hcju/isabelleps huggingface dataset:
allocationProperty: fixes a :: "('c set \<times> 'd) set"   and G :: "'c set"   and N :: "'d set" shows "(a \<in> possible_allocations_rel G N) = (runiq a \<and> runiq (a\<inverse>) \<and> Domain a partitions G \<and> Range a \<subseteq> N)"
similar_mat_witI: fixes P :: "'a mat"   and Q :: "'a mat"   and n :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "P * Q = 1\<^sub>m n"   and "Q * P = 1\<^sub>m n"   and "A = P * B * Q"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "P \<in> carrier_mat n n"   and "Q \<in> carrier_mat n n" shows "similar_mat_wit A B P Q"
star_of_norm: fixes x :: "'a" shows "hypreal_of_real (norm x) = hnorm (star_of x)"
smult_dvd_cancel: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" assumes "Polynomial.smult a p dvd q" shows "p dvd q"
necessaryPhiI: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and ms :: "'node list"   and z :: "'node"   and v :: "'var" assumes "old.pathsConverge g n ns m ms z"   and "v \<in> oldDefs g n"   and "v \<in> oldDefs g m" shows "necessaryPhi g v z"
unstream_stream: fixes xs :: "'b LList" shows "unstream\<cdot>(stream\<cdot>xs) = xs"
simply_connected_eq_homotopic_paths: fixes S :: "'c set" shows "simply_connected S = (path_connected S \<and> (\<forall>p q. path p \<and> path_image p \<subseteq> S \<and> path q \<and> path_image q \<subseteq> S \<and> pathstart q = pathstart p \<and> pathfinish q = pathfinish p \<longrightarrow> homotopic_paths S p q))"
map_disc_iff: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and a :: "('c, 'd) tllist" shows "is_TNil (tmap f1 f2 a) = is_TNil a"
coll_scale_pair: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "coll 0 (a, b) (c, d)"   and "(a, b) \<noteq> 0" shows "\<exists>x. (c, d) = x *\<^sub>R (a, b)"
primitive_part_smult_int: fixes d :: "int"   and f :: "int poly" shows "primitive_part (Polynomial.smult d f) = Polynomial.smult (sgn d) (primitive_part f)"
bprv_\<phi>L_eqv: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Loeb_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (Syntax_with_Connectives.eqv cnj imp (Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi>) (imp (HBL1.PP var subst P (enc (Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi>))) \<phi>))"
igSwapIPresIGWlsSTR_imp_igSwapIPresIGWls: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" assumes "igSwapIPresIGWlsSTR MOD" shows "igSwapIPresIGWls MOD"
continuous_proj: fixes s :: "('c \<Rightarrow>\<^sub>F 'd) set"   and i :: "'c" shows "continuous_on s (\<lambda>x. (x)\<^sub>F i)"
iMOD_cut_less: fixes r :: "nat"   and m :: "nat"   and t :: "nat" shows "[ r, mod m ] \<down>< t = (if t \<le> r then {} else [ r, mod m, (t - Suc r) div m ])"
finite_less_Inf_iff: fixes X :: "'a set"   and a :: "'a" assumes "finite X"   and "X \<noteq> {}" shows "(a < \<Sqinter> X) = (\<forall>x\<in>X. a < x)"
assms: shows "dep (prob_proj PROB (prob_dom PROB - vs)) v v'"
mk_monoid_simps_2: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
chart_image_eq_vimage: fixes X :: "'a set"   and c :: "('a, 'b) chart" assumes "X \<subseteq> domain c" shows "apply_chart c ` X = inv_chart c -` X \<inter> codomain c"
pdt_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.pdt prob = {(x, y). x \<noteq> y \<and> IFC_def.is_pd prob y x}"
not_MInfty_nonneg: fixes x :: "ereal" assumes "0 \<le> x" shows "x \<noteq> - \<infinity>"
