hcju/isabelleps huggingface dataset:
GreenThm_typeI_divisible: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeI_chain i j F two_chain s"   and "gen_division s (cubeImage ` two_chain)" shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} (two_chain_boundary two_chain)"
ref_simps_1: fixes cf :: "CAttId"   and a :: "nat" shows "ref (objLoc cf a) = objV (cls cf) a"
cases: fixes a :: "'node"   and n :: "'node"   and S :: "'node set" assumes "a \<in> obs_intra n S" obtains as :: "'edge list" and n' :: "'node" where "a = n'" and "n -as\<rightarrow>\<^sub>\<iota>* n'" and "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S" and "n' \<in> S"
is_zero_row_eq_row_zero: fixes a :: "'b"   and A :: "(('c, 'd) vec, 'b) vec" shows "is_zero_row a A = (row a A = 0)"
eval_monom_smallomega: fixes basis :: "(real \<Rightarrow> real) list"   and es :: "real list"   and e :: "real" assumes "basis \<noteq> []"   and "basis_wf basis"   and "length es = length basis"   and "e < hd es" shows "eval_monom (1, es) basis \<in> \<omega>(\<lambda>x. hd basis x powr e)"
ga_pid_system_automaton: shows "ga_automaton = pid_system_automaton"
assms: shows "Lfilled k lholed [Label n les es] LI"
some_betw: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "c \<in> Q"   and "a \<noteq> b"   and "a \<noteq> c"   and "b \<noteq> c" shows "[[a b c]] \<or> [[b c a]] \<or> [[c a b]]"
identity_telescopic: fixes x :: "nat \<Rightarrow> 'b"   and l :: "'b"   and n :: "nat" assumes "x \<longlonglongrightarrow> l" shows "(\<lambda>N. \<Sum>k = n..N. x (Suc k) - x k) \<longlonglongrightarrow> l - x n"
minusinf_repeats: fixes p :: "fm"   and d :: "int"   and bbs :: "bool list"   and x :: "int"   and k :: "int"   and bs :: "int list" assumes "d_\<delta> p d"   and "iszlfm p" shows "Ifm bbs ((x - k * d) # bs) (minusinf p) = Ifm bbs (x # bs) (minusinf p)"
take_butlast: fixes n :: "nat"   and xs :: "'e list" assumes "n < length xs" shows "take n (butlast xs) = take n xs"
intros_2: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "\<phi> \<omega>"   and "(\<phi> suntil \<psi>) (stl \<omega>)" shows "(\<phi> suntil \<psi>) \<omega>"
map_poly_inj_zero_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_inj_zero_hom hom \<equiv> inj_zero_hom hom"
assms: fixes x :: "int poly" assumes "x \<in># factors_of_factor_tree t" shows "Mp x = x"
assms_2: assumes "otherbot \<noteq> default_value" shows "\<exists>G nP i F. wf_graph G \<and> \<not> sinvar G nP \<and> F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<and> sinvar (delete_edges G F) nP \<and> i \<in> snd ` F \<and> sinvar G (nP(i := otherbot))"
Un_Int_eq_4: fixes T :: "'e set"   and S :: "'e set" shows "T \<inter> (S \<union> T) = T"
list_update_append1: fixes i :: "nat"   and xs :: "'g list"   and ys :: "'g list"   and x :: "'g" assumes "i < length xs" shows "(xs @ ys)[i := x] = xs[i := x] @ ys"
inj_vimage_image_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj f" shows "f -` f ` A = A"
tau_proj_X: shows "\<tau> \<upharpoonleft> X = x # xs"
supp_restrict0: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "supp (f \<down> A) \<subseteq> A"
intro: fixes ginit :: "('a, 'b, 'c) Gen_Scheduler.global_config set"   and wf_local :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>gc. gc \<in> ginit \<Longrightarrow> sched_typing.wf_gglobal wf_local gc" shows "sched_typing_init_axioms ginit wf_local"
simps_1: fixes P :: "pi" shows "resChain [] P = P"
jxzP: fixes i :: "'b" assumes "i \<in> {j}" shows "x \<^bsub>(P i)\<^esub>\<prec> z"
