists A n"
ide_def: fixes a :: "'b" shows "C'.ide a \<equiv> C' a a \<noteq> C'.null \<and> (\<forall>f. (C' f a \<noteq> C'.null \<longrightarrow> C' f a = f) \<and> (C' a f \<noteq> C'.null \<longrightarrow> C' a f = f))"
between_commute: fixes a :: "'a"   and b :: "'a" shows "between (a, b) = between (b, a)"
l_comb_def: fixes R :: "('f, 'g) Ring_scheme"   and M :: "('e, 'f, 'h) Module_scheme"   and n :: "nat"   and s :: "nat \<Rightarrow> 'f"   and m :: "nat \<Rightarrow> 'e" shows "l_comb R M n s m = \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s\<^bsub>M\<^esub> m j) n"
Fadmit_Not: fixes \<sigma> :: "('a, 'b, 'c) subst"   and \<phi> :: "('a, 'b, 'c) formula" assumes "Fadmit \<sigma> \<phi>" shows "Fadmit \<sigma> (! \<phi>)"
SI_def: fixes s :: "'s" shows "SI s = (S s \<and> (\<forall>x\<in>set_of s. I x))"
that: fixes L :: "real" assumes "\<And>t. t \<in> T \<Longrightarrow> L-lipschitz_on X (f t)" shows "thesis"
intros_7: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c\<^sub>1 :: "com"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and c\<^sub>2 :: "com" assumes "bval b s"   and "\<pi>: (c\<^sub>1, s) \<Rightarrow> t" shows "\<pi>: (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
DEP_REFL: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" shows "reflexive (dep PROB)"
thetaWhileW0_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaWhileW0 tval aval indis \<equiv> {(while tst do c, while tst do d) |tst c d. PL_Indis.compatTst tval indis tst \<and> (c, d) \<in> PL_Indis.BisT tval aval indis} \<union> {(c1 ;; while tst do c, d1 ;; while tst do d) |tst c1 d1 c d. PL_Indis.compatTst tval indis tst \<and> (c1, d1) \<in> PL_Indis.BisT tval aval indis \<and> (c, d) \<in> PL_Indis.BisT tval aval indis}"
hyps: fixes n :: "nat"   and k :: "nat" assumes "n < p ^ Suc d"   and "k < p ^ Suc d"   and "prime p" shows "(n choose k) mod p = (\<Prod>i\<le>d. nth_digit_general n i p choose nth_digit_general k i p) mod p"
slp_def: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" shows "CFGExit.same_level_path' n as n' \<equiv> CFGExit.path n as n' \<and> CFGExit.same_level_path as"
admD: fixes P :: "'e \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'e" assumes "adm P"   and "chain Y"   and "\<And>i. P (Y i)" shows "P (\<Squnion>i. Y i)"
bv_to_nat_lower_limit: fixes w :: "bit list" assumes "0 < bv_to_nat w" shows "2 ^ (length (norm_unsigned w) - 1) \<le> bv_to_nat w"
perpendicular_def: fixes H1 :: "circline"   and H2 :: "circline" shows "perpendicular H1 H2 = (cos_angle (of_circline H1) (of_circline H2) = 0)"
aux1: fixes l :: "nat" assumes "\<^bold>|r \<^sup>@ l\<^bold>| \<le> i"   and "i < \<^bold>|r \<^sup>@ l\<^bold>| + \<^bold>|r\<^bold>|" shows "i mod \<^bold>|r\<^bold>| = i - \<^bold>|r \<^sup>@ l\<^bold>|"
simps_4: fixes v :: "Cooper.num" shows "not (Lt v) = NOT (Lt v)"
cos_zero_lemma: fixes x :: "real" assumes "0 \<le> x"   and "cos x = 0" shows "\<exists>n. odd n \<and> x = real n * (pi / 2)"
Pan_synth: fixes A :: "nat"   and H :: "msg set" assumes "Pan A \<in> synth H" obtains "Pan A \<in> H"
not_col_exists: fixes A :: "'p"   and B :: "'p" assumes "A \<noteq> B" shows "\<exists>C. \<not> Col A B C"
absolutely_integrable_drop: fixes S :: "(real, 1) vec set"   and f :: "(real, 1) vec \<Rightarrow> 'a" shows "(f absolutely_integrable_on S) = (f \<circ> vec absolutely_integrable_on (\<lambda>x. x $ 1) ` S)"
