hcju/isabelleps huggingface dataset:
widen_Array: fixes P :: "'a prog"   and T :: "ty"   and U :: "ty" shows "P \<turnstile> T \<le> U\<lfloor>\<rceil> = (T = NT \<or> (\<exists>V. T = V\<lfloor>\<rceil> \<and> P \<turnstile> V \<le> U))"
degree_mod_less': fixes b :: "'c poly"   and a :: "'c poly" assumes "b \<noteq> 0"   and "a mod b \<noteq> 0" shows "degree (a mod b) < degree b"
subst_id_fv: fixes x :: "nat"   and e :: "exp"   and v :: "exp" assumes "x \<notin> FV e" shows "subst x v e = e"
euclidean_representation_sum: fixes f :: "'a \<Rightarrow> real"   and b :: "'a" shows "((\<Sum>i\<in>Basis. f i *\<^sub>R i) = b) = (\<forall>i\<in>Basis. f i = inner b i)"
quadratic_shape1a: fixes a :: "real"   and x :: "real"   and y :: "real"   and b :: "real"   and c :: "real"   and z :: "real" assumes "0 < a"   and "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"   and "x < z \<and> z < y" shows "a * z\<^sup>2 + b * z + c < 0"
sinvar_spec_impl: fixes default_node_properties :: "'c"   and sinvar_spec :: "'d graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> bool"   and sinvar_impl :: "'d list_graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> bool"   and receiver_violation :: "bool"   and offending_flows_impl :: "'d list_graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<times> 'd) list list"   and node_props_impl :: "('d, 'c) TopoS_Params \<Rightarrow> 'd \<Rightarrow> 'c"   and eval_impl :: "'d list_graph \<Rightarrow> ('d, 'c) TopoS_Params \<Rightarrow> bool"   and G :: "'d list_graph"   and nP :: "'d \<Rightarrow> 'c" assumes "TopoS_List_Impl default_node_properties sinvar_spec sinvar_impl receiver_violation offending_flows_impl node_props_impl eval_impl"   and "wf_list_graph G" shows "sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP"
cnt_altdef: fixes i :: "nat" assumes "i < n" shows "cnt i = linorder_rank R (set xs) (xs ! i)"
rectangle_eq: fixes x :: "'a" shows "is_rectangle x = (x ; 1 ; x = x)"
dom_simp: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.arr f" shows "S.VV.dom f = S.VxV.dom f"
atm_imp_pos_or_neg_lit: fixes A :: "'b"   and C :: "'b literal multiset" assumes "A \<in> atms_of C" shows "Pos A \<in># C \<or> Neg A \<in># C"
Keys_to_list_sorted_wrt: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "sorted_wrt (\<succ>\<^sub>t) (Keys_to_list ps)"
monom_one_mult: fixes n :: "nat"   and m :: "nat" shows "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
communicating_iff: fixes x :: "'s option"   and y :: "'s option" shows "((x, y) \<in> K2.communicating) = ((\<exists>n. 0 < K2.p x y n) \<and> (\<exists>n. 0 < K2.p y x n))"
Ints_power: 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 a :: "'c"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus" shows "power.power one times a n \<in> ring_1.Ints one plus zero minus"
mod_add_if_z: fixes x :: "int"   and z :: "int"   and y :: "int" assumes "x < z"   and "y < z"   and "0 \<le> y"   and "0 \<le> x"   and "0 \<le> z" shows "(x + y) mod z = (if x + y < z then x + y else x + y - z)"
hyps_4: shows "roomk s r1 \<in> {k1, k2}"
analz_Un_implSecureSet_2_aux_3: fixes H :: "msg set"   and Ag :: "(agent \<times> agent) set"   and G :: "msg set" assumes "Enc_keys_clean H"   and "Ag \<inter> broken (parts H \<inter> range LtK) = {}" shows "analz (implSecureSet_aux Ag G \<union> H) \<subseteq> synth (analz H) \<union> - payload"
