hcju/isabelleps huggingface dataset:
edges_restrict_edges: fixes g :: "'b ugraph"   and E :: "('b \<times> 'b) set" shows "edges (restrict_edges g E) = edges g \<inter> (E \<union> E\<inverse>)"
at_within_codomain: fixes x :: "'a"   and c :: "('b, 'a) chart" assumes "x \<in> codomain c" shows "at x within codomain c = at x"
ENFnrSR_fsts_weakrefl_instance: fixes P :: "'a \<Rightarrow> 'v \<Rightarrow> 'a \<Rightarrow> 'v \<Rightarrow> bool"   and default_node_properties :: "'a"   and nP :: "'v \<Rightarrow> 'a"   and f :: "('v \<times> 'v) set"   and G :: "'v graph"   and i :: "'v" assumes "sinvar_all_edges_normal_form_not_refl_SR P"   and "\<forall>s r. P default_node_properties s default_node_properties r"   and "\<forall>s r. nP r \<noteq> default_node_properties \<longrightarrow> \<not> P (nP s) s (nP r) r \<longrightarrow> \<not> P default_node_properties s (nP r) r"   and "\<forall>s r. P (nP s) s default_node_properties r"   and "f \<in> set_offending_flows G nP"   and "i \<in> fst ` f" shows "\<not> sinvar G (nP(i := default_node_properties))"
ZassenhausTr6_1: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and H1 :: "'e set"   and K :: "'e set"   and K1 :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "\<natural>\<^bsub>G\<^esub>K \<triangleright> K1" shows "Group (\<natural>\<^bsub>G\<^esub>H \<inter> K / (H1 \<inter> K \<diamondop>\<^bsub>G\<^esub> H \<inter> K1))"
HFST: fixes a :: "tm"   and b :: "tm"   and c :: "tm"   and d :: "tm" shows "{HPair a b EQ HPair c d} \<turnstile> a EQ c"
dim_vec_of_basis_enum': fixes v :: "'b" shows "dim_vec (vec_of_basis_enum v) = length canonical_basis"
continuous_within_poly: fixes z :: "'a"   and s :: "'a set"   and p :: "'a poly" shows "continuous (at z within s) (poly p)"
concat_tl_basis: fixes x :: "'b list"   and xs :: "'b list list"   and G :: "'b list set" assumes "x # xs \<in> lists \<BB> G" shows "concat xs \<in> \<langle>G\<rangle>"
nhds_metric: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.metric_space dist uniformity open" shows "topological_space.nhds open x = (INF e\<in>{0<..}. principal {y. dist y x < e})"
hom_mult: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "Eval (x \<otimes>\<^bsub>P\<^esub> y) = Eval x \<otimes>\<^bsub>S\<^esub> Eval y"
rep_eq: fixes x :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "'b"   and xb :: "'c fset" shows "ffold x xa xb = Finite_Set.fold x xa (fset xb)"
state_\<alpha>_conv_2: fixes s :: "('l, 't) locks \<times> ('m_t \<times> 'm) \<times> 'm_w \<times> 's_i" shows "thr (state_\<alpha> s) = thr_\<alpha> (thr s)"
dtor2_ctor2: fixes x :: "('b, 'b JF1, 'b JF2) F2" shows "dtor2 (ctor2 x) = x"
simps_4: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c" shows "(case Infinity of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = f1"
SUCCEED_rule: fixes \<Phi> :: "'a \<Rightarrow> bool" shows "SUCCEED \<le> SPEC \<Phi>"
sel_rews_2: shows "stail\<cdot>\<bottom> = \<bottom>"
diamond_conjugation_var_1: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "d p = p"   and "d q = q" shows "( |x\<rangle> p \<le> ad q) = (\<langle>x| q \<le> ad p)"
elims_1: fixes x :: "'at set"   and xa :: "'at Literal set set"   and y :: "bool" assumes "x \<Turnstile> xa = y" obtains I :: "'at set" and S :: "'at Literal set set" where "x = I" and "xa = S" and "y = (\<forall>C. C \<in> S \<longrightarrow> I \<Turnstile> C)"
prod_star_closure: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<le> z\<^sup>\<star>"   and "y \<le> z\<^sup>\<star>" shows "x ; y \<le> z\<^sup>\<star>"
safe_def: fixes Quorum :: "process set set"   and s :: "v_state"   and r :: "nat"   and v :: "val" assumes "quorum_process Quorum" shows "quorum_process.safe Quorum s r v = (\<forall>r'<r. \<forall>Q w. quorum_process.quorum_for Quorum Q w (votes s r') \<longrightarrow> v = w)"
