hcju/isabelleps huggingface dataset:
BuildSeq_insf: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and z :: "hf"   and m :: "hf"   and n :: "hf"   and y :: "hf" assumes "BuildSeq B C s k z"   and "m \<^bold>\<in> succ k"   and "n \<^bold>\<in> succ k"   and "B y \<or> C y (app s m) (app s n)" shows "BuildSeq B C (insf s (succ k) y) (succ k) y"
homeomorphism_on_extension_sequentially': fixes S :: "'c set"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>u b. \<lbrakk>\<forall>n. u n \<in> S; b \<in> T; u \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> convergent (\<lambda>n. f (u n))"   and "\<And>u c. \<lbrakk>\<forall>n. u n \<in> S; c \<in> f ` T; (\<lambda>n. f (u n)) \<longlonglongrightarrow> c\<rbrakk> \<Longrightarrow> convergent u"   and "\<And>b. b \<in> T \<Longrightarrow> \<exists>u. (\<forall>n. u n \<in> S) \<and> u \<longlonglongrightarrow> b \<and> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b" shows "homeomorphism_on T f"
Times_empty: fixes A :: "'a set"   and B :: "'b set" shows "(A \<times> B = {}) = (A = {} \<or> B = {})"
Keys_bad_Ltk: fixes IK :: "msg set"   and Bad :: "agent set" shows "Keys_bad (IK \<inter> range LtK) Bad = Keys_bad IK Bad"
ExQ: fixes xs :: "real list" shows "eval (clearQuantifiers \<phi>) xs = eval \<phi> xs"
LeastFieldDecl_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and Cs :: "char list list" shows "P \<turnstile> C has least F:T via Cs \<equiv> (Cs, T) \<in> FieldDecls P C F \<and> (\<forall>(Cs', T')\<in>FieldDecls P C F. P,C \<turnstile> Cs \<sqsubseteq> Cs')"
list_all2_antisym: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y"   and "list_all2 P xs ys"   and "list_all2 Q ys xs" shows "xs = ys"
assms_3: shows "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
prefixes_def: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and t :: "nat" assumes "r1_bc s" shows "r1_bc.prefixes s j t \<equiv> the (eval (r1_bc.r_prefixes s) [t, j])"
le_F_T: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq>\<^sub>F Q" shows "P \<sqsubseteq>\<^sub>T Q"
union_mono_aux: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A \<subseteq> B" shows "A \<union> C \<subseteq> B \<union> C"
Inf_fin_Inf: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A"   and "A \<noteq> {}" shows "semilattice_inf.Inf_fin inf A = Inf A"
coeff_linear_poly_power: fixes i :: "nat"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "i \<le> n" shows "coeff ([:a, b:] ^ n) i = of_nat (n choose i) * b ^ i * a ^ (n - i)"
copr_inf: shows "algebraic_semidom_class.coprime f [:real_of_int (- crb (prod_list fss)), 1:]"
Rel_AND_def: shows "Rel_AND = {((x0, x1), w0, w1). (x0, w0) \<in> Rel0 \<and> (x1, w1) \<in> Rel1}"
prems_5: shows "(t # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"
induct: fixes x :: "'a fm"   and P :: "'a fm \<Rightarrow> bool" assumes "\<turnstile> x"   and "\<And>p. tautology p \<Longrightarrow> P p"   and "\<And>i p q. P (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i q)"   and "\<And>p q. \<lbrakk>\<turnstile> p; P p; \<turnstile> (p \<^bold>\<longrightarrow> q); P (p \<^bold>\<longrightarrow> q)\<rbrakk> \<Longrightarrow> P q"   and "\<And>p i. \<lbrakk>\<turnstile> p; P p\<rbrakk> \<Longrightarrow> P (K i p)" shows "P x"
absolutely_integrable_on_1_iff: fixes S :: "'a set"   and f :: "'a \<Rightarrow> (real, 1) vec" shows "(f absolutely_integrable_on S) = ((\<lambda>x. f x $ 1) absolutely_integrable_on S)"
axioms_1: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
length_rotate: fixes n :: "nat"   and xs :: "'a list" shows "|rotate n xs| = |xs|"
