hcju/isabelleps huggingface dataset:
face_cycle_sets_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "pre_digraph_map.face_cycle_sets (app_iso hom G) (map_iso hom) = (`) (iso_arcs hom) ` face_cycle_sets"
induct: fixes P :: "event list \<Rightarrow> bool"   and a0 :: "event list" assumes "P []"   and "\<And>i j k A B X evs. P evs \<Longrightarrow> P (Says i j k A B X # evs)" shows "P a0"
Nset_injTr2: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "\<forall>l\<le>Suc n. f l \<le> Suc n"   and "inj_on f {i. i \<le> Suc n}"   and "f (Suc n) = Suc n" shows "\<forall>l\<le>n. f l \<le> n"
reduced_GB_D3: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_reduced_GB G" shows "is_monic_set G"
nn_integral_K_cfg: fixes cfg :: "'a cfg"   and f :: "'a cfg \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf (K_cfg cfg)) f = \<integral>\<^sup>+ s. f (cont cfg s) \<partial>measure_pmf (action cfg)"
singleActionChain_3: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P'"
arr_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "partial_magma.arr (free_monoidal_category.comp C) f = (f \<noteq> {} \<and> (\<forall>t. t \<in> f \<longrightarrow> f = free_monoidal_category.mkarr C t))"
Eps_cong: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "Eps P = Eps Q"
finite_Nats_le_real: fixes x :: "real" shows "finite {n. 0 < n \<and> real n \<le> x}"
l7_20: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and M :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A M B"   and "Cong M A M B" shows "A = B \<or> Tarski_neutral_dimensionless.Midpoint Bet Cong M A B"
c_sequences: fixes xs :: "'a list" shows "local.c_sequences xs \<le> length xs - 1"
rep_eq: fixes x :: "'b upred"   and xa :: "'b upred" shows "\<lbrakk>x \<Rightarrow> xa\<rbrakk>\<^sub>e = (\<lambda>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<longrightarrow> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
image_mono: fixes A :: "'q set"   and B :: "'q set"   and f :: "'q \<Rightarrow> 'r" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
restrict_modes_def: fixes mdss :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list"   and X :: "'Var\<^sub>C set" shows "conc.restrict_modes mdss X \<equiv> map (\<lambda>mds m. mds m \<inter> X) mdss"
atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum g {..Suc n} = g 0 + (\<Sum>i\<le>n. g (Suc i))"
assms_3: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set"   and v :: "'v" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. k \<le> j"   and "\<forall>j\<in>dom (\<alpha> m) - it. j \<le> k"   and "\<alpha> m k = Some v"   and "it \<subseteq> dom (\<alpha> m)"   and "I it \<sigma>" shows "I (it - {k}) (f (k, v) \<sigma>)"
sccs_conv_sccs_verts: shows "sccs = (\<restriction>) G ` sccs_verts"
integer_mod_group_def: fixes n :: "nat" shows "integer_mod_group n \<equiv> if n = 0 then integer_group else \<lparr>carrier = {0..<int n}, mult = \<lambda>x y. (x + y) mod int n, one = 0\<rparr>"
leg1_simps_4: shows "trg (tab\<^sub>1 s) = trg s"
In0_def: fixes M :: "('c, 'd) Old_Datatype.node set" shows "In0 M \<equiv> Scons (Numb 0) M"
1: shows "\<guillemotleft>corestr f : local.dom f \<rightarrow> img f\<guillemotright>"
know1s: fixes C :: "specID"   and m2 :: "Secrets" shows "knows C [KS2Expression (sKS m2)] = know C (sKS m2)"
subst_Var: fixes x :: "name"   and y :: "name"   and s :: "lam" shows "Var x[y::=s] = (if x = y then s else Var x)"
