hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "complex mat"   and x2 :: "com"   and x3 :: "complex mat"   and P :: "complex mat \<Rightarrow> com \<Rightarrow> complex mat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>p {x1} x2 {x3}"   and "\<And>P. is_quantum_predicate P \<Longrightarrow> P P SKIP P"   and "\<And>P U. is_quantum_predicate P \<Longrightarrow> P (adjoint U * P * U) (Utrans U) P"   and "\<And>P Q R S1 S2. \<lbrakk>is_quantum_predicate P; is_quantum_predicate Q; is_quantum_predicate R; \<turnstile>\<^sub>p {P} S1 {Q}; P P S1 Q; \<turnstile>\<^sub>p {Q} S2 {R}; P Q S2 R\<rbrakk> \<Longrightarrow> P P (S1;; S2) R"   and "\<And>n P Q S M. \<lbrakk>\<And>k. k < n \<Longrightarrow> is_quantum_predicate (P k); is_quantum_predicate Q; \<And>k. k < n \<Longrightarrow> \<turnstile>\<^sub>p {P k} S ! k {Q}; \<And>k. k < n \<Longrightarrow> P (P k) (S ! k) Q\<rbrakk> \<Longrightarrow> P (matrix_sum d (\<lambda>k. adjoint (M k) * P k * M k) n) (Measure n M S) Q"   and "\<And>P Q S M. \<lbrakk>is_quantum_predicate P; is_quantum_predicate Q; \<turnstile>\<^sub>p {Q} S {adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1}; P Q S (adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1)\<rbrakk> \<Longrightarrow> P (adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1) (While M S) P"   and "\<And>P Q P' Q' S. \<lbrakk>is_quantum_predicate P; is_quantum_predicate Q; is_quantum_predicate P'; is_quantum_predicate Q'; P \<le>\<^sub>L P'; \<turnstile>\<^sub>p {P'} S {Q'}; P P' S Q'; Q' \<le>\<^sub>L Q\<rbrakk> \<Longrightarrow> P P S Q" shows "P x1 x2 x3"
"case": fixes f :: "t list \<Rightarrow> t list \<Rightarrow> 'a"   and x1 :: "t list"   and x2 :: "t list" shows "(case x1 _> x2 of x _> xa \<Rightarrow> f x xa) = f x1 x2"
total_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "Relation.total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
partitioning1_empty0: fixes ts :: "'b set list"   and ss :: "'b set list" assumes "{} \<notin> set ts" shows "{} \<notin> set (partitioning1 ss ts)"
simps_11: fixes p :: "fm"   and q :: "fm" shows "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
clinear: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_clinear f" shows "clinear f"
stut_not: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "STUTINV F" shows "STUTINV \<not> F"
real_vector_space_on_axioms: shows "real_vector_space_on S"
assms_3: fixes P :: "ccs"   and Q :: "ccs" assumes "(P, Q) \<in> X" shows "(Q, P) \<in> X"
left_one_impl: fixes a :: "'b" shows "(1::'b) r\<rightarrow> a = a"
preserves_hom: fixes f :: "'a \<times> 'a \<times> 'a"   and a :: "'a \<times> 'a \<times> 'a"   and b :: "'a \<times> 'a \<times> 'a" assumes "VVV.in_hom f a b" shows "\<guillemotleft>\<alpha>' f : HoVH a \<Rightarrow> HoHV b\<guillemotright>"
redT_updT'_convert_new_thread_action_eq_None: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and f :: "'k \<Rightarrow> 'i"   and ta :: "('h, 'k, 'j) new_thread_action"   and t :: "'h" shows "(redT_updT' ts (convert_new_thread_action f ta) t = None) = (redT_updT' ts ta t = None)"
finite_unfinished_dnodes: fixes G_invar :: "'a \<Rightarrow> bool"   and G_succ :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and G_empty :: "'a"   and G_add :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> nat option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat"   and g :: "'a"   and G_\<alpha> :: "'a \<Rightarrow> 'b \<times> 'b \<Rightarrow> enat"   and s :: "'b"   and S :: "'b set" assumes "Dijkstra_Impl G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin g G_\<alpha>" shows "finite (Dijkstra.unfinished_dnodes (G_\<alpha> g) s S)"
