hcju/isabelleps huggingface dataset:
transversal_segment_neg: fixes x :: "'a"   and y :: "'a"   and w :: "'a" assumes "transversal_segment x y"   and "w \<in> {x--y}"   and "f w \<bullet> rot (x - y) < 0" shows "\<forall>z\<in>{x--y}. f z \<bullet> rot (x - y) < 0"
term_brackets_dist: fixes term_bracket :: "nat \<Rightarrow> dB \<Rightarrow> dB option"   and vs :: "nat list"   and t :: "dB"   and t' :: "dB" assumes "bracket_abstraction term_bracket"   and "bracket_abstraction.term_brackets term_bracket vs t = Some t'" shows "var_dist vs t' \<leftrightarrow> t"
Rep_formula: fixes x :: "'a formula" shows "Rep_formula x \<in> fba"
d: shows "\<exists>x\<in>set x. xa \<in> set x"
awalk_empty_ends: fixes u :: "'a"   and v :: "'a" assumes "S.awalk u [] v" shows "u = v"
paths_are_winning_for_one_player: fixes P :: "'a llist"   and p :: "Player" assumes "G'.valid_path P" shows "G'.winning_path p P = (\<not> G'.winning_path p** P)"
ws_status: fixes i :: "nat"   and fn :: "'f \<times> nat" assumes "i \<in> set (\<sigma> fn)" shows "simple_arg_pos NS fn i"
transp_rel_generatI: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "transp A"   and "transp B"   and "transp C" shows "transp (rel_generat A B C)"
Var: fixes x :: "'var" assumes "x \<in> var" shows "Var x \<in> trm"
N_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "grover_state n f" shows "grover_state.N n = 2 ^ n"
rel_def_internal: fixes Rv :: "('V \<times> 'a) set"   and Rw :: "('W \<times> 'b) set" shows "rel Rv Rw \<equiv> br \<alpha> invar O \<langle>Rv, Rw\<rangle>graph_rel"
cross_ratio_real: fixes z :: "complex"   and u :: "complex"   and v :: "complex"   and w :: "complex" assumes "is_real z"   and "is_real u"   and "is_real v"   and "is_real w"   and "z \<noteq> u \<and> v \<noteq> w \<or> z \<noteq> w \<and> u \<noteq> v" shows "is_real (to_complex (cross_ratio (of_complex z) (of_complex u) (of_complex v) (of_complex w)))"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "openin X (T i)"
Zp_int_mult_closed: fixes x :: "nat \<Rightarrow> int"   and n :: "int" assumes "x \<in> carrier Zp" shows "[n] \<cdot> x \<in> carrier Zp"
is_sig_redD2: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail F p" shows "ordered_powerprod_lin.is_le_rel top_tail"
chine_in_hom: shows "\<guillemotleft>\<mu>\<nu>\<pi>.chine : \<mu>\<nu>\<pi>.apex \<rightarrow>\<^sub>C \<mu>\<nu>\<pi>.cod.apex\<guillemotright>"
fps_cutoff_fps_const: fixes n :: "nat"   and c :: "'a" shows "fps_cutoff n (fps_const c) = (if n = 0 then 0 else fps_const c)"
code: fixes b :: "bool"   and acc :: "real"   and m :: "real" shows "sin_series_stream_aux b acc m = MSSCons (if b then - inverse acc else inverse acc) (sin_series_stream_aux (\<not> b) (acc * (m + 1) * (m + 2)) (m + 2))"
set_sum_list_element: fixes x :: "'a"   and As :: "'a set list" assumes "x \<in> (\<Sum>A\<leftarrow>As. A)" shows "\<exists>as\<in>listset As. x = (\<Sum>a\<leftarrow>as. a)"
boundedE: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b \<inter># c" obtains "a \<subseteq># b" and "a \<subseteq># c"
setops\<^sub>s\<^sub>s\<^sub>t_are_pairs: fixes t :: "('fun, 'b) Term.term"   and A :: "('fun, 'b) stateful_strand_step list" assumes "t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t A" shows "\<exists>s s'. t = pair (s, s')"
WHILEI_lfp_conv: fixes m :: "'c nres"   and I :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres" shows "m \<bind> WHILE\<^bsup>I\<^esup> c f = lfp (WHILEI_lfp_body I m c f) \<bind> (\<lambda>s. ASSERT (I s) \<bind> (\<lambda>_. ASSUME (\<not> c s) \<bind> (\<lambda>_. RETURN s)))"