unit_vec_carrier: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i \<in> carrier_vec n"
rreach_def: fixes s :: "Store"   and y :: "Value"   and x :: "Value" shows "s\<turnstile>Ref y reachable_from x = (\<exists>l. s\<turnstile> l reachable_from x \<and> y = ref l)"
y: shows "B y \<or> C y (app s m) (app s n)"
a_7: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "d x \<cdot> ad (d y + d z) = d x \<cdot> ad y \<cdot> ad z"
asympartpI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "r x y"   and "\<not> r y x" shows "asympartp r x y"
cnj_complex_def: shows "cnj_class.cnj = Complex.cnj"
simps_3: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b pmf"   and x :: "'c" assumes "finite A"   and "A \<noteq> {}" shows "fold_bind_random_permutation f g x A = pmf_of_set A \<bind> (\<lambda>a. fold_bind_random_permutation f g (f a x) (A - {a}))"
Suc_Suc_Suc_Suc_induct: fixes i :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i < Suc (Suc (Suc (Suc 0)))"   and "i = 0 \<Longrightarrow> P i"   and "i = 1 \<Longrightarrow> P i"   and "i = 2 \<Longrightarrow> P i"   and "i = 3 \<Longrightarrow> P i" shows "P i"
key_gen_def: shows "key_gen = sample_uniform (order \<G>) \<bind> (\<lambda>x. let h = \<^bold>g [^] x in return_spmf (h, h))"
lCdom2: fixes a :: "('a, 'b) Combinators list"   and b :: "('a, 'b) Combinators list"   and c :: "('a, 'b) Combinators list" shows "list2FWpolicy (a @ b @ c) = list2FWpolicy ((a @ b) @ c)"
function_ring_defs_1: shows "carrier F = S \<rightarrow>\<^sub>E carrier R"
l_span_cont_H: fixes H :: "'a set" assumes "H \<subseteq> carrier M" shows "H \<subseteq> linear_span R M (carrier R) H"
LTL_prop_entailment_restrict_to_propos: fixes S :: "'b ltl set"   and \<phi> :: "'b ltl" shows "S \<Turnstile>\<^sub>P \<phi> = (S \<inter> propos \<phi>) \<Turnstile>\<^sub>P \<phi>"
Suc: fixes x :: "real" shows "cos (x + real n * (2 * pi)) = cos x"
pairE: fixes A :: "'b ptrm"   and B :: "'b ptrm"   and Y :: "'b ptrm" assumes "PPair A B \<approx> Y" obtains B :: "'b ptrm" and D :: "'b ptrm" where "Y = PPair B D" and "A \<approx> B" and "Ba__ \<approx> D"
less_eq_dual_iff: fixes x :: "'a"   and y :: "'a" shows "(dual x \<le> dual y) = (y \<le> x)"
alternate_list_of: fixes xs :: "'b llist" assumes "lfinite xs" shows "alternate (list_of xs) = list_of (lalternate xs)"
WT_code_into_WT: fixes wf_md :: "'a J_prog \<Rightarrow> String.literal \<Rightarrow> (String.literal list \<times> 'a expr) mdecl \<Rightarrow> bool"   and P :: "'a J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'a expr"   and T :: "ty" assumes "wf_prog wf_md P"   and "P,E \<turnstile> e ::' T"   and "ran E \<subseteq> types P" shows "P,E \<turnstile> e :: T"
wf_measures: fixes fs :: "('b \<Rightarrow> nat) list" shows "wf (measures fs)"
mem_val_alt_8_unchanged0: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 (pa AND 68719476732) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 8 (pa AND 68719476732) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
meet_compl_bot_left2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter> (\<^bold>- x \<sqinter> y) = \<bottom>"
surjective: fixes r :: "('c, 'd, 'e) pre_digraph_scheme" shows "r = \<lparr>pre_digraph.verts = pre_digraph.verts r, arcs = pre_digraph.arcs r, tail = tail r, head = head r, \<dots> = pre_digraph.more r\<rparr>"
minus_real_alg: fixes x :: "real_alg"   and y :: "real_alg" shows "real_of x - real_of y = real_of (x - y)"