all_normalized_and_ex_tcsigs_def: fixes sub :: "(String.literal \<times> String.literal) set"   and tcs :: "'a \<Rightarrow> ('b \<Rightarrow> String.literal set list option) option" shows "all_normalized_and_ex_tcsigs sub tcs \<equiv> \<forall>ars\<in>ran tcs. \<forall>ss\<in>ran ars. \<forall>s\<in>set ss. wf_sort sub s"
unrest_ex_in: fixes y :: "'a \<Longrightarrow> 'b"   and x :: "'c \<Longrightarrow> 'b"   and P :: "'b upred" assumes "mwb_lens y"   and "x \<subseteq>\<^sub>L y" shows "x \<sharp> (\<exists> y \<bullet> P)"
odlist_Abs_1: shows "dxs = odlist_Abs xs"
E: shows "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s"
diff_diff_right: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "c - (b - a) = c + a - b"
lens_compat_def: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" shows "(X ##\<^sub>L Y) = (\<forall>s\<^sub>1 s\<^sub>2. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2)"
intros_1: shows "rel_12_23 \<^bold>1 \<^bold>2"
comp_arr_inv: fixes f :: "'a"   and g :: "'a" assumes "A.inverse_arrows f g" shows "f \<cdot>\<^sub>A g = A.dom g"
repcs_eq_rept: fixes st :: "'s \<times> ('c \<Rightarrow> real)"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and st'' :: "'s \<times> ('c \<Rightarrow> real)"   and cfg'' :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "repcs st cfg = repcs st'' cfg''" shows "rept st (action cfg) = rept st'' (action cfg'')"
L_transform_formula_hereditarily_fs: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "Formula.hereditarily_fs (L_transform_formula x)"
bs1: fixes M :: "lam" shows " M \<longrightarrow>\<^sub>\<beta>\<^sup>* M"
appL: fixes s :: "dB"   and t :: "dB"   and u :: "dB" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "s \<degree> u \<rightarrow>\<^sub>\<beta> t \<degree> u"
hoare_cnvalid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
mult_le_cancel_left2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (times c a) c = ((less zero c \<longrightarrow> less_eq a one) \<and> (less c zero \<longrightarrow> less_eq one a))"
less_imp_le: fixes y :: "('d, 'e) Term.term"   and x :: "('d, 'e) Term.term" assumes "y \<rhd> x" shows "y \<unrhd> x"
jordan_nf_diag_block_mat: fixes Ms :: "('b mat \<times> (nat \<times> 'b) list) list" assumes "\<And>A jbs. (A, jbs) \<in> set Ms \<Longrightarrow> jordan_nf A jbs" shows "jordan_nf (diag_block_mat (map fst Ms)) (concat (map snd Ms))"
simps_8: fixes \<C> :: "t_context"   and t :: "t"   and va :: "relop_f"   and ts :: "checker_type" shows "check_single \<C> (Relop_f t va) ts = (if is_float_t t then type_update ts [TSome t, TSome t] (Type [T_i32]) else Bot)"
head_nz: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "(head p = 0\<^sub>p) = (p = 0\<^sub>p)"
Im_def: fixes complex :: "complex" shows "Im complex = (case complex of Complex x1 x2 \<Rightarrow> x2)"
assms_11: fixes xs :: "'a set list" assumes "xs \<noteq> []"   and "xs \<in> lists G" shows "emeasure M (scylinder (space S) xs) = emeasure N (scylinder (space S) xs)"
inf_llist_def: fixes f :: "nat \<Rightarrow> 'a" shows "inf_llist f = lmap f (iterates Suc 0)"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'd"   and sum :: "'e + 'f" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Inr x2 \<longrightarrow> P (f2 x2)))"
simps_2: fixes n :: "nat" shows "enumerate_regs (V (R n)) = {n}"
continuous_on_open_invariant: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
get_disconnected_document_ok: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None" shows "h \<turnstile> ok get_disconnected_document node_ptr"
