hcju/isabelleps huggingface dataset:
nb: shows "length \<psi> = length (\<phi>\<^sub>P \<Psi> List_Supplement.embed \<psi>)"
F1: shows "\<forall>x\<^sub>1. analz x\<^sub>1 \<union> synth (analz x\<^sub>1) = analz (synth (analz x\<^sub>1))"
coeffs_of_int_poly: fixes ff_ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c mod_ring \<Rightarrow> bool"   and p :: "int"   and x :: "int poly" assumes "mod_ring_gen ff_ops R p" shows "coeffs (of_int_poly (poly_mod.Mp p x)) = map of_int (coeffs (poly_mod.Mp p x))"
id_func_preserves_cod: shows "preserves_cod (id_func AA)"
Negative_def: shows "Negative \<equiv> Direction (\<lambda>x y. y < x) \<B>\<^sub>i\<^sub>u \<B>\<^sub>i\<^sub>l \<B>\<^sub>u \<B>\<^sub>l \<I>\<^sub>u \<I>\<^sub>l \<B>\<^sub>i\<^sub>l_update Geq Leq (\<lambda>x y. y \<le> x)"
MkElem_in_hom: fixes \<AA> :: "('a \<times> 'a) set"   and f :: "('b set, 'b \<Rightarrow> 'b) concrete_category.arr"   and x :: "'b" assumes "setcat \<AA>"   and "partial_magma.arr (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)) f"   and "x \<in> concrete_category.Dom f" shows "partial_magma.in_hom (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)) (concrete_category.MkArr {setcat.DOWN (set_category_data.unity (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)))} (concrete_category.Dom (partial_magma.dom (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)) f)) (\<lambda>_\<in>{setcat.DOWN (set_category_data.unity (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)))}. setcat.DOWN (setcat.UP x))) (set_category_data.unity (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose))) (partial_magma.dom (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)) f)"
anonymous_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "anonymous_sds agents alts sds \<equiv> social_decision_scheme agents alts sds \<and> anonymous_sds_axioms agents alts sds"
snth_szip: fixes s1 :: "'c stream"   and s2 :: "'d stream"   and n :: "nat" shows "szip s1 s2 !! n = (s1 !! n, s2 !! n)"
star_unfoldl: fixes x :: "'a" shows "(1::'a) + x ; x\<^sup>; \<le> x\<^sup>;"
append_vec_le: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec"   and v' :: "'b vec"   and w' :: "'b vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "(v @\<^sub>v v' \<le> w @\<^sub>v w') = (v \<le> w \<and> v' \<le> w')"
fast_uniform_def: fixes n :: "nat" shows "Fast_Dice_Roll.fast_uniform n = Fast_Dice_Roll.fast_dice_roll n 1 0"
prognosis_ap: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "const_on (prognosis ae as a (\<Gamma>, e, S)) (ap S) many"
intro: fixes V\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>B :: "'r \<Rightarrow> 'r"   and src\<^sub>B :: "'r \<Rightarrow> 'r"   and trg\<^sub>B :: "'r \<Rightarrow> 'r"   and V\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>C :: "'s \<Rightarrow> 's"   and src\<^sub>C :: "'s \<Rightarrow> 's"   and trg\<^sub>C :: "'s \<Rightarrow> 's"   and V\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and H\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and \<a>\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't \<Rightarrow> 't"   and \<i>\<^sub>D :: "'t \<Rightarrow> 't"   and src\<^sub>D :: "'t \<Rightarrow> 't"   and trg\<^sub>D :: "'t \<Rightarrow> 't"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and H :: "'s \<Rightarrow> 't"   and \<Phi>\<^sub>H :: "'s \<times> 's \<Rightarrow> 't"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" assumes "pseudonatural_transformation_whisker_left V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B 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 H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1"   and "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudonatural_equivalence_whisker_left V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B 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 H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1"