mod_diff_mult_self2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m * k \<le> n" shows "(n - m * k) mod m = n mod m"
Int_Ico: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.linorder less_eq less" shows "ord.atLeast less_eq a \<inter> ord.atLeast less_eq b = ord.atLeast less_eq (ord.max less_eq a b)"
addrow_mat: fixes A :: "'c mat"   and n :: "nat"   and nc :: "nat"   and l :: "nat"   and a :: "'c"   and k :: "nat" assumes "A \<in> carrier_mat n nc"   and "l < n" shows "addrow a k l A = addrow_mat n a k l * A"
seqSubstInputChain: fixes xvec :: "name list"   and \<sigma> :: "(name list \<times> 'a list) list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<sigma>" shows "seqSubs' (inputChain xvec N P) \<sigma> = inputChain xvec (N[<\<sigma>>]) (P[<\<sigma>>])"
poly_eval_focus_in_Polys: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and a :: "'b \<Rightarrow> 'c"   and Y :: "'b set" assumes "p \<in> P[X]" shows "poly_eval (\<lambda>x. monomial (a x) 0) (focus Y p) \<in> P[X - Y]"
mult_cong_r: fixes b :: "'a"   and b' :: "'a"   and a :: "'a" assumes "b \<sim> b'"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b' \<in> carrier G" shows "a \<otimes> b \<sim> a \<otimes> b'"
comm_semiring_0_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "comm_semiring_0_ow U plus zero times \<equiv> comm_semiring_ow U plus times \<and> comm_monoid_add_ow U plus zero \<and> mult_zero_ow U times zero"
index_correct_1: fixes idx_f :: "('L, ('Q, 'L) ta_rule dlist) HashMap.hashmap" assumes "hta_idx_f H = Some idx_f" shows "hll_idx.is_index rhsl (ls.\<alpha> \<delta>) idx_f"
cpt_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and P :: "'c set"   and env :: "'c \<Rightarrow> nat"   and \<pi>l :: "'d list" assumes "Shallow S s0 \<delta> AP L AP'" shows "cpt P env \<pi>l \<equiv> \<forall>p\<in>P. env p < length \<pi>l"
Knuth_ex_4_6_2_22_main_axioms: shows "Knuth_ex_4_6_2_22_main a b u v w"
B: fixes x :: "'a" assumes "x \<in> X" shows "dist x (g (f x)) \<le> 3 * C * lambda"
1: shows "bp (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> wrap {} \<union> P\<^sub>2 \<union> wrap (B\<^sub>2 \<union> {u}) \<union> {{v} |v. v \<in> V - {u}})"
ln_primorial: fixes x :: "real" shows "ln (real (primorial x)) = \<theta> x"
subst_type_IT: fixes t :: "dB"   and e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and U :: "LambdaType.type"   and T :: "LambdaType.type"   and u :: "dB" assumes "IT t"   and "e\<langle>i:U\<rangle> \<turnstile> t : T"   and "IT u"   and "e \<turnstile> u : U" shows "IT (t[u/i])"
erf_remainder_integral_def: fixes n :: "nat"   and x :: "real" shows "erf_remainder_integral n x = lim (\<lambda>m. integral {x..x + real m} (\<lambda>t. exp (- t\<^sup>2) / t ^ (2 * n)))"
is_regular_sequence_remdups: fixes fs :: "('c \<Rightarrow>\<^sub>0 'd) list" assumes "is_regular_sequence fs" shows "is_regular_sequence (rev (remdups (rev fs)))"
to_hma\<^sub>v_def: fixes v :: "'b Matrix.vec" shows "Mod_Type_Connect.to_hma\<^sub>v v = (\<chi>i. v $v mod_type_class.to_nat i)"
eventually_nhds_in_open: fixes s :: "'a set"   and x :: "'a" assumes "open s"   and "x \<in> s" shows "\<forall>\<^sub>F y in nhds x. y \<in> s"
4: shows "submatrix A I UNIV *\<^sub>v v = 0\<^sub>v (card {i. i < n \<and> i \<in> I})"
DAcc_def: fixes i :: "AbstrLevelsID"   and C :: "CSet" shows "DAcc i C \<equiv> {Z. \<exists>x. x \<in> OUT C \<and> x \<in> IN Z \<and> Z \<in> AbstrLevel i \<and> C \<in> AbstrLevel i}"
DSourcesA6_L0: shows "DSources level0 sA6 = {}"
pred_inject_2: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and aa :: "'c llist" shows "llist_all P (LCons a aa) = (P a \<and> llist_all P aa)"
