hcju/isabelleps huggingface dataset:
bisim1Cond1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,if (e) e1 else e2,h \<turnstile> (if (e') e1 else e2, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
snd_sym_preproc_dgrad_set_le: fixes d :: "'a \<Rightarrow> nat"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "dickson_grading d" shows "dgrad_set_le d (pp_of_term ` Keys (set (snd (sym_preproc gs fs)))) (pp_of_term ` Keys (set gs \<union> set fs))"
MonotModel_axioms: shows "MonotModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
mignotte_bound_main: fixes f :: "int poly"   and g :: "int poly"   and n :: "nat"   and k :: "nat" assumes "f \<noteq> 0"   and "g dvd f"   and "degree g \<le> n" shows "\<bar>poly.coeff g k\<bar> \<le> \<lfloor>real (n - 1 choose k) * mahler_measure f\<rfloor> + int (min k 1 * (n - 1 choose (k - 1))) * \<bar>lead_coeff f\<bar>"
caratheodory_lemma: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "sigma_algebra \<Omega> M"   and "outer_measure_space M f" shows "measure_space \<Omega> (lambda_system \<Omega> M f) f"
d_future_consistent_def: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" shows "d_future_consistent P D R \<equiv> \<forall>u\<in>range D. \<forall>xs ys. (xs, ys) \<in> R u \<longrightarrow> (xs \<in> traces P) = (ys \<in> traces P) \<and> next_dom_events P D u xs = next_dom_events P D u ys"
iface_packet_check_def: fixes ifs :: "interface list"   and p :: "('b, 'c) simple_packet_ext_scheme" shows "iface_packet_check ifs p \<equiv> find (\<lambda>i. iface_name i = p_iiface p \<and> iface_mac i = p_l2dst p) ifs"
nn_integral_measure_spmf: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_spmf p) f = \<integral>\<^sup>+ x. ennreal (spmf p x) * f x \<partial>count_space UNIV"
sum_simp_setup_5: shows "\<forall>v\<in>cf.V. sum f' (cf.incoming v) = (\<Sum>u\<in>cf.V. f' (u, v))"
G_completeness: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "('b, 'c) epsym \<Rightarrow> univ list \<Rightarrow> bool" assumes "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP" shows "CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> (ProblemIkTpartG_GEModel.intT prot protFw eintT eintP) (ProblemIkTpartG_GEModel.intF eintF) (ProblemIkTpartG_GEModel.intP eintP)"
induct: fixes x1 :: "com"   and x2 :: "state"   and P :: "com \<Rightarrow> state \<Rightarrow> bool" assumes "x1 \<down> x2"   and "\<And>f s. f s \<noteq> {} \<Longrightarrow> P (Do f) s"   and "\<And>c1 s0 c2. \<lbrakk>c1 \<down> s0; P c1 s0; \<And>s1. s0 -c1\<rightarrow> s1 \<Longrightarrow> c2 \<down> s1; \<And>s1. s0 -c1\<rightarrow> s1 \<Longrightarrow> P c2 s1\<rbrakk> \<Longrightarrow> P (c1; c2) s0"   and "\<And>b s c1 c2. \<lbrakk>b s; c1 \<down> s; P c1 s\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) s"   and "\<And>b s c2 c1. \<lbrakk>\<not> b s; c2 \<down> s; P c2 s\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) s"   and "\<And>b s c. \<not> b s \<Longrightarrow> P (WHILE b DO c) s"   and "\<And>b s c. \<lbrakk>b s; c \<down> s; P c s; \<And>t. s -c\<rightarrow> t \<Longrightarrow> WHILE b DO c \<down> t; \<And>t. s -c\<rightarrow> t \<Longrightarrow> P (WHILE b DO c) t\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s"   and "\<And>p s. \<lbrakk>body p \<down> s; P (body p) s\<rbrakk> \<Longrightarrow> P (CALL p) s"   and "\<And>c f s g. \<lbrakk>c \<down> f s; P c (f s)\<rbrakk> \<Longrightarrow> P (LOCAL f; c; g) s" shows "P x1 x2"
upd_cols_upd_col_swap: fixes i :: "'c"   and I :: "'c set"   and M :: "('d, 'c) sq_matrix"   and x :: "('d, 'c) vec"   and f :: "'c \<Rightarrow> ('d, 'c) vec" assumes "i \<notin> I" shows "upd_cols (upd_col M i x) I f = upd_col (upd_cols M I f) i x"
ABinds_lookup_fresh: fixes v :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "atom v \<sharp> \<Gamma>" shows "(ABinds \<Gamma>\<cdot>ae) v = \<bottom>"