intro: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and \<T> :: "'b stateful_policy" assumes "wf_graph G"   and "valid_reqs M"   and "all_security_requirements_fulfilled M G"   and "wf_stateful_policy \<T>"   and "hosts \<T> = nodes G"   and "flows_fix \<T> \<subseteq> edges G"   and "all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<T>)"   and "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>)" shows "stateful_policy_compliance \<T> G M"
1: shows "(g \<star> \<epsilon>) \<cdot> (\<eta> \<star> g) = g"
continuous_real_root: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real"   and n :: "nat" assumes "continuous F f" shows "continuous F (\<lambda>x. root n (f x))"
elem_nonzeroval_gen_proper: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "v x \<noteq> 0" shows "Vr K v \<diamondsuit>\<^sub>p x \<noteq> carrier (Vr K v)"
effect_apply_eqvt': fixes p :: "perm"   and f :: "'effect"   and P :: "'state" shows "p \<bullet> \<langle>f\<rangle>P = \<langle>p \<bullet> f\<rangle>(p \<bullet> P)"
perp_cong: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and R :: "'a"   and X :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A \<noteq> P"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B P A"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B R B"   and "Cong A P B R"   and "Tarski_neutral_dimensionless.Col Bet A B X"   and "Bet P X R" shows "Cong A R P B"
lconsts_freshen: fixes a :: "nat"   and f :: "'form" shows "lconsts (freshen a f) = freshenLC a ` lconsts f"
case_prod_eta: fixes f :: "'c \<times> 'd \<Rightarrow> 'e" shows "(\<lambda>(x, y). f (x, y)) = f"
gcsort_count_input: fixes xs :: "'c list" shows "count_inv (count (mset xs)) (0, [length xs], xs)"
min_group_size: fixes G :: "'a set" assumes "G \<in> \<G>" shows "1 \<le> card G"
T_on'_append: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "Move_to_Front.T_on' A s (xs @ ys) = Move_to_Front.T_on' A s xs + Move_to_Front.T_on' A (Partial_Cost_Model.config' A s xs) ys"
edom_env: fixes S :: "'c set"   and m :: "'c \<Rightarrow> 'd" shows "edom (m f|` S) = edom m \<inter> S"
xs: shows "set xs \<subseteq> A \<times> UNIV"
topological_basis_def: fixes "open" :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "class.topological_space open" shows "topological_space.topological_basis open B = ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union> B' = x)))"
simps_5: fixes f1 :: "'g"   and f2 :: "nat \<Rightarrow> 'g"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
Union: fixes a :: "nat \<Rightarrow> 'b set"   and A :: "'b set set" assumes "\<And>i. a i \<in> sigma A" shows "\<Union> (range a) \<in> sigma A"
fmfilter_alt_defs_2: fixes A :: "'d set" shows "fmdrop_set A = fmfilter (\<lambda>a. a \<notin> A)"
eru2: shows "u\<^sub>2 \<equiv>\<^sub>E ru2"
u_sums_U: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" shows "MC_syntax.u K x y sums MC_syntax.U K x y"
degree_aform_err_min_range_mono: fixes p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and D :: "floatarith \<Rightarrow> floatarith"   and l :: "float"   and u :: "float"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "min_range_mono p F D l u X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
of_bl_drop: fixes n :: "nat"   and xs :: "bool list" shows "of_bl (drop n xs) = of_bl xs && mask (length xs - n)"
total_lexord: fixes r :: "('a \<times> 'a) set" assumes "total r" shows "total (lexord r)"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "vc (x ::= a) Q = True"
iIN_iTILL_subset_conv: fixes n' :: "nat"   and d' :: "nat"   and n :: "nat" shows "([n'\<dots>,d'] \<subseteq> [\<dots>n]) = (n' + d' \<le> n)"
simps_1: fixes \<zeta> :: "('a, 'b) hd" shows "hsize (Hd \<zeta>) = 1"