assms_2: shows "\<forall>Q\<in>X. dom Q = dom P"
d_def: shows "d = (\<lambda>n. if even n then 2 * real n else 0)"
is_sig_GB_inI_triv: fixes d :: "'a \<Rightarrow> nat"   and u :: "'t"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "\<not> d (pp_of_term u) \<le> dgrad_max d \<or> \<not> component_of_term u < length fs" shows "is_sig_GB_in d G u"
mat_of_cols_carrier_3: fixes n :: "nat"   and vs :: "'a vec list" shows "dim_col (mat_of_cols n vs) = length vs"
cod_simps_2: shows "fgh.\<nu>\<pi>.cod.leg0 = fgh.\<nu>\<pi>.leg0"
scale_zero_left: fixes x :: "'a poly" shows "smult (0::'a) x = 0"
plus_le_conv: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "(x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z) = (x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z)"
iterates_def: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.ccpo Sup less_eq less" shows "ccpo.iterates Sup less_eq \<equiv> \<lambda>f. {x. ccpo.iteratesp Sup less_eq f x}"
Hash_synth_analz: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<notin> synth (analz H)" shows "(Hash \<lbrace>X, Y\<rbrace> \<in> synth (analz H)) = (Hash \<lbrace>X, Y\<rbrace> \<in> analz H)"
word_log_esimps_7: fixes x :: "'a word" shows "0 && x = 0"
init_item_rule: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "item_rule (init_item r k) = r"
ne_lists_2: fixes xs :: "'b list"   and A :: "'b set" assumes "xs \<noteq> []"   and "xs \<in> lists A" shows "tl xs \<in> lists A"
times_eint_simps_4: fixes m :: "int" shows "eint m * \<infinity> = \<infinity>"
of_seq_snth: fixes xs :: "'a stream" shows "of_seq ((!!) xs) = xs"
Bignorabimus_single: fixes I :: "interp"   and \<alpha> :: "game"   and X :: "(variable \<Rightarrow> real) set"   and \<omega> :: "variable \<Rightarrow> real"   and M :: "variable set" shows "game_sem I \<alpha> (selectlike X \<omega> M) \<subseteq> game_sem I \<alpha> X"
additive_subgroup_a_kernel: shows "additive_subgroup (a_kernel R S h) R"
Lex_inject: fixes x :: "'b list"   and y :: "'b list" assumes "x \<in> {xs. True}"   and "y \<in> {xs. True}" shows "(Lex x = Lex y) = (x = y)"
bilinear_eq: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'a set"   and B :: "'a set"   and T :: "'b set"   and C :: "'b set"   and x :: "'a"   and y :: "'b" assumes "bilinear f"   and "bilinear g"   and "S \<subseteq> span B"   and "T \<subseteq> span C"   and "x \<in> S"   and "y \<in> T"   and "\<And>x y. \<lbrakk>x \<in> B; y \<in> C\<rbrakk> \<Longrightarrow> f x y = g x y" shows "f x y = g x y"
inconsistent_iarrays_def: fixes A :: "'d iarray iarray"   and b :: "'d iarray" shows "inconsistent_iarrays A b = (\<not> consistent_iarrays A b)"
algebra_iff_Int: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "algebra \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<Omega> - a \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<inter> b \<in> M))"
idx_valid: fixes bound :: "'a bound"   and x :: "'a" assumes "in_bound less_eq0 less0 bound x" shows "idx0 bound x < size0 bound"
RHS: fixes f :: "'a \<Rightarrow> complex" assumes "continuous_on S f \<and> f ` S \<subseteq> sphere 0 1" shows "\<exists>g. continuous_on S (complex_of_real \<circ> g) \<and> (\<forall>x\<in>S. f x = exp (\<i> * complex_of_real (g x)))"
osum_Preorder: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Preorder r"   and "Preorder r'" shows "Preorder (r +o r')"
allocationUniverseProperty: fixes x :: "'a"   and X :: "'b set" shows "{(x, X)} - {(x, {})} \<in> allocationsUniverse"