assms_1: shows "f \<approx>>\<^bsub>SET\<^esub> g"
rep_eq: fixes x :: "'a pmf" shows "pmf x = (\<lambda>xa. measure_pmf.prob x {xa})"
\<I>_full_sel_2: fixes x :: "'c" shows "responses_\<I> \<I>_full x = UNIV"
exhaust: fixes y :: "typ" obtains x1 :: "nat" where "y = TVar x1"   | x21 :: "typ" and x22 :: "typ" where "y = x21 -> x22"
ev_nxt: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "ev \<phi> = (\<lambda>xs. \<phi> xs \<or> nxt (ev \<phi>) xs)"
natural_transformation_axioms: shows "natural_transformation A2xA1.comp (\<cdot>\<^sub>B) F.sym F.sym F.sym"
mult2_s_locally_cancel: fixes s :: "('b \<times> 'b) set"   and ns :: "('b \<times> 'b) set"   and X :: "'b multiset"   and Z :: "'b multiset"   and Y :: "'b multiset" assumes "s O ns \<subseteq> s"   and "ns O s \<subseteq> s"   and "refl ns"   and "trans ns"   and "locally_irrefl s (X + Z)"   and "locally_irrefl s (Y + Z)"   and "trans s"   and "(X + Z, Y + Z) \<in> mult2_s ns s" shows "(X, Y) \<in> mult2_s ns s"
gga_to_list_correct: shows "graph_to_list gbm_\<alpha> gbm_invar gga_to_list"
IV: assumes "k < l - 1" shows "\<exists>a. is_kth_obs \<pi> k a"
assms_1: shows "x \<in>\<^sub>\<circ> vrestriction D"
of_int_dvd: fixes a :: "int"   and b :: "int" assumes "of_int a / of_int b \<in> \<int>"   and "b \<noteq> 0" shows "b dvd a"
fdg: shows "(f12, f21, g'') = splitFace g' v a f' ws"
valid_sop: fixes sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and \<theta> :: "nat \<Rightarrow> nat option" assumes "valid_sop sop"   and "sop = (D, f)"   and "D \<subseteq> dom \<theta>" shows "f \<theta> = f (\<theta> |` D)"
m_closed: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and y :: "'e" assumes "H \<le> G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
null_eqI: fixes n :: "'a" assumes "\<And>f. n \<cdot>\<^sub>S f = n \<and> f \<cdot>\<^sub>S n = n" shows "n = S.null"
fps_lr_inverse_one_minus_fps_X_2: shows "fps_right_inverse (1 - fps_X) (1::'b) = Abs_fps (\<lambda>n. 1::'b)"
lex_ord_raw_EqI: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> order option"   and ko :: "'o" assumes "oalist_inv xs"   and "oalist_inv ys"   and "\<And>k. k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys) \<Longrightarrow> f k (lookup_raw xs k) (lookup_raw ys k) = Some Eq" shows "lex_ord_raw ko f xs ys = Some Eq"
path_prepend: fixes u :: "'b"   and v :: "'b"   and E :: "('b \<times> 'b) set"   and l :: "'b list"   and w :: "'b" assumes "(u, v) \<in> E"   and "path E v l w" shows "path E u (u # l) w"
hyp_CondR: fixes b :: "expr"   and bv :: "val"   and e2 :: "expr"   and e1 :: "expr" assumes "constVal b = Some bv"   and "\<not> the_Bool bv"   and "P b"   and "P e2" shows "P (b ? e1 : e2)"
r_2: shows "(u \<circ> r) \<longlonglongrightarrow> liminf u"
tuple_ABC_eq_ABC'_2: shows "tuple_ABC = tuple_ABC'"
next_subseqs_foldr: fixes state :: "'state"   and out :: "(int \<times> int poly list) list"   and state' :: "'state"   and base :: "int \<times> int poly list"   and elements :: "int poly list"   and n :: "nat" assumes "next_subseqs_foldr sl_impl state = (out, state')"   and "sli base elements n state" shows "sli base elements (Suc n) state' \<and> set out = S base elements (Suc n)"
prems_1: shows "LeftDerivationLadder a D L \<alpha>"
base_case_invertible_mat_R: shows "invertible_mat (M_mat_R [[1], [0], [- 1]] [([], []), ([0], []), ([], [0])])"