vsv_Limit_vrange_in_VsetI: fixes \<alpha> :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "Limit \<alpha>"   and "\<R>\<^sub>\<circ> (VLambda A f) \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "vfinite (\<D>\<^sub>\<circ> (VLambda A f))" shows "\<R>\<^sub>\<circ> (VLambda A f) \<in>\<^sub>\<circ> Vset \<alpha>"
sign_simps_7: fixes al :: "'b"   and bl :: "'b"   and cj :: "'b" shows "al - bl - cj = al - (bl + cj)"
subset_closed_def: fixes C :: "'a set set" shows "subset_closed C = (\<forall>S'\<in>C. \<forall>S\<subseteq>S'. S \<in> C)"
freshEnv_def2: fixes xs :: "'b"   and x :: "'c"   and rho :: "'b \<Rightarrow> 'c \<Rightarrow> ('d, 'e, 'b, 'c, 'f) qTerm set option" shows "freshEnv xs x rho = qFreshEnv xs x (pickE rho)"
H2_idem: shows "H2 \<circ> H2 = H2"
bij_om: shows "bij_betw (ordermap A VWF) A (elts \<alpha>)"
distinct_5: fixes x2 :: "char list"   and x31 :: "expr"   and x32 :: "bop"   and x33 :: "expr" shows "Var x2 \<noteq> x31 \<guillemotleft>x32\<guillemotright> x33"
upt_rec_numeral: fixes m :: "num"   and n :: "num" shows "[numeral m..<numeral n] = (if numeral m < numeral n then numeral m # [Suc (numeral m)..<numeral n] else [])"
preal_exists_greater: fixes A :: "rat set"   and y :: "rat" assumes "Dedekind_Real.cut A"   and "y \<in> A" shows "\<exists>u\<in>A. y < u"
carrier_open: fixes T :: "'b set set" assumes "topology T" shows "carrier\<^bsub>T\<^esub> open\<^bsub>T\<^esub>"
prems_1: shows "\<forall>aa. ListMem aa (a # as) \<longrightarrow> agree (fst aa) s \<and> agree (snd aa) s"
heap_read_conf: fixes h :: "'heap"   and a :: "'addr"   and al :: "addr_loc"   and v :: "'addr val"   and T :: "ty" assumes "heap_read h a al v"   and "P,h \<turnstile> a@al : T"   and "hconf h" shows "P,h \<turnstile> v :\<le> T"
f_the_inv_into_f: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and y :: "'g" assumes "inj_on f A"   and "y \<in> f ` A" shows "f (the_inv_into A f y) = y"
sup_absorb2: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "less_eq_prod x y" shows "sup_prod x y = y"
sublist_imp_subseq: fixes xs :: "'a list"   and ys :: "'a list" assumes "sublist xs ys" shows "subseq xs ys"
tm_floatarith_eq_Some_num_params: fixes prec :: "nat"   and ord :: "nat"   and a :: "float interval list"   and b :: "float list"   and f :: "floatarith"   and tf :: "taylor_model" assumes "tm_floatarith prec ord a b f = Some tf" shows "num_params (tm_poly tf) \<le> 1"
ri: fixes y :: "'b" assumes "y \<in> T" shows "r (i y) = y"
FL_logically_equivalent_def: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state" shows "FL_logically_equivalent F P Q \<equiv> \<forall>x. x \<in> \<A>[F] \<longrightarrow> P \<Turnstile> x = Q \<Turnstile> x"
iso_hcomp: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.iso \<mu>"   and "C.iso \<nu>"   and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" shows "C.iso (\<nu> \<star>\<^sub>C \<mu>)"
map_indets_plus: fixes f :: "'d \<Rightarrow> 'b"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "map_indets f (p + q) = map_indets f p + map_indets f q"
insert_index_def: fixes i :: "nat" shows "insert_index i \<equiv> \<lambda>i'. if i' < i then i' else Suc i'"
assms_2: shows "sgn (poly r x) = - sgn (poly p x)"
cet_var: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat" shows "\<Gamma> \<turnstile>\<^sub>c CVar x : \<Gamma> x"
