hcju/isabelleps huggingface dataset:
preserves_modes_mem_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "preserves_modes_mem \<R> \<equiv> \<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>x\<^sub>A. mem\<^sub>A x\<^sub>A = mem\<^sub>C (var\<^sub>C_of x\<^sub>A)) \<and> (\<forall>m. var\<^sub>C_of ` mds\<^sub>A m = range var\<^sub>C_of \<inter> mds\<^sub>C m)"
Diff_insert: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A - (a \<triangleright> B) = A - B - {a}"
ut: shows "term_variants_pred ((\<lambda>_. [])(a := [b])) u t"
mult_less_cancel_right2: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (times a c) c = ((less_eq zero c \<longrightarrow> less a one) \<and> (less_eq c zero \<longrightarrow> less one a))"
negligible_sphere: fixes a :: "'b"   and e :: "real" shows "negligible (sphere a e)"
compile_cn_gs_correct: fixes gs :: "recf list"   and xs :: "nat list"   and ft :: "nat"   and ffp :: "nat"   and anything :: "nat list" assumes "\<forall>g\<in>set gs. terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"   and "ft = max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))" shows "{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci gs) ft {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> Suc (length xs) @ anything}"
bigoE_bounded_real_fun: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and x0 :: "real"   and cg :: "real" assumes "f \<in> O(g)"   and "\<And>x. x0 \<le> x \<Longrightarrow> cg \<le> \<bar>g x\<bar>"   and "0 < cg"   and "\<And>b. x0 \<le> b \<Longrightarrow> bounded (f ` {x0..b})" shows "\<exists>c>0. \<forall>x\<ge>x0. \<bar>f x\<bar> \<le> c * \<bar>g x\<bar>"
TopoS_List_Impl_def: 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" shows "TopoS_List_Impl default_node_properties sinvar_spec sinvar_impl receiver_violation offending_flows_impl node_props_impl eval_impl \<equiv> (SecurityInvariant sinvar_spec default_node_properties receiver_violation \<and> (\<forall>G nP. wf_list_graph G \<longrightarrow> sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP)) \<and> (\<forall>G nP. wf_list_graph G \<longrightarrow> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar_spec (list_graph_to_graph G) nP = set ` set (offending_flows_impl G nP)) \<and> (\<forall>P. SecurityInvariant.node_props_formaldef default_node_properties P = node_props_impl P) \<and> (\<forall>G P. (distinct (nodesL G) \<and> distinct (edgesL G) \<and> SecurityInvariant.eval sinvar_spec default_node_properties (list_graph_to_graph G) P) = eval_impl G P)"
lconcat_lfilter_neq_LNil: fixes xss :: "'c llist llist" shows "lconcat (lfilter (\<lambda>xs. \<not> lnull xs) xss) = lconcat xss"
generate_in_carrier: fixes H :: "int set"   and h :: "int" assumes "H \<subseteq> carrier (Z n)"   and "h \<in> generate (Z n) H" shows "h \<in> carrier (Z n)"
g2_path_continuous: shows "continuous_on (cbox a b) (\<lambda>x. (g2 x, x))"
graeffe_poly_impl: fixes f :: "'a poly"   and c :: "'a"   and as :: "'a list"   and m :: "nat" assumes "f = Polynomial.smult c (\<Prod>a\<leftarrow>as. [:- a, 1::'a:])" shows "graeffe_poly_impl f m = graeffe_poly c as m"
mono_Retr_1: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "Bisim.mono (PL_Indis.Sretr tval aval indis)"
hyps_3: assumes "\<And>xsa ys x y. \<lbrakk>xs = xsa @ ys; x \<in> set xsa; y \<in> set ys\<rbrakk> \<Longrightarrow> x \<le> y" shows "sorted xs"
MEM_statelist_FDOM: fixes s0 :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and h :: "('a, 'b) fmap" assumes "s0 \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "ListMem h (state_list s0 as)" shows "fmdom' h = fmdom' s0"
n: shows "\<nexists>\<sigma>. D \<cdot> \<sigma> \<subseteq># E"
assms_3: shows "src\<^sub>C a = trg\<^sub>C b"
assms: shows "s \<in> reachable (closed (pnet np n)) TT"