R: fixes S :: "'a set"   and T :: "'b set" assumes "P S \<and> Q T" shows "R (S \<times> T)"
AssertAllState'_precond: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "\<triangle> (\<T> (assert_all_state t as)) \<and> \<nabla> (assert_all_state t as) \<and> (\<not> \<U> (assert_all_state t as) \<longrightarrow> \<Turnstile> (assert_all_state t as))"
good_item_simps_49: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binpa :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and ys :: "'varSort"   and y :: "'var"   and delta :: "'c" assumes "good Y"   and "goodInp inp"   and "goodBinp binpa" shows "Op delta inp binpa #[Y / y]_ys = Op delta (inp %[Y / y]_ys) (binpa %%[Y / y]_ys)"
submonoid_inverse_closed: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> K" shows "inverse u \<in> K"
in_closed_iff_infdist_zero: fixes A :: "'c set"   and x :: "'c" assumes "closed A"   and "A \<noteq> {}" shows "(x \<in> A) = (infdist x A = 0)"
osn_rreq': fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg (\<lambda>m. rreq_rrep_sn m \<and> msg_zhops m) \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:4, PAodv-:5} \<union> {PRreq-:n |n. True} \<longrightarrow> 1 \<le> osn \<xi>)"
DSourcesA71_L1: shows "DSources level1 sA71 = {sA6}"
term_powerprod_gd_term: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and cmp_term :: "'b nat_term_order" assumes "term_powerprod pair_of_term term_of_pair"   and "\<And>v. fst (rep_nat_term v) = rep_nat_pp (fst (pair_of_term v))"   and "\<And>t. snd (rep_nat_term (term_of_pair (t, the_min))) = 0"   and "\<And>v w. snd (pair_of_term v) \<le> snd (pair_of_term w) \<Longrightarrow> snd (rep_nat_term v) \<le> snd (rep_nat_term w)"   and "\<And>s t k. term_of_pair (s + t, k) = splus (term_of_pair (s, k)) (term_of_pair (t, k))"   and "\<And>t v. term_powerprod.splus pair_of_term term_of_pair t v = splus (term_of_pair (t, the_min)) v" shows "gd_term pair_of_term term_of_pair (\<lambda>s t. le_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term)"
insert_spec_nonex: fixes ref :: "'b"   and xs :: "'b list"   and oid :: "'b" assumes "ref \<notin> set xs" shows "insert_spec xs (oid, Some ref) = xs"
simps_6: shows "TernaryUnknown \<noteq> TernaryFalse"
bij_betw_totatives: fixes m1 :: "nat"   and m2 :: "nat" assumes "1 < m1"   and "1 < m2"   and "coprime m1 m2" shows "bij_betw (\<lambda>x. (x mod m1, x mod m2)) (totatives (m1 * m2)) (totatives m1 \<times> totatives m2)"
blocksReds: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   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 Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and vs' :: "val list"   and e :: "expr"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option" assumes "wf_prog wf_md P"   and "length Vs = length Ts"   and "length vs = length Ts"   and "distinct Vs"   and "P \<turnstile> Ts Casts vs to vs' "   and "P,E(Vs [\<mapsto>] Ts) \<turnstile> \<langle>e,(h\<^sub>0, l\<^sub>0(Vs [\<mapsto>] vs'))\<rangle> \<rightarrow>* \<langle>e',(h\<^sub>1, l\<^sub>1)\<rangle>" shows "\<exists>vs''. P,E \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow>* \<langle>blocks (Vs, Ts, vs'', e'),(h\<^sub>1, override_on l\<^sub>1 l\<^sub>0 (set Vs))\<rangle> \<and> (\<exists>ws. P \<turnstile> Ts Casts vs'' to ws ) \<and> length vs = length vs''"