THE_defaultI2: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and d :: "'a" assumes "\<exists>!x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE_default d P)"
cod_inv: fixes f :: "'a" assumes "S.iso f" shows "S.cod (S.inv f) = S.dom f"
stutter_extend_simps_3: fixes G :: "('b, 'd) graph_rec_scheme" shows "g_V0 (stutter_extend G) = g_V0 G"
equiv_class_member: fixes x :: "'a list"   and A :: "'a list set"   and y :: "'a list" assumes "x \<in> A"   and "\<approx>A `` {x} = \<approx>A `` {y}" shows "y \<in> A"
Card_order_empty: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "|{}| \<le>o r"
is_dghmD_8: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv (\<FF>\<lparr>ObjMap\<rparr>)"
smult_smult_assoc: fixes a :: "'a"   and b :: "'a"   and v :: "'a Matrix.vec" shows "a \<cdot>\<^sub>v (b \<cdot>\<^sub>v v) = a * b \<cdot>\<^sub>v v"
returning: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and m' :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile> rs\<^sub>1 \<Rightarrow>\<^sub>r Undecided"   and "matches \<gamma> m' p" shows "\<Gamma>,\<gamma>,p\<turnstile> rs\<^sub>1 @ Rule m' Return # rs\<^sub>2 \<Rightarrow>\<^sub>r Undecided"
disc_eq_case_1: fixes tm :: "('c, 'd) tm" shows "is_Hd tm = (case tm of Hd uu_ \<Rightarrow> True | App uu_ uua_ \<Rightarrow> False)"
commute: fixes a :: "'k"   and b :: "'k" shows "a * b = b * a"
len: shows "eval (Cn 1 r_length [r_pdec1]) [arg] \<down>= e_length b"
Struct_def: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and arOf :: "'b \<Rightarrow> 'd list"   and resOf :: "'b \<Rightarrow> 'd"   and intT :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and intF :: "'b \<Rightarrow> 'e list \<Rightarrow> 'e"   and intP :: "'c \<Rightarrow> 'e list \<Rightarrow> bool" shows "Struct wtFsym wtPsym arOf resOf intT intF intP \<equiv> Signature TYPE('d) wtFsym wtPsym \<and> Tstruct intT \<and> Struct_axioms wtFsym arOf resOf intT intF intP"
\<rho>: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> f\<^sup>*) \<rho> f g"
closure_of_topspace: fixes X :: "'c topology" shows "X closure_of topspace X = topspace X"
TimesLR_ZeroR: fixes r :: "'a language" shows "TimesLR r Zero = Zero"
observable_moves_preserves_stack: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cs :: "'edge list"   and rs :: "'edge list" assumes "S,f \<turnstile> (m # ms,s) =as\<Rightarrow> (m' # ms',s')"   and "valid_node m"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "ms = targetnodes rs" obtains cs' :: "'edge list" and rs' :: "'edge list" where "valid_node m'" and "valid_call_list cs' m'" and "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)" and "valid_return_list rs' m'" and "length rs' = length cs'" and "ms' = targetnodes rs'" and "upd_cs cs as = cs'"
eexE: fixes F :: "(state \<Rightarrow> 'a) \<Rightarrow> behavior \<Rightarrow> bool"   and sigma :: "behavior"   and vs :: "state \<Rightarrow> 'b"   and G :: "behavior \<Rightarrow> bool" assumes "(\<exists>\<exists> x. F x) sigma"   and "basevars vs"   and "\<And>x. \<lbrakk>basevars (x, vs); F x sigma\<rbrakk> \<Longrightarrow> G sigma" shows "G sigma"
A\<^sub>i: fixes x :: "i"   and y :: "i"   and z :: "i" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
p: shows "p = rat_of_int (z1 * n2) / rat_of_int (n1 * n2)"
subset_Collect_conv: fixes S :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "(S \<subseteq> Collect P) = (\<forall>x\<in>S. P x)"
join_idem: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> x = x"