eFresh_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var"   and eX :: "'c withERR" assumes "\<not> eFresh MOD ys y eX" shows "\<exists>s X. eX = OK X \<and> \<not> igFresh MOD ys y X \<and> igWls MOD s X"
simps_8: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and s :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (select\<langle>t,s\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> fv t \<union> fv s) S"
HStartBallot_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv6 s"   and "Inv2c s" shows "HInv6 s'"
1: fixes t :: "'a \<times> 'a" assumes "t \<in> cl ts" shows "t \<in> cl (cl ts)"
asubg_pOp_closed: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set"   and x :: "'c"   and y :: "'c" assumes "aGroup A"   and "A +> H"   and "x \<in> H"   and "y \<in> H" shows "x \<plusminus>\<^bsub>A\<^esub> y \<in> H"
diff_delta_arctan_upper_43_def: shows "diff_delta_arctan_upper_43 \<equiv> \<lambda>x. 256 * x ^ 14 / ((35 * x ^ 6 + 315 * x ^ 4 + 693 * x\<^sup>2 + 429)\<^sup>2 * (1 + x\<^sup>2))"
size_neq_size_imp_neq: fixes x :: "'c"   and y :: "'c" assumes "size x \<noteq> size y" shows "x \<noteq> y"
min_pm1: fixes a :: "nat"   and b :: "nat" shows "a - b + min a b = a"
n_intersect_number_def: fixes b1 :: "'c set"   and n :: "nat"   and b2 :: "'c set" shows "b1 |\<inter>|\<^sub>n b2 \<equiv> int (card {x \<in> Pow (b1 \<inter> b2). card x = n})"
subdg_dghm_inc_op_dg_is_dghm: shows "dghm_inc (op_dg \<BB>) (op_dg \<CC>) : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> op_dg \<CC>"
vint_uminus_uminus: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x = -\<^sub>\<int> (-\<^sub>\<int> x)"
fds_abs_converges_imp_converges: fixes f :: "'a fds"   and s :: "'a" assumes "fds_abs_converges f s" shows "fds_converges f s"
partitions_iff_Abs_multiset: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" shows "(p partitions n) = (finite {x. 0 < p x} \<and> number_partition n (Abs_multiset p))"
simps_2: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and \<mu> :: "'a" assumes "bicategorical_language V src trg" shows "bicategorical_language.Nml V src trg \<^bold>\<langle>\<mu>\<^bold>\<rangle> = partial_magma.arr V \<mu>"
h1_def: fixes x :: "'a" shows "h1 x = (if g x = (0::'b) then if f x = (0::'b) then if h x = (0::'b) then h x else (1::'b) else f x else h x / g x)"
v_equals_sum_group_sizes: shows "\<v> = int (sum card \<G>)"
Rep_unit: fixes x :: "Linear_Aggregation_Algebras.unit" shows "Linear_Aggregation_Algebras.unit.Rep_unit x \<in> {0..1}"
apex_composite: shows "gf.apex = g.leg0 \<down>\<down> f.leg1"
ccHeap_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "ccHeap \<Gamma> = (if nonrec \<Gamma> then case hd \<Gamma> of (x, xa) \<Rightarrow> ccHeap_nonrec x xa else ccHeap_rec \<Gamma>)"
join_restrict_table: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set"   and B :: "nat set"   and Y :: "'b option list set"   and b :: "bool"   and v :: "'b option list" assumes "table n A X"   and "table n B Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A" shows "(v \<in> join X b Y) = (wf_tuple n (A \<union> B) v \<and> restrict A v \<in> X \<and> (if b then restrict B v \<in> Y else restrict B v \<notin> Y))"
top_join: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<top> \<squnion> x = \<top>"
path_inf_priorities_def: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist" assumes "ParityGame G" shows "ParityGame.path_inf_priorities G P \<equiv> {k. \<forall>n. k \<in> lset (ldropn n (lmap \<omega>\<^bsub>G\<^esub> P))}"
nextFreeBlockIncreases: fixes cfile1 :: "CFile"   and index :: "nat"   and "value" :: "byte"   and cfile2 :: "CFile" assumes "nextFreeBlockInvariant cfile1"   and "cfWrite cfile1 index value = Some cfile2" shows "nextFreeBlock cfile1 \<le> nextFreeBlock cfile2"
