hcju/isabelleps huggingface dataset:
f13: shows "word_of_int (uint (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) * uint (get_operand2 (snd instr) s1)) \<noteq> (if fst instr = arith_type UMUL \<or> fst instr = arith_type UMULcc then word_of_int (uint (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) * uint (get_operand2 (snd instr) s1)) else word_of_int (sint (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) * sint (get_operand2 (snd instr) s1))) \<or> word_of_int (uint (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) * uint (get_operand2 (snd instr) s2)) \<noteq> (if fst instr = arith_type UMUL \<or> fst instr = arith_type UMULcc then word_of_int (uint (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) * uint (get_operand2 (snd instr) s2)) else word_of_int (sint (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) * sint (get_operand2 (snd instr) s2))) \<or> low_equal (snd (fst (write_cpu (ucast ((if fst instr = arith_type UMUL \<or> fst instr = arith_type UMULcc then word_of_int (uint (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) * uint (get_operand2 (snd instr) s1)) else word_of_int (sint (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) * sint (get_operand2 (snd instr) s1))) >> 32)) Y (snd (fst (get_curr_win () s1)))))) (snd (fst (write_cpu (ucast ((if fst instr = arith_type UMUL \<or> fst instr = arith_type UMULcc then word_of_int (uint (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) * uint (get_operand2 (snd instr) s2)) else word_of_int (sint (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) * sint (get_operand2 (snd instr) s2))) >> 32)) Y (snd (fst (get_curr_win () s2))))))"
specKeysSecrets_def: fixes C :: "specID" shows "specKeysSecrets C \<equiv> {kKS x |x. x \<in> specKeys C} \<union> {sKS s |s. s \<in> specSecrets C}"
pmatch_list_length_neq: fixes vs :: "v list"   and ps :: "pat list"   and cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and s :: "v store_v list"   and m :: "(char list \<times> v) list match_result" assumes "length vs \<noteq> length ps" shows "fold2 (\<lambda>p v m. case m of No_match \<Rightarrow> No_match | Match_type_error \<Rightarrow> Match_type_error | Match env \<Rightarrow> pmatch_single cenv s p v env) Match_type_error ps vs m = Match_type_error"
rel_inject_2: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x2 :: "'e"   and y2 :: "'f" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
universal_denial_def: fixes A :: "'b set"   and B :: "'b set" shows "A E B \<equiv> \<forall>b\<in>B. b \<notin> A"
iso_is_retraction: fixes f :: "'a" assumes "C.iso f" shows "C.retraction f"
ennsqrt_mono2: fixes x :: "ennreal"   and y :: "ennreal" assumes "x \<le> y" shows "ennsqrt x \<le> ennsqrt y"
trancl_trans: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set"   and z :: "'c" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
refine: fixes S :: "('a, unit) RBT.rbt"   and P :: "'a \<Rightarrow> bool" assumes "rs.invar S" shows "RETURN (find_max_impl P S) \<le> find_max P (rs.\<alpha> S)"
step_1_main_dims_main: fixes n :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_1_main n i j A) = dim_row A \<and> dim_col (??.Jordan_Normal_Form_Existence.step_1_main n i j A) = dim_col A"
rel_S_def: fixes L1 :: "'h \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'i \<Rightarrow> 'p \<Rightarrow> bool"   and Co1 :: "'j \<Rightarrow> 'q \<Rightarrow> bool"   and Co2 :: "'k \<Rightarrow> 'r \<Rightarrow> bool"   and Contra1 :: "'l \<Rightarrow> 's \<Rightarrow> bool"   and Contra2 :: "'m \<Rightarrow> 't \<Rightarrow> bool" shows "rel_S L1 L2 Co1 Co2 Contra1 Contra2 = vimage2p Rep_S Rep_S (rel_G L1 L2 Co1 Co2 Contra1 Contra2)"