cs_inj: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and n' :: "nat" assumes "is_path \<pi>"   and "\<pi> n \<noteq> local.return"   and "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>\<^esup> n'" shows "n = n'"
lcmI: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c"   and "b dvd c"   and "\<And>d. \<lbrakk>a dvd d; b dvd d\<rbrakk> \<Longrightarrow> c dvd d"   and "normalize c = c" shows "c = lcm a b"
the_inv_into_f_f: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e" assumes "inj_on f A"   and "x \<in> A" shows "the_inv_into A f (f x) = x"
assms: shows "s' = snd (fst (logical_instr_sub1 instr_name result s)) \<and> get_S (cpu_reg_val PSR s) = 0"
rep_eq: fixes x :: "'a poly \<times> 'a poly" shows "Rep_ratfps (local.quot_to_ratfps x) = (case x of (x, y) \<Rightarrow> let (x', y') = normalize_quot (x, y) in if constant_term y' = (0::'a) then 0 else quot_to_fract (x', y'))"
singlPart_def: fixes I :: "'a set" shows "singlPart I \<equiv> {{i} |i. i \<in> I}"
less_not_sym: fixes x :: "'a word"   and y :: "'a word" assumes "x <s y" shows "\<not> y <s x"
lemma_termdiff1: fixes z :: "'a"   and h :: "'a"   and m :: "nat" shows "(\<Sum>p<m. (z + h) ^ (m - p) * z ^ p - z ^ m) = (\<Sum>p<m. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))"
rtrancl_Image_in_Field: fixes R :: "('a \<times> 'a) set"   and V :: "'a set" shows "R\<^sup>* `` V \<subseteq> Field R \<union> V"
axioms_1: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
repNodes_eq_def: fixes p :: "ref"   and q :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and rep :: "ref \<Rightarrow> ref" shows "repNodes_eq p q low high rep \<equiv> (rep \<propto> high) p = (rep \<propto> high) q \<and> (rep \<propto> low) p = (rep \<propto> low) q"
ltakeWhile_LCons: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist" shows "ltakeWhile P (LCons x xs) = (if P x then LCons x (ltakeWhile P xs) else LNil)"
meas_N_eq: shows "(\<rightarrow>\<^sub>M) (dens_ctxt_measure \<Y> \<rho>) = (\<rightarrow>\<^sub>M) (state_measure (V \<union> V') \<Gamma>)"
\<UU>\<bb>\<xx>_invertible: fixes opinl :: "'opinl"   and ts :: "Unboxed.type option list"   and opubx :: "'opubx" assumes "\<UU>\<bb>\<xx> opinl ts = Some opubx" shows "\<BB>\<oo>\<xx> opubx = opinl"
assms_2: shows "InvariantClCurrentLevel (getCl state) (getM state)"
isomorphic_group_triviality: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "G \<cong> H"   and "Group.group G"   and "Group.group H" shows "trivial_group G = trivial_group H"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "max b (max a c) = max a (max b c)"
sep_disj_nat_def: fixes m1 :: "nat"   and m2 :: "nat" shows "m1 ## m2 \<equiv> True"
unsigned_Hahn_decomposition: fixes N :: "'b measure"   and M :: "'b measure"   and A :: "'b set" assumes "sets N = sets M"   and "A \<in> sets M"   and "emeasure M A \<noteq> top"   and "emeasure N A \<noteq> top" shows "\<exists>Y\<in>sets M. Y \<subseteq> A \<and> (\<forall>X\<in>sets M. X \<subseteq> Y \<longrightarrow> emeasure N X \<le> emeasure M X) \<and> (\<forall>X\<in>sets M. X \<subseteq> A \<longrightarrow> X \<inter> Y = {} \<longrightarrow> emeasure M X \<le> emeasure N X)"
nth_prime_nonzero: fixes n :: "nat" shows "nth_prime n \<noteq> 0"
Ds_mono: fixes A :: "'b set option"   and A' :: "'b set option"   and es :: "'b exp list" assumes "A \<sqsubseteq> A'"   and "\<D>s es A" shows "\<D>s es A'"
single_valued_monomial_rel': shows "IS_LEFT_UNIQUE monomial_rel"
