hcju/isabelleps huggingface dataset:
succ_min_rvar: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and xs :: "nat"   and xr :: "nat"   and eq :: "nat \<times> linear_poly"   and dir :: "('i, 'a) Direction" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "xs \<in> lvars (\<T> s)"   and "xs \<in> rvars (\<T> s')"   and "xr \<in> rvars (\<T> s)"   and "xr \<in> lvars (\<T> s')"   and "eq = eq_for_lvar (\<T> s) xs"   and "dir = Positive \<or> dir = Negative" shows "\<not> \<unrhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> xs) (LB dir s xs) \<longrightarrow> (0 < coeff (rhs eq) xr \<and> \<lhd>\<^sub>u\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> xr) (UB dir s xr) \<or> coeff (rhs eq) xr < 0 \<and> \<rhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> xr) (LB dir s xr)) \<and> (\<forall>x\<in>rvars_eq eq. x < xr \<longrightarrow> \<not> (0 < coeff (rhs eq) x \<and> \<lhd>\<^sub>u\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x) (UB dir s x) \<or> coeff (rhs eq) x < 0 \<and> \<rhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x) (LB dir s x)))"
subset_eq_diff_conv: fixes A :: "'k multiset"   and C :: "'k multiset"   and B :: "'k multiset" shows "(A - C \<subseteq># B) = (A \<subseteq># B + C)"
AG_gfp: fixes p :: "'b set" shows "\<^bold>A\<^bold>G p = gfp (\<lambda>s. p \<inter> \<^bold>A\<^bold>X s)"
x1: shows "flow0 x t1 \<in> {a<--<b}"
continuous_on_cong: fixes s :: "'c set"   and t :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "s = t"   and "\<And>x. x \<in> t \<Longrightarrow> f x = g x" shows "continuous_on s f = continuous_on t g"
separatedNC: fixes p :: "('b set set, 'c) Combinators list" assumes "OnlyTwoNets p"   and "NetsCollected2 p"   and "NetsCollected p"   and "noDenyAll1 p"   and "allNetsDistinct p" shows "separated p"
summable: fixes x :: "'a" assumes "x \<in> A" shows "f x abs_summable_on B x"
Rep_cfun_strict: shows "Rep_cfun \<bottom> = \<bottom>"
nth_map_matrix: fixes f :: "'d \<Rightarrow> 'a"   and x :: "(('d, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" shows "map_matrix f x $ i $ j = f (x $ i $ j)"
simps_4: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c" shows "(case \<bottom> of \<bottom> \<Rightarrow> f1 | Def x \<Rightarrow> f2 x) = f1"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Cod (t \<^bold>\<otimes> u) = Cod t \<^bold>\<otimes> Cod u"
homeomorphic_path_connected_space_imp: fixes X :: "'b topology"   and Y :: "'c topology" assumes "path_connected_space X"   and "X homeomorphic_space Y" shows "path_connected_space Y"
many_calls_valid_list: fixes xs :: "Vars.var list"   and x :: "Vars.var" assumes "xs \<in> valid_lists {x} ({x} G\<times> {x})" shows "xs \<in> range (\<lambda>n. replicate n x)"
mcont2mcont': fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub' :: "'c set \<Rightarrow> 'c"   and ord' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b \<Rightarrow> 'c" assumes "class.ccpo Sup less_eq less"   and "\<And>x. mcont lub' ord' Sup less_eq (f x)"   and "\<And>y. mcont lub ord Sup less_eq (\<lambda>x. f x y)"   and "mcont lub ord lub' ord' t" shows "mcont lub ord Sup less_eq (\<lambda>x. f x (t x))"
not_int_div_2: fixes k :: "int" shows "NOT k div 2 = NOT (k div 2)"
distinct_61: fixes x7 :: "char list" shows "Return \<noteq> Goto x7"