simps: fixes a1 :: "call_kind"   and a2 :: "typeop"   and a3 :: "'a OCL_Types.type"   and a4 :: "'a OCL_Types.type"   and a5 :: "'a OCL_Types.type" shows "typeop_type a1 a2 a3 a4 a5 = ((\<exists>\<sigma> \<tau>. a1 = DotCall \<and> a2 = OclAsTypeOp \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<sigma> \<and> (\<sigma> < \<tau> \<or> \<tau> < \<sigma>)) \<or> (\<exists>\<sigma> \<tau>. a1 = DotCall \<and> a2 = OclIsTypeOfOp \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = Boolean[1] \<and> \<sigma> < \<tau>) \<or> (\<exists>\<sigma> \<tau>. a1 = DotCall \<and> a2 = OclIsKindOfOp \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = Boolean[1] \<and> \<sigma> < \<tau>) \<or> (\<exists>\<tau> \<rho> \<sigma> \<upsilon>. a1 = ArrowCall \<and> a2 = SelectByKindOp \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<upsilon> \<and> element_type \<tau> \<rho> \<and> \<sigma> < \<rho> \<and> update_element_type \<tau> \<sigma> \<upsilon>) \<or> (\<exists>\<tau> \<rho> \<sigma> \<upsilon>. a1 = ArrowCall \<and> a2 = SelectByTypeOp \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<upsilon> \<and> element_type \<tau> \<rho> \<and> \<sigma> < \<rho> \<and> update_element_type \<tau> \<sigma> \<upsilon>))"
SUP_constant: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and c :: "'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>y. c) ` A) = (if A = {} then bot else c)"
f_shrink_nth_eq_sublist_list: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div> k ! n = last_message (sublist_list xs [n * k..<n * k + k])"
size_neq: fixes x :: "'b ltln" shows "size x \<noteq> 0"
Fvars_tms: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "FvarsT (tms t1 t2) = FvarsT t1 \<union> FvarsT t2"
x: shows "isFDERIV DIM('a) [0..<DIM('a)] fas (list_of_eucl x)"
converse_rtranclp_induct: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P b"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>*\<^sup>* z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
Cons_4: shows "xs = ys @ j # concat (map (\<lambda>xs. xs @ [j]) (zs # xss)) @ as"
codomains_char: fixes f :: "('o, 'a) arr" shows "codomains f = (if arr f then {cod f} else {})"
f_borel: shows "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
greatest_fixpoint_same: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "is_greatest_fixpoint f x" shows "x = \<nu> f"
list_update_overwrite: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b"   and y :: "'b" shows "xs[i := x, i := y] = xs[i := y]"
add_left_zero: fixes a :: "'a" shows "(0::'a) + a = a"
right_imp_options: fixes opt :: "game"   and g :: "game" assumes "zin opt (right_options g)" shows "zin opt (options g)"
of_rat_of_int_eq: fixes z :: "int" shows "of_rat (rat_of_int z) = of_int z"
hyps: fixes S :: "(('a, 'b) fm list \<times> 'b) set"   and used :: "'b set" assumes "consistent A S"   and "(ps, a) \<in> S"   and "finite used"   and "A \<union> \<Union> (block_nominals ` S) \<subseteq> used" shows "consistent A ({(witness_list ps used, a)} \<union> S)"
Int_commute: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> B = B \<inter> A"
conj_left_commute: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "(P \<and> Q \<and> R) = (Q \<and> P \<and> R)"
square_one_cong_one: fixes x :: "'a"   and n :: "'a" assumes "[x = 1::'a] (mod n)" shows "[x\<^sup>2 = 1::'a] (mod n)"
is_acc_def: fixes G :: "('a, 'b) b_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'a" assumes "b_graph G" shows "b_graph.is_acc G r \<equiv> \<exists>\<^sub>\<infinity>i. r i \<in> bg_F G"