indRelRTPO_to_indRelRSTPO: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q"   and "refl SRel" shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q"
order_less_subst1: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c"   and c :: "'c" assumes "a < f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
fmlookup_ran_iff: fixes y :: "'c"   and m :: "('d, 'c) fmap" shows "(y |\<in>| fmran m) = (\<exists>x. m $$ x = Some y)"
elims_2: fixes x :: "('a, unit) RBT.rbt"   and xa :: "('a \<times> 'a) list" assumes "wf_list_graph_impl_rs x xa" obtains V :: "('a, unit) RBT.rbt" where "x = V__" and "xa = []"   | V :: "('a, unit) RBT.rbt" and v1 :: "'a" and v2 :: "'a" and Es :: "('a \<times> 'a) list" where "x = V" and "xa = (v1, v2) # Es" and "rbt_contains v1 V \<and> rbt_contains v2 V \<and> wf_list_graph_impl_rs V Es"
empty_eqvt: fixes pi :: "('b \<times> 'b) list" shows "pi \<bullet> {} = {}"
map_le_same_dom_imp_same_map: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "dom f = dom g"   and "f \<subseteq>\<^sub>m g" shows "f = g"
stateinterpol_diff: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real" shows "stateinterpol \<nu> \<omega> (statediff \<nu> \<omega>) = \<nu>"
Ri_limit_Sup: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and Ns :: "'b literal multiset set llist" assumes "sat_preserving_redundancy_criterion \<Gamma> Rf Ri"   and "chain (redundancy_criterion.derive \<Gamma> Rf) Ns" shows "Ri (Liminf_llist Ns) = Ri (Sup_llist Ns)"
simps_89: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x81 :: "'c formula"   and x82 :: "\<I>"   and x83 :: "'c formula" shows "(case Since x81 x82 x83 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f8 x81 x82 x83"
wellformed_init_item: fixes r :: "symbol \<times> symbol list"   and k :: "nat" assumes "r \<in> \<RR>"   and "k \<le> length Doc" shows "wellformed_item (init_item r k)"
yun_erel_pow: fixes F :: "int poly"   and f :: "rat poly"   and n :: "nat" assumes "yun_erel F f" shows "yun_erel (F ^ n) (f ^ n)"
tailR_def: fixes t :: "('c, 'd) FingerTreeStruc" shows "tailR t = (case viewR t of Some (x, m) \<Rightarrow> m)"
interval_bounds_real_2: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "Inf {a..b} = a"
Some_eq_map_of_iff: fixes xys :: "('a \<times> 'b) list"   and y :: "'b"   and x :: "'a" assumes "distinct (list.map fst xys)" shows "(Some y = map_of xys x) = ((x, y) \<in> set xys)"
wf_permute_alts: fixes \<sigma> :: "'b \<Rightarrow> 'b" assumes "\<sigma> permutes alts" shows "is_pref_profile (permute_profile \<sigma> R)"
replicate_eq_replicate: fixes m :: "nat"   and x :: "'g"   and n :: "nat"   and y :: "'g" shows "(replicate m x = replicate n y) = (m = n \<and> (m \<noteq> 0 \<longrightarrow> x = y))"
infinitesimal_linear'_helper: fixes At :: "atomUni"   and p :: "real \<times> real \<times> real"   and B :: "real"   and C :: "real"   and x :: "real" assumes "At = LessUni p \<or> At = EqUni p"   and "B \<noteq> 0" shows "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. aEvalUni At x) = evalUni (substInfinitesimalLinearUni B C At) x"
fold_compose1: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "(\<lambda>x. f (g x)) = f \<circ> g"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>l2. P [] l2"   and "\<And>v va. P (v # va) []"   and "\<And>x1 l1 x2 l2. \<lbrakk>x1 < x2 \<Longrightarrow> P l1 (x2 # l2); \<lbrakk>\<not> x1 < x2; x1 = x2\<rbrakk> \<Longrightarrow> P l1 l2; \<lbrakk>\<not> x1 < x2; x1 \<noteq> x2\<rbrakk> \<Longrightarrow> P (x1 # l1) l2\<rbrakk> \<Longrightarrow> P (x1 # l1) (x2 # l2)" shows "P a0 a1"