sim_trans: fixes as\<^sub>0 :: "'a"   and alstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and bs\<^sub>0 :: "'c"   and blstep :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and R\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and cs\<^sub>0 :: "'d"   and clstep :: "'b \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and R\<^sub>2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "simulation as\<^sub>0 alstep bs\<^sub>0 blstep R\<^sub>1"   and "simulation bs\<^sub>0 blstep cs\<^sub>0 clstep R\<^sub>2" shows "simulation as\<^sub>0 alstep cs\<^sub>0 clstep (R\<^sub>1 OO R\<^sub>2)"
while_eval: fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>While e c \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>Stmt.If e (c ;; While e c) Stop, mds \<oplus> annos, mem\<rangle>"
bangTauE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> !P \<longmapsto> \<tau> \<prec> P'"   and "guarded P" obtains Q :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<parallel> P \<longmapsto> \<tau> \<prec> Q" and "P' \<sim> Q \<parallel> !P"
splitAt_dist_ram21: fixes vs :: "'a list"   and a :: "'a list"   and ram1 :: "'a"   and b :: "'a list"   and ram2 :: "'a"   and c :: "'a list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(a, b) = splitAt ram1 (fst (splitAt ram2 vs))"
univ_poly_is_principal: fixes K :: "'a list set" assumes "subfield K (poly_ring R)" shows "principal_domain (K [X]\<^bsub>poly_ring R\<^esub>)"
ivl_splice_Un: fixes n :: "nat" shows "{0..<2 * n} = (*) 2 ` {0..<n} \<union> (\<lambda>i. Suc (2 * i)) ` {0..<n}"
wf_jvm_prog_phi_def: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list" shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> \<equiv> wf_prog (\<lambda>P C (M, Ts, T\<^sub>r, mxs, mxl\<^sub>0, is, xt). wt_method P C Ts T\<^sub>r mxs mxl\<^sub>0 is xt (\<Phi> C M))"
substT_Var: fixes x :: "'var"   and y :: "'var"   and t :: "'trm" assumes "x \<in> var"   and "y \<in> var"   and "t \<in> trm" shows "substT (Var x) t y = (if x = y then t else Var x)"
rep_eq: fixes x :: "('sf, 'sc, 'sz) good_interp"   and xa :: "('sf, 'sz) strm"   and xb :: "(real, 'sz) vec" shows "blinfun_apply (blin_frechet x xa xb) = frechet (raw_interp x) (raw_term xa) xb"
cases: fixes a1 :: "'c"   and a2 :: "'c list"   and a3 :: "'c" assumes "trace a1 a2 a3" obtains final :: "'c" where "a1 = final__" and "a2 = []" and "a3 = final__"   | conf' :: "'c" and T :: "'c list" and final :: "'c" and conf :: "'c" where "a1 = conf" and "a2 = conf' # T" and "a3 = final" and "trace conf' T final" and "conf \<Rightarrow> conf'"
cases: fixes a :: "(name \<times> ty) list" assumes "valid a" obtains "a = []"   | \<Gamma> :: "(name \<times> ty) list" and aa :: "name" and T :: "ty" where "a = (aa, T) # \<Gamma>" and "valid \<Gamma>" and "aa \<sharp> \<Gamma>"
set_r_ar_cos: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "set_rcs (b_ag R) I = set_ar_cos R I"
addN_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and g :: "'g"   and n :: "'node" shows "uninst.addN u g n \<equiv> fold (\<lambda>v. Mapping.map_default v {} (insert n)) (sorted_list_of_set (u g n))"
igSwapAllIPresIGWlsAll_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSwapAllIPresIGWlsAll MOD \<equiv> igSwapIPresIGWls MOD \<and> igSwapAbsIPresIGWlsAbs MOD"
simps_9: fixes v :: "nat \<Rightarrow> 'a"   and l1 :: "linear_poly"   and l2 :: "linear_poly" shows "v \<Turnstile>\<^sub>c GEQPP l1 l2 = (l2 \<lbrace> v \<rbrace> \<le> l1 \<lbrace> v \<rbrace>)"
span_Un: fixes S :: "'b set"   and T :: "'b set" shows "local.span (S \<union> T) = {x + y |x y. x \<in> local.span S \<and> y \<in> local.span T}"
newton_seq_fact3: fixes l :: "nat" shows "val_Zp (f' \<bullet> ns l) \<le> val_Zp (f \<bullet> ns l)"
lessThan_atLeast0: fixes n :: "nat" shows "{..<n} = {0..<n}"