1: shows "typ_of (mk_eq (Abs \<tau> (t $ Bv 0)) (decr 0 t)) = Some propT"
Scheduler_L4: fixes C :: "Config"   and activation :: "nat \<Rightarrow> nat list"   and t :: "nat" assumes "Scheduler C activation"   and "t mod cycleLength C mem schedule C" shows "activation t = [t mod cycleLength C]"
map_const_spmf_of_set: fixes A :: "'a set"   and c :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "map_spmf (\<lambda>_. c) (spmf_of_set A) = return_spmf c"
stop_doesnt_access: fixes x :: "'Var" shows "doesnt_modify Stop x \<and> doesnt_read Stop x"
inject_1: fixes x11 :: "name"   and x12 :: "value list"   and y11 :: "name"   and y12 :: "value list" shows "(Vconstr x11 x12 = Vconstr y11 y12) = (x11 = y11 \<and> x12 = y12)"
emeasure_eq_measure: fixes M :: "'h pmf"   and A :: "'h set" shows "emeasure (measure_pmf M) A = ennreal (measure_pmf.prob M A)"
that: fixes s1' :: "'x1 \<times> 'm1" assumes "t \<turnstile> s1' \<approx> s2'" shows "thesis"
Im1_sum: fixes f :: "'a \<Rightarrow> quat"   and S :: "'a set" shows "Im1 (sum f S) = (\<Sum>x\<in>S. Im1 (f x))"
fg_freeTr1: fixes M :: "('e, 'a, 'f) Module_scheme"   and A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'a \<Rightarrow> 'e \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'e"   and n :: "nat" assumes "R module M"   and "free_generator R M A"   and "R module fgmodule R A z i f s"   and "free_generator R (fgmodule R A z i f s) A"   and "g \<in> mHom R M (fgmodule R A z i f s)"   and "\<forall>x\<in>A. g x = x" shows "\<forall>fa sa. fa \<in> {j. j \<le> n} \<rightarrow> A \<and> sa \<in> {j. j \<le> n} \<rightarrow> carrier R \<longrightarrow> l_comb R (fgmodule R A z i f s) n sa (cmp g fa) = l_comb R (fgmodule R A z i f s) n sa fa"
prems_2: shows "(v, a) \<in> set (x # l)"
exec_gpv_plus_oracle_left: fixes oracle1 :: "'b \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'b) spmf"   and oracle2 :: "'b \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'b) spmf"   and gpv :: "('a, 'c, 'e) gpv"   and s :: "'b" shows "exec_gpv (oracle1 \<oplus>\<^sub>O oracle2) (left_gpv gpv) s = exec_gpv oracle1 gpv s"
commutes_with_unit: shows "\<psi> V \<one>\<^bsub>V\<^esub> = one"
Inf_pres_topf: shows "Inf_pres (\<lambda>x. \<top>)"
hbase_tl: fixes ns :: "C list"   and b :: "nat" assumes "C ns \<in> hbase b"   and "ns \<noteq> []" shows "C (tl ns) \<in> hbase b"
primorial_conv_primorial': fixes x :: "real" shows "primorial x = primorial' (nat \<lfloor>primes_pi x\<rfloor>)"
m_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier M"   and "y \<in> carrier M" shows "x \<oplus>\<^bsub>M\<^esub> y \<in> carrier M"
inj_onI: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
simps_10: fixes v :: "'c match_expr"   and rs :: "'c rule list" shows "process_ret (Rule v Unknown # rs) = Rule v Unknown # process_ret rs"
poly_div_mult_right: fixes x :: "'a poly"   and y :: "'a poly"   and z :: "'a poly" shows "x div (y * z) = x div y div z"
prc_refl: fixes f :: "'f \<times> nat" shows "prc f f = (False, True)"
fun_comp_eq_conv: fixes f :: "'j \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and fg :: "'h \<Rightarrow> 'i" shows "(f \<circ> g = fg) = (\<forall>x. f (g x) = fg x)"