mQmpTr4: fixes H :: "'a set"   and N :: "'a set"   and a :: "'a" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N"   and "a \<in> N" shows "mr_coset a (mdl M N) H = mr_coset a M H"
set_in_convex_hull: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n" shows "A \<subseteq> gram_schmidt.convex_hull n A"
ID_distr_global_1: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and ls :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" shows "RID\<^sub>G (s(r \<mapsto> ls)) = insert r (RID\<^sub>G (s(r := None)) \<union> RID\<^sub>L ls)"
invertible: fixes u :: "'a" assumes "u \<in> R" shows "comm.additive.invertible u"
update_Some_unfold: fixes k :: "'b"   and v :: "'a"   and al :: "('b \<times> 'a) list"   and x :: "'b"   and y :: "'a" shows "(map_of (AList.update k v al) x = Some y) = (x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y)"
upper_min_interval: fixes x :: "'a interval"   and y :: "'a interval" shows "upper (min_interval x y) = min (upper x) (upper y)"
integral_subset_negligible: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<subseteq> T"   and "negligible (T - S)" shows "integral S f = integral T f"
ring_one: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "1\<^sub>r\<^bsub>R\<^esub> \<in> carrier R"
rinf_\<Upsilon>: fixes p :: "fm"   and x :: "real"   and bs :: "real list" assumes "isrlfm p"   and "\<not> Ifm (x # bs) (minusinf p)"   and "\<not> Ifm (x # bs) (plusinf p)"   and "\<exists>x. Ifm (x # bs) p" shows "\<exists>(l, n)\<in>set (\<Upsilon> p). \<exists>(s, m)\<in>set (\<Upsilon> p). Ifm ((Inum (x # bs) l / real_of_int n + Inum (x # bs) s / real_of_int m) / 2 # bs) p"
inorder_update_splay: fixes t :: "('c \<times> 'd) tree"   and x :: "'c"   and y :: "'d" assumes "sorted1 (inorder t)" shows "inorder (update x y t) = upd_list x y (inorder t)"
weakly_seq_sentences_none: fixes P :: "'c option process"   and xs :: "'c option list" assumes "weakly_sequential P"   and "xs \<in> sentences P" shows "None \<notin> set xs"
initial_def: fixes a :: "'a" shows "initial a \<equiv> ide a \<and> (\<forall>b. ide b \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>))"
Span_subgroup_props_1: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "Span K Us \<subseteq> carrier R"
dom: shows "\<alpha> \<in> S \<rightarrow>\<^sub>E S"
step_preserves_type: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E,hp s \<turnstile> e : T" shows "P,E,hp s' \<turnstile> e' :\<^bsub>NT\<^esub> T"
gOpIndif_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gOpIndif MOD \<equiv> \<forall>delta inp1' inp2' inp binp1' binp2' binp. wlsInp delta inp1' \<and> wlsBinp delta binp1' \<and> wlsInp delta inp2' \<and> wlsBinp delta binp2' \<and> gWlsInp MOD delta inp \<and> gWlsBinp MOD delta binp \<longrightarrow> gOp MOD delta inp1' inp binp1' binp = gOp MOD delta inp2' inp binp2' binp"
pow_carrier_mod: fixes g :: "'a"   and k :: "nat" assumes "g \<in> carrier \<G>" shows "g [^] (k mod order \<G>) = g [^] k"
simps_2: fixes x :: "nat"   and xs :: "nat list" shows "lenc (x # xs) = penc (Suc x) (lenc xs)"
par_by_merge_commute: fixes M :: "(('d, 'd, 'd) mrg, 'd) urel"   and P :: "'d hrel"   and Q :: "'d hrel" assumes "M is (;;) swap\<^sub>m" shows "P \<parallel>\<^bsub>M\<^esub> Q = Q \<parallel>\<^bsub>M\<^esub> P"