monad_state_alt_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and alt :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_alt return bind alt" shows "monad_state_alt (return_state return) (bind_state bind) get_state put_state (alt_state alt)"
nb\<^sub>2: shows "cnf \<^bold>\<Or>fs = {{literal_formula_to_literal f |f. f \<in> set fs}}"
right_neutral: fixes a :: "'a" shows "a \<cdot> 1 = a"
conjunctProtoD: fixes a :: "protocol"   and b :: "8 word"   and x :: "protocol" assumes "simple_proto_conjunct a (Proto b) = Some x" shows "x = Proto b \<and> (a = ProtoAny \<or> a = Proto b)"
sorted_smaller: fixes j :: "nat"   and j' :: "nat" assumes "sorted j"   and "j' \<le> j" shows "sorted j'"
that: fixes s :: "nat \<Rightarrow> real" assumes "(f \<circ> s) \<longlonglongrightarrow> l"   and "\<And>i. s i \<in> {a..}"   and "filterlim s at_top sequentially" shows "thesis"
fg: shows "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n"
rotate3: fixes A3 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A3 (insert A1 (insert A2 H)) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 H)) \<turnstile> B"
semiring_normalization_rules_3: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus m (times a m) = times (plus a one) m"
bounded_drift: fixes s :: "real"   and t :: "real"   and C :: "nat \<Rightarrow> real \<Rightarrow> real"   and D :: "nat \<Rightarrow> real \<Rightarrow> real"   and p :: "nat"   and q :: "nat" assumes "s \<le> t"   and "rho_bound1 C"   and "rho_bound2 C"   and "rho_bound1 D"   and "rho_bound2 D"   and "correct p t"   and "correct q t" shows "\<bar>C p t - D q t\<bar> \<le> \<bar>C p s - D q s\<bar> + 2 * \<rho> * (t - s)"
simps: fixes p :: "'b list" shows "awalk_to_apath p = (if (\<nexists>u. distinct (awalk_verts u p)) \<and> (\<exists>u. Ex (awalk u p)) then let (q, r, s) = awalk_cyc_decomp p in awalk_to_apath (q @ s) else p)"
maximum: fixes C :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite C" shows "\<exists>n. \<forall>l\<in>C. f l \<le> n"
injflip: fixes A :: "('c \<times> 'd) set" shows "inj_on flip A"
WhileTRedsThrow: fixes b :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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 E :: "char list \<Rightarrow> ty option"   and c :: "expr"   and r :: "nat \<times> char list list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>b,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Throw r,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>while (b) c,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw r,s\<^sub>2\<rangle>"
fst_in_eclose: fixes x :: "hf"   and y :: "hf" shows "x \<^bold>\<in> eclose \<langle>x, y\<rangle>"
less_eq_option_None_is_None: fixes x :: "'b option" assumes "x \<le> None" shows "x = None"
star_right_induct_mult: fixes y :: "'a"   and x :: "'a" assumes "y \<sqinter> x \<le> y" shows "y \<sqinter> top \<le> y"
nip_eq: fixes x :: "i"   and y :: "i" shows "PP x y = (P x y \<and> x \<noteq> y)"
bij_betw_Suc: fixes M :: "nat set"   and N :: "nat set" shows "bij_betw Suc M N = (Suc ` M = N)"
hoare_conseq: fixes P' :: "'a set"   and R :: "('a \<times> 'a) set"   and Q' :: "'a set"   and P :: "'a set"   and Q :: "'a set" assumes "{P'} R {> Q'}"   and "P \<subseteq> P'"   and "Q' \<subseteq> Q" shows "{P} R {> Q}"
select_convs_3: fixes precondition_of :: "'c list"   and add_effects_of :: "'c list"   and delete_effects_of :: "'c list"   and more :: "'d" shows "delete_effects_of \<lparr>strips_operator.precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = delete_effects_of"
range_constant: fixes x :: "'a" shows "range (\<lambda>_. x) = {x}"
