hcju/isabelleps huggingface dataset:
infiniteTr_strong_coind: fixes phi :: "'a treeFI \<Rightarrow> bool"   and tr :: "'a treeFI" assumes "phi tr"   and "\<And>tr. phi tr \<Longrightarrow> \<exists>tr'\<in>set (sub tr). phi tr' \<or> infiniteTr tr'" shows "infiniteTr tr"
T_Sub: fixes \<Gamma> :: "binding list"   and t :: "trm"   and S :: "ty"   and T :: "ty" assumes "\<Gamma> \<turnstile> t : S"   and "\<Gamma>\<turnstile>S<:T" shows "\<Gamma> \<turnstile> t : T"
assms_3: shows "x1 1 \<in> carrier_mat d d"
upper: shows "inverse (aform_val e X) \<le> alpha * aform_val e X + zeta + delta"
analytic_on_linear: fixes c :: "complex"   and S :: "complex set" shows "(*) c analytic_on S"
cases: fixes a1 :: "'h"   and a2 :: "'i"   and r :: "('h \<times> 'j) set"   and s :: "('j \<times> 'i) set" assumes "(a1, a2) \<in> r O s" obtains a :: "'h" and b :: "'j" and c :: "'i" where "a1 = a" and "a2 = c" and "(a, b) \<in> r" and "(b, c) \<in> s"
a_get_parent_def: fixes node_ptr :: "(_) node_ptr" shows "a_get_parent node_ptr = Heap_Error_Monad.bind (check_in_heap (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) (\<lambda>_. Heap_Error_Monad.bind (Heap_Error_Monad.bind object_ptr_kinds_M (filter_M (\<lambda>ptr. Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. return (node_ptr \<in> set children))))) (\<lambda>parent_ptrs. if parent_ptrs = [] then return None else return (Some (hd parent_ptrs))))"
hequiv_sym_rel: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set" assumes "Hintikka A H" shows "sym (hequiv_rel A H)"
simps_25: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Set_local i) ts = (if i < length (local \<C>) then type_update ts [TSome (local \<C> ! i)] (Type []) else Bot)"
inline1_def: shows "local.inline1 \<equiv> curry (spmf.fixp_fun (\<lambda>inline1 (gpv, s). the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))))"
asymp_equiv_imp_bigomega: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "f \<in> \<Omega>[F](g)"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "(case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
inf_sup_ord_2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) y"
integrable_alt_subset: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "(f integrable_on S) = ((\<forall>a b. (\<lambda>x. if x \<in> S then f x else (0::'d)) integrable_on cbox a b) \<and> (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball (0::'c) B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow> norm (integral (cbox a b) (\<lambda>x. if x \<in> S then f x else (0::'d)) - integral (cbox c d) (\<lambda>x. if x \<in> S then f x else (0::'d))) < e))"
inject_3: fixes x31 :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool"   and x32 :: "'i"   and x33 :: "'j"   and x34 :: "(('g \<Rightarrow> 'h option) \<Rightarrow> 'h option) list"   and y31 :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool"   and y32 :: "'i"   and y33 :: "'j"   and y34 :: "(('g \<Rightarrow> 'h option) \<Rightarrow> 'h option) list" shows "(x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 = y31:y32\<hookrightarrow>\<^bsub>y33\<^esub>y34) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33 \<and> x34 = y34)"
mset_subset_eqI: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "\<And>a. count A a \<le> count B a" shows "A \<subseteq># B"
Max_add_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and k :: "'a" assumes "class.linordered_ab_semigroup_add plus less_eq less"   and "finite S"   and "S \<noteq> {}" shows "linorder.Max less_eq ((\<lambda>x. plus (f x) k) ` S) = plus (linorder.Max less_eq (f ` S)) k"
subst_dsj: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "subst (dsj \<phi> \<chi>) t x = dsj (subst \<phi> t x) (subst \<chi> t x)"
set_cond_spmf: fixes p :: "'c spmf"   and A :: "'c set" shows "set_spmf (cond_spmf p A) = set_spmf p \<inter> A"