fls_subdegree_integral_ge: fixes f :: "'b fls" assumes "fls_integral f \<noteq> 0" shows "fls_subdegree f + 1 \<le> fls_subdegree (fls_integral f)"
produced_by_scan_step_trans: fixes k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat"   and x :: "item" assumes "indexle k' u' k u"   and "produced_by_scan_step x k' u'" shows "produced_by_scan_step x k u"
sum_list_neutral: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x = (0::'b)" shows "sum_list xs = (0::'b)"
less_floor_iff: fixes z :: "int"   and x :: "'c" shows "(z < \<lfloor>x\<rfloor>) = (of_int z + (1::'c) \<le> x)"
z_eq_v1_solves: assumes "z = v1" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
payload_inductive_cases_2: fixes X :: "ltkey"   and P :: "bool" assumes "LtK X \<in> payload" shows "P"
inj: fixes f :: "'b \<Rightarrow> 'b" assumes "bijection f" shows "inj f"
hv_add: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "Hv (x + y) = Hv x + Hv y"
THE_default_none: fixes P :: "'b \<Rightarrow> bool"   and d :: "'b" assumes "\<nexists>!x. P x" shows "THE_default d P = d"
multisets_union: fixes M :: "'b multiset"   and A :: "'b set"   and N :: "'b multiset" assumes "M \<in> multisets A"   and "N \<in> multisets A" shows "M + N \<in> multisets A"
mod_ring_plus64: shows "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (plus_p64 pp) (+)"
pop_def: fixes q :: "('c, 'd) pq" shows "PQ.pop q = (if PQ.is_empty q then None else Some (PQ.min q, remove_min q))"
disc2_noSrcPorts: shows "\<forall>a. \<not> disc2 (Src_Ports a)"
induct: fixes P :: "'a formula \<Rightarrow> bool"   and a0 :: "'a formula" assumes "\<And>k. P (Atom k)"   and "\<And>k. P (\<^bold>\<not> (Atom k))"   and "P \<bottom>"   and "P (\<^bold>\<not> \<bottom>)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G)"   and "\<And>F G. \<lbrakk>P F; \<And>x. x \<in> set (cnf_lists F) \<Longrightarrow> P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>va. P (\<^bold>\<not> (\<^bold>\<not> va))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<and> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<or> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<rightarrow> vb))"   and "\<And>v va. P (v \<^bold>\<rightarrow> va)" shows "P a0"
splitTree_aux_def: fixes p :: "'d \<Rightarrow> bool"   and i :: "'d"   and t :: "('c, 'd) FingerTree" shows "splitTree_aux p i t \<equiv> if \<not> p i \<and> p (i + annot t) then Abs_splitres (FingerTreeStruc.splitTree p i (Rep_FingerTree t)) else Abs_splitres (Empty, undefined, Empty)"
nth_image_precond_def: fixes X :: "'e list set"   and n :: "nat" shows "nth_image_precond X n = (X \<noteq> {} \<and> (\<forall>xs\<in>X. n < length xs \<and> env_len X (length xs)))"
min_product_set_min_set: fixes X :: "'b fset set set" assumes "finite X" shows "min_set (\<Otimes>\<^sub>m X) = \<Otimes>\<^sub>m X"
iMin_Min_conv: fixes I :: "'a set" assumes "finite I"   and "I \<noteq> {}" shows "iMin I = Min I"
gfp_omega_upperbound: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "mono f"   and "A \<le> Omega_fun f g A" shows "A \<le> star_fun f \<circ> g"
product_integral_prod: fixes I :: "'i set"   and f :: "'i \<Rightarrow> 'b \<Rightarrow> 'd" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> integrable (distr M borel (Y i)) (f i)" shows "LINT x|Pi\<^sub>M I (\<lambda>i. distr M borel (Y i)). (\<Prod>i\<in>I. f i (x i)) = (\<Prod>i\<in>I. Y.expectation i (f i))"
one_prod_def: shows "one_prod = (1::'c, 1::'d)"
rev_eq_Cons_iff: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "(rev xs = y # ys) = (xs = rev ys @ [y])"
tendsto_iff_Liminf_eq_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and f0 :: "'c" assumes "F \<noteq> bot" shows "(f \<longlongrightarrow> f0) F = (Liminf F f = f0 \<and> Limsup F f = f0)"
finite_dimensional_vector_space_on_axioms: shows "finite_dimensional_vector_space_on (tangent_space p) (*\<^sub>R) basis"