y_axis_integral_measurable: shows "(\<lambda>x. integral UNIV (\<lambda>y. if (y, x) \<in> Dx_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> j) i (y, x) else 0)) \<in> borel_measurable lborel"
de_Morgan_disj: fixes P :: "bool"   and Q :: "bool" shows "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)"
is_iso_class_def: fixes F :: "'a set" shows "is_iso_class F \<equiv> \<exists>f. f \<in> F \<and> F = \<lbrakk>f\<rbrakk>"
elem_set: fixes x :: "'a"   and xo :: "'a option" shows "(x \<in> set_option xo) = (xo = Some x)"
A: fixes r :: "'d rexp"   and r' :: "'d rexp"   and s :: "'d rexp"   and s' :: "'d rexp" assumes "r ~ r'"   and "s ~ s'" shows "Alt r s ~ Alt r' s'"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and u :: "'c"   and w :: "'c"   and w' :: "'c"   and \<theta> :: "'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and \<theta>' :: "'c"   and \<beta> :: "'c"   and g :: "'c"   and r :: "'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<rho> :: "'c" assumes "partial_magma.ide V u \<and> partial_magma.ide V w \<and> partial_magma.ide V w' \<and> partial_magma.in_hom V \<theta> (H f w) u \<and> partial_magma.in_hom V \<theta>' (H f w') u \<and> partial_magma.in_hom V \<beta> (H g w) (H g w') \<and> V (H r \<theta>) (V (\<a> r f w) (H \<rho> w)) = V (H r \<theta>') (V (\<a> r f w') (V (H \<rho> w') \<beta>))" shows "uw\<theta>w'\<theta>'\<beta>_axioms V H \<a> r \<rho> f g u w \<theta> w' \<theta>' \<beta>"
lim_bdd: fixes n :: "nat" shows "lim (\<lambda>k. f (d k) (r n)) \<in> {- M..M}"
\<theta>_def: shows "\<theta> \<equiv> Var(z := s)"
67: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> - - y \<squnion> x \<sqinter> - (- y \<sqinter> z) = x \<sqinter> - (- y \<sqinter> z)"
subseqs_of_subseq: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<in> set (subseqs xs)" shows "set (subseqs ys) \<subseteq> set (subseqs xs)"
lincomb_list_as_mat_mult: fixes ws :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "\<forall>w\<in>set ws. dim_vec w = n" shows "lincomb_list c ws = mat_of_cols n ws *\<^sub>v vec (length ws) c"
enforce_spmf_top: shows "enforce_spmf \<top> = id"
OR4: fixes evs4 :: "event list"   and B :: "agent"   and X :: "msg"   and M :: "nat"   and A :: "agent"   and X' :: "msg"   and NB :: "nat"   and KAB :: "nat" assumes "evs4 \<in> orb"   and "B \<noteq> Server"   and "\<forall>p q. X \<noteq> \<lbrace>p, q\<rbrace>"   and "Says B Server \<lbrace>Nonce M, Agent A, Agent B, X', Crypt (shrK B) \<lbrace>Nonce NB, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs4"   and "Gets B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>X, Nonce NB, Key KAB\<rbrace>\<rbrace> \<in> set evs4" shows "Says B A \<lbrace>Nonce M, X\<rbrace> # evs4 \<in> orb"
init_match: fixes X :: "'c"   and G :: "('c \<times> ('c, 'd) RHS) list"   and a :: "'d" shows "(X \<in> set (init_match G a)) = ((X, Leaf a) \<in> set G)"
defAssD: fixes g :: "'g"   and m :: "'node"   and v :: "'val"   and ns :: "'node list" assumes "defAss g m v"   and "g \<turnstile> Entry g-ns\<rightarrow>m" shows "\<exists>n\<in>set ns. v \<in> allDefs g n"
adj_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<sim> D" shows "f ` C \<sim> f ` D"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "less_eq0 a b"   and "less_eq0 b c" shows "less_eq0 a c"
LI_soundness: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"   and "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto>\<^sup>* (S\<^sub>2, \<theta>\<^sub>2)"   and "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>2,\<theta>\<^sub>2\<rangle>" shows "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>1,\<theta>\<^sub>1\<rangle>"