rewriteL_comp_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and l :: "'b \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'b" assumes "f \<circ> g = l" shows "f \<circ> (g \<circ> h) = l \<circ> h"
wf_eq_minimal: fixes r :: "('a \<times> 'a) set" shows "wf r = (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q))"
U_3: fixes i :: "nat"   and x :: "'a" shows "U i x < top"
SReal_hrabs: fixes x :: "real star" assumes "x \<in> \<real>" shows "\<bar>x\<bar> \<in> \<real>"
mIH_2: fixes ni2 :: "nat"   and n2 :: "nat ifex" assumes "Rmi_g ni2 n2 bdd"   and "snd (snd (pm_pth (dpm bdd) n)) \<noteq> ni2" shows "e \<noteq> n2"
tr_s_var_assignI: fixes var :: "char list"   and T :: "x \<Rightarrow> x option"   and var' :: "char list"   and x :: "x"   and x' :: "x" assumes "(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'"   and "(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'" shows "tr_s T (s_ass var x) (s_ass var' x')"
ww: shows "w \<in> fst ` C \<inter> pwalk_verts v"
ts_ord_anti_iff: fixes x :: "'a"   and y :: "'a" shows "(!x \<sqsubseteq> !y) = (t y \<sqsubseteq> t x)"
has_integral_vec1_I_cbox: fixes f :: "(real, 1) vec \<Rightarrow> 'c"   and y :: "'c"   and a :: "(real, 1) vec"   and b :: "(real, 1) vec" assumes "(f has_integral y) (cbox a b)" shows "(f \<circ> vec has_integral y) {a $ 1..b $ 1}"
cases: fixes x :: "'b trm" obtains r :: "'b rexp" where "x = Lam r__"   | X :: "'b list set" and r :: "'b rexp" where "x = Trn X r"
matrix_eq: fixes A :: "(('d, 'e) vec, 'f) vec"   and B :: "(('d, 'e) vec, 'f) vec" shows "(A = B) = (\<forall>x. A *v x = B *v x)"
w_strictly_prefers_def: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and w :: "'b"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw" shows "StableMarriage.w_strictly_prefers Pw w \<mu> = {m' \<in> Field (Pw w). \<forall>m\<in>{m. (m, w) \<in> \<mu>}. (m, m') \<in> Pw w \<and> (m', m) \<notin> Pw w}"
det_row_span: fixes x :: "('a, 'b) vec"   and A :: "(('a, 'b) vec, 'b) vec"   and i :: "'b" assumes "x \<in> vec.span {row j A |j. j \<noteq> i}" shows "det (\<chi>k. if k = i then row i A + x else row k A) = det A"
ipSuc: fixes I :: "'d set"   and R :: "('d, 'e) Ring_scheme"   and n :: "nat" shows "I \<^bsup>\<diamondsuit>R Suc n\<^esup> = I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> I \<^bsup>\<diamondsuit>R n\<^esup>"
sWlsBinp_def: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and delta :: "'opSym"   and sbinp :: "'bindex \<Rightarrow> ('varSort, 'f) sAbs option" shows "sWlsBinp SEM delta sbinp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) sbinp \<and> liftAll2 (sWlsAbs SEM) (barOf delta) sbinp"
le_less: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "ko.le ko x y = (ko.lt ko x y \<or> x = y)"
ws_widen_trans: fixes G :: "prog"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>U"   and "G\<turnstile>U\<preceq>T"   and "ws_prog G" shows "G\<turnstile>S\<preceq>T"
cs_true: shows "INTERP N \<Turnstile>m mset CAs"
assms_8: shows "negligible (\<lambda>\<eta>. 1 / real (card (prf_domain \<eta>)))"
pos_decomp_concat: fixes pss :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list list" shows "(concat pss)\<^sub>+ = concat (map pos_decomp pss)"
rel_set_image: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a set"   and B :: "'b set"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "rel_set P A B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B; P a b\<rbrakk> \<Longrightarrow> Q (f a) (g b)" shows "rel_set Q (f ` A) (g ` B)"
not_p2: fixes t :: "real" assumes "0 < t"   and "t \<le> return_time p1 x0" shows "flow0 x0 t \<notin> p2"
code: fixes t :: "'b tree" shows "mirror t = Node (root t) (mirror (right t)) (mirror (left t))"
erlang_CDF_at0: fixes k :: "nat"   and l :: "real" shows "erlang_CDF k l 0 = 0"
slice_Union: fixes S :: "'c set set"   and y :: "'b" shows "slice (\<Union> S) y = (\<Union>x\<in>S. slice x y)"