next_plane0_def: fixes p :: "nat"   and g :: "graph" shows "next_plane0\<^bsub>p\<^esub> g \<equiv> if final g then [] else concat (map (\<lambda>f. concat (map (\<lambda>v. concat (map (\<lambda>i. generatePolygon i v f g) [3..<Suc (maxGon p)])) (vertices f))) (nonFinals g))"
effective_well_defined_Tensor: fixes M1 :: "int poly fract list list"   and M2 :: "int poly fract list list" assumes "mat (rat_poly.row_length M1) (length M1) M1"   and "mat (rat_poly.row_length M2) (length M2) M2" shows "mat (rat_poly.row_length M1 *n rat_poly.row_length M2) (length M1 *n length M2) (M1 \<otimes> M2)"
vpairs_eqI: fixes r :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> r \<Longrightarrow> \<exists>a b. x = \<langle>a, b\<rangle>" shows "vpairs r = r"
of_nat_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero n = (plus one ^^ n) zero"
weakly_fair_def4: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "weakly_fair enabled taken = \<box>\<diamond>(\<lambda>s. enabled s \<longrightarrow> taken s)"
fls_base_factor_to_fps_mult': fixes f :: "'a fls"   and g :: "'a fls" assumes "fls_nth f (fls_subdegree f) * fls_nth g (fls_subdegree g) \<noteq> (0::'a)" shows "fls_base_factor_to_fps (f * g) = fls_base_factor_to_fps f * fls_base_factor_to_fps g"
finsert_is_funion: fixes a :: "'c"   and A :: "'c fset" shows "finsert a A = {|a|} |\<union>| A"
elims: fixes x :: "(action \<times> nat) list"   and y :: "nat list" assumes "modify_tprog x = y" obtains "x = []" and "y = []"   | ac :: "action" and ns :: "nat" and nl :: "(action \<times> nat) list" where "x = (ac, ns) # nl" and "y = action_map ac # ns # modify_tprog nl"
idempotent_conv_closed: fixes x :: "'a" assumes "idempotent x" shows "idempotent (x\<^sup>T)"
binomial_less_binomial_Suc: fixes k :: "nat"   and n :: "nat" assumes "k < n div 2" shows "n choose k < n choose Suc k"
abs_eq: fixes x :: "int" shows "the_signed_int (Word.Word x) = signed_take_bit (LENGTH('c) - Suc 0) x"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "Pds'.Cd d X = greatest (Pd_shuffle_to_top ds' Y d) (X \<inter> Field (Pd_shuffle_to_top ds' Y d))"
inv_FactGroup: fixes X :: "'a set" assumes "X \<in> carrier (G Mod H)" shows "inv\<^bsub>G Mod H\<^esub> X = set_inv X"
Compl_subset_Compl_iff: fixes A :: "'d set"   and B :: "'d set" shows "(- A \<subseteq> - B) = (B \<subseteq> A)"
prod_coprime: fixes A :: "'a set"   and g :: "'a \<Rightarrow> nat" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> coprime (g x) (g y)" shows "divisor_count (prod g A) = (\<Prod>x\<in>A. divisor_count (g x))"
homeomorphism_moving_point: fixes S :: "'b set"   and T :: "'b set"   and a :: "'b"   and b :: "'b" assumes "openin (top_of_set (affine hull S)) S"   and "S \<subseteq> T"   and "T \<subseteq> affine hull S"   and "connected S"   and "a \<in> S"   and "b \<in> S" obtains f :: "'b \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'b" where "homeomorphism T T f g" and "f a = b" and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" and "bounded {x. \<not> (f x = x \<and> g x = x)}"
valid_indexI: fixes "is" :: "nat list"   and ds :: "nat list" assumes "length is = length ds"   and "\<And>m. m < length ds \<Longrightarrow> is ! m < ds ! m" shows "is \<lhd> ds"