1113: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "- (x \<squnion> (y \<squnion> z) \<sqinter> u) \<sqinter> (x \<squnion> z \<sqinter> u) = bot"
LVar: fixes E :: "env"   and vn :: "lname"   and T :: "ty"   and dt :: "loc \<Rightarrow> ty option" assumes "lcl E vn = Some T"   and "is_acc_type (prg E) (pkg E) T" shows "E,dt\<Turnstile>LVar vn\<Colon>=T"
1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
Lim_Un: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "(f \<longlongrightarrow> l) (at x within S)"   and "(f \<longlongrightarrow> l) (at x within T)" shows "(f \<longlongrightarrow> l) (at x within S \<union> T)"
Shuffle_assoc: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Shuffle (Shuffle r s) t = Shuffle r (Shuffle s t)"
opeS: shows "openin (top_of_set S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
CEMD_def: shows "CEMD \<equiv> CEM \<and> CMD"
inner_prod_is_linear: fixes u :: "complex vec"   and v :: "nat \<Rightarrow> complex vec"   and l :: "nat \<Rightarrow> complex" assumes "\<forall>i\<in>{0, 1}. dim_vec u = dim_vec (v i)" shows "\<langle>u|l 0 \<cdot>\<^sub>v v 0 + l 1 \<cdot>\<^sub>v v 1\<rangle> = (\<Sum>i\<le>1. l i * \<langle>u|v i\<rangle>)"
SA_States_disjunct: fixes A :: "('d, 'e, 'f) seqauto"   and A' :: "('d, 'e, 'f) seqauto" assumes "States A \<inter> States A' = {}" shows "A' \<noteq> A"
normal_density_nonneg: fixes \<mu> :: "real"   and \<sigma> :: "real"   and x :: "real" shows "0 \<le> normal_density \<mu> \<sigma> x"
dghm_ObjMap_app_in_HomCod_Obj: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
finite_reach: shows "finite (reach \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>))"
abs_eq: shows "0 = Abs_poly_mapping (\<lambda>k. 0::'c)"
Assert'Precond_1: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<triangle> (\<T> (assert_bound a s))"
cdens_ctxt_invarI: fixes vs :: "nat list"   and vs' :: "nat list"   and \<delta> :: "cexpr"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "distinct (vs @ vs')"   and "free_vars \<delta> \<subseteq> set (vs @ vs')"   and "\<Gamma> \<turnstile>\<^sub>c \<delta> : REAL"   and "nonneg_cexpr (set vs \<union> set vs') \<Gamma> \<delta>"   and "subprob_cexpr (set vs) (set vs') \<Gamma> \<delta>" shows "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"
ack_less_mono2: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j < k" shows "ack i j < ack i k"
orderE: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" obtains "a = min a b"
FvarsT: fixes t :: "'trm" assumes "t \<in> trm" shows "FvarsT t \<subseteq> var"
iso_map\<^sub>1_ide: fixes f :: "'c" assumes "C.ide f" shows "D.iso (\<sigma>\<^sub>1 f)"
simps_3: fixes a :: "name"   and x :: "name"   and P :: "pi" assumes "a \<sharp> x" shows "depth (a<x>.P) = 1 + depth P"
pdevs_apply_add_pdevs: fixes X :: "'b pdevs"   and Y :: "'b pdevs"   and n :: "nat" shows "pdevs_apply (add_pdevs X Y) n = pdevs_apply X n + pdevs_apply Y n"
Icc_eq_Icc: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and h :: "'a"   and l' :: "'a"   and h' :: "'a" assumes "class.order less_eq less" shows "(ord.atLeastAtMost less_eq l h = ord.atLeastAtMost less_eq l' h') = (l = l' \<and> h = h' \<or> \<not> less_eq l h \<and> \<not> less_eq l' h')"
abs_eq: fixes xa :: "'a"   and x :: "'b" shows "single_pp xa x = PP (Poly_Mapping.single xa x)"
graph_ext_def: fixes nodes :: "'a set"   and edges :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons graph_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple nodes edges) more"
N_def: fixes i :: "'n" shows "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < r / real CARD('n))"
