hcju/isabelleps huggingface dataset:
Cnext_solve_kills: fixes K\<^sub>1 :: "clock"   and n :: "nat"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and K\<^sub>2 :: "clock"   and \<Phi> :: "'c TESL_atomic list" shows "{K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>, K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi>"
eval_elim_cases_15: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (Expr e)\<succ>\<rightarrow> (x, s')" obtains v :: "val" where "x = \<diamondsuit>" and "G\<turnstile>Norm s \<midarrow>e-\<succ>v\<rightarrow> s'"
assms_2: shows "\<forall>ip\<in>dom dests. ip \<in> vD rt \<and> the (dests ip) = inc (sqn rt ip)"
columns_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "columns A = {column i A |i. i \<in> UNIV}"
F_eq: shows "F = domain_and_range_permutation A B `` {f}"
that: fixes g :: "'a set \<Rightarrow> 'c" assumes "g \<in> hom (G Mod N) H"   and "\<And>x. x \<in> carrier G \<Longrightarrow> g (N #> x) = h x" shows "thesis"
igAbsCongU_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igAbsCongU MOD \<equiv> \<forall>xs x x' y s X X'. isInBar (xs, s) \<and> igWls MOD s X \<and> igWls MOD s X' \<longrightarrow> igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSubst MOD xs (igVar MOD xs y) x X = igSubst MOD xs (igVar MOD xs y) x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
assms_1: shows "x \<in> set (remdups_wrt f xs)"
QP: shows "(\<mu> or\<^sub>r \<eta> \<in> old (fst (n, ns)) \<longrightarrow> {\<mu>, \<eta>} \<inter> (old (fst (n, ns)) \<union> new (fst (n, ns))) \<noteq> {}) \<and> (\<forall>nd\<in>ns. \<mu> or\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<inter> old nd \<noteq> {})"
R10_bc_1: shows "pmf (sds R10) b = 0"
fid_on_vsingleton: fixes a :: "V" shows "fid_on (set {a}) = set {[a, a]\<^sub>\<circ>}"
root_sgn_power: fixes n :: "nat"   and y :: "real" assumes "0 < n" shows "root n (sgn y * \<bar>y\<bar> ^ n) = y"
rep_eq: fixes x :: "('b, 'a) uexpr"   and xa :: "('b, 'a) uexpr" shows "\<lbrakk>x \<sqinter> xa\<rbrakk>\<^sub>e = (\<lambda>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<sqinter> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
step_invariantE: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<times> 'f \<times> 'e \<Rightarrow> bool"   and s :: "'e"   and a :: "'f"   and s' :: "'e" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" obtains "P (s, a, s')"
sign_inverse: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "sign (inv p) = sign p"
tr_step: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c'' :: "('a, 'b, 'c) configuration" assumes "c \<turnstile> ev \<mapsto> c'"   and "trace c' t c''" shows "trace c (ev # t) c''"
setPair_noteq_allow: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b"   and P :: "bool" assumes "{a, b} \<noteq> {c, d}" shows "\<not> (a = c \<and> b = d \<and> P)"
Compl_disjoint: fixes A :: "'b set" shows "A \<inter> - A = {}"
mult_eq_def: fixes r :: "('a \<times> 'a) set" shows "mult_eq r = (mult1 r)\<^sup>*"
rassoc: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<cdot> ys) \<cdot> zs = xs \<cdot> ys \<cdot> zs"
adapt_bexp_is_subst: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>s\<^sub>y\<^sub>m :: "'c \<times> nat \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" assumes "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m s" shows "adapt_bexp e s \<sigma>\<^sub>s\<^sub>y\<^sub>m = e \<sigma>"
assms_3: shows "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"
inv: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> H" shows "local.inverse a \<in> generate H"
axioms_1: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" assumes "normal_series G \<GG>" shows "Group.group G"
C_eq_until_separated: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l))) = Cp p"
otherassms_1: shows "l \<in> labels \<Gamma> p"
graph_path_axioms: shows "graph_path \<alpha>e \<alpha>n invar inEdges'"
lead_coeff_prod_mset: fixes A :: "'a poly multiset" shows "Polynomial.lead_coeff (\<Prod>\<^sub># A) = \<Prod>\<^sub># (image_mset Polynomial.lead_coeff A)"
