hcju/isabelleps huggingface dataset:
subseteq_code: shows "(\<subseteq>) = Set_Impl.subset_eq"
merge_discrete_def: fixes x :: "'c"   and y :: "'c" shows "merge_discrete x y = (if x = y then Some y else None)"
seq_concat_2: fixes ss1 :: "'a \<times> ('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss2 :: "'a \<times> ('b \<times> 'a) list" assumes "ss1 \<in> seq ars"   and "ss2 \<in> seq ars"   and "lst ss1 = fst ss2" shows "lst (fst ss1, snd ss1 @ snd ss2) = lst ss2"
prems_2: fixes m :: "nat" assumes "m < length ds" shows "(a # is) ! m < ds ! m"
gc_invar_gc_step: fixes i :: "nat"   and Ns :: "('f \<times> 'l) set llist" assumes "enat (Suc i) < llength Ns"   and "gc_invar Ns (enat i)"   and "lnth Ns i \<leadsto>GC lnth Ns (Suc i)" shows "gc_invar Ns (enat (Suc i))"
UnI2: fixes c :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "(c::'b) \<in> (B::'b set)" shows "(c::'b) \<in> (A::'b set) \<union> (B::'b set)"
pell_valuation_neg_neg: fixes x :: "int"   and y :: "int" assumes "x < 0"   and "y < 0"   and "solution (x, y)" shows "pell_valuation (x, y) < - 1"
distinct_1: fixes x2 :: "'k" shows "None \<noteq> Some x2"
dioph_ineq_dfs_def: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "dioph_ineq_dfs n ks l = dioph_ineq_dfs.dfs n ks (dioph_empt ks l) [l]"
assms: fixes c' :: "real" assumes "ereal c' < c" shows "\<exists>s. s \<bullet> (1::'a) = c' \<and> \<not> fds_abs_converges f s"
assms_3: shows "G\<^sub>2 \<sqsubseteq> G\<^sub>3"
hcomp_closed: fixes f :: "'a"   and g :: "'a" assumes "Arr f"   and "Arr g"   and "trg\<^sub>B f = src\<^sub>B g" shows "Arr (g \<star>\<^sub>B f)"
simps_16: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com"   and x21 :: "char list"   and x22 :: "aexp" shows "IF x41 THEN x42 ELSE x43 \<noteq> x21 ::= x22"
ld_sum_inequality: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "log 2 x + log 2 y + 2 \<le> 2 * log 2 (x + y)"
in_eq_out_degree: fixes v :: "'a" shows "in_degree G v = out_degree G v"
assms_3: shows "\<forall>p. local_peak ars p \<longrightarrow> (\<exists>\<gamma>1 \<gamma>2 \<gamma>3 \<delta>1 \<delta>2 \<delta>3. LT ars r (fst p, snd p, \<gamma>1, \<gamma>2, \<gamma>3, \<delta>1, \<delta>2, \<delta>3))"
subset_insertI2: fixes A :: "'b set"   and B :: "'b set"   and b :: "'b" assumes "A \<subseteq> B" shows "A \<subseteq> insert b B"
s_TS_distinct: fixes init :: "nat list"   and i :: "nat"   and qs :: "nat list"   and h :: "nat list" assumes "distinct init"   and "i < length qs" shows "distinct (fst (TSdet init h qs i))"
postcond_equiv: fixes P :: "('b, 'c) urel" shows "(P = true ;; P) = (in\<alpha> \<sharp> P)"
proj5_eq_dhops: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>5 (the (rt dip)) = the (dhops rt dip)"
less_imp_inverse_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 a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less zero a" shows "less (inverse b) (inverse a)"
that_3: fixes g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" assumes "x = \<bottom>"   and "y = CFn\<cdot>g" shows "thesis"
replay''_sound_pre_hyps: 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 "wf_theory thy"   and "\<And>x. x \<in> set (hyps P) \<Longrightarrow> term_ok thy x"   and "\<And>x. x \<in> set (hyps P) \<Longrightarrow> typ_of x = Some propT"   and "replay'' thy [] (fst ` (fv_Proof P \<union> FV (set (hyps P)))) (hyps P) P = Some res" shows "thy,set (hyps P) \<turnstile> res"
basis_no_fact: fixes u :: "'b list"   and G :: "'b list set"   and v :: "'b list" assumes "u \<in> \<langle>G\<rangle>"   and "v \<in> \<langle>G\<rangle>"   and " u \<cdot> v \<in>B G " shows "u = \<epsilon> \<or> v = \<epsilon>"
noFault_start: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "t \<noteq> Fault f" shows "s \<noteq> Fault f"
IH: fixes ds :: "bool list"   and T :: "tree" assumes "path p (delete ds T)" shows "path p T"
openin_contains_ball: fixes T :: "'b set"   and S :: "'b set" shows "openin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x\<in>S. \<exists>e>0. ball x e \<inter> T \<subseteq> S))"