doesnt_read_or_modify_doesnt_modify: fixes dma :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and \<C>_vars :: "'c \<Rightarrow> 'c set"   and eval :: "((('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and c :: "'e"   and x :: "'c" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.doesnt_read_or_modify \<C>_vars eval c x" shows "sifum_security_init.doesnt_modify dma eval c x"
simps: fixes ant :: "'b form multiset"   and suc :: "'b form multiset" shows "seq_size ( ant \<Rightarrow>* suc) = size ant + size suc"
white_def: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" shows "white r s \<equiv> obj_at (\<lambda>obj. obj_mark obj \<noteq> sys_fM s) r s"
nn_integral_count_space_indicator: fixes X :: "'d set"   and f :: "'d \<Rightarrow> ennreal" assumes "NO_MATCH UNIV X" shows "integral\<^sup>N (count_space X) f = set_nn_integral (count_space UNIV) X f"
divide_le_0_abs_iff: 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 a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a (abs b)) zero = (less_eq a zero \<or> b = zero)"
f_Exec_Stream_Init_eq_f_Exec_Stream_Cons_output: fixes output_fun :: "'e \<Rightarrow> 'd message_af"   and c :: "'e"   and trans_fun :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f list" assumes "output_fun c = NoMsg" shows "map output_fun (f_Exec_Comp_Stream_Init trans_fun xs c) = NoMsg # map output_fun (f_Exec_Comp_Stream trans_fun xs c)"
closedin_trans: fixes T :: "'b set"   and S :: "'b set"   and U :: "'b set" assumes "closedin (top_of_set T) S"   and "closedin (top_of_set U) T" shows "closedin (top_of_set U) S"
faceG: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> \<G>"   and "Y \<in> \<G>" shows "X \<inter> Y face_of X"
b: fixes k :: "nat" shows "\<exists>P\<subseteq>M k. infinite P \<and> (\<forall>j i. i \<le> j \<longrightarrow> j \<le> k \<longrightarrow> b k (j, i) = fst (bf P (ediff d) (j, i)))"
SC_discr_imp_SC_BisT: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('c, 'b) com"   and tval :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "PL_Indis indis"   and "PL_Indis.SC_discr aval indis c" shows "PL_Indis.SC_BisT tval aval indis c"
pt_fresh_perm_app: fixes a :: "'e"   and pi :: "('e \<times> 'e) list"   and x :: "'d" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)"   and "a \<sharp> pi"   and "a \<sharp> x" shows "a \<sharp> pi \<bullet> x"
leq_method_subtypes: fixes D :: "char list"   and C :: "char list"   and 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 M :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "char list list \<times> expr"   and Ds :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> D has least M = (Ts', T', m') via Ds"   and "wf_prog wf_md P" shows "\<forall>Ts T m Cs. P \<turnstile> C has M = (Ts, T, m) via Cs \<longrightarrow> Ts = Ts' \<and> P \<turnstile> T' \<le> T"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "i < length (take n xs)" shows "take n xs ! i = xs ! i"
map_spmf_conv_ap: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "map_spmf f p = Applicative.ap (return_spmf f) p"
state_sim_mono: fixes s :: "'b"   and A :: "('b \<Rightarrow> bool) set"   and t :: "'b"   and B :: "('b \<Rightarrow> bool) set" assumes "s ~A~ t"   and "B \<subseteq> A" shows "s ~B~ t"
simps_6: fixes v1 :: "'b"   and l1 :: "value" shows "AExp_Lexorder.less_aexp_aux (V v1) (L l1) = False"
S2: shows "set_to_nat (D \<union> {x}) = set_to_nat D + 2 ^ x"
simps_5: shows "get_axiom Achoice = choice_axiom"
bij_inv: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "bij (inv f)"
is_unifier_def: fixes \<sigma> :: "'s"   and AA :: "'a set" shows "is_unifier \<sigma> AA = (card (AA \<cdot>as \<sigma>) \<le> 1)"
eventually_elim2: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F"   and "\<And>i. \<lbrakk>P i; Q i\<rbrakk> \<Longrightarrow> R i" shows "eventually R F"
unit_factor_mod_ring_def: fixes x :: "'b mod_ring" shows "unit_factor x = x"