ord_eq_less_trans: fixes a :: "'a"   and b :: "'a"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a" assumes "a = b"   and "less b c" shows "less a c"
assms_4: shows "path_image g \<subseteq> cbox a b"
is_run_def: fixes r :: "nat \<Rightarrow> 'v" shows "G'.is_run r \<equiv> r 0 \<in> G'.V0 \<and> ipath G'.E r"
neutral_map: fixes A :: "'c set" shows "(\<lambda>y. \<top>\<^sub>o\<^sub>w) ` A \<subseteq> U"
simps: fixes a :: "'b set"   and sp :: "'b set"   and A :: "'b set set" shows "(a \<in> sigma_sets sp A) = ((\<exists>aa. a = aa \<and> aa \<in> A) \<or> a = {} \<or> (\<exists>aa. a = sp - aa \<and> aa \<in> sigma_sets sp A) \<or> (\<exists>aa. a = \<Union> (range aa) \<and> (\<forall>x. aa x \<in> sigma_sets sp A)))"
P_mod_diffxxx2: fixes t :: "'a"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and m :: "nat" assumes "Idomain S"   and "t \<in> carrier S"   and "t \<noteq> \<zero>\<^bsub>S\<^esub>"   and "maximal_ideal S (S \<diamondsuit>\<^sub>p t)"   and "PolynRg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y"   and "f \<in> carrier R"   and "g \<in> carrier R"   and "h \<in> carrier R"   and "deg R S X g \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g)"   and "deg R S X h + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g) \<le> deg R S X f"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g)"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) h)"   and "rel_prime_pols R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g) (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) h)"   and "P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (f \<plusminus> -\<^sub>a g \<cdot>\<^sub>r h)"   and "0 < m" shows "\<exists>g1 h1. g1 \<in> carrier R \<and> h1 \<in> carrier R \<and> deg R S X g1 \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g1) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (g \<plusminus> -\<^sub>a g1) \<and> deg R S X h1 + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) g1) \<le> deg R S X f \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (h \<plusminus> -\<^sub>a h1) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) (f \<plusminus> -\<^sub>a g1 \<cdot>\<^sub>r h1)"
lm065: fixes f :: "('c \<times> 'd) set"   and x :: "'c"   and A :: "'d set" assumes "runiq f"   and "x \<notin> Domain f" shows "{f \<union> {(x, y)} |y. y \<in> A} \<subseteq> runiqs"
pnTimes_PLUS: fixes xs :: "'a rexp list"   and r :: "'a rexp" assumes "xs \<noteq> []"   and "\<forall>x\<in>set xs. \<not> is_Zero x \<and> \<not> is_Plus x" shows "pnTimes (PLUS xs) r = flatten PLUS (Timess (set xs) r)"
in_set_spmf: fixes x :: "'c"   and p :: "'c spmf" shows "(x \<in> set_spmf p) = (Some x \<in> set_pmf p)"
mset_ran_subst_outside: fixes i :: "int"   and r :: "int set"   and a :: "int \<Rightarrow> 'a"   and x :: "'a" assumes "i \<notin> r" shows "mset_ran (a(i := x)) r = mset_ran a r"
simps_1: fixes bdd :: "bdd" shows "Rmi_g 0 Falseif bdd = True"
PBD_axioms: shows "PBD \<V> \<B> \<K>"
ar_def: fixes x :: "'a" shows "ar x = (1::'a) \<sqinter> - r x"
da_elim_cases_58: fixes Env :: "env"   and B :: "lname set"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In3 []\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
finite_deriv_prove: fixes s :: "(nat \<times> form) list" shows "finite (deriv s) = prove s"
assms: fixes "is" :: "nat list" assumes "is \<lhd> dims A" shows "lookup A is = e is"
ex_pivot_fun: shows "\<exists>f. pivot_fun A f (dim_col A)"
finprod_finprod: fixes A :: "'e set"   and B :: "'f set"   and g :: "'e \<Rightarrow> 'f \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> carrier G" shows "(\<Otimes>a\<in>A. finprod G (g a) B) = (\<Otimes>(a, b)\<in>A \<times> B. g a b)"