nat_power_mult_distrib: fixes m :: "nat"   and n :: "nat"   and a :: "'a" assumes "0 < m"   and "0 < n" shows "nat_power (m * n) a = nat_power m a * nat_power n a"
prems: shows "(a ## l) @@ s \<in> A\<^sup>\<omega>"
induct: fixes P :: "'b heap list \<Rightarrow> bool"   and a0 :: "'b heap list" assumes "P []"   and "\<And>x lhs rhs. \<lbrakk>P lhs; P rhs\<rbrakk> \<Longrightarrow> P (Hp x lhs # rhs)"   and "\<And>va. P (heap.Empty # va)" shows "P a0"
surjective: fixes r :: "('d, 'e, 'f) LSCategory_scheme" shows "r = \<lparr>Obj = obj\<^bsub>r\<^esub>, Mor = mor\<^bsub>r\<^esub>, Dom = Dom r, Cod = Cod r, Id = Category.Id r, Comp = (;;\<^bsub>r\<^esub>), mor2ZF = mor2ZF r, \<dots> = LSCategory.more r\<rparr>"
PORI_4: fixes f :: "Three \<Rightarrow> ValD" assumes "f One = \<bottom>" shows "f \<in> unlr POR.delta"
Maxr_lemma: fixes rf :: "recf"   and xs :: "nat list"   and w :: "nat" assumes "primerec rf (Suc (length xs))" shows "rec_exec (rec_maxr rf) (xs @ [w]) = Maxr (\<lambda>args. 0 < rec_exec rf args) xs w"
in_results_gpv_restrict_gpvD: fixes x :: "'d"   and \<I> :: "('e, 'f) \<I>"   and \<I>' :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, 'f) gpv" assumes "x \<in> results_gpv \<I> (restrict_gpv \<I>' gpv)" shows "x \<in> results_gpv \<I> gpv"
DFA_def: shows "MN.DFA = \<lparr>states = h ` UNIV // R, init = h (R `` {[]}), final = {h (R `` {u}) |u. u \<in> A}, nxt = \<lambda>q x. h (\<Union>u\<in>MN.hinv q. R `` {u @ [x]})\<rparr>"
right_segment_congruence: fixes c :: "'p"   and d :: "'p"   and p :: "'p"   and q :: "'p"   and a :: "'p"   and b :: "'p" assumes "{c, d} = {p, q}"   and "a b \<equiv> p q" shows "a b \<equiv> c d"
rQuot_def: fixes x :: "'b"   and A :: "'b list set" shows "rQuot x A = {xs. xs @ [x] \<in> A}"
map_replicate_const: fixes k :: "'c"   and lst :: "'d list" shows "map (\<lambda>x. k) lst = replicate (List.length lst) k"
prj_simps_3: shows "C.arr f_src_src.Prj\<^sub>0"
separation_t1: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.t1_space open" shows "(x \<noteq> y) = (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
mono_neutral_cong_right: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'b)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g T = sum h S"
lowerPowers_eq: fixes b :: "nat"   and i :: "nat"   and p :: "nat \<Rightarrow>\<^sub>0 'a"   and x :: "nat" shows "lookup (lowerPowers b i p) x = (if b \<le> x \<and> x < b + i then 0::'a else lookup p (dec_above b i x))"
simps_5: fixes master :: "clock"   and \<delta>\<tau> :: "'b tag_const"   and measuring :: "clock"   and slave :: "clock" shows "\<lbrakk> master time-delayed by \<delta>\<tau> on measuring implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<forall>n. hamlet (Rep_run \<rho> n master) \<longrightarrow> (let measured_time = time (Rep_run \<rho> n measuring) in \<forall>m\<ge>n. first_time \<rho> measuring m (measured_time + \<delta>\<tau>) \<longrightarrow> hamlet (Rep_run \<rho> m slave))}"
pre_snoc: fixes \<Pi> :: "('a, 'b) chain"   and x :: "'b"   and P :: "'a" shows "pre (cSnoc \<Pi> x P) = pre \<Pi>"
stable_on_CD_on: fixes ds :: "'b set"   and X :: "'a set" assumes "ManiPref.stable_on ds X" shows "ManiPref.CD_on ds X = X"
Box_def: shows "Box \<equiv> \<Lambda> u. Box_abs\<cdot>u"