assms_1: fixes x :: "'a" assumes "x \<in> A - {x'}" shows "y alg_dvd f x"
identity_pseudonatural_transformation_axioms: shows "identity_pseudonatural_transformation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F"
prems: fixes C :: "('a, 'b) ctxt" assumes "C \<noteq> \<box>"   and "Fun f ss = C\<langle>t\<rangle>" shows "thesis"
simps_5: fixes n :: "nat"   and k :: "nat"   and fs :: "(char list \<times> POPLmarkRecord.type) list" shows "\<up>\<^sub>\<tau> n k (RcdT fs) = RcdT (\<up>\<^sub>r\<^sub>\<tau> n k fs)"
E_valid_1: shows "fst ` E \<subseteq> V"
simps_2: fixes \<mu> :: "'a" shows "Nml \<^bold>\<langle>\<mu>\<^bold>\<rangle> = arr \<mu>"
rot_def: fixes v :: "'a" shows "rot v = eucl_of_list [v $\<^sub>e 1, - v $\<^sub>e 0]"
subgroup_indicator_pos: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> carrier G" shows "0 < subgroup_indicator G H a"
id_rsp: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun R R id id"
assms_1: shows "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)"
floor_power: fixes x :: "'a"   and n :: "nat" assumes "x = of_int \<lfloor>x\<rfloor>" shows "\<lfloor>x ^ n\<rfloor> = \<lfloor>x\<rfloor> ^ n"
B: shows "\<forall>x\<in>set xs. mi \<le> key x \<and> key x \<le> ma"
DIV: fixes p :: "nat" assumes "p \<in># prime_factorization n" shows "p - 1 dvd n - 1"
assumerational: shows "(\<lambda>n. real_of_int (b (n + 1)) / real_of_int (a (n + 1))) sums (real_of_int p / real_of_int q)"
simps_9: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'c"   and f3 :: "'c" shows "(case NoAction of Action x \<Rightarrow> f1 x | NoAction \<Rightarrow> f2 | Undefined \<Rightarrow> f3) = f2"
both_assoc: fixes t :: "'b ttree"   and t' :: "'b ttree"   and t'' :: "'b ttree" shows "t \<otimes>\<otimes> (t' \<otimes>\<otimes> t'') = t \<otimes>\<otimes> t' \<otimes>\<otimes> t''"
intro: fixes f :: "'b \<Rightarrow> 'a" assumes "\<exists>g. g \<circ> f = id \<and> f \<circ> g = id" shows "invertible_lf_axioms f"
cases: fixes x :: "(natural \<Rightarrow> 'a) \<times> natural \<times> natural" obtains f :: "natural \<Rightarrow> 'a" and n :: "natural" and m :: "natural" where "x = (f, n, m)"
axioms_1: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and empty :: "'b" assumes "mem_correct_empty lookup update P empty" shows "mem_correct lookup update P"
f2: shows "(\<forall>v0 v1. truncate_up v0 v1 + truncate_down v0 (- 1 * v1) = 0) = (\<forall>v0 v1. truncate_up v0 v1 = - 1 * truncate_down v0 (- 1 * v1))"
simps_4: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and t :: "char list \<Rightarrow> int option" shows "Fold.fold (IF b THEN c1 ELSE c2) t = IF b THEN Fold.fold c1 t ELSE Fold.fold c2 t"
b: shows "K .\<bottom>. \<phi> \<noteq> {}"
dis_log_adv2: fixes \<G> :: "'a cyclic_group"   and x :: "nat"   and \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf" assumes "dis_log_alt_reductions \<G>" shows "Discrete_Log.dis_log_alt.advantage2 \<G> x \<A> = Discrete_Log.dis_log.advantage \<G> (Discrete_Log.dis_log_alt.adversary2 x \<A>)"
simps_2: fixes X :: "(('fun, 'atom) Term.term \<times> nat) list"   and F :: "(('fun, ('fun, 'atom) Term.term \<times> nat) Term.term \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term) list" shows "tfr\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t = ((\<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. \<exists>a. \<Gamma> (Var x) = Var a) \<or> (\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<longrightarrow> T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` set X)))"