not_less_iff_gr_or_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "linorder less_eq less" shows "(\<not> less x y) = (less y x \<or> x = y)"
encode_problem_with_operator_interference_exclusion_complete_i: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list" assumes "is_valid_problem_strips \<Pi>"   and "is_parallel_solution_for_problem \<Pi> \<pi>"   and "\<forall>k<length \<pi>. are_all_operators_non_interfering (\<pi> ! k)" shows "valuation_for_plan \<Pi> \<pi> \<Turnstile> encode_interfering_operator_exclusion \<Pi> (length \<pi>)"
moebius_mu_squarefree_eq: fixes n :: "nat" assumes "squarefree n" shows "moebius_mu n = (- (1::'b)) ^ card (prime_factors n)"
ccw_vector_space_axioms: shows "ccw_vector_space ccw S"
ford_fulkerson_1: shows "isMaxFlow f = (\<not> Ex isAugmentingPath)"
A7: shows "\<forall>a b c p q. B\<^sub>\<real> a p c \<and> B\<^sub>\<real> b q c \<longrightarrow> (\<exists>x. B\<^sub>\<real> p x b \<and> B\<^sub>\<real> q x a)"
Ana_var: fixes x :: "('fun, 'atom) Term.term \<times> nat" shows "Ana (Var x) = ([], [])"
apply_one: shows "(\<langle>$\<rangle>) 1 = id"
one_prod_def: shows "1 = (1::'a, 1::'b)"
permutation_conv_induced_automorph: fixes X :: "'b set set"   and C0 :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "ThinChamberComplexManyFoldings X C0"   and "(f, g) \<in> ThinChamberComplexManyFoldings.fundfoldpairs X C0" shows "(\<rightarrow>) (Abs_permutation (OpposedThinChamberComplexFoldings.induced_automorphism X f g)) = OpposedThinChamberComplexFoldings.induced_automorphism X f g"
P_ps_def: shows "P_ps = (SOME x. valid_decomp X (snd x) \<and> standard_decomp (poly_deg f) (snd x) \<and> exact_decomp X 0 (snd x) \<and> cone_decomp (fst x) (snd x) \<and> hom_decomp (snd x) \<and> direct_decomp (ideal F \<inter> P[X]) [ideal {f} \<inter> P[X], fst x])"
is_len_cong: fixes A :: "'p"   and B :: "'p"   and l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and C :: "'p"   and D :: "'p" assumes "TarskiLen A B l"   and "TarskiLen C D l" shows "Cong A B C D"
id_dom_cod_1: fixes A :: "'o" assumes "A \<in> Ob" shows "Dom (Id A) = A"
wf_Ord_ordertype: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "wf r"   and "trans r" shows "Ord (ordertype A r)"
indoc_def: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and u :: "'a list" assumes "traces ind" shows "traces.indoc ind a u \<equiv> \<exists>u\<^sub>1 u\<^sub>2. u = u\<^sub>1 @ [a] @ u\<^sub>2 \<and> a \<notin> set u\<^sub>1 \<and> (\<forall>a\<in>{a}. \<forall>b\<in>set u\<^sub>1. ind a b)"
observable_moves_singular_slice_edge: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')"   and "tl ms = targetnodes rs"   and "length rs = length cs"   and "\<forall>i<length cs. call_of_return_node (tl ms ! i) (sourcenode (cs ! i))" shows "slice_edges S cs as = [last as]"
cr_cblinfun_def: shows "cr_cblinfun \<equiv> \<lambda>x y. x = cblinfun_apply y"
sorted_lr_of_tran_s3_hlp: fixes f :: "('b \<times> 32 simple_match \<times> char list \<times> simple_action) list"   and a :: "'b"   and b :: "'b \<times> of_match_field set \<times> of_action list"   and s :: "char list list" assumes "\<forall>x\<in>set f. fst x \<le> a"   and "b \<in> set (lr_of_tran_s3 s f)" shows "fst b \<le> a"
spref_ext: fixes r :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "r <p u" shows "r <p u \<cdot> v"
a: shows "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
that: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "flow \<Delta> f"   and "value_flow \<Delta> f = v" shows "thesis"
a_consistent_stackD: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, e, S)" shows "Astack S \<sqsubseteq> a"
B: shows "n \<notin> bad_shakey \<inter> (bad_pwd \<union> bad_prikey)"