init_triv_filt_def: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" shows "init_triv_filt M F = (Filtration.filtration M F \<and> sets (F \<bottom>) = {{}, space M})"
in_pairsI': fixes aps :: "bool \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and flag :: "bool"   and a :: "('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and xs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and b :: "('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "\<And>gs bs h ps. set (aps flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))"   and "a \<in> set xs"   and "b \<in> set xs"   and "a \<noteq> b" shows "(a, b) \<in>\<^sub>p snd ` set (pairs aps flag xs)"
exhaust: fixes y :: "'c \<times> 'd" obtains a :: "'c" and b :: "'d" where "y = (a, b)"
thm_relation_negation_3: fixes v :: "i"   and p :: "\<o>" shows "[p\<^sup>- \<^bold>\<equiv> \<^bold>\<not>p in v]"
homeomorphic_refl: fixes s :: "'b set" shows "s homeomorphic s"
power_divides_special: fixes a :: "'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "a \<noteq> (0::'a)" shows "a ^ (i * j) / a ^ (k * i) = (a ^ j / a ^ k) ^ i"
spr_jview_tStep_eq_inv: fixes a :: "'a"   and t' :: "('a, 'es, 'as) BEState Trace"   and t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState" assumes "spr_jview a t' = spr_jview a (t \<leadsto> s)" shows "\<exists>t'' s'. t' = t'' \<leadsto> s'"
transitive_closure_trans_2: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and z :: "'a" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
order_refl: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k" assumes "class.preorder less_eq less" shows "less_eq x x"
65: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<sqinter> x \<squnion> - (x \<squnion> y) \<sqinter> y = bot"
intro: fixes C1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "category C1"   and "category C2" shows "product_category C1 C2"
defined6: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<delta> \<upsilon> X = true"
simps_5: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and s :: "char list \<Rightarrow> int" shows "postQ (IF b THEN C\<^sub>1 ELSE C\<^sub>2) s = (if bval b s then postQ C\<^sub>1 s else postQ C\<^sub>2 s)"
val_in_image: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set"   and x :: "'a" assumes "valuation K v"   and "ideal (Vr K v) I"   and "x \<in> I" shows "v x \<in> v ` I"
left_commute: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "sup b (sup a c) = sup a (sup b c)"
ord_of_nat_vempty: shows "0 = 0\<^sub>\<nat>"
inject_2: fixes x2 :: "'d"   and x1 :: "('d, 'e, 'f) input"   and y2 :: "'d"   and y1 :: "('d, 'e, 'f) input" shows "(x2\<lparr>x1 = y2\<lparr>y1) = (x2 = y2 \<and> x1 = y1)"
additive_in_ivp_sols: fixes s :: "'a"   and t :: "real" assumes "s \<in> S"   and "\<P> (\<lambda>\<tau>. \<tau> + t) T \<subseteq> T" shows "(\<lambda>\<tau>. \<phi> (\<tau> + t) s) \<in> Sols (\<lambda>t. f) (\<lambda>s. T) S 0 (\<phi> (0 + t) s)"
Let_4: fixes \<rho> :: "Vars.var \<Rightarrow> Value" assumes "fv (as @ \<Gamma>, body) \<subseteq> set L \<union> domA (as @ \<Gamma>)" shows "\<lbrakk> body \<rbrakk>\<^bsub>\<lbrace>as @ \<Gamma>\<rbrace>\<rho>\<^esub> = \<lbrakk> v \<rbrakk>\<^bsub>\<lbrace>\<Delta>\<rbrace>\<rho>\<^esub>"
surjective: fixes r :: "('a, 'b, 'c) sas_plus_operator_scheme" shows "r = \<lparr>precondition_of = precondition_of r, effect_of = effect_of r, \<dots> = sas_plus_operator.more r\<rparr>"
f: fixes k :: "nat" shows "f k \<subseteq> A \<and> infinite (f k) \<and> card (A - f k) = k"
min_dist_less: fixes src :: "nat"   and v :: "nat"   and d :: "nat"   and d' :: "nat" assumes "connected src v"   and "min_dist src v = d"   and "d' < d" shows "\<exists>v'. connected src v' \<and> min_dist src v' = d'"