rHom_def: fixes A :: "('e, 'g) Ring_scheme"   and R :: "('f, 'h) Ring_scheme" shows "rHom A R = {f \<in> aHom A R. (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. f (x \<cdot>\<^sub>r\<^bsub>A\<^esub> y) = f x \<cdot>\<^sub>r\<^bsub>R\<^esub> f y) \<and> f 1\<^sub>r\<^bsub>A\<^esub> = 1\<^sub>r\<^bsub>R\<^esub>}"
le_of_int_ceiling: fixes x :: "'b" shows "x \<le> of_int \<lceil>x\<rceil>"
g_merge: fixes x :: "'a"   and y :: "'a" assumes "f x = f y" shows "g [x, y] = g [merge x y]"
verit_connective_def_3: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<not> (\<forall>x. \<not> P x))"
distinct_3: fixes x1 :: "agent"   and x3 :: "nonce_t" shows "cAgent x1 \<noteq> cNonce x3"
takefill_alt: fixes fill :: "'b"   and n :: "nat"   and l :: "'b list" shows "takefill fill n l = take n l @ replicate (n - length l) fill"
substBinp_def: fixes xs :: "'varSort"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and x :: "'var" shows "substBinp xs X x \<equiv> psubstBinp (idEnv [x \<leftarrow> X]_xs)"
a_def: shows "a \<equiv> leftrec u v leftcut rightcut"
Image_iff: fixes b :: "'c"   and r :: "('d \<times> 'c) set"   and A :: "'d set" shows "(b \<in> r `` A) = (\<exists>x\<in>A. (x, b) \<in> r)"
cop_correct_nterm: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "obs \<sigma> k \<noteq> obs \<sigma>' k"   and "obs \<sigma> k \<noteq> None"   and "obs \<sigma>' k \<noteq> None"   and "\<sigma> =\<^sub>L \<sigma>'" shows "\<exists>k k'. ((\<sigma>, k), \<sigma>', k') \<in> cop \<or> ((\<sigma>', k'), \<sigma>, k) \<in> cop"
assms_2: shows "\<forall>n. \<exists>m>n. P (s m)"
zero_le_divide_ereal: fixes a :: "ereal"   and b :: "ereal" assumes "0 \<le> a"   and "0 \<le> b" shows "0 \<le> a / b"
frag_closure_minus_cmul: fixes P :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> bool"   and c :: "'b \<Rightarrow>\<^sub>0 int"   and k :: "int" assumes "P 0"   and "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> P (x - y)"   and "P c" shows "P (frag_cmul k c)"
m_closed: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and y :: "'e" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
isometry_on_continuous: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "isometry_on X f" shows "continuous_on X f"
prems_1: assumes "rtrancl_path r x ((a # as) @ [y]) y"   and "rtrancl_path r y ys z" shows "thesis"
Max_eqI: fixes A :: "'b word set"   and x :: "'b word" assumes "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> y \<le>s x"   and "x \<in> A" shows "signed.Max A = x"
simps: fixes a1 :: "bexp"   and a2 :: "com"   and a3 :: "char list \<Rightarrow> int"   and a4 :: "nat" shows "Its a1 a2 a3 a4 = ((\<exists>b s c. a1 = b \<and> a2 = c \<and> a3 = s \<and> a4 = 0 \<and> \<not> bval b s) \<or> (\<exists>b s c s' n. a1 = b \<and> a2 = c \<and> a3 = s \<and> a4 = Suc n \<and> bval b s \<and> (c, s) \<Rightarrow> s' \<and> Its b c s' n))"
part_equivD2: fixes R :: "('b \<times> 'b) set" assumes "part_equiv R" shows "trans R"
cexpr_typing_pairE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c <e1, e2>\<^sub>c : t" obtains t1 :: "pdf_type" and t2 :: "pdf_type" where "t = PRODUCT t1 t2" and "\<Gamma> \<turnstile>\<^sub>c e1 : t1" and "\<Gamma> \<turnstile>\<^sub>c e2 : t2"
simps: fixes n :: "nat" shows "counterAssign (X n) = obj n"
lmember_code_2: fixes x :: "'c"   and y :: "'c"   and ys :: "'c llist" shows "lmember x (LCons y ys) = (x = y \<or> lmember x ys)"