poly_shift_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'e"   and n :: "nat" assumes "UP_cring R" shows "poly_shift f n = f (Suc n)"
intros_17: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e" shows "\<Gamma>\<turnstile>c \<down> Abrupt s"
Ax: fixes k :: "'a"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "Atom k \<in># \<Gamma>"   and "Atom k \<in># \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
field_divide_inverse: fixes a :: "'a"   and b :: "'a" shows "a / b = a * inverse b"
AbstrLevels_A7_A72: fixes i :: "AbstrLevelsID" assumes "sA7 \<in> AbstrLevel i" shows "sA72 \<notin> AbstrLevel i"
nb\<^sub>3: shows "\<forall>op\<in>set ops. map_of (sas_plus_operator.precondition_of op) \<subseteq>\<^sub>m I"
wf_prog_cdecl: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "class G C = Some c"   and "wf_prog G" shows "wf_cdecl G (C, c)"
steps_z_norm_induct: fixes x1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and x2 :: "'d"   and x3 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and x7 :: "'d"   and x8 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and P :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint) \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>\<^sub>\<N>* \<langle>x7, x8\<rangle>"   and "\<And>A l Z. P A l Z l Z"   and "\<And>A l Z l' Z' l'' Z''. \<lbrakk>A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', Z'\<rangle>; P A l Z l' Z'; A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<N> \<langle>l'', Z''\<rangle>\<rbrakk> \<Longrightarrow> P A l Z l'' Z''" shows "P x1 x2 x3 x7 x8"
ord_eq_le_trans: fixes a :: "'k"   and b :: "'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and c :: "'k" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
used_parts_subset_parts: fixes X :: "msg"   and evs :: "event list" assumes "X \<in> used evs" shows "parts {X} \<subseteq> used evs"
assms: shows "\<forall>\<^sub>F n in sequentially. l * real n \<le> real (card (A \<inter> {..<n}))"
is_closedI: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set" assumes "equivalence S"   and "\<And>x y. \<lbrakk>x .=\<^bsub>S\<^esub> y; x \<in> A; y \<in> carrier S\<rbrakk> \<Longrightarrow> y \<in> A"   and "A \<subseteq> carrier S" shows "is_closed\<^bsub>S\<^esub> A"
iF_in: shows "\<iota>F \<in> Inf_F"
num_params_tm_comp_le: fixes tf :: "taylor_model"   and X :: "nat"   and tg :: "taylor_model"   and prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and ga :: "float" assumes "num_params (tm_poly tf) \<le> max 1 X"   and "num_params (tm_poly tg) \<le> X" shows "num_params (tm_poly (tm_comp prec ord I a ga tf tg)) \<le> X"
insertList_queue_invar: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.queue_invar q" shows "SkewBinomialHeapStruc.queue_invar (SkewBinomialHeapStruc.insertList ts q)"
function_ring_defs_4: shows "\<one>\<^bsub>F\<^esub> = function_one S R"
wf_lenlex: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "wf (lenlex r)"
tso_pending_mw_mutate_cong: fixes s :: "'f process_name \<Rightarrow> ('a, 'd, 'b, 'c, 'e) local_state_scheme"   and p :: "'d process_name"   and s' :: "'h process_name \<Rightarrow> ('a, 'd, 'b, 'c, 'g) local_state_scheme"   and P :: "'c \<Rightarrow> 'a \<Rightarrow> 'c option \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'a \<Rightarrow> 'c option \<Rightarrow> bool" assumes "filter is_mw_Mutate (sys_mem_store_buffers p s) = filter is_mw_Mutate (sys_mem_store_buffers p s')"   and "\<And>r f r'. P r f r' = Q r f r'" shows "(\<forall>r f r'. mw_Mutate r f r' \<in> set (sys_mem_store_buffers p s) \<longrightarrow> P r f r') = (\<forall>r f r'. mw_Mutate r f r' \<in> set (sys_mem_store_buffers p s') \<longrightarrow> Q r f r')"
image_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and r :: "'e set" shows "f ` g ` r = (f \<circ> g) ` r"
equality: fixes r :: "'a thread_t_scheme"   and r' :: "'a thread_t_scheme" assumes "ev_counter r = ev_counter r'"   and "thread_t.more r = thread_t.more r'" shows "r = r'"