word_less_sub_1: fixes x :: "'b word"   and y :: "'b word" assumes "x < y" shows "x \<le> y - 1"
tail_invar_cons_up: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tail_invar (t # bq)"   and "SkewBinomialHeapStruc.rank t' < SkewBinomialHeapStruc.rank t"   and "SkewBinomialHeapStruc.tree_invar t'" shows "SkewBinomialHeapStruc.tail_invar (t' # t # bq)"
homeomorphism_on_continuous: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphism_on S f" shows "continuous_on S f"
star_false_left: fixes P :: "assn" shows "false * P = false"
indicator_times_eq_if_1: fixes S :: "'d set"   and x :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "indicator S x * f x = (if x \<in> S then f x else (0::'c))"
pdevs_val_upd: fixes e :: "nat \<Rightarrow> real"   and n :: "nat"   and e' :: "real"   and X :: "real pdevs" shows "pdevs_val (e(n := e')) X = pdevs_val e X - e n * pdevs_apply X n + e' * pdevs_apply X n"
test1: fixes \<tau> :: "'b state \<times> 'b state"   and A :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> A" shows "\<tau> \<Turnstile> A and B \<triangleq> B"
SC_rel_def: shows "SC_rel = {(X, Y) |X Y. SC X Y}"
nAct_cnf2proj_Suc_dist: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<not> (\<exists>i>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)" shows "Suc (the_enat \<langle>c #\<^bsub>enat n\<^esub>inf_llist t\<rangle>) = \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>Suc \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
read_mem_pc_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "fst (memory_read 8 (cpu_reg_val PC s1) s1) = fst (memory_read 8 (cpu_reg_val PC s2) s2)"
elims: fixes x :: "real poly"   and xa :: "real poly"   and y :: "real poly list" assumes "sturm_aux x xa = y" obtains p :: "real poly" and q :: "real poly" where "x = p" and "xa = q" and "y = (if degree q = 0 then [p, q] else p # sturm_aux q (- (p mod q)))"
varDTrue: fixes b :: "BExpr"   and s :: "Var \<Rightarrow> Val"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and w :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "evalB b s"   and "A s w"   and "(b, A, w, t) \<in> varD" shows "(b, A, s, t) \<in> varD"
simps_2: fixes x21 :: "'b"   and x22 :: "'c \<Rightarrow> ('a, 'b, 'c) env option"   and y21 :: "'b"   and y22 :: "'c \<Rightarrow> ('a, 'b, 'c) env option" shows "(Env x21 x22 = Env y21 y22) = (x21 = y21 \<and> x22 = y22)"
set_bind_pmf: fixes M :: "'c pmf"   and N :: "'c \<Rightarrow> 'b pmf" shows "set_pmf (M \<bind> N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
emp_pref: fixes u :: "'b list" shows "\<epsilon> \<le>p u"
subst_typ'_preserves_term_ok': fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and insts :: "((variable \<times> String.literal set) \<times> typ) list"   and t :: "Core.term" assumes "wf_theory \<Theta>"   and "inst_ok \<Theta> insts"   and "term_ok' (sig \<Theta>) t" shows "term_ok' (sig \<Theta>) (subst_typ' insts t)"
retract_of_connected: fixes T :: "'c set"   and S :: "'c set" assumes "connected T"   and "S retract_of T" shows "connected S"
open_sq_mtx_def: fixes U :: "'a sq_mtx set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
vc_node_found: fixes S :: "'v set"   and V :: "'v set" assumes "rpre S (V, tgt)" shows "rpost S (V, tgt) (insert tgt V, True)"
left_totalI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. \<exists>y. R x y" shows "left_total R"