injective_def: fixes size :: "nat"   and to_index :: "'a \<Rightarrow> nat" shows "injective size to_index \<equiv> \<forall>a b. to_index a = to_index b \<and> to_index a < size \<and> to_index b < size \<longrightarrow> a = b"
cont_id': fixes Sup :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "cont Sup ord Sup ord (\<lambda>x. x)"
Ri_def: fixes \<Gamma> :: "'b Inference_System.inference set"   and N :: "'b literal multiset set" shows "standard_redundancy_criterion.Ri \<Gamma> N = {\<gamma> \<in> \<Gamma>. standard_redundancy_criterion.redundant_infer N \<gamma>}"
Z_Inter: shows "local.Z_inj \<inter> local.Z_not_inj = {}"
DIAnd_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAnd vid1 pid1 pid2 pid3 pid4 = ([([], [ids.DIAndSG1 vid1 pid1 pid3]), ([], [ids.DIAndSG2 vid1 pid2 pid4])], ids.DIAndConcl vid1 pid1 pid2 pid3 pid4)"
set_double_filter_hd: fixes vs :: "'a list"   and v :: "'a"   and v' :: "'a" assumes "set vs = {v, v'}"   and "v \<noteq> v'" shows "hd (filter (\<lambda>v'. v' \<noteq> v) vs) = v'"
iMOD_inext: fixes t :: "nat"   and r :: "nat"   and m :: "nat" assumes "t \<in> [ r, mod m ]" shows "inext t [ r, mod m ] = t + m"
codomains_def: fixes f :: "'a" shows "C.codomains f \<equiv> {b. C.ide b \<and> b \<cdot> f \<noteq> C.null}"
assoc_simps_4: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.dom \<a>\<^sub>D[f, g, h] = (D.dom f \<star>\<^sub>D D.dom g) \<star>\<^sub>D D.dom h"
correct: fixes ts :: "((nat \<times> nat) \<times> int) list"   and x :: "nat \<Rightarrow> int" assumes "distinct (map fst ts)"   and "\<forall>((r, c), uu_)\<in>set ts. r < n" shows "alg ts x = local.pr (\<alpha> ts) x"
ffunion_empty: shows "ffUnion \<lbrace>\<rbrace> = \<lbrace>\<rbrace>"
bottom_right_bottom: fixes a :: "'a" shows "a \<^bold>* \<^bold>\<bottom> = \<^bold>\<bottom>"
Cod_Dom: fixes t :: "'a local.term" assumes "Arr t" shows "Cod (Dom t) = Dom t"
triangle_left: shows "(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) = \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]"
bound_main_lemma_2: fixes s :: "nat set"   and k :: "nat" assumes "s \<noteq> {} \<and> (\<forall>x. x \<in> s \<longrightarrow> x \<le> k)" shows "Sup s \<le> k"
p: shows "\<Turnstile>\<^sub>p {Q} S {P}"
word_log_esimps_7: fixes x :: "'b word" shows "0 AND x = 0"
filter_mset_add_mset: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a multiset" shows "filter_mset P (add_mset x A) = (if P x then add_mset x (filter_mset P A) else filter_mset P A)"
units_group_def: fixes R :: "('c, 'd) ring_scheme" shows "units_group R = \<lparr>carrier = Units R, mult = (\<otimes>\<^bsub>R\<^esub>), one = \<one>\<^bsub>R\<^esub>\<rparr>"
powr_complexpow: fixes x :: "complex"   and n :: "nat" assumes "x \<noteq> 0" shows "x powr of_nat n = x ^ n"
e_0_empty: shows "nat_to_ce_set e_0 = {}"
rel_cset_def: shows "rel_cset \<equiv> map_fun id (map_fun rcset (map_fun rcset id)) rel_set"
span_eq_real: shows "module_on.span (*\<^sub>R) = ??.Real_Vector_Spaces.span"
HRB_slice_is_SDG_path_HRB_slice: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n'' :: "'b SDG_node"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"   and "n'' \<in> SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses {n}"   and "n' \<in> S" shows "n'' \<in> SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S"
prems: shows "\<forall>i<Suc k. \<Gamma>\<turnstile> p i \<rightarrow> p (Suc i)"
rep_eq: fixes x :: "uint32" shows "bit x = bit (Rep_uint32 x)"