P_k_decomp: shows "P_k = P_k_pre @ y # P_k_post"
collapse_2: fixes "term" :: "('e, 'f) Term.term" assumes "is_Fun term" shows "Fun (the_Fun term) (args term) = term"
that: fixes A :: "'a multiset"   and C :: "'a zmultiset"   and B :: "'a multiset" assumes "M = zmset_of A + C"   and "N = zmset_of B + C"   and "A < B" shows "thesis"
s\<^sub>2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [e1]))"
zero_ts_ord: fixes x :: "'a" shows "o \<sqsubseteq> !x"
cINF_lower2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'d"   and u :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_below less_eq (f ` A)"   and "x \<in> A"   and "less_eq (f x) u" shows "less_eq (Inf (f ` A)) u"
mat_diag_diag: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "mat_diag n f * mat_diag n g = mat_diag n (\<lambda>i. f i * g i)"
Mul_Propagate_Black: fixes n :: "nat" shows "\<turnstile> Mul_Propagate_Black n  \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> (\<acute>Safe \<or> \<acute>obc \<subset> Blacks \<acute>M \<or> \<acute>l < \<acute>Queue \<and> (\<acute>l \<le> \<acute>Queue \<or> \<acute>obc \<subset> Blacks \<acute>M))\<rbrace>"
monotoneD: fixes orda :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ordb :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "monotone orda ordb f"   and "orda x y" shows "ordb (f x) (f y)"
simps_1: fixes \<rho> :: "complex mat" shows "denote SKIP \<rho> = \<rho>"
Abs_swap1_1: fixes a :: "atom"   and x :: "'e"   and bs :: "atom set"   and b :: "atom" assumes "a \<notin> supp x - bs"   and "b \<notin> supp x - bs" shows "[bs]set. x = [(a \<rightleftharpoons> b) \<bullet> bs]set. (a \<rightleftharpoons> b) \<bullet> x"
ext: shows "ext = (\<lambda>ps ps'. ps' \<noteq> [] \<and> butlast ps' = ps \<and> adj_path v ps')"
total_finite_trans_max: fixes X :: "'a set"   and r :: "('a \<times> 'a) set" assumes "X \<noteq> {}"   and "finite X"   and "total_on X r"   and "trans r" shows "\<exists>x\<in>X. \<forall>y\<in>X. x \<noteq> y \<longrightarrow> (y, x) \<in> r"
ctwo_not_czero: shows "(ctwo, czero) \<notin> ordIso"
apply_bcontfun_inject: fixes x :: "'c \<Rightarrow>\<^sub>C 'd"   and y :: "'c \<Rightarrow>\<^sub>C 'd" shows "(apply_bcontfun x = apply_bcontfun y) = (x = y)"
intros_32: fixes \<C> :: "t_context"   and n :: "nat" assumes "memory \<C> = Some n" shows "\<C> \<turnstile> [Grow_memory] : [T_i32] _> [T_i32]"
p\<^sub>1_def: shows "p\<^sub>1 = find_closest_bf p\<^sub>0 (take 7 (p\<^sub>2 # ps))"
distinct2: shows "distinct (cols (submatrix (submatrix A UNIV J) I UNIV))"
numer_cf7_deriv: fixes x :: "real" shows "(numer_cf7 has_real_derivative numer_cf7_deriv x) (at x)"
list_succ_simps_2: fixes xs :: "'b list"   and x :: "'b" assumes "index xs x + 1 < length xs" shows "list_succ xs x = xs ! (index xs x + 1)"
cn_2: fixes q :: "int"   and p :: "int" assumes "0 < q" shows "c / real_of_int q ^ n < \<bar>x - real_of_int p / real_of_int q\<bar>"
IH: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "(X, Y) \<in> qSwapped" shows "Y #= Y"
credentials_def: fixes lxl :: "char list set \<times> char list set" shows "credentials lxl \<equiv> fst lxl"
simple_ruleset_tail: fixes r :: "'b rule"   and rs :: "'b rule list" assumes "simple_ruleset (r # rs)" shows "simple_ruleset rs"
measurable_snth_PiM: fixes M :: "'b measure" shows "(!!) \<in> stream_space M \<rightarrow>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)"
augment_list_prefix: fixes n :: "nat"   and m :: "nat"   and f :: "'b list \<Rightarrow> 'b"   and p :: "'b list" assumes "n \<le> m" shows "prefix ((augment_list f ^^ n) p) ((augment_list f ^^ m) p)"
C: shows "U \<subseteq> unaffected_domains I D (D ` set ys) zs"
binop_progress: fixes h :: "'heap"   and v1 :: "'addr val"   and T1 :: "ty"   and v2 :: "'addr val"   and T2 :: "ty"   and bop :: "bop"   and T :: "ty" assumes "typeof\<^bsub>h\<^esub> v1 = \<lfloor>T1\<rfloor>"   and "typeof\<^bsub>h\<^esub> v2 = \<lfloor>T2\<rfloor>"   and "compP2 P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T" shows "\<exists>va. binop bop v1 v2 = \<lfloor>va\<rfloor>"