all_formulas_entailed: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N2 :: "'a set"   and N1 :: "'a set" assumes "consequence_relation Bot entails"   and "\<forall>C\<in>N2. entails N1 {C}" shows "entails N1 N2"
defAss'E: fixes g :: "'g"   and m :: "'node"   and v :: "'var \<times> 'node \<times> Def"   and ns :: "'node list" assumes "braun_ssa.defAss' g m v"   and "g \<turnstile> Entry g-ns\<rightarrow>m" obtains n :: "'node" where "n \<in> set ns" and "v \<in> defs' g n"
partition_on_remove_singleton: fixes A :: "'c set"   and P :: "'c set set"   and X :: "'c set"   and A' :: "'c set" assumes "partition_on A P"   and "X \<in> P"   and "A - X = A'" shows "partition_on A' (P - {X})"
\<B>\<^sub>i\<^sub>l_def: fixes state :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>l state = (case state of State x1 x2 x3 x4 x5 x6 \<Rightarrow> x2)"
if_split: fixes P :: "'a \<Rightarrow> bool"   and Q :: "bool"   and x :: "'a"   and y :: "'a" shows "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
mult_eq_0_not_zero_divisor_right: fixes b :: "'b"   and a :: "'b" assumes "\<not> zero_divisor b" shows "(a * b = (0::'b)) = (a = (0::'b))"
Afix_edom: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "edom (Afix \<Gamma>\<cdot>ae) \<subseteq> fv \<Gamma> \<union> edom ae"
IH: fixes gpv :: "('b, 'a, 'c) gpv" shows "interaction_bound' (map_gpv' f g h gpv) \<le> ib (consider \<circ> g) gpv"
soundness: fixes s :: "form list" assumes "finite (deriv (ns_of_s s))" shows "Svalid s"
finvimage_frrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>r\<^sub>\<bullet> A) -`\<^sub>\<bullet> B = r -`\<^sub>\<bullet> (A \<inter>\<^sub>\<circ> B)"
mult_imp_div_pos_less: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less x (times z y)" shows "less (divide x y) z"
fds_converges_uminus: fixes f :: "'b fds"   and s :: "'b" assumes "fds_converges f s" shows "fds_converges (- f) s"
relcompp_in_rel: fixes R :: "('d \<times> 'f) set"   and S :: "('f \<times> 'e) set" shows "in_rel R OO in_rel S = in_rel (R O S)"
IH2: shows "real (fib (Suc n)) = (\<phi> ^ Suc n - \<psi> ^ Suc n) / sqrt 5"
nonregular_event: fixes ev :: "('a, 'b, 'c) event" shows "regular_event ev \<noteq> (isSnapshot ev \<or> isRecvMarker ev)"
assms_4: shows "subst_domain I = \<V> \<and> ground (subst_range I)"
atMost_Int_atLeast: fixes n :: "'a" shows "{..n} \<inter> {n..} = {n}"
Ex1p_Ex1_eq: shows "Ex1p = (\<lambda>x. x \<in> CofGroups.Ex1)"
f_def: shows "f = (d + (2::'a) * b ^ 3 / ((27::'a) * a\<^sup>2) - b * c / ((3::'a) * a)) / a"
val_ifex: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and val :: "'d \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'d"   and s :: "'e \<Rightarrow> bool" assumes "reduced_bdt_checkers ifex_of val" shows "val_ifex (ifex_of b) s = val b s"
insert_absorb: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "insert a A = A"
simps_1: fixes \<mu> :: "'a" shows "E.Nml \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = B.obj \<mu>"
assms: shows "\<Gamma> \<turnstile> App A B : \<sigma>"