inf_sup_aci_7: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.lattice inf less_eq less sup" shows "sup x (sup y z) = sup y (sup x z)"
delay_bisimulation_diverge_final_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool" assumes "delay_bisimulation_diverge_final trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 final1 final2" shows "delay_bisimulation_diverge_final trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1 final2 final1"
fold_invariant: fixes xs :: "'a list"   and Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and s :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x"   and "P s"   and "\<And>x s. \<lbrakk>Q x; P s\<rbrakk> \<Longrightarrow> P (f x s)" shows "P (fold f xs s)"
P_def: shows "P \<equiv> {{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A i} |A j. j \<in> J \<and> A \<in> Pi j E}"
fin: shows "finite ((\<lambda>c. (case lec_rel c of Leq_Rel \<Rightarrow> LEQ | Lt_Rel \<Rightarrow> LT) (lec_poly c) (lec_const c)) ` cs)"
cs_not_nil_implies_postrecording_event: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "fst (cs (distributed_system.s channel trans send recv init t i) cid) \<noteq> []"   and "i \<le> length t"   and "channel cid = Some (p, q)" shows "\<exists>j<i. computation.postrecording_event channel trans send recv init t j"
primes_1000_def: shows "primes_1000 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]"
mat_mult2_10: fixes U :: "'b mat"   and V :: "'b mat"   and A :: "'b mat" assumes "U \<in> carrier_mat 2 2"   and "V \<in> carrier_mat 2 2"   and "A = U * V" shows "A $$ (1, 0) = U $$ (1, 0) * V $$ (0, 0) + U $$ (1, 1) * V $$ (1, 0)"
arr_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C" shows "partial_magma.arr C f \<equiv> partial_magma.domains C f \<noteq> {} \<or> partial_magma.codomains C f \<noteq> {}"
init_fin_silent_movesD: fixes t :: "'t"   and s :: "status"   and x :: "'x1"   and m :: "'m1"   and s' :: "status"   and x' :: "'x1"   and m' :: "'m1" assumes "(\<tau>trsys.silent_move (r1.init_fin t) r1.init_fin_\<tau>move)\<^sup>*\<^sup>* ((s, x), m) ((s', x'), m')" shows "r1.silent_moves t (x, m) (x', m') \<and> s = s'"
Asm: fixes P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and p :: "pname"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set" assumes "(P, CALL p, Q) \<in> C" shows "C \<turnstile>\<^sub>t {P} CALL p {Q}"
dist_sup_mono: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" assumes "dist_over_sup (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "mono (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"
Ks_3: shows "set Ks \<subseteq> h ` K"
fin: fixes i :: "nat" shows "emeasure M (A i) \<noteq> \<infinity>"
K2_isometry_cross_ratio_endpoints_in_S: fixes a :: "proj2"   and b :: "proj2"   and J :: "cltn2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "is_K2_isometry J"   and "a \<noteq> b" shows "cross_ratio (apply_cltn2 (endpoint_in_S a b) J) (apply_cltn2 (endpoint_in_S b a) J) (apply_cltn2 a J) (apply_cltn2 b J) = cross_ratio (endpoint_in_S a b) (endpoint_in_S b a) a b"
v11I: fixes r :: "V" assumes "vsv r"   and "vsv (r\<inverse>\<^sub>\<circ>)" shows "v11 r"
measure_measure_pmf_finite: fixes S :: "'c set"   and M :: "'c pmf" assumes "finite S" shows "measure_pmf.prob M S = sum (pmf M) S"
refl: fixes x :: "'c" assumes "x \<in> B" shows "bob x x"
Cons_8: shows "set (clause # Wl') \<inter> set newWl = {}"
Fract_less_zero_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(Rat.Fract a b < 0) = (a < 0)"
I_R_additive: fixes a :: "atom"   and xs :: "real list"   and b :: "atom" assumes "I\<^sub>R a xs"   and "I\<^sub>R b xs" shows "I\<^sub>R (a + b) xs"
cfun_eq_iff: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b" shows "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