intro: fixes F :: "('e, 'g, 'd, 'h, 'f, 'i, 'j) Functor_scheme" assumes "\<And>f X Y. f maps\<^bsub>CatDom F\<^esub> X to Y \<Longrightarrow> F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y" shows "FunctorM_axioms F"
singleCombinatorsConc: fixes x :: "('c, 'd) Combinators"   and xs :: "('c, 'd) Combinators list" assumes "singleCombinators (x # xs)" shows "singleCombinators xs"
SUP_identity_eq: fixes Sup :: "'c set \<Rightarrow> 'c"   and A :: "'c set" shows "Sup ((\<lambda>x. x) ` A) = Sup A"
set_ubounds: fixes as :: "atom list" shows "set (ubounds as) = {i. Less 0 (Suc i) \<in> set as}"
assms_3: shows "\<rho> \<in> space (state_measure V' \<Gamma>)"
ide_in_hom_2: fixes f :: "'d" assumes "D.ide f" shows "\<guillemotleft>f : f \<Rightarrow>\<^sub>D f\<guillemotright>"
that: fixes \<Gamma>' :: "(Vars.var \<times> exp) list"   and S' :: "stack_elem list" assumes "(\<Gamma>, e, S) \<Rightarrow>\<^sup>* (\<Gamma>', e, S')"   and "to_gc_conf r (\<Gamma>, e, S) = to_gc_conf r (\<Gamma>', e, S')"   and "\<not> isVal e \<or> safe_hd S' = safe_hd (restr_stack (- set r) S')" shows "thesis"
restriction_within_domain: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "P || X = P || (X \<inter> Domain P)"
rAlpha: fixes A\<^sub>P :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and C :: "'d"   and \<Psi>\<^sub>P :: "'b" assumes "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* (p \<bullet> A\<^sub>P)"   and "A\<^sub>P \<sharp>* C"   and "set p \<subseteq> set A\<^sub>P \<times> set (p \<bullet> A\<^sub>P)"   and "distinctPerm p"   and "Prop C \<Psi> P P' A\<^sub>P \<Psi>\<^sub>P" shows "Prop C \<Psi> P P' (p \<bullet> A\<^sub>P) (p \<bullet> \<Psi>\<^sub>P)"
galois_property: fixes x :: "'c"   and y :: "'a" assumes "x \<in> carrier \<X>\<^bsub>F\<^esub>"   and "y \<in> carrier \<Y>\<^bsub>F\<^esub>" shows "(\<pi>\<^sup>*\<^bsub>F\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>F\<^esub>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^bsub>F\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>F\<^esub> y)"
S_imp_wpo_s: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" assumes "(s, t) \<in> S" shows "s \<succ> t"
prv_imp_ldsj: fixes \<chi> :: "'fmla"   and \<phi>s :: "'fmla list"   and \<phi> :: "'fmla" assumes "\<chi> \<in> fmla"   and "set \<phi>s \<subseteq> fmla"   and "\<phi> \<in> set \<phi>s"   and "prv (imp \<chi> \<phi>)" shows "prv (imp \<chi> (ldsj \<phi>s))"
assms: fixes xs :: "('a \<times> 'b) list"   and x :: "'a" assumes "set xs \<subseteq> A \<times> UNIV"   and "x \<in> A" shows "trace_callee callee1 p xs x = trace_callee callee2 q xs x"
S_\<theta>_disj: shows "\<forall>v\<in>vars\<^sub>s\<^sub>t S. \<theta> v = Var v"
sp_lift_instr: fixes Finca_empty :: "'l"   and Finca_get :: "'l \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef option"   and Finca_add :: "'l \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef \<Rightarrow> 'l"   and Finca_to_list :: "'l \<Rightarrow> ('m \<times> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef) list"   and Fubx_empty :: "'r"   and Fubx_get :: "'r \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef option"   and Fubx_add :: "'r \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef \<Rightarrow> 'r"   and Fubx_to_list :: "'r \<Rightarrow> ('m \<times> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef) list"   and heap_empty :: "'v"   and heap_get :: "'v \<Rightarrow> 'o \<times> 'n \<Rightarrow> 'n option"   and heap_add :: "'v \<Rightarrow> 'o \<times> 'n \<Rightarrow> 'n \<Rightarrow> 'v"   and heap_to_list :: "'v \<Rightarrow> (('o \<times> 'n) \<times> 'n) list"   and is_true :: "'n \<Rightarrow> bool"   and is_false :: "'n \<Rightarrow> bool"   and box_ubx1 :: "'t \<Rightarrow> 'n"   and unbox_ubx1 :: "'n \<Rightarrow> 't option"   and box_ubx2 :: "'u \<Rightarrow> 'n"   and unbox_ubx2 :: "'n \<Rightarrow> 'u option"   and \<OO>\<pp> :: "'p \<Rightarrow> 'n list \<Rightarrow> 