assms_7: shows "p' \<in> sterms \<Gamma> p"
le_less: fixes x :: "'a list"   and y :: "'a list" shows "subseq x y = (strict_subseq x y \<or> x = y)"
pathfinish_linepath: fixes a :: "'c"   and b :: "'c" shows "pathfinish (linepath a b) = b"
simps: fixes a :: "ind" shows "Nat a = (a = Zero_Rep \<or> (\<exists>i. a = Suc_Rep i \<and> Nat i))"
simps_7: shows "[0]\<^sub>2 +\<^sub>2 \<infinity>\<^sub>2 = \<infinity>\<^sub>2"
dist_triangle_eq: fixes x :: "'b"   and z :: "'b"   and y :: "'b" shows "(dist x z = dist x y + dist y z) = (norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y))"
ys_zs_in_mult1: shows "(Ys, Zs) \<in> mult1 {(x, y). x \<in> A \<and> y \<in> A \<and> gt y x}"
awalk_induce: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and S :: "'a set" assumes "H.awalk u p v"   and "set (H.awalk_verts u p) \<subseteq> S" shows "pre_digraph.awalk ((G \<restriction> S) \<restriction> S) u p v"
R_consts_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "\<not> w \<Turnstile>\<^sub>n \<phi> R\<^sub>n false\<^sub>n"
u: shows "u \<equiv> real_of_1 (p, l, r)"
Max_insert2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and a :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> less_eq b a" shows "linorder.Max less_eq (insert a A) = a"
swapped_by_mtf2: fixes q :: "'c"   and xs :: "'c list"   and x :: "'c"   and y :: "'c"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "x < y in xs"   and "y < x in mtf2 n q xs" shows "y = q"
D_inv: fixes r :: "('b \<times> 'b) set"   and \<tau>' :: "'b list"   and \<sigma> :: "'b list"   and \<tau> :: "'b list"   and \<sigma>' :: "'b list" assumes "trans r"   and "irrefl r"   and "(r|\<tau>'| -s r \<down>l \<sigma>, r|\<tau>|) \<in> mul_eq r"   and "(r|\<sigma>'| -s r \<down>l \<tau>, r|\<sigma>|) \<in> mul_eq r" shows "D r \<tau> \<sigma> \<sigma>' \<tau>'"
LSM_sub_submodule: fixes M :: "('g, 'a, 'h) Module_scheme"   and T :: "'g set"   and N :: "'g set" assumes "R module M"   and "T \<subseteq> carrier M"   and "submodule R M N"   and "T \<subseteq> N" shows "LSM\<^bsub>R\<^esub> M T \<subseteq> N"
x: shows "x \<in> elts (\<alpha> \<up> \<gamma>)"
assms_2: shows "\<forall>c\<in>clk_set A. v c \<le> n"
Tf_def: shows "Tf = T \<circ> (\<lambda>(x, y). f x y)"
ccw'_contra: fixes t :: "real \<times> real"   and r :: "real \<times> real"   and q :: "real \<times> real" assumes "ccw' t r q" shows "ccw' t q r = False"
pNF: shows "(x, some_NF r x) \<in> r\<^sup>* \<and> some_NF r x \<in> NF r"
clss_of_interp_def: fixes I :: "'b set" shows "clss_of_interp I = {{#(if A \<in> I then Pos else Neg) A#} |A. True}"
sqnf_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat" shows "sqnf rt dip \<equiv> case rt dip of None \<Rightarrow> unk | Some r \<Rightarrow> \<pi>\<^sub>3 r"
Hoare_dgr_def: fixes P :: "'b \<Rightarrow> 'e"   and D :: "'b \<times> 'b \<Rightarrow> 'e \<Rightarrow> 'e"   and Q :: "'b \<Rightarrow> 'e" shows "\<turnstile> P{| D |}Q  \<equiv> \<exists>X. (\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (pair u i) j) \<and> P = \<Squnion>range X \<and> Q = \<Squnion>range X \<sqinter> - grd (step D)"
same_laws_2: fixes R :: "'a ring"   and a :: "'a"   and b :: "'a" assumes "R \<in> C"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<oplus>\<^bsub>union_ring C\<^esub> b = a \<oplus>\<^bsub>R\<^esub> b"
l_def: shows "l = proj2_line_through (Rep_hyp2 a) (Rep_hyp2 b)"
hyps: shows "real n * x + 1 \<le> (x + 1) ^ n"
alw_destL: 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 \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.alw \<phi> \<pi>l" shows "\<phi> \<pi>l"
3: shows "length M2 \<le> j \<longrightarrow> (v # M \<otimes> M2) ! j = (M \<otimes> M2) ! (j - length M2)"
Lim_at_infinityI: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>B. \<forall>x. B \<le> norm x \<longrightarrow> dist (f x) l \<le> e" shows "(f \<longlongrightarrow> l) at_infinity"