pfp_wn_def: fixes f :: "'a option acom \<Rightarrow> 'a option acom"   and c :: "com" shows "pfp_wn f c = (case iter_widen f (\<bottom>\<^sub>c c) of None \<Rightarrow> None | Some c' \<Rightarrow> iter_narrow f c')"
divide_divide_eq_left: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide a b) c = divide a (times b c)"
num_params_polypow: fixes n :: "nat"   and p :: "'a poly" shows "num_params (p ^\<^sub>p n) \<le> num_params p"
SUP_id_eq: fixes Sup :: "'f set \<Rightarrow> 'f"   and A :: "'f set" shows "Sup (id ` A) = Sup A"
vector_derivative_diff_chain_within: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and S :: "real set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_vector_derivative f') (at x within S)"   and "(g has_derivative g') (at (f x) within f ` S)" shows "(g \<circ> f has_vector_derivative g' f') (at x within S)"
subst_preserves_fresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "fresh zs z X"   and "fresh zs z Y" shows "fresh zs z (X #[Y / y]_ys)"
order_power_n_n: fixes a :: "'a"   and n :: "nat" shows "order a ([:- a, 1::'a:] ^ n) = n"
univalent_insert: fixes a :: "'c"   and R :: "('c \<times> 'b) set"   and b :: "'b" assumes "\<And>c. (a, c) \<notin> R" shows "univalent (insert (a, b) R) = univalent R"
baIANow: fixes \<phi> :: "'a \<Rightarrow> bool"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'a" assumes "\<phi> (tCMP c (t n))"   and "active c (t n)" shows "dynamic_component.eval tCMP active c t t' n (dynamic_component.ba \<phi>)"
loose_bvar1_decr'': fixes t :: "Core.term"   and lev :: "nat"   and lev' :: "nat" assumes "loose_bvar1 t lev"   and "lev < lev'" shows "loose_bvar1 (decr lev' t) lev"
C_arr_expansion_1: fixes f :: "'c" assumes "C.arr f" shows "\<epsilon> (C.cod f) \<cdot>\<^sub>C F (G f) \<cdot>\<^sub>C C.inv (\<epsilon> (C.dom f)) = f"
eq_card_imp_inj_on: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "card (f ` A) = card A" shows "inj_on f A"
Deq_visible: fixes v :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'c list"   and ic :: "state \<Rightarrow> 'c"   and oc :: "state \<Rightarrow> 'c" assumes "\<turnstile> Unchanged v \<longrightarrow> Unchanged q" shows "|~ \<langle>Deq ic q oc\<rangle>_v = Deq ic q oc"
prefixes_nth_take: fixes i :: "nat"   and p :: "'b list" assumes "i \<le> length p" shows "prefixes p ! i = take i p"
Fvars_instInp_ptrm_1: fixes \<tau> :: "'fmla"   and t :: "'trm" assumes "\<tau> \<in> ptrm (Suc 0)"   and "t \<in> trm" shows "Fvars (instInp \<tau> t) = insert out (FvarsT t)"
smult_prod: fixes a :: "'a"   and b :: "'a poly" shows "smult a b = monom a 0 * b"
\<nu>: shows "\<guillemotleft>\<nu> : S.UP g \<Rightarrow>\<^sub>S S.UP g'\<guillemotright> \<and> S.iso \<nu>"
preserves_dom: fixes f :: "'d" assumes "D.arr f" shows "D.dom (trg\<^sub>D f) = trg\<^sub>D (D.dom f)"
card_verts: shows "card (verts H) = card (verts G) + 1"
secure_implies_trace_consistency: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "secure P I D"   and "(xs, ys) \<in> rel_ipurge_aux P I D (D ` set zs)"   and "xs @ zs \<in> traces P" shows "ys @ zs \<in> traces P"
in_hom_char: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr"   and a :: "('a, 'b) concrete_category.arr"   and b :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp" shows "partial_magma.in_hom (concrete_category.COMP Obj Hom Comp) f a b = (partial_magma.arr (concrete_category.COMP Obj Hom Comp) f \<and> partial_magma.ide (concrete_category.COMP Obj Hom Comp) a \<and> partial_magma.ide (concrete_category.COMP Obj Hom Comp) b \<and> concrete_category.Dom f = concrete_category.Dom a \<and> concrete_category.Cod f = concrete_category.Dom b)"