STSteps_steps_1: fixes S :: "'procS"   and P' :: "('procS, 'procT) Proc" shows "SourceTerm S \<longmapsto>ST* P' = (\<exists>S'. S' \<in>S P' \<and> S \<longmapsto>Source* S')"
ipd_impl_not_cd: fixes j :: "nat"   and k :: "nat"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n" assumes "j \<in> {k..i}"   and "\<pi> j = ipd (\<pi> k)" shows "\<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
subsetD: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
converse_in_Lin: fixes L :: "(alt \<times> alt) set" shows "(L\<inverse> \<in> Lin) = (L \<in> Lin)"
A: shows "v = u \<or> v \<in> sinks I D u xs"
drop_res_eq_f: fixes n :: "nat"   and f :: "nat \<Rightarrow> int"   and k :: "nat" assumes "maps_to_n (int (Suc n)) f"   and "\<not> (\<forall>m. \<exists>n>m. f n = int (Suc k))" shows "\<exists>N. \<forall>n>N. f n = drop_res (int (Suc k)) f n"
Cons_3: shows "i \<notin> list_nominals (p # ps)"
self_append_conv: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs = append xs ys) = (ys = [])"
fresh_star_def: fixes as :: "atom set"   and x :: "'a" shows "as \<sharp>* x \<equiv> \<forall>a\<in>as. a \<sharp> x"
map_update_def: fixes x :: "'d"   and y :: "'e option"   and f :: "'d \<Rightarrow> 'e option" shows "map_update x y f = f(x := y)"
in_q63_code: fixes x :: "nat" shows "(x mod 63 \<in> q63) = q63_array !! (x mod 63)"
compat_mult2_2: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "s O ns \<subseteq> s"   and "refl ns"   and "trans ns" shows "mult2_s ns s O mult2_ns ns s \<subseteq> mult2_s ns s"
igSubstIGAbs_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSubstIGAbs MOD \<equiv> \<forall>ys y Y xs x s X. isInBar (xs, s) \<and> igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
3: shows "Left_a.ide f \<and> Left_a.ide g"
Int_Diff: fixes A :: "'k set"   and g :: "'k \<Rightarrow> 'l"   and B :: "'k set" assumes "finite A" shows "sum g A = sum g (A \<inter> B) + sum g (A - B)"
last_index_le_size: fixes xs :: "'b list"   and x :: "'b" shows "last_index xs x \<le> length xs"
multi_arith_prog_mono: fixes m :: "nat"   and c :: "nat \<Rightarrow> nat"   and c' :: "nat \<Rightarrow> nat"   and start :: "int"   and steps :: "nat \<Rightarrow> nat" assumes "\<And>i. i < m \<Longrightarrow> c i \<le> c' i" shows "multi_arith_prog m start steps c \<le> multi_arith_prog m start steps c'"
Pd_refl: fixes x :: "'a"   and d :: "'b" assumes "x \<in> Field (BPd' d)" shows "(x, x) \<in> BPd' d"
netlistaux: fixes x :: "'b"   and a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "x \<in> set (net_list_aux (a # p))" shows "x \<in> set (net_list_aux [a]) \<or> x \<in> set (net_list_aux p)"
inv_lc_q_1: shows "inv lead_coeff q \<in> K - {\<zero>}"
indices_state_def: fixes s :: "('a, 'b) state" shows "indices_state s = {i. \<exists>x b. Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b)}"
cnvalidI: fixes \<Theta> :: "('e set \<times> 'g \<times> 'e set \<times> 'e set) set"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'g, 'f) com option"   and n :: "nat"   and F :: "'f set"   and c :: "('e, 'g, 'f) com"   and P :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "\<And>s t. \<lbrakk>\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P Call p Q,A; \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
at_fun_def: fixes P :: "'a \<Rightarrow> 'b"   and i :: "nat" shows "P !! i = (\<lambda>x. P (x[i ..]) !! i)"
simps: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lunstream g s = (case g s of Done \<Rightarrow> LNil | Skip s' \<Rightarrow> lunstream g s' | Yield x s' \<Rightarrow> LCons x (lunstream g s'))"