phiDefs_finite: fixes g :: "'g"   and n :: "'node" shows "finite (phiDefs g n)"
g2: shows "\<forall>c\<in>C. \<exists>b\<in>B. map g b \<cong> c"
regular_event_cannot_induce_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and c :: "('d, 'e, 'f) configuration"   and p :: "'d"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "\<not> ps c p \<noteq> None"   and "distributed_system.next channel trans send recv c ev c'" shows "regular_event ev \<longrightarrow> \<not> ps c' p \<noteq> None"
strictly_preserves_\<alpha>: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and h :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g"   and "\<F>C.arr h" shows "F (\<F>C.assoc f g h) = \<a>\<^sub>D[F f, F g, F h]"
hyps: fixes i :: "nat"   and S :: "nat \<Rightarrow> bool option" assumes "i < length ls"   and "ls ! i = (p, Write volatile a (D, f) A L R W # is, \<theta>, sb, \<D>, \<O>)" shows "share_all_until_volatile_write (ls[i := (p, is, \<theta>, sb @ [Write\<^sub>s\<^sub>b volatile a t (f \<theta>) A L R W], \<D>', \<O>)]) S = share_all_until_volatile_write ls S"
that: fixes c' :: "String.literal" assumes "c' \<in> normalize_sort cs s"   and "class_les cs c' c" shows "thesis"
hom_boundary_restrict: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "hom_boundary p X (topspace X \<inter> S) = hom_boundary p X S"
rep_eq: fixes x :: "'a interval" shows "set_of x = {fst (bounds_of_interval x)..snd (bounds_of_interval x)}"
UN_from_nat_into: fixes I :: "'b set"   and N :: "'b \<Rightarrow> 'c set" assumes "countable I"   and "I \<noteq> {}" shows "\<Union> (N ` I) = (\<Union>i. N (from_nat_into I i))"
vs_confD: fixes P :: "'b prog"   and h :: "'heap"   and vs :: "'addr \<times> addr_loc \<Rightarrow> 'addr val set"   and v :: "'addr val"   and ad :: "'addr"   and al :: "addr_loc" assumes "h.vs_conf P h vs"   and "v \<in> vs (ad, al)" shows "\<exists>T. h.addr_loc_type P h ad al T \<and> h.conf P h v T"
Proper_Edges_def: shows "Proper_Edges \<equiv> \<lambda>(M, E). \<forall>i<length E. fst (E ! i) < length M \<and> snd (E ! i) < length M"
subgroup_intersect: fixes H :: "'c set"   and H' :: "'c set" assumes "subgroup H H"   and "subgroup H' H" shows "subgroup (H \<inter> H') H"
derived_is_subgroup: fixes H :: "(nat \<Rightarrow> nat) set" assumes "H \<subseteq> carrier (alt_group n)" shows "subgroup (derived (alt_group n) H) (alt_group n)"
fls_lr_inverse_eq_0_iff_1: fixes f :: "'a fls"   and x :: "'a" shows "(fls_left_inverse f x = 0) = (x = (0::'a))"
simps_2: fixes C :: "char list" shows "getbase (Shares C) = C"
homeomorphism_on_sequentially: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x u. \<lbrakk>x \<in> S; \<forall>n. u n \<in> S\<rbrakk> \<Longrightarrow> u \<longlonglongrightarrow> x = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f x" shows "homeomorphism_on S f"
abs_Bonk_Schramm_extension_def: shows "abs_Bonk_Schramm_extension \<equiv> quot_type.abs (\<lambda>x y. x \<in> extended_distance_set \<and> y \<in> extended_distance_set \<and> extended_distance x y = 0) Abs_Bonk_Schramm_extension"
nAct_less_llength_active: fixes x :: "nat"   and c :: "'id"   and t :: "cnf llist"   and n' :: "nat" assumes "enat x < llength (\<pi>\<^bsub>c\<^esub>t)"   and "enat x = \<langle>c #\<^bsub>enat n'\<^esub>t\<rangle>" shows "\<exists>i\<ge>n'. enat i < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
ascP_def: fixes f :: "'b list \<Rightarrow> 'b list" shows "ascP f = (\<forall>xs ys. f (xs @ ys) = f xs @ ys)"