satisfies_bounded_fold_rderiv: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i"   and x :: "'x"   and k :: "nat" assumes "local.wf idx \<phi>"   and "#\<^sub>V \<AA> = idx"   and "size x = idx" shows "(\<AA> \<Turnstile>\<^sub>b fold rderiv (replicate k x) \<phi>) = ((SNOC x ^^ k) \<AA> \<Turnstile>\<^sub>b \<phi>)"
match_sound: fixes t :: "('e, 'd) Term.term"   and p :: "('e, 'd) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "match t p = Some \<sigma>" shows "\<sigma> \<in> matchers {(p, t)}"
fadd: fixes x :: "'a"   and y :: "'a" shows "f (x + y) = f x + f y"
simps: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a option"   and a2 :: "'b option" shows "ord_option ord a1 a2 = ((\<exists>x. a1 = None \<and> a2 = x) \<or> (\<exists>x y. a1 = Some x \<and> a2 = Some y \<and> ord x y))"
NUM_I: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set" assumes "\<psi> \<in> \<R>\<^sup>2"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>i. \<psi> i = f" shows "U \<in> NUM"
matchC_ZO_def2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchC_ZO theta d c = (\<forall>s t d' t'. s \<approx> t \<and> (d, t) \<rightarrow>c (d', t') \<longrightarrow> s \<approx> t' \<and> (d', c) \<in> theta \<or> (\<exists>c' s'. (c, s) \<rightarrow>c (c', s') \<and> s' \<approx> t' \<and> (d', c') \<in> theta) \<or> (\<exists>s'. (c, s) \<rightarrow>t s' \<and> s' \<approx> t' \<and> discr d'))"
M': shows "\<forall>l. \<forall>m\<in>M l. m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
Units_r_inv: fixes x :: "cltn2" assumes "x \<in> Units \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "x \<otimes>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> inv\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = \<one>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"
in_homE: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hom f a b" obtains "Span.arr f" and "Span.dom f = a" and "Span.cod f = b"
fst_eqD: fixes x :: "'c"   and y :: "'d"   and a :: "'c" assumes "fst (x, y) = a" shows "x = a"
simps_29: fixes x2 :: "'b" shows "set1_sum3 (In2 x2) = {}"
ipset_from_netmask_prefix: shows "ipset_from_netmask (pfxm_prefix pfx) (~~ pfxm_mask pfx) = ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
between_triv1: fixes a :: "'a"   and b :: "'a" shows "between (a, b) a"
Ana_subterm: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list" assumes "Ana t = (K, T)" shows "set T \<subset> subterms t"
rtrancl_beta_App: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'"   and "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<degree> t'"
residue_add: fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = (x + y) mod p ^ k"
equivalence_mapI: fixes f :: "'d"   and g :: "'d" assumes "D.quasi_inverses f g" shows "D.equivalence_map f"
pmdl_idI: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "0 \<in> B"   and "\<And>b1 b2. \<lbrakk>b1 \<in> B; b2 \<in> B\<rbrakk> \<Longrightarrow> b1 + b2 \<in> B"   and "\<And>c t b. b \<in> B \<Longrightarrow> monom_mult c t b \<in> B" shows "pmdl B = B"
expr_typing_code: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type" shows "(\<Gamma> \<turnstile> e : t) = (expr_type \<Gamma> e = Some t)"
\<Phi>o_def: fixes a' :: "'c" shows "\<Phi>o a' = S.mkArr (Hom.set (a', a)) (\<iota> ` D.cones a') (\<lambda>x. \<iota> (D.cones_map (\<psi> (a', a) x) \<chi>))"
pdom_minus: fixes g :: "('a, 'b) pfun"   and f :: "('a, 'b) pfun" assumes "g \<subseteq>\<^sub>p f" shows "pdom (f - g) = pdom f - pdom g"
functor_category_axioms: shows "functor_category J (\<cdot>)"
v4: shows "atom v4 \<sharp> (t, t', u, X1, v2, v3)"
continuous_on_const: fixes s :: "'a set"   and c :: "'b" shows "continuous_on s (\<lambda>x. c)"
individually_rational_onI: fixes ds :: "'d set"   and X :: "'x set" assumes "CD_on ds X = X"   and "CH X = X" shows "individually_rational_on ds X"