Sup_action: fixes S :: "('a, 'b) action set" shows "Sup S \<equiv> if S = {} then bot else action_of (\<lambda>A. A \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` S})"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H H'" shows "H \<subseteq> carrier H'"
sterms_psimps_2: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and pn :: "'e" assumes "wellformed \<Gamma>" shows "sterms \<Gamma> (call(pn)) = sterms \<Gamma> (\<Gamma> pn)"
lock_ok_las'_def: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions" shows "lock_ok_las' ls t las \<equiv> \<forall>l. lock_actions_ok' (ls $ l) t (las $ l)"
unitary_exH_k: fixes k :: "nat" assumes "k < n" shows "unitary (exH_k k)"
A_def: shows "A = isolate_variable_sparse p2 var 0"
nAct_finite: fixes n :: "enat"   and c :: "'id"   and t :: "cnf llist" assumes "n \<noteq> \<infinity>" shows "\<exists>n'. \<langle>c #\<^bsub>n\<^esub>t\<rangle> = enat n'"
simps_3: fixes c1 :: "com"   and c2 :: "com"   and t :: "char list \<Rightarrow> int option" shows "defs (c1;; c2) t = (defs c2 \<circ> defs c1) t"
\<I>_common_def: shows "\<I>_common \<equiv> \<lambda>\<eta>. \<I>_uniform (nlists UNIV \<eta>) UNIV \<oplus>\<^sub>\<I> \<I>_uniform UNIV (insert None (Some ` nlists UNIV \<eta>))"
apply_operations_empty: fixes s :: "'state" shows "hb.apply_operations [] s = Some s"
configuration_monotonic_Suc: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "configuration (token_run x n) n \<subseteq> configuration (token_run x (Suc n)) (Suc n)"
StaticCastFail: 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 :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs),s\<^sub>1\<rangle>"   and "(last Cs, C) \<notin> (subcls1 P)\<^sup>*"   and "C \<notin> set Cs" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw (addr_of_sys_xcpt ClassCast, [ClassCast]),s\<^sub>1\<rangle>"
append_eq_append_conv2: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list"   and ts :: "'e list" shows "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)"
EpriK_synth: fixes K :: "nonce_t"   and H :: "msg set" assumes "epriK K \<in> synth H" shows "epriK K \<in> H \<or> (\<exists>N. epriK K = epriKA N)"
expr_sem_op_eq_distr: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and oper :: "pdf_operator"   and e :: "expr"   and t' :: "pdf_type"   and \<rho> :: "nat \<Rightarrow> val" assumes "density_context V V' \<Gamma> \<delta>"   and "\<Gamma> \<turnstile> oper $$ e : t'"   and "free_vars e \<subseteq> V \<union> V'"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> (oper $$ e)) = distr (dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure t') (op_sem oper)"
vec_to_iarray_nth: fixes i :: "nat"   and A :: "('d, 'c) vec" assumes "i < CARD('c)" shows "vec_to_iarray A !! i = A $ mod_type_class.from_nat i"
param_union: fixes R :: "('a \<times> 'b) set" shows "((\<union>), (\<union>)) \<in> \<langle>R\<rangle>set_rel \<rightarrow> \<langle>R\<rangle>set_rel \<rightarrow> \<langle>R\<rangle>set_rel"
assign_global_def: fixes upd :: "('e \<Rightarrow> 'e) \<Rightarrow> 'd control_state_scheme \<Rightarrow> 'd control_state_scheme"   and rhs :: "'d control_state_scheme \<Rightarrow> 'e" shows "assign_global upd rhs = assign (\<lambda>\<sigma>. upd (\<lambda>_. rhs \<sigma>) \<sigma>)"
induct: fixes P :: "'b List.list \<Rightarrow> bool"   and list :: "'b List.list" assumes "P list.Nil"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (list.Cons x1 x2)" shows "P list"
connected_ivt_component_cart: fixes s :: "(real, 'a) vec set"   and x :: "(real, 'a) vec"   and y :: "(real, 'a) vec"   and k :: "'a"   and a :: "real" assumes "connected s"   and "x \<in> s"   and "y \<in> s"   and "x $ k \<le> a"   and "a \<le> y $ k" shows "\<exists>z\<in>s. z $ k = a"