Un_Int_eq_1: fixes S :: "'d set"   and T :: "'d set" shows "(S \<union> T) \<inter> S = S"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog" assumes "known_ptr = DocumentClass.known_ptr"   and "create_character_data = l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_character_data set_val get_disconnected_nodes set_disconnected_nodes" shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_val create_character_data known_ptr"
leq_antisym: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x"   and "x \<le> y" shows "x = y"
gp_expr: shows "LINT x|M. f x * enn2real (nn_transfer_operator gp x) = LINT x|M. f (T x) * enn2real (gp x)"
d_s_subid: fixes x :: "'a" shows "(d x = x) = (x \<le> 1\<^sub>\<sigma>)"
E_App1: fixes t :: "trm"   and t' :: "trm"   and u :: "trm" assumes "t \<longmapsto> t'" shows "t \<cdot> u \<longmapsto> t' \<cdot> u"
productive_in_N: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C \<noteq> {}" shows "C \<in> N"
numeral_Bit0_div_2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and n :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 num.One)) = numeral.numeral one plus n"
eliminate_swap_store_lid: fixes l :: "'a"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and l' :: "'a" assumes "l \<notin> LID\<^sub>S \<sigma>"   and "l' \<notin> LID\<^sub>S \<sigma>" shows "\<R>\<^sub>S id (id(l := l', l' := l)) \<sigma> = \<sigma>"
strict_implies_order: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subset># b" shows "a \<subseteq># b"
simps_4: fixes g :: "char list \<Rightarrow> bool"   and h :: "'b fm \<Rightarrow> bool"   and p :: "'b fm"   and q :: "'b fm" shows "eval g h (p \<^bold>\<and> q) = (eval g h p \<and> eval g h q)"
cong: fixes K :: "'a \<Rightarrow> 'a pmf" shows "MC_syntax.T' K \<equiv> MC_syntax.T' K"
AB_def: shows "AB = (\<lambda>x. (x, f x)) ` A"
finfun_dom_update: fixes f :: "('a, 'b) finfun"   and a :: "'a"   and b :: "'b" shows "finfun_dom (finfun_update f a b) = finfun_update (finfun_dom f) a (b \<noteq> finfun_default f)"
sum_list_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "A (0::'b) (0::'c)"   and "rel_fun A (rel_fun A A) (+) (+)" shows "rel_fun (list_all2 A) A sum_list sum_list"
Im_Ln_of_real: fixes r :: "real" assumes "0 < r" shows "Im (Ln (complex_of_real r)) = 0"
Vinv1_invariant: shows "reach v_TS \<subseteq> Vinv1"
fds_subseries_0: fixes P :: "nat \<Rightarrow> bool" shows "fds_subseries P 0 = 0"
mod_dist_2: fixes P :: "assn"   and b :: "bool"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> P * \<up> b) = (h \<Turnstile> P \<and> b)"
oalist_tc_eq_iff: fixes xs :: "('a, 'b) oalist_tc"   and ys :: "('a, 'b) oalist_tc" shows "(xs = ys) = (list_of_oalist_tc xs = list_of_oalist_tc ys)"
angelic_disjunctive: fixes R :: "'b \<Rightarrow> 'a" shows "{: R :} \<in> Apply.Disjunctive"
fmrel_on_fset_fmrel_restrict: fixes S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'d \<rightharpoonup>\<^sub>f 'e"   and n :: "'d \<rightharpoonup>\<^sub>f 'f" shows "fmrel_on_fset S P m n = fmrel P (fmrestrict_fset S m) (fmrestrict_fset S n)"
step: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and m :: "nat" assumes "r \<in> R"   and "fst r \<noteq> []"   and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R" shows "(snd r, m + 1) \<in> derivable R"
countable_SIGMA: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (Sigma I A)"
arr_pr0_iff: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr (pr0 a b) = (ide a \<and> ide b)"
intY1_func: shows "restrict f intY1 \<in> intY1 \<rightarrow> intY1"
always_imp_mono_2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and P' :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "(\<diamond>P) \<sigma>"   and "(\<box>(\<lambda>s. P s \<longrightarrow> P' s)) \<sigma>" shows "(\<diamond>P') \<sigma>"