step_3_c_dims_main: fixes x :: "'b"   and l :: "nat"   and k :: "nat"   and i :: "(nat \<times> nat) list"   and A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_3_c x l k i A) = dim_row A \<and> dim_col (??.Jordan_Normal_Form_Existence.step_3_c x l k i A) = dim_col A"
at_ds1: fixes a :: "'a" assumes "at TYPE('a)" shows " [(a, a)] \<triangleq> [] "
f_nonneg: fixes x :: "real" assumes "x0 \<le> x" shows "0 \<le> f x"
sigma_sets_UNION: fixes B :: "'b set set"   and X :: "'b set"   and A :: "'b set set" assumes "countable B"   and "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A" shows "\<Union> B \<in> sigma_sets X A"
wp_loop_unfold: fixes P :: "'a \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "sound P"   and "healthy (wp body)" shows "wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P = (\<lambda>s. \<guillemotleft> \<N> G \<guillemotright> s * P s + \<guillemotleft> G \<guillemotright> s * wp body (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P) s)"
simps_2: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and t :: "('d, 'e) bintree"   and ts :: "('d, 'e) bintree list" shows "bts_dfs f (t # ts) = bt_dfs f t @ bts_dfs f ts"
Sup_upto_llist_infinity: fixes Xs :: "'a set llist" shows "Sup_upto_llist Xs \<infinity> = Sup_llist Xs"
map_poly_hom_smult: fixes c :: "int"   and p :: "int poly" shows "of_int_poly (Polynomial.smult c p) = Polynomial.smult (of_int c) (of_int_poly p)"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Can (t \<^bold>\<star> u) = (Can t \<and> Can u \<and> Src t = Trg u)"
less_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less" shows "less x y = (less_eq x y \<and> x \<noteq> y)"
quasi_inverse_in_hom_1: fixes f :: "'a" assumes "equivalence_map f" shows "\<guillemotleft>f\<^sup>~ : trg f \<rightarrow> src f\<guillemotright>"
map_to_set_map_of: fixes l :: "('d \<times> 'e) list" assumes "distinct (map fst l)" shows "map_to_set (map_of l) = set l"
closed_injective_linear_image_eq: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "linear f"   and "inj f" shows "closed (f ` s) = closed s"
map_poly_rebase: fixes p :: "'b mod_ring poly" shows "map_poly rebase p = #p"
algebraic_complex_iff: fixes x :: "complex" shows "algebraic x = (algebraic (Re x) \<and> algebraic (Im x))"
zero_less_floor: fixes x :: "'a" shows "(0 < \<lfloor>x\<rfloor>) = ((1::'a) \<le> x)"
not_less: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
normalize_manual_orderQ_def: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" shows "normalize_manual_orderQ p l = removeAllDuplicates (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l)))"
Par2F: fixes Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P \<parallel> Q'"
set_nebset: fixes x :: "'d set!['e]" shows "set_nebset x \<in> {A. A \<noteq> {} \<and> |A| <o natLeq +c |UNIV|}"
left_null_space_orthogonal_complement_col_space: fixes A :: "((real, 'b) vec, 'a) vec" shows "left_null_space A = orthogonal_complement (col_space A)"
finite_Cvars: fixes C :: "'a acom" shows "finite (vars C)"
cop_F_increasing: fixes x :: "'x set"   and ds :: "'d set" shows "x \<subseteq> cop_F ds x"
upto_simps_1: fixes n :: "int"   and m :: "int" assumes "n < m" shows "upto\<cdot>(MkI\<cdot>m)\<cdot>(MkI\<cdot>n) = []"
sigma_sets_mono': fixes A :: "'c set set"   and B :: "'c set set"   and X :: "'c set" assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