obs_consistent_behaviours: fixes R :: "('a \<times> 'b) set"   and pi :: "'c \<Rightarrow> 'd"   and Sa :: "('a, 'd) spec"   and Sc :: "('b, 'c) spec"   and bc :: "'b list"   and ba :: "'a list" assumes "obs_consistent R pi Sa Sc"   and "bc \<in> beh Sc"   and "ba \<in> beh Sa"   and "(ba, bc) \<in> seq_lift R" shows "map pi (map (obs Sc) bc) = map (obs Sa) ba"
prod_mset_multiset_prime_factorization_nat: fixes x :: "nat" assumes "x \<noteq> 0" shows "\<Prod>\<^sub># (prime_factorization x) = x"
set_iterator_foldli_conv: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'e set" shows "set_iterator iti S = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> iti = foldli l0)"
half_chamber_system_disjoint_union_1: shows "folding_g.\<C> = f \<turnstile> folding_g.\<C> \<union> g \<turnstile> folding_g.\<C>"
conjugate_conjugate_sprod: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (conjugate v \<bullet> w) = v \<bullet>c w"
commute_succs_of_clause: fixes l :: "lit" shows "comp_fun_commute (succs_of_clause l)"
face_cycle_succ_inD: fixes x :: "'b"   and y :: "'b" assumes "face_cycle_succ x \<in> face_cycle_set y" shows "x \<in> face_cycle_set y"
prj1_in_hom: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "cospan f g"   and "a = ECP.pbdom f g"   and "b = local.dom f" shows "\<guillemotleft>prj1 f g : a \<rightarrow> b\<guillemotright>"
sum_nonpos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) zero" shows "less_eq (comm_monoid_add.sum plus zero f A) zero"
tOp_mHom_closed: fixes N :: "('i, 'b, 'j) Module_scheme"   and f :: "'a \<Rightarrow> 'i"   and g :: "'a \<Rightarrow> 'i" assumes "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N" shows "tOp_mHom R M N f g \<in> mHom R M N"
ltree_node: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" shows "ltree (node l a r) = (ltree l \<and> ltree r)"
f_2: fixes x :: "'a" assumes "x \<in> S" shows "g (f x) = x"
add_zerol: fixes x :: "'a" shows "(0::'a) + x = x"
ge_ge_tpoly_trans: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and r :: "('a pvar, hmultiset) tpoly"   and q :: "('a pvar, hmultiset) tpoly"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym r q"   and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym q p" shows "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym r p"
mkbot_Lin: fixes L :: "(alt \<times> alt) set"   and x :: "alt" assumes "L \<in> Lin" shows "mkbot L x \<in> Lin"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs" shows "l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
tag_name_eq2_h2: fixes doc_ptr :: "(_) element_ptr" shows "|h2 \<turnstile> get_tag_name doc_ptr|\<^sub>r = |h' \<turnstile> get_tag_name doc_ptr|\<^sub>r"
bridge_block_nominals: fixes v :: "nat"   and xs :: "(nat \<times> nat) set"   and k :: "'b"   and j :: "'b"   and ps :: "('c, 'b) fm list"   and a :: "'b" shows "block_nominals (mapi_block (bridge k j xs v) (ps, a)) \<union> {k, j} = block_nominals (ps, a) \<union> {k, j}"
intro: fixes F :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "F (\<lambda>x. True)"   and "\<And>P Q. \<lbrakk>F P; F Q\<rbrakk> \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"   and "\<And>P Q. \<lbrakk>\<forall>x. P x \<longrightarrow> Q x; F P\<rbrakk> \<Longrightarrow> F Q" shows "is_filter F"
dom_comp: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "seq g f" shows "local.dom (g \<cdot> f) = local.dom f"
rep_cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "free_monoidal_category.rep C (partial_magma.cod (free_monoidal_category.comp C) f) = free_monoidal_category.norm C (monoidal_language.Cod C (free_monoidal_category.rep C f))"
intro: fixes \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<Sigma> :: "'b set set" assumes "Only_G \<G>"   and "range w \<subseteq> \<Sigma>"   and "finite \<Sigma>" shows "ltl_FG_to_rabin_opt \<Sigma> \<G> w"
simps: fixes a :: "nat" shows "the_Addr (Addr a) = a"