containID: fixes i :: "nat"   and \<Gamma> :: "'a form multiset"   and R :: "('a sequent list \<times> 'a sequent) set" assumes "At i \<in># \<Gamma>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* At i, 0) \<in> derivable R*"
union_eq_empty: fixes M :: "'k multiset"   and N :: "'k multiset" shows "(M + N = {#}) = (M = {#} \<and> N = {#})"
f_the_inv_into_f: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and y :: "'h" assumes "inj_on f A"   and "y \<in> f ` A" shows "f (the_inv_into A f y) = y"
Compl_FP: fixes F :: "'b program" shows "- FP F = (\<Union>act\<in>Acts F. - {s. act `` {s} \<subseteq> {s}})"
le_imp_power_dvd: fixes m :: "nat"   and n :: "nat"   and a :: "'a" assumes "m \<le> n" shows "a ^ m dvd a ^ n"
nodes_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "gnodes A w V \<equiv> {x. transition_system_initial.nodesp (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V) (\<lambda>v. v \<in> ginitial A \<inter> V) x}"
s\<^sub>1: shows "s\<^sub>1 = (h\<^sub>1, l\<^sub>1, sh\<^sub>1)"
sup_Un_eq: fixes R :: "'d set"   and S :: "'d set" shows "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
V_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a" shows "Graph.V c \<equiv> {u. \<exists>v. (u, v) \<in> Graph.E c \<or> (v, u) \<in> Graph.E c}"
ord_spmf_bindI: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'d spmf"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e spmf"   and g :: "'d \<Rightarrow> 'e spmf" assumes "ord_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> ord_spmf P (f x) (g y)" shows "ord_spmf P (p \<bind> f) (q \<bind> g)"
incl_incl_of_2: fixes a :: "'s"   and b :: "'s" assumes "incl_in a b" shows "\<guillemotleft>incl_of a b : a \<rightarrow> b\<guillemotright>"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" shows "\<guillemotleft>G f : G a \<rightarrow>\<^sub>B G b\<guillemotright>"
commutes_with_composition: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> kernel.Partition"   and "y \<in> kernel.Partition" shows "induced (x [\<cdot>] y) = induced x \<cdot>' induced y"
inpt_str_comp_aux: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and r' :: "'b \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'a \<times> 'b"   and x :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "(\<forall>n. (\<forall>i<n. inpt_st (\<lambda>(u, v) (u', v'). r u u' OO r' v v') (u i) (u (Suc i)) (x i)) \<longrightarrow> inpt_st r (fst (u n)) (fst (u (Suc n))) (x n) \<and> (\<forall>y. r (fst (u n)) (fst (u (Suc n))) (x n) y \<longrightarrow> inpt_st r' (snd (u n)) (snd (u (Suc n))) y)) \<longrightarrow> (\<forall>i<n. inpt_st r ((fst \<circ> u) i) ((fst \<circ> u) (Suc i)) (x i) \<and> (\<forall>y. r (fst (u i)) (fst (u (Suc i))) (x i) y \<longrightarrow> inpt_st r' (snd (u i)) (snd (u (Suc i))) y))"
assms_2: shows "eval (Mn n f) xs \<down>= z"
no_inf_Throw: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "('b, 'c) xstate" shows "\<not> \<Gamma>\<turnstile> (Throw, s) \<rightarrow> \<dots>(\<infinity>)"
IH_1: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "equivalent_on \<sigma> \<eta> (vars_of t1) I" shows "I (t1 \<lhd> \<sigma>) (t1 \<lhd> \<eta>)"
Times_Un_distrib1: fixes A :: "'b set"   and B :: "'b set"   and C :: "'c set" shows "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C"
simps_3: fixes e :: "'a"   and a :: "'b"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.insert e a (t # t' # bq) = (if SkewBinomialHeapStruc.rank t \<noteq> SkewBinomialHeapStruc.rank t' then SkewBinomialHeapStruc.Node e a 0 [] # t # t' # bq else SkewBinomialHeapStruc.skewlink e a t t' # bq)"
inf_sup_aci_7: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
If: fixes b :: "bexp"   and l :: "nat"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "max (sec b) l \<turnstile> c\<^sub>1"   and "max (sec b) l \<turnstile> c\<^sub>2" shows "l \<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2"