coprime_mult_right_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one c (times a b) = (algebraic_semidom.coprime times one c a \<and> algebraic_semidom.coprime times one c b)"
case_two_not_learn_V: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat" assumes "total_cons s"   and "\<exists>t. eval (total_cons.r_prefixes s) [t, j] \<up>"   and "2 \<le> j"   and "\<phi> j = (\<lambda>x. eval (total_cons.r_psi s) [j, x])" shows "\<not> learn_total \<phi> V_constotal s"
Enc_partsE: fixes SK :: "nat"   and K :: "nat"   and EK :: "nat"   and M :: "msg"   and H :: "msg set" assumes "Enc SK K EK M \<in> parts H" obtains "Crypt K (sign SK M) \<in> parts H" and "Crypt EK (Key K) \<in> parts H"
nfoldli_alt: shows "nfoldli = nfoldli_alt"
con_elim: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "con \<phi> \<psi> \<pi>l" obtains "\<phi> \<pi>l" and "\<psi> \<pi>l"
complex_norm_eq_1_exp_eq: fixes z :: "complex" shows "(cmod z = 1) = (exp (\<i> * complex_of_real (Arg z)) = z)"
comm_ring_hom_axioms: shows "comm_ring_hom hom"
CardSecret_secrecy_lemma: fixes i :: "nat"   and evs :: "event list"   and K :: "nat"   and p :: "nat"   and CardSecret :: "nat" assumes "CA i \<notin> bad"   and "evs \<in> set_cr"   and "Key K \<notin> analz (knows Spy evs)"   and "Crypt (pubEK (CA i)) \<lbrace>Key K, Pan p, Nonce CardSecret\<rbrace> \<in> parts (knows Spy evs)" shows "Nonce CardSecret \<notin> analz (knows Spy evs)"
divmod_integer_code: fixes k :: "integer"   and l :: "integer" shows "divmod_integer k l = (if k = 0 then (0, 0) else if 0 < l then if 0 < k then Code_Numeral.divmod_abs k l else case Code_Numeral.divmod_abs k l of (r, s) \<Rightarrow> if s = 0 then (- r, 0) else (- r - 1, l - s) else if l = 0 then (0, k) else apsnd uminus (if k < 0 then Code_Numeral.divmod_abs k l else case Code_Numeral.divmod_abs k l of (r, s) \<Rightarrow> if s = 0 then (- r, 0) else (- r - 1, - l - s)))"
subst_cong: fixes T :: "'c trm"   and \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm" assumes "\<And>x. x \<in> vars T \<Longrightarrow> \<pi>1 x = \<pi>2 x" shows "subst \<pi>1 T = subst \<pi>2 T"
add_right_shift: assumes "x - y AND mask n = 0"   and "y AND mask n = 0"   and "x - y \<le> x - y + y" shows "x - y + y >> n = (x - y >> n) + (y >> n)"
reduction_exists: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and polarity :: "sign"   and t :: "'b trm"   and s :: "'b trm"   and C :: "'b eclause"   and S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list"   and L1 :: "'b literal" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "polarity = neg \<or> \<not> basic_superposition.equivalent_eq_exists trm_ord t s (cl_ecl C) (basic_superposition.int_clset trm_ord sel S) \<sigma> L1 \<or> basic_superposition.proper_subterm_red trm_ord sel t S \<sigma>"   and "\<forall>x y. x \<in> vars_of_cl (cl_ecl C) \<longrightarrow> occurs_in y (Var x \<lhd> \<sigma>) \<longrightarrow> basic_superposition.trm_rep trm_ord sel y S = y"   and "basic_superposition.eligible_literal trm_ord sel L1 C \<sigma>"   and "basic_superposition.trm_rep trm_ord sel (t \<lhd> \<sigma>) S \<noteq> t \<lhd> \<sigma>"   and "L1 \<in> cl_ecl C"   and "basic_superposition.orient_lit_inst trm_ord L1 t s polarity \<sigma>"   and "\<forall>x\<in>S. finite (cl_ecl x)"   and "ground_clause (subst_cl (cl_ecl C) \<sigma>)"   and "fo_interpretation (basic_superposition.same_values (\<lambda>t. basic_superposition.trm_rep trm_ord sel t S))"   and "C \<in> S"   and "Ball S (basic_superposition.well_constrained trm_ord)"   and "all_trms_irreducible (subst_set (trms_ecl C) \<sigma>) (\<lambda>t. basic_superposition.trm_rep trm_ord sel t S)"   and "\<not> validate_ground_clause (basic_superposition.int_clset trm_ord sel S) (subst_cl (cl_ecl C) \<sigma>)"   and "closed_under_renaming S" shows "\<exists>\<sigma>' u u' p v D L2. basic_superposition.reduction trm_ord sel pos_ord L1 C \<sigma>' t s polarity L2 u u' p v D (basic_superposition.same_values (\<lambda>t. basic_superposition.trm_rep trm_ord sel t S)) S \<sigma> \<and> variable_disjoint C D"
drop_one: fixes x :: "'a list" shows "drop (Suc 0) x = tl x"
d: fixes p :: "int poly" shows "roots_of_2_irr p = (if degree p = 1 then [Rational (Rat.Fract (- coeff p 0) (coeff p 1))] else let ri = root_info p; cr = root_info.l_r ri; B = root_bound p in roots_of_2_main p ri cr [(- B, B)] [])"
simps: fixes q :: "'b"   and n :: "nat" shows "oldest_token q n = (if configuration q n \<noteq> {} then Some (Min (configuration q n)) else None)"
simps_2: fixes at :: "('a, 'b) atm" shows "varsL (Neg at) = varsA at"
dom_cod: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.dom (B.VV.cod f) = B.VV.cod f"
K_cfg_map_repcs: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg" shows "K_cfg (repcs s cfg) = map_pmf (\<lambda>cfg'. repcs (THE s'. s' \<in> set_pmf (rept s (action cfg)) \<and> abss s' = state cfg') cfg') (K_cfg cfg)"
of_nat_0: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 0 = zero"
G_adj_def: shows "G_adj \<equiv> wgraph_by_map.adj lookup"
unit_vecs_first: fixes n :: "nat" shows "unit_vecs n = unit_vecs_first n n"
lemma_4_13_ii: fixes b :: "'a"   and a :: "'a" assumes "b r\<rightarrow> a = a" shows "a = (1::'a) \<or> b = (1::'a)"
distrib_imp1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<And>x y z. x \<cdot> (y + z) = x \<cdot> y + x \<cdot> z" shows "x + y \<cdot> z = (x + y) \<cdot> (x + z)"
simps_1: fixes \<Sigma> :: "'b list"   and n :: "nat"   and a :: "'b"   and m :: "nat" assumes "formula \<Sigma>" shows "formula.rexp_of n (FQ a m) = rexp.Inter (TIMES [Full, Atom (AQ m a), Full]) (formula.ENC n {m})"
complex_cnj_zero_iff: fixes z :: "complex" shows "(cnj z = 0) = (z = 0)"
rel_inject_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x2 :: "'b"   and y2 :: "'c" shows "rel_ref R (Ref x2) (Ref y2) = R x2 y2"
Ri_mono: fixes N :: "'b literal multiset set"   and N' :: "'b literal multiset set"   and \<Gamma> :: "'b inference set" assumes "N \<subseteq> N'" shows "standard_redundancy_criterion.Ri \<Gamma> N \<subseteq> standard_redundancy_criterion.Ri \<Gamma> N'"
intros: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and ad :: "'b"   and al :: "addr_loc"   and v :: "'b val" assumes "a \<in> actions E"   and "action_obs E a = NormalAction (ReadMem ad al v)" shows "a \<in> read_actions E"
vcompI: fixes b :: "V"   and c :: "V"   and r :: "V"   and a :: "V"   and s :: "V" assumes "\<langle>b, c\<rangle> \<in>\<^sub>\<circ> r"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s" shows "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<circ> s"
hyps: fixes \<sigma>' :: "'state"   and \<sigma> :: "'state"   and n' :: "nat" assumes "\<sigma>' \<in> big P \<sigma>"   and "\<sigma>' \<in> small_nstep P \<sigma> n1"   and "\<sigma>' \<in> small_nstep P \<sigma> n'" shows "n1 = n'"