subst_dom_update2: fixes t :: "('a, 'b) Term.term"   and v :: "'b"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t \<noteq> Var v" shows "subst_domain (\<sigma>(v := t)) = insert v (subst_domain \<sigma>)"
zip_def: shows "zip = zipWith\<cdot>\<langle>,\<rangle>"
repc_games: fixes I :: "interp"   and f :: "char"   and d :: "real" shows "Games (repc I f d) = Games I"
hyps_3: shows "p\<^sub>4 = Point x\<^sub>4 y\<^sub>4"
mr_is_allow: shows "applied_rule_rev Cp x p = \<lfloor>AllowPortFromTo a b po\<rfloor>"
isCont_arcsin: fixes x :: "real" assumes "- 1 < x"   and "x < 1" shows "isCont arcsin x"
Quotient3_abs_rep: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and a :: "'c" assumes "Quotient3 R Abs Rep" shows "Abs (Rep a) = a"
scheme_axioms_def: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f" shows "scheme_axioms R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str \<equiv> \<forall>x. x \<in> X \<longrightarrow> (\<exists>U. x \<in> U \<and> is_open U \<and> (\<exists>f \<phi>\<^sub>f. affine_scheme R addition multiplication zero unit U (ind_topology.ind_is_open X is_open U) (ind_sheaf.ind_sheaf \<O>\<^sub>X U) (ind_sheaf.ind_ring_morphisms \<rho> U) b (ind_sheaf.ind_add_str add_str U) (ind_sheaf.ind_mult_str mult_str U) (ind_sheaf.ind_zero_str zero_str U) (ind_sheaf.ind_one_str one_str U) f \<phi>\<^sub>f))"
monoid_axioms: shows "Group_Theory.monoid (\<eta> ` M) (\<cdot>') \<one>'"
balance_cl2: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "cl_inv l"   and "cl_inv' r" shows "cl_inv (balance l k v r)"
assms_3: shows "is_const (fst (strip_comb f))"
bet_neq23__neq: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "B \<noteq> C" shows "A \<noteq> C"
cases: fixes a1 :: "'node"   and a2 :: "'var option"   and a3 :: "'edge list"   and a4 :: "'node" assumes "DynPDG_edge a1 a2 a3 a4" obtains n :: "'node" and n' :: "'node" and as :: "'edge list" where "a1 = na__" and "a2 = None" and "a3 = asa__" and "a4 = n'a__" and "na__ controls n'a__ via asa__"   | n :: "'node" and V :: "'var" and n' :: "'node" and as :: "'edge list" where "a1 = n" and "a2 = Some V" and "a3 = as" and "a4 = n'" and "n influences V in n' via as"
Int_mono: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b set"   and D :: "'b set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<inter> B \<subseteq> C \<inter> D"
Vinv3_def: fixes Quorum :: "process set set" assumes "quorum_process Quorum" shows "quorum_process.Vinv3 Quorum = {s. ran (decisions s) \<subseteq> quorum_process.locked Quorum s}"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>nat. P (TVar nat)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)" shows "P ty"
Sim: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> Rel" shows "P \<leadsto><Rel> Q"
cod_comp: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" shows "S.cod (g \<cdot> f) = S.cod g"
MOST_ge_nat: fixes m :: "nat" shows "\<forall>\<^sub>\<infinity>n. m \<le> n"
altc_sample_writer_def: fixes altc_sample :: "'b \<Rightarrow> ('d \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'d \<Rightarrow> ('w, 'a, 'm) writerT" shows "local.altc_sample_writer altc_sample p f = WriterT (altc_sample p (\<lambda>p. run_writer (f p)))"
dom_subid_aux2: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> y \<le> y"
bf_vars_def: fixes bf :: "('b \<Rightarrow> bool) \<Rightarrow> bool" shows "bf_vars bf = {v. \<exists>as. bf_restrict v True bf as \<noteq> bf_restrict v False bf as}"
composition_closed: fixes a :: "('c set \<times> 'd) set"   and b :: "('c set \<times> 'd) set" assumes "a \<in> stfx.carrier_direct_lim"   and "b \<in> stfx.carrier_direct_lim" shows "stfx.mult_rel a b \<in> stfx.carrier_direct_lim"
extended_chop: assumes "chop t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "P"
integrable_bound2: shows "interval_lebesgue_integrable lborel (ereal (exp (- 1))) 1 g2"