ov: fixes p :: "'b"   and q :: "'b" shows "((p, q) \<in> ov) = (\<exists>k l u v t. (k \<parallel> p \<and> p \<parallel> u \<and> u \<parallel> v) \<and> (k \<parallel> l \<and> l \<parallel> q \<and> q \<parallel> v) \<and> l \<parallel> t \<and> t \<parallel> u)"
groupI: fixes G :: "('e, 'f) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
assms_3: shows "u \<noteq> 0\<^sub>h"
faces: fixes x :: "'b set"   and y :: "'b set" assumes "x \<in> Y"   and "order.greater_eq x y" shows "y \<in> Y"
assms: fixes x :: "'a"   and y :: "'a" assumes "Q x"   and "P x y" shows "Q y"
algebraic_times: fixes x :: "'a"   and y :: "'a" assumes "algebraic x"   and "algebraic y" shows "algebraic (x * y)"
assms: fixes i :: "nat" assumes "i < n" shows "\<exists>j. suffix (index i + j) w \<Turnstile>\<^sub>n af (formula i) (w [index i \<rightarrow> index i + j])[X]\<^sub>\<nu>"
mini_lb: fixes x :: "'c"   and xs :: "'c list"   and key :: "'c \<Rightarrow> 'd" assumes "x \<in> set xs" shows "key (xs ! mini xs key) \<le> key x"
strategyproofness: shows "(pmf (sds R15) d < pmf (sds R7) d \<or> pmf (sds R15) d + (pmf (sds R15) b + pmf (sds R15) a) < pmf (sds R7) d + (pmf (sds R7) b + pmf (sds R7) a)) \<or> pmf (sds R15) d = pmf (sds R7) d \<and> pmf (sds R15) d + (pmf (sds R15) b + pmf (sds R15) a) = pmf (sds R7) d + (pmf (sds R7) b + pmf (sds R7) a)"
comp_FSwImorph: fixes h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'h, 'j, 'm) model_scheme"   and h' :: "'h \<Rightarrow> 'n"   and hA' :: "'j \<Rightarrow> 'o"   and MOD'' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'n, 'o, 'p) model_scheme" assumes "FSwImorph h hA MOD MOD'"   and "FSwImorph h' hA' MOD' MOD''" shows "FSwImorph (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
pairs_subst_comp: fixes F :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) list"   and \<delta> :: "'e \<Rightarrow> ('c, 'f) Term.term"   and \<theta> :: "'f \<Rightarrow> ('c, 'd) Term.term" shows "F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta> \<circ>\<^sub>s \<theta> = (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>"
induct: fixes G :: "prog"   and x1 :: "state"   and x2 :: "obj_tag"   and x3 :: "loc"   and x4 :: "state"   and P :: "state \<Rightarrow> obj_tag \<Rightarrow> loc \<Rightarrow> state \<Rightarrow> bool" assumes "G\<turnstile>x1 \<midarrow>halloc x2\<succ>x3\<rightarrow> x4"   and "\<And>x s oi. P (Some x, s) oi undefined (Some x, s)"   and "\<And>s a x oi' oi. \<lbrakk>new_Addr (heap s) = Some a; (x, oi') = (if atleast_free (heap s) (Suc (Suc 0)) then (None, oi) else (Some (Xcpt (Loc a)), CInst (SXcpt OutOfMemory)))\<rbrakk> \<Longrightarrow> P (Norm s) oi a (x, init_obj G oi' (Inl a) s)" shows "P x1 x2 x3 x4"
ord_eq_less_subst: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and c :: "'b" assumes "a = f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
IH: assumes "odd x"   and "1 < x" shows "Jacobi (- 1) x = (- 1) ^ nat ((x - 1) div 2)"
homeomorphism_sym: fixes S :: "'c set"   and t :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "homeomorphism S t f g = homeomorphism t S g f"
pdoW1k: fixes n :: "nat" shows "partial_density_operator (denote_while_n_comp M0 M1 DS n \<rho>)"
CHAOS_all_tickfree_traces1: fixes A :: "'a set" shows "T (CHAOS A) = {s. set s \<subseteq> ev ` A}"
word_of_int_inverse: fixes r :: "int"   and a :: "'b word" assumes "word_of_int r = a"   and "0 \<le> r"   and "r < 2 ^ LENGTH('b)" shows "uint a = r"
left_add_twice: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_numeral one times plus" shows "plus a (plus a b) = plus (times (numeral.numeral one plus (num.Bit0 num.One)) a) b"