Cons_2: shows "\<forall>z\<in>set (z # zs). \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"
fmran'_def: shows "fmran' \<equiv> (\<lambda>x. \<Union> (set_option ` range x)) \<circ> fmlookup"
bot_xor: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "\<bottom>\<^bsub>L\<^esub> \<oplus>\<^bsub>L\<^esub> x = x"
is_ord_delete_root: fixes t :: "'b tree" assumes "avl t"   and "is_ord t"   and "t \<noteq> ET" shows "is_ord (delete_root t)"
nat_mod_eq_iff: fixes x :: "nat"   and n :: "nat"   and y :: "nat" shows "(x mod n = y mod n) = (\<exists>q1 q2. x + n * q1 = y + n * q2)"
span_Un: fixes S :: "'b set"   and T :: "'b set" shows "vs1.span (S \<union> T) = {x + y |x y. x \<in> vs1.span S \<and> y \<in> vs1.span T}"
index_row_1: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "row A i $ j = A $$ (i, j)"
splitFace_edges_g'_help: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and g' :: "graph" assumes "pre_splitFace g ram1 ram2 f vs"   and "(f12, f21, g') = splitFace g ram1 ram2 f vs"   and "vs \<noteq> []" shows "\<E> g' = \<E> g \<union> \<E> f \<union> Edges vs \<union> Edges (rev vs) \<union> {(ram2, last vs), (hd vs, ram1), (ram1, hd vs), (last vs, ram2)}"
unique: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X"   and "P x"   and "P y" shows "x = y"
Ostrowski_elem_nonzero: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "'a" assumes "vals_nonequiv K (Suc n) vv"   and "x \<in> carrier K"   and "Ostrowski_elem K (Suc n) vv x" shows "x \<noteq> \<zero>"
ord_p_lt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<preceq>\<^sub>p q" shows "lt p \<preceq>\<^sub>t lt q"
scalar_prod: fixes v1 :: "'a list"   and v2 :: "'a list" shows "scalar_prod v1 v2 = sum_list (map2 (*) v1 v2)"
code_blue_dfs_correct: fixes G :: "'a b_graph_rec"   and Gi :: "('a \<Rightarrow> bool, 'a \<Rightarrow> 'a list, 'a list, 'a \<Rightarrow> bool) gen_bg_impl" assumes "b_graph G"   and "finite ((g_E G)\<^sup>* `` g_V0 G)"   and "(Gi, G) \<in> bg_impl_rel_ext unit_rel Id" shows "RETURN (code_blue_dfs Gi) \<le> SPEC (\<lambda>r. case r of None \<Rightarrow> \<forall>prpl. \<not> b_graph.is_lasso_prpl G prpl | Some L \<Rightarrow> b_graph.is_lasso_prpl G L)"
ground_subst_ground_lit: fixes \<sigma> :: "'s"   and L :: "'a literal" assumes "is_ground_subst \<sigma>" shows "is_ground_lit (L \<cdot>l \<sigma>)"
unwinding_implies_view_partitioned_ind: shows "view_partitioned_ind"
identify_blocks_main_cong: fixes k :: "nat"   and n :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and xs :: "(nat \<times> nat) list" assumes "k < n"   and "same_upto n k A B" shows "??.Jordan_Normal_Form_Existence.identify_blocks_main A k xs = ??.Jordan_Normal_Form_Existence.identify_blocks_main B k xs"
effect_lookupE: fixes r :: "'b ref"   and h :: "heap"   and h' :: "heap"   and x :: "'b" assumes "effect (!r) h h' x" obtains "h' = h" and "x = Ref.get h r"
igSubstIPresIGWlsSTR_imp_igSubstIPresIGWls: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" assumes "igSubstIPresIGWlsSTR MOD" shows "igSubstIPresIGWls MOD"
h1: shows "(\<forall>x. eval (allZero p var) (L[var := x])) = (\<forall>x y. (\<Sum>i\<le>n. k i x * y ^ i) = 0)"
simplex_likeD_iso: fixes P :: "'a set" assumes "domain.simplex_like P" shows "\<exists>f A. OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A"
subset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "(A \<subseteq> insert x B) = (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