minus_mult_distrib_mat: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and B :: "'c mat"   and C :: "'c mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat nr n"   and "C \<in> carrier_mat n nc" shows "(A - B) * C = A * C - B * C"
toplevel_summands_ACI_norm_alt: fixes r :: "'c rexp" shows "toplevel_summands (ACI_norm_alt r) = ACI_norm_alt ` toplevel_summands r"
Union_rel_br: fixes l :: "'e \<Rightarrow> 'g set"   and lI :: "'e \<Rightarrow> bool"   and s :: "'g \<Rightarrow> 'f set"   and sI :: "'g \<Rightarrow> bool" shows "\<langle>br l lI, br s sI\<rangle>Union_rel = br (\<lambda>a. \<Union> (s ` l a)) (\<lambda>a. lI a \<and> (\<forall>c\<in>l a. sI c))"
deriv: fixes za :: "complex" assumes "cmod za < K" shows "a * f za = (1 + za) * f' za"
even_sign_changes_iff: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "hd xs \<noteq> (0::'a)"   and "last xs \<noteq> (0::'a)" shows "even (sign_changes xs) = (sgn (hd xs) = sgn (last xs))"
1: shows "direct_decomp T [sum_list ` listset ss, sum_list ` listset (map cone ps)]"
x1_nin_l1: shows "x1 \<notin> set l1"
check_ipresIGFreshAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model" shows "ipresIGFreshAbs check (errMOD MOD) MOD"
sync: shows "\<forall>a t t'. clock_jview a t = clock_jview a t' \<longrightarrow> tLength t = tLength t'"
get_host_ok: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_host shadow_root_ptr"
div_lt: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m < n * k" shows "m div n < k"
Omega_def: fixes x :: "'a"   and y :: "'a" shows "\<Omega> x y = d y \<cdot> ad (x \<cdot> y)"
FW'_def: shows "FW' = uncurry \<circ>\<circ> FW \<circ> curry"
filterlim_at_top_mult_tendsto_pos: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c"   and "filterlim g at_top F" shows "LIM x F. g x * f x :> at_top"
of_name_name_def: fixes x :: "name" shows "of_name x = x"
hyps: assumes "k < i" shows "column (mod_type_class.from_nat i) (Gram_Schmidt_upt_k A k) = column (mod_type_class.from_nat i) A"
infsetsum_cmult_right: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "c \<noteq> (0::'a) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. c * f x) = c * infsetsum f A"
subt: fixes u :: "('d, 'e) Term.term"   and ss :: "('d, 'e) Term.term list"   and t :: "('d, 'e) Term.term"   and f :: "'d" assumes "u \<in> set ss"   and "u \<unrhd> t" shows "Fun f ss \<unrhd> t"
is_valid_fun_call_def: fixes get :: "'c \<Rightarrow> 'd fundef option"   and fd2 :: "('e, 'f, 'c, 'g, 'h, 'i, 'j, 'k) Ubx.instr fundef"   and n :: "nat"   and \<Sigma>2 :: "('l, 'm, 'n) unboxed list"   and g :: "'c" shows "is_valid_fun_call get fd2 n \<Sigma>2 g \<equiv> n < length (body fd2) \<and> body fd2 ! n = Ubx.instr.ICall g \<and> (\<exists>gd. get g = Some gd \<and> list_all is_dyn_operand (take (arity gd) \<Sigma>2))"
exists_wt_sym: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and \<zeta> :: "('s, 'v) hd" assumes "legal_zpassign A" shows "\<exists>f\<in>local.ground_heads \<zeta>. zhmset_of (wt_sym f + \<delta>\<^sub>h * arity_sym\<^sub>h f) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (Hd \<zeta>)))"
cover: shows "\<Union> S \<subseteq> A \<union> B"
intro: fixes A :: "'d set" assumes "csubspace A"   and "closed A" shows "closed_csubspace A"
simps_1: fixes E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "char list"   and ts :: "fterm list" shows "eval\<^sub>l E F G (Pos p ts) = G p (eval\<^sub>t\<^sub>s E F ts)"
add_iso_r: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<le> z" shows "x + y \<le> x + z"
dvd_lcm_I1_int: fixes i :: "int"   and m :: "int"   and n :: "int" assumes "i dvd m" shows "i dvd lcm m n"