assms_1: shows "matrix_idempotent_semiring.coreflexive f"
scale_cancel_left: fixes a :: "'b"   and x :: "('b, 'c) Finite_Cartesian_Product.vec"   and y :: "('b, 'c) Finite_Cartesian_Product.vec" shows "(a *s x = a *s y) = (x = y \<or> a = (0::'b))"
Rep_stone_phi_pair_inverse: fixes x :: "'c stone_phi_pair" shows "Abs_stone_phi_pair (Rep_stone_phi_pair x) = x"
Card_order_Times_singl1: fixes r :: "('c \<times> 'c) set"   and b :: "'d" assumes "Card_order r" shows "r =o |Field r \<times> {b}|"
RModHom: shows "RModuleHom R (\<cdot>) M (\<cdot>) T"
SystemDCCut_def: shows "SystemDCCut = (([[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) TT]]Geq (f0 fid1) (Const 0)) \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) TT]]Geq (Differential (Var vid1)) (Differential (Const 0)) \<leftrightarrow> ([[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) (TT && Geq (f0 fid1) (Const 0))]]Geq (Differential (Var vid1)) (Differential (Const 0))))"
OUTfromV_VARto_lemma: fixes x :: "chanID" assumes "OUTfromV x \<noteq> {}"   and "OUTfromV_VARto" shows "\<exists>v. x \<in> VARto v"
Broadcast_Deliver_prefix_closed: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and r :: "'a"   and ix :: "'a set"   and e :: "'b"   and j :: "nat"   and i :: "'a" assumes "xs @ [Broadcast (r, Delete ix e)] prefix of j"   and "i \<in> ix" shows "Deliver (i, Create i e) \<in> set xs \<or> Deliver (i, Append i e) \<in> set xs \<or> (\<exists>mo. Deliver (i, Expunge e mo i) \<in> set xs) \<or> (\<exists>mo. Deliver (i, Store e mo i) \<in> set xs)"
assms_1: shows "f \<midarrow>a\<rightarrow> b"
hyps_2: shows "\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1"
polyneg_norm: fixes p :: "'b poly" assumes "isnpoly p" shows "isnpoly (~\<^sub>p p)"
reduce_column_aux_index': fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P_init :: "'a mat"   and K_init :: "'a mat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "nat list" assumes "H \<in> carrier_mat m n"   and "P_init \<in> carrier_mat m m"   and "K_init \<in> carrier_mat m n"   and "P_init * H = K_init"   and "(P, K) = local.reduce_column_aux div_op xs H (P_init, K_init)"   and "0 < m"   and "invertible_mat P_init"   and "0 \<notin> set xs"   and "\<forall>x\<in>set xs. x < m"   and "distinct xs" shows "\<forall>i\<in>set xs. Matrix.row K i = Matrix.row (addrow (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 K_init) i"
n_in_obs_intra: fixes n :: "'node"   and S :: "'node set" assumes "valid_node n"   and "n \<in> S" shows "obs_intra n S = {n}"
rep_eq: fixes x :: "'a Matrix.vec" shows "($) x = snd (Rep_vec x)"
assms_2: shows "u \<in> verts (app_iso hom G)"
Int_Un_distrib: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<inter> (B \<union> C) = A \<inter> B \<union> A \<inter> C"
measurable_product_then_coordinatewise: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'b measure"   and i :: "'c" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. f x i) \<in> borel_measurable M"
fs_int_d_Suc: fixes n :: "nat"   and fs :: "int vec list"   and k :: "nat" assumes "fs_int_indpt n fs"   and "k < length fs" shows "rat_of_int (fs_int.d n fs (Suc k)) = \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) k\<parallel>\<^sup>2 * rat_of_int (fs_int.d n fs k)"
type_ok_def: fixes P :: "'addr J_prog" shows "type_ok P \<equiv> \<lambda>(E, T) e c. \<exists>T'. P,E,c \<turnstile> e : T' \<and> P \<turnstile> T' \<le> T"
s_super: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and CDef :: "classDef"   and D :: "nat" assumes "CT C = Some CDef"   and "cSuper CDef = D" shows "CT \<turnstile> C <: D"