elims_3: fixes x :: "'a basic_type"   and xa :: "'a basic_type" assumes "\<not> basic_subtype_fun x xa" obtains \<sigma> :: "'a basic_type" where "x = OclAny" and "xa = \<sigma>__"   | \<sigma> :: "'a basic_type" where "x = OclVoid" and "xa = \<sigma>'__" and "\<not> \<sigma>'__ \<noteq> OclVoid"   | \<sigma> :: "'a basic_type" where "x = Boolean" and "xa = \<sigma>''__" and "\<sigma>''__ \<noteq> OclAny"   | \<sigma> :: "'a basic_type" where "x = Real" and "xa = \<sigma>'''__" and "\<sigma>'''__ \<noteq> OclAny"   | \<sigma> :: "'a basic_type" where "x = Integer" and "xa = \<sigma>''''__" and "\<not> (\<sigma>''''__ = Real \<or> \<sigma>''''__ = OclAny)"   | \<sigma> :: "'a basic_type" where "x = UnlimitedNatural" and "xa = \<sigma>'''''__" and "\<not> (\<sigma>'''''__ = Integer \<or> \<sigma>'''''__ = Real \<or> \<sigma>'''''__ = OclAny)"   | \<sigma> :: "'a basic_type" where "x = String" and "xa = \<sigma>''''''__" and "\<sigma>''''''__ \<noteq> OclAny"   | \<C> :: "'a" and \<sigma> :: "'a basic_type" where "x = \<langle>\<C>\<rangle>\<^sub>\<T>" and "xa = \<sigma>'''''''__" and "\<not> (case \<sigma>'''''''__ of OclAny \<Rightarrow> True | \<langle>x\<rangle>\<^sub>\<T> \<Rightarrow> \<C> < x | _ \<Rightarrow> False)"   | uu_ :: "('a, String.literal) phantom" and \<sigma> :: "'a basic_type" where "x = Enum uu_" and "xa = \<sigma>" and "\<sigma> \<noteq> OclAny"
Min_in: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq A \<in> A"
hom_one: fixes Ra :: "('c, 'd) ring_scheme"   and Sa :: "('e, 'f) ring_scheme"   and ha :: "'c \<Rightarrow> 'e"   and h :: "'g \<Rightarrow> 'h"   and R :: "('g, 'i) ring_scheme"   and S :: "('h, 'j) ring_scheme" assumes "ring_hom_ring Ra Sa ha"   and "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
preallocatedD: fixes h :: "heap"   and C :: "char list" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "\<exists>fs. h (addr_of_sys_xcpt C) = \<lfloor>(C, fs)\<rfloor>"
finite_juncts_2: fixes \<phi> :: "('a, 'b) aformula" shows "finite (conjuncts \<phi>)"
merge_monoms_merge_monoms2: fixes a :: "String.literal list"   and b :: "char list list"   and a' :: "String.literal list"   and b' :: "char list list" assumes "(a, b) \<in> monom_rel"   and "(a', b') \<in> monom_rel" shows "(merge_monoms a a', merge_monoms b b') \<in> monom_rel"
perm_simps_1: fixes p :: "perm" shows "p \<bullet> Unit = Unit"
crel_vs_return: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "Rel R x y" shows "Rel (crel_vs R) (Wrap x) \<langle>y\<rangle>"
is_aligned_replicate: fixes w :: "'b word"   and n :: "nat" assumes "is_aligned w n"   and "n \<le> LENGTH('b)" shows "to_bl w = take (LENGTH('b) - n) (to_bl w) @ replicate n False"
proj_def: fixes r :: "('d \<times> 'c) set"   and x :: "'d" shows "Equiv_Relations.proj r x = r `` \<eta> x"
constructive_security_sim_obsf: shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common (absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr))) adv"
ass: shows "\<exists>s'. s (t1 @ t2)\<Longrightarrow>\<^bsub>SES\<^esub> s'"
int_simps_9: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> P = #True = P"
pos_invariant_iff_compl_neg_invariant: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.positively_invariant f X M = auto_ll_on_open.negatively_invariant f X (X - M)"
appliedBackjump_def: fixes stateA :: "State"   and stateB :: "State" shows "appliedBackjump stateA stateB \<equiv> \<exists>l level. getConflictFlag stateA = True \<and> isBackjumpLevel level l (getC stateA) (getM stateA) \<and> getF stateB = getF stateA \<and> getM stateB = prefixToLevel level (getM stateA) @ [(l, False)] \<and> getConflictFlag stateB = False \<and> getC stateB = []"