'n"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'p \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'q \<Rightarrow> 'n list \<Rightarrow> 'n"   and \<II>\<nn>\<ll> :: "'p \<Rightarrow> 'n list \<Rightarrow> 'q option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'q \<Rightarrow> 'n list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'q \<Rightarrow> 'p"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'s \<Rightarrow> ('n, 't, 'u) unboxed list \<Rightarrow> ('n, 't, 'u) unboxed option"   and \<UU>\<bb>\<xx> :: "'q \<Rightarrow> Unboxed.type option list \<Rightarrow> 's option"   and \<BB>\<oo>\<xx> :: "'s \<Rightarrow> 'q"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'s \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and get_arity :: "'m \<Rightarrow> nat option"   and instr :: "('w, 'x, 'm, 'p, 'q) Inca.instr"   and \<Sigma>1 :: "Unboxed.type option list"   and instr' :: "('w, 'x, 'm, 'p, 'q, 's, 'y, 'z) Ubx.instr"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'m \<Rightarrow> 'aa fundef option" assumes "inca_to_ubx_simulation Finca_empty Finca_get Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>"   and "inca_to_ubx_simulation.lift_instr \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> get_arity instr \<Sigma>1 = Ok (instr', \<Sigma>2)"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "ubx_sp.sp_instr \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> F instr' \<Sigma>1 = Ok \<Sigma>2"
ide_char: fixes f :: "(bool, unit) arr" shows "ide f = (f \<noteq> Null \<and> Dom f \<in> UNIV \<and> Cod f = Dom f \<and> Map f = ())"
ex_TCons_iff: fixes Y :: "('c, 'b) tllist set" shows "local.ex_TCons Y = (\<exists>xs\<in>Y. \<not> is_TNil xs)"
induct: fixes P :: "(nat \<times> form) list \<Rightarrow> form \<Rightarrow> bool"   and a0 :: "(nat \<times> form) list"   and a1 :: "form" assumes "\<And>s v. P s (Var v [])"   and "\<And>s v va vb. P s (Var v (va # vb))"   and "\<And>s n. P s (LC n)"   and "\<And>s n xs. (\<And>x. x \<in> set xs \<Longrightarrow> P s x) \<Longrightarrow> P s (Op n xs)"   and "\<And>s n v f. \<lbrakk>\<And>x. \<lbrakk>v \<in> fv_subst1 s; x = fresh_for (fv_subst1 s)\<rbrakk> \<Longrightarrow> P ((v, Var x []) # s) f; v \<notin> fv_subst1 s \<Longrightarrow> P s f\<rbrakk> \<Longrightarrow> P s (Quant n v f)" shows "P a0 a1"
assms_2: shows "reachable ((f, xs, ls) # rest, None) (rest, eval f xs)"
fresh_permute_iff: fixes p :: "perm"   and a :: "atom"   and x :: "'a" shows "p \<bullet> a \<sharp> p \<bullet> x = a \<sharp> x"
wit_LCons2: fixes x :: "'a"   and y :: "'a"   and xs :: "'a llist" shows "local.wit (LCons x (LCons y xs)) = LCons x (prepend (local.pick x y) (local.wit (LCons y xs)))"
ExitBlockAbrupt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and abrs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and nrms :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile>abrs,css \<Down> Normal s" shows "\<Gamma>\<turnstile>[],(nrms, abrs) # css \<Down> Abrupt s"
image_r_vimage_l: fixes Q :: "'a set"   and ob :: "'b \<Rightarrow> 'a"   and P :: "'b set" assumes "Q \<subseteq> ob ` P"   and "observable ob P" shows "ob -` Q \<subseteq> P"
in_br_conv: fixes c :: "'a"   and a :: "'b"   and \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "((c, a) \<in> br \<alpha> I) = (a = \<alpha> c \<and> I c)"
rbd_rfb_dual: fixes R :: "('c \<times> 'd) set" shows "bd\<^sub>\<R> R = \<partial>\<^sub>F (fb\<^sub>\<R> (\<smile> R))"
finite_cone_carrier: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n" shows "gram_schmidt.finite_cone n A \<subseteq> carrier_vec n"
netmap_fst_netgmap_rel: fixes sr :: "'e \<Rightarrow> 'd \<times> 'f"   and s :: "'e net_state" shows "(\<lambda>i. map_option (fst \<circ> sr) (netmap s i)) = fst (netgmap sr s)"
normed_vectorspace_def: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real" shows "normed_vectorspace V norm \<equiv> vectorspace V \<and> Normed_Space.norm V norm"
ncs_def: shows "ncs = map (map_prod id normalize_ns_constraint) cs"