simps_124: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x3 :: "48 word" shows "(case EtherDst x3 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f3 x3"
s3: fixes f :: "face" assumes "f \<in> set (facesAt g a)"   and "|vertices f| \<le> 4" shows "\<V> f \<inter> V \<subseteq> {a}"
extremum_unique: fixes a :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(a \<le> \<delta>) = (a = \<delta>)"
p'_def: shows "p' = sig_trd bs (poly_of_pair p)"
nb\<^sub>1: shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). range_of \<Psi> v \<noteq> None"
prod_memb_def: shows "prod_memb = (\<lambda>(i, j) (tab, ps). tab ! i ! j \<noteq> None)"
wfLexLessRestricted: fixes Vbl :: "nat set" assumes "finite Vbl" shows "wf (lexLessRestricted Vbl)"
integral_powr: fixes y :: "real"   and a :: "real"   and b :: "real" assumes "y \<noteq> - 1"   and "a \<le> b"   and "0 < a" shows "integral {a..b} (\<lambda>x. x powr y) = inverse (y + 1) * (b powr (y + 1) - a powr (y + 1))"
alw_valid_iff_valid: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "(\<turnstile> \<box>F) = (\<turnstile> F)"
assms: fixes t :: "real" assumes "0 < t"   and "t < 1" shows "g t = c"
cSup_cInf: fixes S :: "'c multiset set" assumes "S \<noteq> {}"   and "subset_mset.bdd_above S" shows "Sup S = Inf {x. \<forall>s\<in>S. s \<subseteq># x}"
ntr2ntrp: fixes c :: "'n list multiset"   and w :: "('p, 'ba) label list list"   and c' :: "'n list multiset"   and s :: "'n list"   and cl :: "'n list multiset" assumes "(c, w, c') \<in> trcl (ntr fg)"   and "mon_c fg (add_mset s cl) \<inter> (mon_c fg c \<union> mon_ww fg w) = {}" shows "((s, cl + c), map ENV w, s, cl + c') \<in> trcl (ntrp fg)"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
degree_primitive_part: fixes p :: "'a poly" shows "degree (primitive_part p) = degree p"
distinct_2: fixes x2 :: "('e, 'f, 'g) Tree"   and x1 :: "('e, 'f, 'g) Tree set['e]" shows "tNot x2 \<noteq> tConj x1"
converse_subset_swap: fixes r :: "('a \<times> 'b) set"   and s :: "('b \<times> 'a) set" shows "(r \<subseteq> s\<inverse>) = (r\<inverse> \<subseteq> s)"
expl_cond_expect_point_meas: fixes Y :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> real" assumes "disc_fct Y"   and "point_measurable M (space N) Y" shows "point_measurable M UNIV (expl_cond_expect M Y X)"
fr: shows "\<forall>n. (f \<circ> r) n \<in> S"
simps: fixes a :: "nat \<times> nat" shows "zerozero' a = (\<exists>x y. a = (x, y) \<and> equals (x, y) (0, 0))"
wffp: fixes sl :: "('St, 'U, 'C) state list" shows "wffp sl = (sl \<noteq> [] \<and> (\<forall>i. Suc i < length sl \<longrightarrow> sl ! Suc i \<in> \<delta> (sl ! i)))"
simps: fixes i :: "nat"   and u :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'c Heap"   and s :: "'c" shows "imp_for' i u f s = (if u \<le> i then return s else f i s \<bind> imp_for' (i + 1) u f)"
\<psi>'_def: shows "\<psi>' \<equiv> absp \<psi>"
D_DIM_wdD: assumes "wd TYPE('c)" shows "D = DIM('c)"
induct: fixes x :: "fmla"   and P :: "fmla \<Rightarrow> bool" assumes "x \<in> special_axioms"   and "\<And>A i x. P (A(i::=x) IMP exi i A)" shows "P x"
atm_of_subst_lit: fixes L :: "'a literal"   and \<sigma> :: "'s" shows "atm_of (L \<cdot>l \<sigma>) = atm_of L \<cdot>a \<sigma>"
assms_2: fixes vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option"   and ln :: "'l \<Rightarrow>f nat" shows "ta_seq_consist P vs (llist_of (convert_RA ln))"
coprime_self: fixes a :: "'a" shows "coprime a a = is_unit a"
proves_eq_reflexive: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq' (the (typ_of t)) t t"