one_notin_nonpos_Ints: shows "(1::'a) \<notin> \<int>\<^sub>\<le>\<^sub>0"
ag_l_zero: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d" assumes "aGroup A"   and "x \<in> carrier A" shows "\<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> x = x"
i_shrink_last_i_take_mult: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "0 < k" shows "f \<Down> (n * k) \<div>\<^sub>l k = f \<div>\<^sub>l k \<Down> n"
doesnt_have_mode: fixes x :: "Eg1.var"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and m :: "Mode" shows "(x \<notin> mds\<^sub>A_of mds\<^sub>C m) = (Eg2_var\<^sub>C_of_Eg1 x \<notin> mds\<^sub>C m)"
simps_1: fixes cx :: "int"   and cy :: "int"   and more :: "'b" shows "cx \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = cx"
type_definition_mod_ring: shows "type_definition Rep_mod_ring Abs_mod_ring {0..<int CARD('b)}"
inj: fixes ya :: "real" assumes "root_cond (abs_int_poly (poly_uminus p), - r, - l) ya" shows "- real_of_1 x = ya"
lcons: fixes l :: "'a llist"   and a :: "'a" assumes "l \<in> A\<^sup>\<star>"   and "P l"   and "a \<in> A" shows "P (a ## l)"
mono: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
op_type_alt_simps_1: fixes \<tau> :: "'a OCL_Types.type"   and op :: "metaop"   and \<sigma> :: "'a OCL_Types.type" shows "mataop_type \<tau> op \<sigma> = (op = AllInstancesOp \<and> \<sigma> = Set \<tau>)"
rep_eq: fixes x :: "integer" shows "lsb x = lsb (int_of_integer x)"
abs_div_pos: 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" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y" shows "divide (abs x) y = abs (divide x y)"
mono_ndet_FD_right: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<le> Q" shows "(S |-| P) \<le> Q"
VLambda_iff2: fixes a :: "V"   and b :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" shows "(\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VLambda A f) = (b = f a \<and> a \<in>\<^sub>\<circ> A)"
alphaAbs_ex_equal_or_qFresh_imp_alphaAbs_qAbs: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "alphaAbs_ex_equal_or_qFresh xs x X xs' x' X'" shows "qAbs xs x X $= qAbs xs' x' X'"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) R local.map \<rr>.map"
b: shows "steps0 (1, l, r) B n = (s'', l'', r'') \<and> 0 < s''"
iso_cancel_right: fixes g :: "'a"   and f :: "'a"   and f' :: "'a" assumes "B.iso g"   and "f \<cdot>\<^sub>B g = f' \<cdot>\<^sub>B g"   and "B.seq f g"   and "B.iso g" shows "f = f'"
real_poly_smult: fixes c :: "'a"   and p :: "'a poly" assumes "c \<in> \<real>"   and "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (smult c p)) \<subseteq> \<real>"
semiring_mat: fixes n :: "nat"   and b :: "'d" shows "semiring (ring_mat TYPE('c) n b)"
nmi: shows "\<not> Ifm (a # bs) (minusinf p)"
is_extensional: fixes f :: "'a" assumes "\<not> B.arr f" shows "trg\<^sub>B f = B.null"
hp_induct: fixes P :: "('b, 'c, 'a) hp \<Rightarrow> bool"   and hp :: "('b, 'c, 'a) hp" assumes "\<And>x. P ($\<alpha> x)"   and "\<And>x1 x2. P (x1 := x2)"   and "\<And>x1 x2. P (DiffAssign x1 x2)"   and "\<And>x. P (? x)"   and "\<And>x1 x2. P (EvolveODE x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<union>\<union> x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 ;; x2)"   and "\<And>x. P x \<Longrightarrow> P x**" shows "P hp"
match_ex_table_stack_xlift: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc' :: "nat"   and d :: "nat"   and n :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', d)\<rfloor>" shows "match_ex_table P C pc (stack_xlift n xt) = \<lfloor>(pc', n + d)\<rfloor>"
hRe_diff: fixes x :: "complex star"   and y :: "complex star" shows "hRe (x - y) = hRe x - hRe y"
one: fixes R :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "\<one>\<^bsub>R\<^esub> \<in> generate_field R H"