gen_model_subset_aux: fixes M :: "('e, 'g, 'f, 'h) KripkeStructure_scheme"   and T :: "'f set"   and M' :: "('e, 'i, 'f, 'j) KripkeStructure_scheme"   and t :: "'f" assumes "\<And>a. Restr (relations M a) T = Restr (relations M' a) T"   and "(\<Union> (range (relations M)))\<^sup>* `` {t} \<subseteq> T" shows "(\<Union> (range (relations M)))\<^sup>* `` {t} \<subseteq> (\<Union> (range (relations M')))\<^sup>* `` {t}"
rv_Low_Use_Low: fixes S :: "'node set"   and n :: "'node"   and as :: "'edge list"   and as' :: "'edge list"   and s :: "'state"   and s' :: "'state" assumes "(_Low_) \<in> S"   and "n -as\<rightarrow>* (_Low_)"   and "n -as'\<rightarrow>* (_Low_)"   and "\<forall>V\<in>rv S n. state_val s V = state_val s' V"   and "preds (slice_kinds S as) s"   and "preds (slice_kinds S as') s'" shows "\<forall>V\<in>Use (_Low_). state_val (transfers (slice_kinds S as) s) V = state_val (transfers (slice_kinds S as') s') V"
Rep_seqauto_cases: fixes y :: "'d set \<times> 'd \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) set \<times> ('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" assumes "y \<in> seqauto" obtains x :: "('d, 'e, 'f) seqauto" where "y = Rep_seqauto x"
TyAll_canonical: fixes v :: "trm"   and T\<^sub>1 :: "POPLmark.type"   and T\<^sub>2 :: "POPLmark.type" assumes "[] \<turnstile> v : (\<forall><:T\<^sub>1. T\<^sub>2)"   and "v \<in> value" shows "\<exists>t S. v = (\<lambda><:S. t)"
Appl: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and u :: "Lambda_Z.term" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "App s u \<rightarrow>\<^sub>\<beta> App t u"
baEANow: fixes c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and \<phi> :: "'KS \<Rightarrow> bool" assumes "sb.eval c t t' n [\<phi>]\<^sub>b"   and "\<parallel>c\<parallel>\<^bsub>t n\<^esub>" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"
awalk_hd_in_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v" shows "u \<in> verts G"
upds_restr_stack: fixes V :: "Vars.var set"   and S :: "stack_elem list" shows "upds (restr_stack V S) = upds S \<inter> V"
mult_le_cancel_left_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less c zero" shows "less_eq (times c a) (times c b) = less_eq b a"
on_tripleD_3: fixes l1 :: "'d"   and v1 :: "'e"   and v2 :: "'e"   and l2 :: "'d"   and v3 :: "'f"   and v4 :: "'f"   and R :: "('e \<times> 'f) set" assumes "((l1, v1, v2), l2, v3, v4) \<in> on_triple R" shows "(v2, v4) \<in> R"
iT_Mult_iEx_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<diamond> t I \<otimes> k. P t) = (\<diamond> t I. P (t * k))"
f: fixes x :: "'a" assumes "x \<in> s" shows "(f has_derivative (\<lambda>x. 0::'b)) (at x)"
rtrancl_to_subtuple: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xm :: "'d \<rightharpoonup>\<^sub>f 'e"   and ym :: "'d \<rightharpoonup>\<^sub>f 'e" assumes "(subtuple r)\<^sup>*\<^sup>* xm ym" shows "subtuple r\<^sup>*\<^sup>* xm ym"
weakly_locally_path_connected_at: fixes x :: "'c"   and X :: "'c topology" shows "weakly_locally_path_connected_at x X = (\<forall>V. openin X V \<and> x \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> U \<subseteq> V \<and> (\<forall>y\<in>U. \<exists>C. path_connectedin X C \<and> C \<subseteq> V \<and> x \<in> C \<and> y \<in> C)))"
match_ex_table_app: fixes xt\<^sub>1 :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and D :: "char list"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list" assumes "\<forall>xte\<in>set xt\<^sub>1. \<not> matches_ex_entry P D pc xte" shows "match_ex_table P D pc (xt\<^sub>1 @ xt) = match_ex_table P D pc xt"