split_pattern: fixes xs :: "[:'r:]"   and pat :: "[:'r:]"   and us :: "[:'r:]"   and vs :: "[:'r:]" assumes "xs \<noteq> \<bottom>"   and "split\<cdot>pat\<cdot>xs = (us, vs)" shows "us :@ vs = pat"
bdd_below_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq A = (\<exists>m. \<forall>x\<in>A. less_eq m x)"
Rf_indep: fixes N' :: "'b literal multiset set"   and N :: "'b literal multiset set" assumes "N' \<subseteq> standard_redundancy_criterion.Rf N" shows "standard_redundancy_criterion.Rf N \<subseteq> standard_redundancy_criterion.Rf (N - N')"
input_stored_in_reg_def: fixes t' :: "transition"   and t :: "transition"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "input_stored_in_reg t' t e = (case input_stored_in_reg_aux t' t (Store_Reuse.total_max_reg e) (max (Arity t) (Arity t')) of None \<Rightarrow> None | Some (i, r) \<Rightarrow> if length (filter (\<lambda>(r', u). r' = r) (Updates t')) = 1 then Some (i, r) else None)"
ih: shows "\<exists>\<sigma> \<zeta> R. s = (\<sigma>, NodeS i \<zeta> R)"
rev_image_eqI: fixes x :: "'c"   and A :: "'c set"   and b :: "'d"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
Joints_msum_aform: fixes X :: "'a \<times> 'a pdevs"   and d :: "nat"   and YS :: "('a \<times> 'a pdevs) list"   and Y :: "'a \<times> 'a pdevs" assumes "degree_aform X \<le> d"   and "\<And>Y. Y \<in> set YS \<Longrightarrow> degree_aform Y \<le> d" shows "Joints (msum_aform d X Y # YS) = {(x + y) # ys |x y ys. y \<in> Affine Y \<and> x # ys \<in> Joints (X # YS)}"
Units_eq: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "Units G = carrier G"
notIn_Un_bij_betw3: fixes b :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and A' :: "'d set" assumes "b \<notin> A"   and "f b \<notin> A'" shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
fds_nth_zeta_pos: fixes n :: "nat" assumes "0 < n" shows "fds_nth fds_zeta n = (1::'a)"
Wbis_prefix: shows "Wbis \<subseteq> Wretr Wbis"
hoare_tripleD_1: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set"   and \<sigma> :: "heap option"   and r :: "'d" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P"   and "run c (Some h) \<sigma> r" shows "\<not> is_exn \<sigma>"
distinct_ExcessNotAtRecList: fixes ps :: "(nat \<times> nat) list"   and g :: "graph" assumes "distinct (map fst ps)" shows "distinct (ExcessNotAtRecList ps g)"
gaugeD_1: fixes \<gamma> :: "'a \<Rightarrow> 'a set"   and x :: "'a" assumes "gauge \<gamma>" shows "x \<in> \<gamma> x"
assms_4: shows "x \<noteq> 0\<^sub>h"
stabilizer_subgroup: fixes x :: "'c" assumes "x \<in> E" shows "subgroup (stabilizer G \<phi> x) G"
cvalidtI: fixes \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and F :: "'e set"   and c :: "('d, 'f, 'e) com"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "\<And>s t. \<lbrakk>\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A; \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A"   and "\<And>s. \<lbrakk>\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A; s \<in> P\<rbrakk> \<Longrightarrow> \<Gamma>\<turnstile>c \<down> Normal s" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
Disj_def: fixes xset :: "('a, 'b, 'c) formula set['a]" shows "Disj xset = Formula.Not (Conj (map_bset Formula.Not xset))"
q_def: shows "q = hom_component q0 (deg_pm (lpp q0))"
accI_pos: fixes x :: "'s"   and y :: "'s"   and n :: "nat" assumes "0 < p x y n" shows "(x, y) \<in> acc"