CX: fixes h' :: "real" assumes "h' \<in> {0<--<h}" shows "(flow0 x0 h', Dflow x0 h' o\<^sub>L d) \<in> CX"
lex_conv: fixes r :: "('b \<times> 'b) set" shows "lex r = {(xs, ys). length xs = length ys \<and> (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y) \<in> r)}"
pair_imageI: fixes a :: "'c"   and b :: "'d"   and A :: "('c \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "(a, b) \<in> A" shows "f a b \<in> (\<lambda>(a, b). f a b) ` A"
of_char_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and c :: "char" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.of_char times one plus zero c = comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) [digit0 c, digit1 c, digit2 c, digit3 c, digit4 c, digit5 c, digit6 c, digit7 c]"
wf_imp_nf_ex: fixes a :: "'a" assumes "wfP (\<rightarrow>)\<inverse>\<inverse>" obtains b :: "'a" where "a \<rightarrow>\<^sup>* b" and "is_final b"
check_proof_really_sound: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and P :: "proofterm"   and res :: "Core.term" assumes "check_proof thy P res" shows "thy,set (hyps P) \<tturnstile> res"
return_time_function_def: fixes A :: "'a set"   and x :: "'a" shows "return_time_function A x = (if x \<in> recurrent_subset A then \<Sqinter> {n \<in> {1..}. (T ^^ n) x \<in> A} else 0)"
add_0_right: fixes a :: "'a" shows "a \<sqinter> top = a"
givenByI: fixes A :: "'d set"   and v :: "'d \<Rightarrow> 'e" assumes "\<And>x y. \<lbrakk>x \<in> A; v x = v y\<rbrakk> \<Longrightarrow> y \<in> A" shows "A \<in> givenBy v"
reachable_neq_reachable1: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c"   and w :: "'c" assumes "wf_digraph G"   and "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"   and "v \<noteq> w" shows "v \<rightarrow>\<^sup>+\<^bsub>G\<^esub> w"
vfsequence_vcard_in_omega: shows "vcard xs \<in>\<^sub>\<circ> \<omega>"
added_files_Deliver_Append_same_collapse: fixes i :: "'c"   and j :: "'c"   and e :: "'b" shows "added_files [Deliver (i, Append j e)] e = [j]"
map_carrier_vec: fixes h :: "'b \<Rightarrow> 'a"   and v :: "'b vec"   and n :: "nat" shows "(map_vec h v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
prj_simps_8: shows "Maps.cod SPN_fgh.Prj\<^sub>0\<^sub>1 = SPN_fgh.\<nu>.apex"
lossless_try_spmf: fixes p :: "'c spmf"   and q :: "'c spmf" shows "lossless_spmf (TRY p ELSE q) = (lossless_spmf p \<or> lossless_spmf q)"
vpair_inject: fixes a :: "V"   and b :: "V"   and a' :: "V"   and b' :: "V" assumes "\<langle>a, b\<rangle> = \<langle>a', b'\<rangle>" obtains "a = a'" and "b = b'"
Syntax_with_Connectives_axioms: shows "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"
Nonce_Pair: fixes X :: "nonce_t"   and X' :: "msg"   and Y' :: "msg" shows "Nonce X \<noteq> \<langle>X', Y'\<rangle>"
good_item_simps_54: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and xa :: "'var"   and y :: "'var" assumes "good X" shows "X #[xa \<and> y]_zs #[xa \<and> y]_zs = X"
arr_iff_in_hom: fixes f :: "'c" shows "C.arr f = \<guillemotleft>f : C.dom f \<Rightarrow>\<^sub>C C.cod f\<guillemotright>"
insert_def: fixes a :: "'b"   and B :: "'b set" shows "??.Set.insert a B = {x. x = a} \<union> B"
distinct_1: fixes x1 :: "'b"   and x21 :: "'b Trace"   and x22 :: "'b" shows "tInit x1 \<noteq> x21 \<leadsto> x22"
relpow_Suc_E2: fixes x :: "'a"   and z :: "'a"   and n :: "nat"   and R :: "('a \<times> 'a) set" assumes "(x, z) \<in> R ^^ Suc n" obtains y :: "'a" where "(x, y) \<in> R" and "(y, z) \<in> R ^^ n"
LstSeq_imp_Ord: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "LstSeq s k y" shows "Ord k"
BV_correct_initial: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and T :: "ty"   and m :: "nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M: []\<rightarrow>T = m in C" shows "P,\<Phi> |- start_state P C M [ok]"
dot_cross7_self_2: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "x \<bullet> y \<times>\<^sub>7 x = 0"