homeomorphic_balls_1: fixes d :: "real"   and e :: "real"   and a :: "'b"   and b :: "'b" assumes "0 < d"   and "0 < e" shows "ball a d homeomorphic ball b e"
assms: shows "\<exists>q r. b = q * a + smult m r"
fine_append: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and c :: "real"   and D' :: "(real \<times> real \<times> real) list" assumes "fine \<delta> (a, b) D"   and "fine \<delta> (b, c) D'" shows "fine \<delta> (a, c) (D @ D')"
decomposition_clause_pos_neg: fixes C :: "'at Literal set" shows "C = negative_part C \<union> positive_part C"
suffix_length_le: fixes xs :: "'d list"   and ys :: "'d list" assumes "suffix xs ys" shows "length xs \<le> length ys"
cont: shows "\<forall>\<^sub>F n in F. continuous_on (cball z r) (f n) \<and> (\<forall>w\<in>ball z r. (f n has_field_derivative f' n w) (at w))"
dbl_simps_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (uminus (numeral.numeral one plus k)) = uminus (neg_numeral.dbl plus (numeral.numeral one plus k))"
LeftRightInvUniq: fixes h :: "'b"   and f :: "'b"   and g :: "'b" assumes "h \<approx>> f"   and "f \<approx>> g"   and "f ;; g = id (dom f)"   and "h ;; f = id (cod f)" shows "h = g"
diff_union_cancelR: fixes M :: "'k multiset"   and N :: "'k multiset" shows "M + N - N = M"
simps_1: fixes x :: "nat"   and ya :: "nat" shows "(LnChain x = LnChain ya) = (x = ya)"
hyps_5: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "rho \<in> envsOf P" shows "freshEnv xs x rho"
25: fixes x :: "'a" shows "bot \<squnion> x = x"
convergent_powser'_geometric: shows "convergent_powser' (msllist_of_msstream (mssalternate 1 (- 1))) (\<lambda>x. inverse (1 + x))"
mod_mult_left_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times (modulo a c) b) c = modulo (times a b) c"
isoE: fixes f :: "'a" assumes "local.iso f" obtains g :: "'a" where "inverse_arrows f g"
diff_add_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ab_group_add plus zero minus uminus" shows "plus (minus a b) c = minus (plus a c) b"
that: fixes ns\<^sub>1 :: "'node list"   and ns\<^sub>2 :: "'node list" assumes "g \<turnstile> n-ns\<^sub>1\<rightarrow>x"   and "g \<turnstile> x-ns\<^sub>2\<rightarrow>m"   and "ns = ns\<^sub>1 @ tl ns\<^sub>2"   and "ns = butlast ns\<^sub>1 @ ns\<^sub>2" shows "thesis"
of_nat_nat_take_bit_eq: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and n :: "nat"   and k :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.of_nat one plus zero (nat (take_bit n k)) = ring_1.of_int one plus zero minus (take_bit n k)"
wa_refine_axioms: shows "wa_refine WAC WAA \<alpha>"
crit_pair_cbelow_imp_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "\<And>p q. \<lbrakk>p \<in> F; q \<in> F; p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m F p q" shows "is_Groebner_basis F"
Utrans_P_is_tensor_P1: fixes A :: "complex mat" shows "Utrans_P vars1 A = Utrans (tensor_P A (1\<^sub>m K))"
K_cfg_def: fixes cfg :: "'a cfg" shows "K_cfg cfg = map_pmf (cont cfg) (action cfg)"
cong_4312: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Cong D C A B"
max_list_non_empty_map: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" assumes "\<And>x y. max (f x) (f y) = f (max x y)"   and "xs \<noteq> []" shows "max_list_non_empty (map f xs) = f (max_list_non_empty xs)"
tensor_relations2_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme"   and MN :: "('e \<times> 'f, 'g, 'i) Module_scheme" shows "TR2 R M N MN = {x. \<exists>m\<in>carrier M. \<exists>n1\<in>carrier N. \<exists>n2\<in>carrier N. x = (m, n1 \<plusminus>\<^bsub>N\<^esub> n2) \<plusminus>\<^bsub>MN\<^esub> -\<^sub>a\<^bsub>MN\<^esub> ((m, n1) \<plusminus>\<^bsub>MN\<^esub> (m, n2))}"
l1': shows "(\<lambda>n. fst (f (r1 (r2 n)))) \<longlonglongrightarrow> l1"