BRPar: fixes R :: "ccs"   and T :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and P :: "ccs"   and Q :: "ccs" assumes "(R, T) \<in> Rel"   and "(P, Q) \<in> bangRel Rel" shows "(R \<parallel> P, T \<parallel> Q) \<in> bangRel Rel"
valid_stepE: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and 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 s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "(s, s') \<in> substitution.steps subst" shows "\<exists>r. substitution.revision_step subst r s s'"
usappconst_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and f :: "char" assumes "usappconst \<sigma> U f \<noteq> undeft" shows "SConst \<sigma> f = undeft \<or> (\<exists>r. SConst \<sigma> f = Aterm r \<and> FVT r \<inter> U = {})"
Interp_def: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset" shows "Interp N C = interp N C \<union> production N C"
metric_LIMSEQ_I: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r" shows "X \<longlonglongrightarrow> L"
something_disjoint: fixes x :: "i" assumes "CEMU"   and "x \<noteq> u" shows "\<exists>v. \<not> O v x"
rev_Nil_wrap_def: shows "rev_Nil_wrap = rev []"
bound: fixes j :: "nat" shows "AE x in M. u j x \<le> w x"
mono_cross_mult_less_hmset: fixes A :: "hmultiset"   and Aa :: "hmultiset"   and B :: "hmultiset"   and Ba :: "hmultiset" assumes "A < Aa"   and "B < Ba" shows "A * Ba + B * Aa < A * B + Aa * Ba"
poly_lift_hom_X_plus_nat_pow: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and a :: "'a"   and n :: "nat" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "a \<in> carrier R" shows "poly_lift_hom R S \<phi> (X_poly_plus R a [^]\<^bsub>UP R\<^esub> n) = X_poly_plus S (\<phi> a) [^]\<^bsub>UP S\<^esub> n"
vec_simps_20: fixes ac :: "'a"   and cb :: "'a"   and bc :: "'a" shows "(ac = cb - bc) = (ac + bc = cb)"
fin_tuples: fixes X :: "complex multiset" shows "finite (tuples X)"
47: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> (- - x \<squnion> y) = top \<squnion> y"
DERIV_D: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a" assumes "(f has_field_derivative D) (at x)" shows "(\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'a\<rightarrow> D"
alg_aOps_lookup_update: fixes ec :: "'rep"   and ec' :: "'rep"   and e :: "'aAct list"   and M :: "'ma" assumes "k_isNode ec"   and "k_isNode ec'" shows "lookup aOps (update aOps ec e M) ec' = (if simAbs ec' = simAbs ec then Some e else lookup aOps M ec')"
smult_1_left: fixes p :: "'b poly" shows "smult (1::'b) p = p"
borel_measurable_ennreal_iff: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "((\<lambda>x. ennreal (f x)) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
history_consistent_hd_prog_p: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and xs :: "'a memref list" assumes "history_consistent \<theta> p xs" shows "p = hd_prog p xs"
eq_below_trans: fixes a :: "'a"   and b :: "'a"   and below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a" assumes "a = b"   and "below b c" shows "below a c"
Gauss_Jordan_column_k_PA_def: fixes A' :: "(('b, 'c) vec, 'c) vec \<times> nat \<times> (('b, 'd) vec, 'c) vec"   and k :: "nat" shows "Gauss_Jordan_column_k_PA A' k = (let P = fst A'; i = fst (snd A'); A = snd (snd A'); from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if (\<forall>m\<ge>from_nat_i. A $ m $ from_nat_k = (0::'b)) \<or> i = nrows A then (P, i, A) else let Gauss = Gauss_Jordan_in_ij_PA (P, A) from_nat_i from_nat_k in (fst Gauss, i + 1, snd Gauss))"
iso_refl: fixes G :: "('c, 'd) monoid_scheme" shows "G \<cong> G"
finite_Collect_op: fixes I :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "finite {i \<in> I. x i \<noteq> (0::'c)}"   and "finite {i \<in> I. y i \<noteq> (0::'c)}" shows "finite {i \<in> I. x i + y i \<noteq> (0::'c)}"
bij_def: fixes f :: "'a \<Rightarrow> 'b" shows "bij f = (inj f \<and> surj f)"