tendsto_snd: fixes f :: "'b \<Rightarrow> 'c \<times> 'd"   and a :: "'c \<times> 'd"   and F :: "'b filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. snd (f x)) \<longlongrightarrow> snd a) F"
ideal_generated_pair_exists_UNIV: fixes 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 a :: "'b"   and b :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "(ring.ideal_generated plus zero uminus times {a, b} = ring.ideal_generated plus zero uminus times {one}) = (\<exists>p q. plus (times p a) (times q b) = one)"
measure_return: fixes X :: "'b set"   and M :: "'b measure"   and x :: "'b" assumes "X \<in> sets M" shows "Sigma_Algebra.measure (return M x) X = indicat_real X x"
that: fixes tk :: "nat" assumes "tki = tk mod N"   and "has_ticket (ts t) tk" shows "thesis"
gamma3_def: shows "gamma3 = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
Nmlize_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Dom t\<^bold>\<rfloor> = Dom \<^bold>\<lfloor>t\<^bold>\<rfloor>"
constant_process_borel_predictable: fixes c :: "'c" shows "borel_predict_stoch_proc F (\<lambda>n w. c)"
7: shows "irreflexive (f\<^sup>\<star> * e * f\<^sup>T\<^sup>\<star> * f\<^sup>\<star>)"
execute_tap: fixes f :: "heap \<Rightarrow> 'a"   and h :: "heap" shows "execute (tap f) h = Some (f h, h)"
learn_totalE_3: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "learn_total \<psi> U s"   and "f \<in> U" shows "\<psi> (the (s (f \<triangleright> n))) \<in> \<R>"
invar_3: shows "\<Gamma> \<turnstile>\<^sub>c \<delta> : REAL"
star_of_simps_3: fixes x :: "'a" shows "star_of (- x) = - star_of x"
abs_eq: fixes x :: "complex mat" shows "cblinfun_of_mat x = CBlinfun (\<lambda>v. if x \<in> carrier_mat (length canonical_basis) (length canonical_basis) then basis_enum_of_vec (x *\<^sub>v vec_of_basis_enum v) else (0::'c))"
less_imp_neq: fixes x :: "nat list"   and y :: "nat list" assumes "x <\<^sub>v y" shows "x \<noteq> y"
int_of_integer_inject: fixes x :: "integer"   and y :: "integer" shows "(int_of_integer x = int_of_integer y) = (x = y)"
eq_\<I>_converter_mono: fixes \<I>1 :: "('y, 'z) \<I>"   and \<I>2 :: "('aa, 'ab) \<I>"   and conv :: "('y, 'z, 'aa, 'ab) converter"   and conv' :: "('y, 'z, 'aa, 'ab) converter"   and \<I>1' :: "('y, 'z) \<I>"   and \<I>2' :: "('aa, 'ab) \<I>" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>1' \<le> \<I>1"   and "\<I>2 \<le> \<I>2'" shows "\<I>1',\<I>2' \<turnstile>\<^sub>C conv \<sim> conv'"
nonzero_pow_nonzero: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp"   and "a \<noteq> \<zero>" shows "a [^] n \<noteq> \<zero>"
is_scalar: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and is_scalar :: "'a itself \<Rightarrow> bool"   and lex_comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and pot_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.nat_term_compare rep_nat_term splus less_eq less is_scalar lex_comp deg_comp pot_comp" shows "is_scalar = (\<lambda>_. \<forall>x. snd (rep_nat_term x) = 0)"
distinct_sortQ: fixes xs :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "singleCombinators xs"   and "all_in_list xs l"   and "distinct xs" shows "distinct (qsort xs l)"
full_abstraction_and_enc_respects_binary_pred_impl_SRel_iff_TRel_preserve: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "enc_respects_binary_pred Pred" shows "rel_preserves_binary_pred (_Collect (P, Q) (\<exists>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<and> (SP, SQ) \<in> SRel)) Pred = rel_preserves_binary_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
less_le_not_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.preorder less_eq less" shows "less x y = strict less_eq x y"
Cons_2: shows "e \<in> set (elements (m # M'))"
discI: fixes option :: "'e option"   and x2 :: "'e" assumes "option = Some x2" shows "option \<noteq> None"