ladder_j_def: fixes L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "ladder_j L index = deriv_j (L ! index)"
MatchOr_def: fixes m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "MatchOr m1 m2 = MatchNot (MatchAnd (MatchNot m1) (MatchNot m2))"
intros: fixes s :: "'b + 'c"   and x :: "'b" assumes "s = Inl x" shows "x \<in> Basic_BNFs.setl s"
pfa_append_grow'_def: shows "pfa_append_grow' \<equiv> \<lambda>(a, n) x. pfa_ensure (a, n) (n + 1) \<bind> (\<lambda>a'. pfa_append a' x \<bind> return)"
prod_2_list_def: fixes x :: "'e \<Rightarrow> 'a decision option"   and y :: "('f \<Rightarrow> 'b decision option) list" shows "x \<Otimes>\<^sub>2\<^sub>L y = (x \<Otimes>\<^sub>L y) (\<Otimes>\<^sub>2)"
finite_cases: fixes F :: "'b set" assumes "finite F" obtains "F = {}"   | A :: "'b set" and x :: "'b" where "F = insert x A" and "x \<notin> A" and "finite A"
assms: shows "(a, b) \<in> R\<^sup>*"
set_plus_intro2: fixes b :: "'b"   and C :: "'b set"   and a :: "'b" assumes "b \<in> C" shows "a + b \<in> a +o C"
zminus_minus: fixes i :: "int"   and j :: "int" shows "i - - j = i + j"
card_face_cycle_sets_da': shows "card H.face_cycle_sets = card G.face_cycle_sets - 1 + card ({segment G.face_cycle_succ a a', segment G.face_cycle_succ a' a, {}} - {{}})"
LHS_def: shows "LHS = (concat css \<bullet>\<currency>\<currency> concat hfvss) ((1::'f) \<delta>\<delta> hs ! i)"
multiplicity_self: fixes p :: "'a" assumes "p \<noteq> (0::'a)"   and "\<not> is_unit p" shows "multiplicity p p = 1"
tfinite_map_tree: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and vertices :: "'e fset"   and nodeOf :: "'e \<Rightarrow> ('b, 'd) graph_node"   and edges :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) set"   and vidx :: "'e \<Rightarrow> nat"   and inst :: "'e \<Rightarrow> 'c"   and f :: "'g \<Rightarrow> 'f"   and t :: "'g tree" assumes "Tasked_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions vertices nodeOf edges vidx inst" shows "tfinite (map_tree f t) = tfinite t"
assms: shows "pairwise (\<lambda>i j. disjnt (S i) (S j)) I"
eqvt_fresh_cong3: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e"   and a :: "atom"   and x :: "'b"   and y :: "'c"   and z :: "'d" assumes "\<And>p x y z. p \<bullet> f x y z = f (p \<bullet> x) (p \<bullet> y) (p \<bullet> z)"   and "a \<sharp> x"   and "a \<sharp> y"   and "a \<sharp> z" shows "a \<sharp> f x y z"
abs_eq: fixes xa :: "'c set"   and x :: "'c \<Rightarrow> 'd set" shows "PiC xa x = Bcontfun ` (Pi xa x \<inter> bcontfun)"
sums_l: shows "(\<lambda>i. (- 1) ^ i * a i) sums l"
degree_finite_fourier_poly: fixes ws :: "complex list" shows "degree (finite_fourier_poly ws) \<le> length ws - 1"
that: fixes tk :: "nat" assumes "tki = tk mod N"   and "locks_ticket (ts t) tk" shows "thesis"
tarski_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski C B \<equiv> tarski_absolute C B \<and> tarski_space C B"
StartBallot_HInv3: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "StartBallot s s' t"   and "HInv3_inner s p q d" shows "HInv3_inner s' p q d"
transpD: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "transp r"   and "r x y"   and "r y z" shows "r x z"
rtranclp_into_tranclp2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
dynlookup: shows "dynlookup G statT (invocation_class mode s a' statT) sig = Some m"