4: shows "restrict ((\<bullet>) (Id B)) (Hom A B) = (\<lambda>g\<in>Hom A B. g)"
comm_monoid_axioms: shows "comm_monoid (lift_option (+)) (Some (0::'b))"
weak_add_group_morphism: fixes f :: "'a \<Rightarrow> 'd"   and I :: "'a set" assumes "weak_ring_morphism f I R" shows "weak_group_morphism f I (add_monoid R)"
perp_in_right_comm: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "X PerpAt A B D C "
run_eval_list: shows "\<exists>run_eval_list. \<forall>env es s. evaluate_list True env s es (run_eval_list env es s)"
polytope_Int_polyhedron: fixes S :: "'a set"   and T :: "'a set" assumes "polytope S"   and "polyhedron T" shows "polytope (S \<inter> T)"
distinctPerm_def: fixes p :: "('a \<times> 'a) list" shows "distinctPerm p \<equiv> distinct (map fst p @ map snd p)"
starfun_ln_HInfinite: fixes x :: "real star" assumes "x \<in> HInfinite"   and "0 < x" shows "(*f* real_ln) x \<in> HInfinite"
adj_iso4: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "mono (g \<circ> f)"
swap_cs_Snapshot_Send: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSnapshot ev"   and "isSend ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
ev_iff_shift: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" shows "ev \<phi> xs = (\<exists>xl xs2. xs = xl @- xs2 \<and> \<phi> xs2)"
axioms_3: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "open_cover_of_subset X is_open U I C" shows "open_cover_of_subset_axioms is_open I C"
set_integral_mono_AE: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "set_integrable M A f"   and "set_integrable M A g"   and "AE x\<in>A in M. f x \<le> g x" shows "set_lebesgue_integral M A f \<le> set_lebesgue_integral M A g"
foldr_fold: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f y \<circ> f x = f x \<circ> f y" shows "foldr f xs = fold f xs"
set_inverse_def: fixes A :: "('a \<Rightarrow> 'b) set" shows "set_inverse A = {\<lambda>x. inverse (f x) |f. f \<in> A}"
rep_nat_pp_pp_def: fixes x :: "('a, 'b) pp" shows "rep_nat_pp x = pp_of_fun (\<lambda>n. rep_nat (lookup_pp x (abs_nat n)))"
doctor_optimal_match_def: fixes ds :: "D2 set"   and Y :: "X5 set" shows "BSI.doctor_optimal_match ds Y = (BSI.stable_on ds Y \<and> (\<forall>X. \<forall>x\<in>X. BSI.stable_on ds X \<longrightarrow> (\<exists>y\<in>Y. (x, y) \<in> PX5d (X5d x))))"
next_action_from_execs: shows "\<forall>s execs. int.next_action s execs \<rightharpoonup> (\<lambda>a. a \<in> actions_in_execution (execs (current s)))"
generate_valid_stateful_policy_IFSACS_2_def: fixes G :: "'b list_graph"   and M :: "'b SecurityInvariant list" shows "TopoS_Stateful_Policy_impl.generate_valid_stateful_policy_IFSACS_2 G M = \<lparr>hostsL = nodesL G, flows_fixL = edgesL G, flows_stateL = inefficient_list_intersect (TopoS_Stateful_Policy_impl.filter_IFS_no_violations G M) (TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS G M)\<rparr>"
diff_apply_chart: fixes \<psi> :: "('a, 'b) chart" assumes "\<psi> \<in> atlas" shows "diff k (charts_submanifold (domain \<psi>)) charts_eucl (apply_chart \<psi>)"
assms_1: shows "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<BB>"
fls_regpart_minus: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_regpart (f - g) = fls_regpart f - fls_regpart g"