RHS: shows "\<guillemotleft>RHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>"
nonneg_cexpr_sem_integrate_vars': fixes vs :: "nat list"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "distinct vs"   and "set vs \<inter> V' = {}"   and "nonneg_cexpr (set vs \<union> V') \<Gamma> e"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> set vs \<union> V'" shows "nonneg_cexpr V' \<Gamma> (integrate_vars \<Gamma> vs e)"
carrier_is_submodule: shows "Module.submodule (carrier_vec n) class_ring V"
counit_simps_5: shows "trg\<^sub>D \<psi> = trg\<^sub>D (F f)"
semiring_1_cancel_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.semiring_1_cancel plus minus zero one times \<equiv> (class.cancel_comm_monoid_add plus minus zero \<and> class.monoid_mult one times) \<and> class.semiring plus times \<and> class.zero_neq_one one zero"
rel_mono: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_gpv R1 R2 \<le> rel_gpv R1a R2a"
hom_dvd: fixes p :: "int poly"   and q :: "int poly" assumes "p dvd q" shows "Ring_Hom_Poly.of_int_poly p dvd Ring_Hom_Poly.of_int_poly q"
Quotient_rel_abs: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'c"   and s :: "'c" assumes "Quotient R Abs Rep T"   and "R r s" shows "Abs r = Abs s"
Chaos_Sim: fixes S :: "'e \<Rightarrow> 'd" shows "Chaos \<preceq>S Chaos"
subspace_add: fixes S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "m2.subspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
a_closed: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)" shows "x \<oplus>\<^bsub>UP INTEG\<^esub> y \<in> carrier (UP INTEG)"
card_0_eq: fixes A :: "'q set" assumes "finite A" shows "(card A = 0) = (A = {})"
tarski_full_lemma: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and Y :: "'b set" assumes "Tarski cl f Y" shows "\<exists>L. isLub Y \<lparr>pset = fix f (pset cl), order = induced (fix f (pset cl)) (order cl)\<rparr> L"
psubst_subst_compose_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good X"   and "good Y"   and "freshEnv ys y rho" shows "X #[Y / y]_ys #[rho] = X #[rho] #[Y #[rho] / y]_ys"
simps_1: fixes a :: "'a" shows "the_default a None = a"
has_vderiv_on_def: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and S :: "real set" shows " D f = f' on S = (\<forall>x\<in>S. (f has_vector_derivative f' x) (at x within S))"
symmetry2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x y. (x, y) \<in> {(x, y). R x y}\<^sup>* \<longrightarrow> (y, x) \<in> {(x, y). R x y}\<^sup>*" shows "symp R\<^sup>*\<^sup>*"
aug_addition_inc_z: fixes z :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set" shows "z \<in> addition_set f (aug_pm_set z i A)"
llength_prepend: fixes xs :: "'b list"   and ys :: "'b llist" shows "llength (prepend xs ys) = enat (length xs) + llength ys"
d_inductive_unwinding: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd" assumes "deterministic P" shows "secure P I D = (\<forall>xs\<in>traces P. \<forall>u\<in>range D \<inter> (- I) `` range D. next_dom_events P D u (ipurge_tr_rev I D u xs) = next_dom_events P D u xs)"
finsum_cong: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'a vec"   and g :: "'b \<Rightarrow> 'a vec" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier_vec n) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum V f A = finsum V g B"
finite_divisors: fixes n :: "nat" assumes "0 < n" shows "finite (divisors n)"
gamma_const: shows "\<forall>x. \<gamma> x \<bullet> i = a"
sum_list_eq_nthI: fixes i :: "nat"   and xs :: "'c list" assumes "i < length xs"   and "\<And>j. \<lbrakk>j < length xs; j \<noteq> i\<rbrakk> \<Longrightarrow> xs ! j = (0::'c)" shows "sum_list xs = xs ! i"
nrows_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "nrows A = CARD('e)"
exhaust: fixes y :: "('b, 'a) Term.term" obtains x1 :: "'a" where "y = Var x1"   | x21 :: "'b" and x22 :: "('b, 'a) Term.term list" where "y = Fun x21 x22"
0: fixes f :: "'fsym" assumes "f \<in> fsymsC c" shows "intF1 f = intF2 f"