has_integral_unique: fixes f :: "'a \<Rightarrow> 'b"   and k1 :: "'b"   and i :: "'a set"   and k2 :: "'b" assumes "(f has_integral k1) i"   and "(f has_integral k2) i" shows "k1 = k2"
fup_Aexp_Var: fixes x :: "Vars.var"   and n :: "Arity\<^sub>\<bottom>" shows "esing x\<cdot>n \<sqsubseteq> \<A>\<^sup>\<bottom>\<^bsub>n\<^esub> (Var x)"
hyps: assumes "- (x \<squnion> - y) = z" shows "- (x \<squnion> - (y \<squnion> k_ \<otimes> (x \<squnion> z))) = z"
find_consistent_signs_at_roots_1_R: fixes p :: "real poly"   and qs :: "real poly list" shows "p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> set (find_consistent_signs_at_roots_R p qs) = set (characterize_consistent_signs_at_roots p qs)"
Proj_open_map: fixes X :: "('b \<times> real) nonzero set" assumes "open X" shows "open (Proj ` X)"
find_non0_1: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and l :: "nat"   and A :: "'a mat"   and m :: "nat" assumes "det_selection_fun sel_fun"   and "find_non0 sel_fun l A = Some m" shows "A $$ (m, l) \<noteq> (0::'a)"
terminates_filter_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and P :: "'f \<Rightarrow> bool" assumes "terminates g" shows "terminates (filter_raw P g)"
hyp3: shows "snd (snd (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k])) \<le> nrows A"
l6_11_uniqueness: fixes A :: "'p"   and X :: "'p"   and R :: "'p"   and B :: "'p"   and C :: "'p"   and Y :: "'p" assumes "A Out X R"   and "Cong A X B C"   and "A Out Y R"   and "Cong A Y B C" shows "X = Y"
inject_1: fixes x21 :: "'a"   and x22 :: "'a"   and y21 :: "'a"   and y22 :: "'a" shows "(DenyAllFromTo x21 x22 = DenyAllFromTo y21 y22) = (x21 = y21 \<and> x22 = y22)"
delete_update: fixes k :: "'c"   and l :: "'c"   and v :: "'d"   and al :: "('c \<times> 'd) list" assumes "k \<noteq> l" shows "delete l (update k v al) = update k v (delete l al)"
divide_le_0_abs_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a (abs b)) zero = (less_eq a zero \<or> b = zero)"
simps: fixes instrs :: "('dyn, 'b, 'fun, 'op, 'opinl) Inca.instr list"   and ar :: "nat" shows "compile_fundef (Fundef instrs ar) = lift 0 instrs (replicate ar None) \<bind> (\<lambda>(xs, \<Sigma>\<^sub>1). (if \<Sigma>\<^sub>1 = [None] then Ok () else Error ()) \<bind> (\<lambda>(). optim 0 xs (replicate ar None) \<bind> (\<lambda>(ys, \<Sigma>\<^sub>2). Ok (Fundef (if \<Sigma>\<^sub>2 = [None] then ys else xs) ar))))"
negligible_outer_le: fixes S :: "'b set" shows "negligible S = (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue T \<le> e)"
sgn2_zero_impl_arg_pos: fixes x :: "nat" assumes "sgn2 x = 0" shows "0 < x"
qSwapAll_involutive: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and A :: "('a, 'b, 'c, 'd, 'e) qAbs" shows "X #[[x \<and> y]]_zs #[[x \<and> y]]_zs = X \<and> A $[[x \<and> y]]_zs $[[x \<and> y]]_zs = A"
f: shows "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M J M"
open_int_def: shows "open = generate_topology (range lessThan \<union> range greaterThan)"
ifupdateirrel: fixes any :: "char list"   and p :: "('a, 'd) simple_packet_scheme" shows "simple_fw fw (p\<lparr>p_oiface := any\<rparr>) = simple_fw fw p"
mem_fsps_fst_sub: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and a :: "'a set"   and b :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "(a, b) \<in> fsps R M N f H" shows "a \<subseteq> H"