frechet_derivative_eq_vector_derivative: fixes f :: "real \<Rightarrow> 'b"   and x :: "real" assumes "f differentiable at x" shows "frechet_derivative f (at x) = (\<lambda>r. r *\<^sub>R vector_derivative f (at x))"
subst_support_def: fixes \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term"   and \<tau> :: "'d \<Rightarrow> ('e, 'f) Term.term" shows "(\<theta> supports \<tau>) = (\<tau> = \<theta> \<circ>\<^sub>s \<tau>)"
simps_1: fixes uu :: "('e \<times> 'e) set"   and uv :: "'f \<Rightarrow> 'e"   and U :: "'e set" shows "c_sources_aux uu uv U [] = U"
Transset_def: fixes x :: "V" shows "Transset x \<equiv> \<forall>y\<in>elts x. y \<subseteq>\<^sub>\<circ> x"
reachable_def: 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" assumes "substitution subst" shows "substitution.reachable subst s \<equiv> \<exists>e s'. substitution.execution subst e s' s"
word_of_integer_def: shows "word_of_integer \<equiv> map_fun int_of_integer id word_of_int"
length_aux: shows "Monitor.progress \<sigma> P (formula.MatchF I r) j + length aux = progress_regex \<sigma> P r j"
mult_numeral_left_semiring_numeral: fixes v :: "num"   and w :: "num"   and z :: "'b" shows "numeral v * (numeral w * z) = numeral (v * w) * z"
no_position_replicate_num_args: fixes t :: "('c, 'd) tm"   and d :: "dir" shows "\<not> position_of t (replicate (num_args t) dir.Left @ [d])"
coeff_0_power: fixes p :: "'b poly"   and n :: "nat" shows "constant_term (p ^ n) = constant_term p ^ n"
respection_of_binary_predicates_and_closures_6: fixes Rel :: "('a \<times> 'a) set"   and Pred :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_respects_binary_pred Rel Pred" shows "rel_respects_binary_pred ((symcl (Rel\<^sup>=))\<^sup>+) Pred"
connected_punctured_universe: fixes a :: "'a" assumes "2 \<le> DIM('a)" shows "connected (- {a})"
SCallInitThrowReds: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and vs :: "val list"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and a :: "char list exp"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),False\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "M \<noteq> clinit"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),False\<rangle> \<rightarrow>* \<langle>throw a,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
all_pairs_def: fixes xs :: "'a list" shows "all_pairs xs \<equiv> concat (map (\<lambda>x. map (Pair x) xs) xs)"
GE_Wit: fixes \<sigma> :: "'tp" assumes "\<not> unprot \<sigma>"   and "\<not> isRes \<sigma> \<or> protCl \<sigma>" shows "eintP (Guard \<sigma>) [eintF (Wit \<sigma>) []]"
fin_2: shows "finite ((+) (- a) ` (B - {a}))"
indRelTEQ_impl_TRel_is_weak_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_simulation (indRelTEQ TRel) (STCal Source Target)" shows "weak_reduction_simulation (TRel\<^sup>*) Target"
echelon_form_of_unit: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "is_unit (fst (echelon_form_of_det A k))"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Inv (t \<^bold>\<star> u) = Inv t \<^bold>\<star> Inv u"
zorder_hurwitz_zeta: fixes a :: "real" assumes "0 < a" shows "zorder (hurwitz_zeta a) 1 = - 1"
assms_4: fixes i :: "nat" assumes "i < nr"   and "Suc i < nr" shows "f i < f (Suc i) \<or> f (Suc i) = nc"
conjugate_square_eq_0_1: fixes x :: "'a" shows "(x * conjugate x = (0::'a)) = (x = (0::'a))"
unat_sub: fixes b :: "'a word"   and a :: "'a word" assumes "b \<le> a" shows "unat (a - b) = unat a - unat b"
elims_3: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b"   and xb :: "'b" assumes "\<not> to_ord x xa xb" obtains r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and x :: "'b" and y :: "'b" where "xa__ = r" and "xa = x" and "xb = y" and "\<not> \<not> r\<^sup>*\<^sup>* y x"
extend_9: shows "Ind (set b\<^sub>2) (sset u')"
injects: fixes a :: "'a"   and slist :: "[:'a:]"   and a' :: "'a"   and slist' :: "[:'a:]" assumes "a \<noteq> \<bottom>"   and "slist \<noteq> \<bottom>" shows "(a :# slist = a' :# slist') = (a = a' \<and> slist = slist')"