invoke_def: fixes xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> 'd \<Rightarrow> ((_) heap, exception, 'c) prog)) list"   and ptr :: "(_) object_ptr"   and args :: "'d" shows "invoke xs ptr args = Heap_Error_Monad.bind (check_in_heap ptr) (\<lambda>_. invoke_rec xs ptr args)"
division_of_def: fixes s :: "'c set set"   and i :: "'c set" shows "(s division_of i) = (finite s \<and> (\<forall>K\<in>s. K \<subseteq> i \<and> K \<noteq> {} \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>K1\<in>s. \<forall>K2\<in>s. K1 \<noteq> K2 \<longrightarrow> interior K1 \<inter> interior K2 = {}) \<and> \<Union> s = i)"
intro: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FGModule G smult V"   and "FinGroupRepresentation_axioms G smult V" shows "FinGroupRepresentation G smult V"
neighborhoods_eq: shows "neighborhoods x = {U. is_open\<^sub>X U \<and> x \<in> U}"
assms_3: shows "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> \<A>\<^sub>d"
is_distinguishing_formula_def: fixes x :: "('idx, 'pred, 'act, 'effect) formula"   and P :: "'state"   and Q :: "'state" shows "x distinguishes P from Q \<equiv> P \<Turnstile> x \<and> \<not> Q \<Turnstile> x"
two_chain_vertical_boundary_def: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "two_chain_vertical_boundary twoChain \<equiv> \<Union> (vertical_boundary ` twoChain)"
embed_measure_count_space': fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "embed_measure (count_space A) f = count_space (f ` A)"
all_args_def: fixes f :: "'d \<Rightarrow> ('a, 'b, 'c) prog" shows "all_args f = (\<Union>arg. {f arg})"
if_True: fixes x :: "'e"   and y :: "'e" shows "(if True then x else y) = x"
ant_sol: fixes a :: "ant"   and b :: "ant"   and c :: "ant" assumes "a \<in> Z\<^sub>\<infinity>"   and "b \<in> Z\<^sub>\<infinity>"   and "c \<in> Z\<^sub>\<infinity>"   and "b \<noteq> \<infinity>"   and "a = b + c" shows "a - b = c"
C: shows "C = four_block_mat A1 (A2 * P') A0 B"
mcalc2: fixes c' :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list"   and \<sigma> :: "'tp" assumes "c' \<in> gPB" shows "mcalc2 \<sigma> c'"
constrains_weaken_L: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set" assumes "F \<in> A co A'"   and "B \<subseteq> A" shows "F \<in> B co A'"
coeff_map_poly_hom: fixes p :: "'a poly"   and i :: "nat" shows "coeff (map_poly hom p) i = hom (coeff p i)"
fun_upd_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'c" shows "f \<circ> g(x := y) = (f \<circ> g)(x := f y)"
B_trusts_OR3: fixes B :: "agent"   and X :: "msg"   and Nb :: "nat"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>X, Nonce Nb, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> orb" shows "\<exists>M. Says Server B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>X, Nonce Nb, Key K\<rbrace>\<rbrace> \<in> set evs"
var_MonotoneInPhi: fixes b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "Monotone (\<lambda>\<Phi> (s, t). (b, A, \<Phi>, s, t) \<in> var)"
assms_1: shows "\<forall>M\<in>carrier. \<forall>x>M. x \<in> carrier"
obtain_shortest_path: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat" assumes "Graph.connected c u v" obtains p :: "(nat \<times> nat) list" where "Graph.isShortestPath c u p v"
eq_neg_iff_add_eq_0: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (plus a b = zero)"
simps: fixes M :: "('a, 'b, 'c) FSM"   and V :: "'a list set" shows "is_det_state_cover M V = (\<exists>f. is_det_state_cover_ass M f \<and> V = f ` d_reachable M (initial M))"