preservation: shows "rel_weakly_preserves_barbs (indRelLSTPO SRel TRel) (STCalWB SWB TWB)"
set_Keys_to_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (Keys_to_list ps) = Keys (set ps)"
abs_div_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y" shows "divide (abs x) y = abs (divide x y)"
WTrt_hext_mono: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and h' :: "heap" assumes "P,E,h,sh \<turnstile> e : T"   and "h \<unlhd> h'" shows "P,E,h',sh \<turnstile> e : T"
rev_transversal_segment: fixes a :: "'a"   and b :: "'a" shows "rev.transversal_segment a b = transversal_segment a b"
entt_emp: fixes A :: "assn" shows "A \<Longrightarrow>\<^sub>t emp"
openin_closedin_eq: fixes U :: "'b topology"   and S :: "'b set" shows "openin U S = (S \<subseteq> topspace U \<and> closedin U (topspace U - S))"
poly_higher_pderiv_aux3: fixes n :: "nat"   and k :: "nat"   and c :: "'a"   and p :: "'a poly" assumes "n \<le> k" shows "\<exists>q. poly ((pderiv ^^ k) ([:- c, 1::'a:] ^ n * p)) c = fact n * poly q c"
anycast_simps_5: fixes P :: "'a \<Rightarrow> bool"   and m :: "'a" shows "anycast P (send m) = True"
intro: fixes K :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "subgroup_of_group K G composition unit"   and "normal_subgroup_axioms K G composition unit" shows "normal_subgroup K G composition unit"
strip_while_idem: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" assumes "no_trailing P xs" shows "strip_while P xs = xs"
chain_imageI: fixes le_a :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Y :: "'f set"   and le_b :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" assumes "Complete_Partial_Order.chain le_a Y"   and "\<And>x y. \<lbrakk>x \<in> Y; y \<in> Y; le_a x y\<rbrakk> \<Longrightarrow> le_b (f x) (f y)" shows "Complete_Partial_Order.chain le_b (f ` Y)"
ngba_acc_def: fixes cs :: "('c \<Rightarrow> bool) list"   and p :: "'c" shows "ngba_acc cs p \<equiv> {k \<in> {0..<length cs}. (cs ! k) p}"
Gr_domain_1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "Domain (BNF_Def.Gr A f) = A"
assms_2: shows "x' \<in> S'.set (S'.dom f')"
scaleR_2: fixes x :: "'b" shows "2 *\<^sub>R x = x + x"
replace_red_stable_is_red: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.is_red F f"   and "punit.red (F - {p}) p q" shows "punit.is_red (insert q (F - {p})) f"
visits_step: fixes target :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and t :: "(String.literal \<times> value list) list" shows "visits target e s r (h # t) = (\<exists>(s', T)|\<in>|possible_steps e s r (fst h) (snd h). visits target e s' (evaluate_updates T (snd h) r) t)"
PRJ\<^sub>0_def: shows "PRJ\<^sub>0 \<equiv> \<lambda>K H. if cospan K H then \<lbrakk>\<lbrakk>B.prj\<^sub>0 (REP H) (REP K)\<rbrakk>\<rbrakk> else null"
disturbed_one_step_axioms: shows "disturbed_one_step t t1 s s0 x incr p B L"
sig_inv_set_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "sig_inv_set' i \<subseteq> sig_inv_set' j"
border_length_r_less: fixes ys :: "'a list" shows "\<forall>xs. strict_border xs ys \<longrightarrow> length xs < length ys"
iterateoi_rule: fixes m :: "'s" assumes "invar m" shows "map_iterator_linord (iterateoi m) (\<alpha> m)"
permute_zero: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pt permute" shows "permute 0 x = x"
ListUpdateE_updates: fixes xs :: "('a \<times> 'b) list"   and x :: "'a \<times> 'b"   and l'' :: "'a \<Rightarrow> 'b" assumes "distinct (map fst xs)"   and "x \<in> set xs" shows "ListUpdateE l'' xs (fst x) = snd x"
matches_not_idem: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchNot (MatchNot m)) a p = matches \<gamma> m a p"
node_ptr_casts_commute: fixes ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr" shows "(cast ptr = Some node_ptr) = (cast node_ptr = ptr)"