G_eq: fixes s :: "'s"   and t :: "'s" shows "G s t = \<integral>\<^sup>+ \<omega>. emeasure (count_space UNIV) {i. (s ## \<omega>) !! i = t} \<partial>T s"
isFault_simps_3: fixes f :: "'f" shows "isFault (Fault f) = True"
discI_2: fixes sum :: "'e + 'f"   and x2 :: "'f" assumes "sum = Inr x2" shows "\<not> isl sum"
assms_1: shows "theta \<subseteq> Sretr (theta \<union> Sbis)"
add_comm: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> y = y \<squnion> x"
Union_disjoint: fixes C :: "'a set set"   and A :: "'a set" shows "(\<Union> C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
csmall_nstep_base: fixes P :: "'prog"   and \<sigma> :: "'state" shows "csmall_nstep P \<sigma> 0 = {(\<sigma>, collect_id)}"
OrdmemD: fixes k :: "V"   and j :: "V" assumes "Ord k"   and "j \<in>\<^sub>\<circ> k" shows "j \<subset>\<^sub>\<circ> k"
f_eq_cos: fixes z :: "complex" assumes "z \<in> S" shows "f z = cos (g z)"
T_soundness: 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 intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "ModelIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw intT intF intP" shows "CM.Model (ProblemIkTpart.TE_wtFsym wtFsym resOf) wtPsym (ProblemIkTpart.TE_arOf arOf) (ProblemIkTpart.TE_resOf resOf) parOf (ProblemIkTpart.tPB wtFsym arOf resOf \<Phi> prot protFw) intT (ModelIkTpart.TE_intF intT intF) intP"
diff1_1: shows "local.rev_curr f \<noteq> local.rev_curr g"
Abs_eq_iff_1: fixes bs :: "atom set"   and x :: "'e"   and bs' :: "atom set"   and y :: "'e" shows "([bs]set. x = [bs']set. y) = (\<exists>p. (bs, x) \<approx>set (=) supp p (bs', y))"
sr_eq: shows "r * rat_of_nat s = rat_of_int n"
interval_integrable_isCont: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "\<And>x. \<lbrakk>min a b \<le> x; x \<le> max a b\<rbrakk> \<Longrightarrow> isCont f x" shows "interval_lebesgue_integrable lborel (ereal a) (ereal b) f"
leg_simps_3: shows "C.arr \<mu>.dom.leg1"
vsubsetI: fixes A :: "V"   and B :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> x \<in>\<^sub>\<circ> B" shows "A \<subseteq>\<^sub>\<circ> B"
ideD_3: fixes a :: "'a" assumes "B.ide a" shows "B.cod a = a"
reduce_system_matrix_signs_helper_aux_R: fixes S :: "nat list"   and signs :: "rat list list"   and p :: "real poly"   and subsets :: "(nat list \<times> nat list) list" assumes "\<forall>x. List.member S x \<longrightarrow> x < length signs"   and "p \<noteq> 0" shows "alt_matrix_A_R (take_indices signs S) subsets = take_cols_from_matrix (alt_matrix_A_R signs subsets) S"
induct: fixes P :: "H2 \<Rightarrow> bool"   and H2 :: "H2" assumes "P H1"   and "P H2" shows "P H2"
fun_diff_def: fixes A :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b" shows "A - B = (\<lambda>x. A x - B x)"
rep: fixes U :: "'a set" assumes "U \<in> carrier (G Mod stabilizer m)" shows "rep U \<in> U"
NextRecvUpd'_next': fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and q :: "'b"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and q' :: "'b" assumes "safe (prop_config c0 q)"   and "InvGlobPointstampsEq c0"   and "cri.InvMsgInGlob (exchange_config c0)"   and "NextRecvUpd' c0 c1 p q" shows "local.next'\<^sup>+\<^sup>+ (prop_config c0 q') (prop_config c1 q')"
step: assumes "(x, n) \<in> steps rel n"   and "(n, y) \<in> steps rel n" shows "thesis"
\<sigma>_\<sigma>'_p: shows "D \<cdot> \<sigma> = D' \<and> D' \<cdot> \<sigma>' = D"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Static of Static \<Rightarrow> f1 | NonStatic \<Rightarrow> f2) = f1"
cfisi: shows "f\<inverse> O s\<inverse> \<subseteq> d\<inverse>"
simps_4: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and P :: "complex mat" shows "wlp (Measure n M S) P = wlp_measure n M (map wlp S) P"
distinct_187: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x141 :: "expr"   and x142 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> while (x141) x142"
dp_p_fun2: fixes x :: "'a"   and y :: "'a" assumes "is_direct_product x y" shows "is_p_fun y"
distinct_1: fixes x2 :: "'a" shows "TypeError \<noteq> Normal x2"