a_closed: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "additive_subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "x \<oplus>\<^bsub>G\<^esub> y \<in> H"
SLN_strip_1: fixes P :: "assn" shows "P * Automation.SLN = P"
substInp_def2: fixes ys :: "'varSort"   and Y :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and y :: "'var" shows "substInp ys Y y = lift (subst ys Y y)"
less_eq_multiset_total: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "\<not> M \<le> N" shows "N \<le> M"
totalize_def: fixes F :: "'b program" shows "totalize F \<equiv> mk_program (Init F, totalize_act ` Acts F, AllowedActs F)"
drop_ext': fixes k :: "nat"   and xs :: "'c list"   and xs' :: "'c list" assumes "\<forall>i. k \<le> i \<and> i < length xs \<longrightarrow> xs' ! i = xs ! i"   and "0 < k"   and "xs \<noteq> []"   and "length xs' = length xs" shows "drop k xs' = drop k xs"
rctxt_induct: fixes x2 :: "trm \<Rightarrow> (char list \<times> trm) list"   and P2 :: "(trm \<Rightarrow> (char list \<times> trm) list) \<Rightarrow> bool" assumes "x2 \<in> rctxt"   and "\<And>E l fs. E \<in> ctxt \<Longrightarrow> P2 (\<lambda>t. (l, E t) \<Colon> fs)"   and "\<And>v E l. \<lbrakk>v \<in> value; E \<in> rctxt; P2 E\<rbrakk> \<Longrightarrow> P2 (\<lambda>t. (l, v) \<Colon> E t)" shows "P2 x2"
cases: fixes a1 :: "'b option"   and a2 :: "('b \<times> 'b option) list" assumes "insert_seq a1 a2" obtains start :: "'b option" and oid :: "'b" where "a1 = starta__" and "a2 = [(oid__, starta__)]"   | start :: "'b option" and list :: "('b \<times> 'b option) list" and prev :: "'b" and ref :: "'b option" and oid :: "'b" where "a1 = start" and "a2 = list @ [(prev, ref), (oid, Some prev)]" and "insert_seq start (list @ [(prev, ref)])"
in_intervalI: fixes X :: "'a interval"   and x :: "'a" assumes "lower X \<le> x"   and "x \<le> upper X" shows "x \<in>\<^sub>i X"
neg_h: shows "trm_rep (t \<cdot> s) S \<noteq> trm_rep (trm_rep t S \<cdot> trm_rep s S) S"
continuous_map_snd: fixes X :: "'c topology"   and Y :: "'d topology" shows "continuous_map (prod_topology X Y) Y snd"
NF_iff_no_step: fixes a :: "'b"   and A :: "('b \<times> 'b) set" shows "(a \<in> NF A) = (\<forall>b. (a, b) \<notin> A)"
sel_1: fixes x1 :: "'a interval"   and x2 :: "nat" shows "idx_interval.int (IdxInterval x1 x2) = x1"
that: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and n\<^sub>x :: "nat"   and n\<^sub>y :: "nat"   and n\<^sub>z :: "nat" assumes "x \<in> X \<and> y \<in> X \<and> z \<in> X \<and> [f[x .. y ..  z]X] \<and> f n\<^sub>x = x \<and> f n\<^sub>y = y \<and> f n\<^sub>z = z" shows "thesis"
evals_induct: fixes x3 :: "(char list \<times> trm) list"   and x4 :: "(char list \<times> trm) list"   and P2 :: "(char list \<times> trm) list \<Rightarrow> (char list \<times> trm) list \<Rightarrow> bool" assumes "x3 [\<longmapsto>] x4"   and "\<And>t t' l fs. t \<longmapsto> t' \<Longrightarrow> P2 ((l, t) \<Colon> fs) ((l, t') \<Colon> fs)"   and "\<And>v fs fs' l. \<lbrakk>v \<in> value; fs [\<longmapsto>] fs'; P2 fs fs'\<rbrakk> \<Longrightarrow> P2 ((l, v) \<Colon> fs) ((l, v) \<Colon> fs')" shows "P2 x3 x4"
not_Some_eq: fixes x :: "'c option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
hom_componentsE: fixes q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "q \<in> hom_components p" obtains t :: "'b \<Rightarrow>\<^sub>0 nat" where "t \<in> keys p" and "q = hom_component p (deg_pm t)"
swap_swap_id: fixes n :: "nat"   and m :: "nat"   and t :: "'b poly" shows "swap n m (swap m n t) = t"