locally_reflects_ide: fixes f :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C a\<guillemotright>"   and "C.ide (C.L f)" shows "C.ide f"
cut_ge_Int_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<ge> t = I \<inter> {t..}"
bind_cong_All: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c measure"   and g :: "'b \<Rightarrow> 'c measure" assumes "\<forall>x\<in>space M. f x = g x" shows "M \<bind> f = M \<bind> g"
Domainp_mod_ring_rel_integer: fixes p :: "int"   and pp :: "integer" assumes "mod_ring_locale TYPE('b) p"   and "p = int_of_integer pp" shows "Domainp (mod_ring_locale.mod_ring_rel_integer p) = (\<lambda>x. 0 \<le> x \<and> x < pp)"
equiv'_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool" assumes "Simulation_Invariant A B sim PA PB" shows "Simulation_Invariant.equiv' sim PA PB \<equiv> \<lambda>a b. sim a b \<and> PA a \<and> PB b"
homotopic_with_id2: fixes X :: "'c topology"   and g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x" shows "homotopic_with (\<lambda>x. True) X X (g \<circ> f) id"
rtc_implies_srtc: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "r\<^sup>*\<^sup>* a b" shows "relation.srtc r a b"
analytic_pre_newman: fixes T :: "complex set" assumes "T \<subseteq> {s. 1 / 2 < Re s}" shows "pre_newman analytic_on T"
unit_ief_lens: shows "ief_lens 0\<^sub>L"
lappend_trans_fusion: fixes g :: "('c, 'd) lgenerator'"   and h :: "'e \<Rightarrow> ('c, 'e) step"   and sh :: "'e"   and sg :: "'d" shows "lunstream (lappend_trans g h sh) (Inl sg) = lappend (lunstream' g sg) (lunstream h sh)"
exhaust: fixes y :: "comb" obtains "y = K"   | "y = S"   | x31 :: "comb" and x32 :: "comb" where "y = x31 \<bullet> x32"
op_ivl_Suc: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {m..<Suc n} = (if n < m then 0::'b else sum g {m..<n} + g n)"
injectivity: fixes x :: "'b"   and y :: "'b" assumes "[:x:] = [:y:]" shows "x = y"
meet_below2: fixes y :: "'c"   and z :: "'c"   and x :: "'c" assumes "y \<sqsubseteq> z" shows "x \<sqinter> y \<sqsubseteq> z"
dmd_def: fixes F :: "'a \<Rightarrow> bool" shows "\<diamond>F \<equiv> \<not> \<box>\<not> F"
assms_2: shows "G \<subseteq> dgrad_p_set d m"
hull_in: fixes S :: "'b set \<Rightarrow> bool"   and s :: "'b set" assumes "\<And>T. Ball T S \<Longrightarrow> S (\<Inter> T)" shows "S (S hull s)"
inorder_update: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and x :: "'c"   and y :: "'d" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (update x y t) = upd_list x y (Tree2.inorder t)"
inj_semigroup_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_semigroup_add_hom hom \<equiv> semigroup_add_hom hom \<and> injective hom"
smult_diff_right: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "smult a (p - q) = smult a p - smult a q"
polyfun_extract_matrix: fixes i :: "nat"   and m :: "nat"   and j :: "nat"   and n :: "nat"   and a :: "nat"   and c :: "nat" assumes "i < m"   and "j < n" shows "polyfun {..<a + (m * n + c)} (\<lambda>f. extract_matrix (\<lambda>i. f (i + a)) m n $$ (i, j))"
singletonize_L4Ports_normalized_generic: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b"   and m' :: "'b match_expr"   and pt :: "ipt_l4_ports" assumes "wf_disc_sel (disc, sel) C"   and "m' \<in> (\<lambda>spt. Match (C spt)) ` set (singletonize_L4Ports pt)" shows "normalized_n_primitive (disc, sel) (case_ipt_l4_ports (\<lambda>x pts. length pts \<le> 1)) m'"