take_bit_set_bit_eq: fixes n :: "nat"   and m :: "nat"   and a :: "'a" shows "take_bit n (set_bit m a) = (if n \<le> m then take_bit n a else set_bit m (take_bit n a))"
portfolio_def: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "portfolio p = finite (support_set p)"
normalize_match_preserves_disc_negated: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and neg :: "bool" assumes "\<exists>m_DNF\<in>set (normalize_match m). has_disc_negated disc neg m_DNF" shows "has_disc_negated disc neg m"
scalar_times_pol_expr: fixes a :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat" assumes "a \<in> carrier S"   and "pol_coeff S c"   and "n \<le> fst c" shows "a \<cdot>\<^sub>r polyn_expr R X n c = polyn_expr R X n (sp_cf S a c)"
append_assoc: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
in_Fvars_substD: fixes y :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "y \<in> Fvars (subst \<phi> t x)"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "y \<in> Fvars \<phi> - {x} \<union> (if x \<in> Fvars \<phi> then FvarsT t else {})"
Tree\<^sub>\<alpha>_rep_abs: fixes t :: "('a, 'b, 'c, 'd) Tree" shows "rep_Tree\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> t) =\<^sub>\<alpha> t"
Wax_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.Wax wtFsym resOf prot protFw protCl \<equiv> {ProblemIkTpartG.wax \<sigma> |\<sigma>. \<not> ProblemIkTpart.unprot prot protFw \<sigma> \<and> (\<not> ProblemIkTpart.isRes wtFsym resOf \<sigma> \<or> protCl \<sigma>)}"
inject_1: fixes x11 :: "nat"   and x12 :: "nat"   and y11 :: "nat"   and y12 :: "nat" shows "(State x11 x12 = State y11 y12) = (x11 = y11 \<and> x12 = y12)"
bres_top_top: shows "\<top> \<rightarrow> \<top> = \<top>"
continuous_on_poly: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and p :: "'b poly" assumes "continuous_on A f" shows "continuous_on A (\<lambda>x. poly p (f x))"
un_B_inject: fixes x :: "'c dual"   and y :: "'c dual" shows "(un_B x = un_B y) = (x = y)"
hom_char: fixes a :: "'a"   and b :: "'a" shows "Left.hom a b = hom a b \<inter> Collect (left (trg \<mu>))"
run_ran_aux: fixes succi :: "'c \<Rightarrow> 'c list"   and E :: "('d \<times> 'd) set"   and Rv :: "('c \<times> 'd) set"   and src :: "'d"   and v :: "'d"   and v0i :: "'c" assumes "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"   and "(src, v) \<in> E\<^sup>*"   and "(v0i, src) \<in> Rv" shows "v \<in> Range Rv"
rank_less_imp_inconsistent: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "rank A < (if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b)) + 1 else 0)" shows "inconsistent A b"
fds_deriv_zeta: shows "fds_deriv fds_zeta = - fds mangoldt * fds_zeta"
open_unresolvable_def: fixes X :: "'b set" shows "open_unresolvable X = (K \<circ> I \<circ> K = K \<circ> I)"
power_divide: 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 a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.field divide inverse times one plus zero minus uminus" shows "power.power one times (divide a b) n = divide (power.power one times a n) (power.power one times b n)"
know2knowsPQ_s: fixes P :: "specID"   and m :: "Secrets"   and Q :: "specID" assumes "know P (sKS m) \<or> know Q (sKS m)" shows "knows P [sE m] \<or> knows Q [sE m]"
intro: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and S :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah" assumes "category C"   and "replete_set_category S"   and "functor C S F" shows "set_valued_functor C S F"
left: fixes i :: "'a" assumes "i \<in> I"   and "a < i"   and "i \<le> b" shows "(f \<longlongrightarrow> l i) (at_left i)"
intro: fixes A :: "('h, 'i, 'j) sa_rec_scheme"   and B :: "('k, 'i, 'l) sa_rec_scheme"   and R :: "('h \<times> 'k) set" assumes "sa A"   and "sa B"   and "lsimulation R A B"   and "lsimulation (R\<inverse>) B A"   and "bisimulation R A B" shows "lbisimulation R A B"
truncate_up_uminus_eq: fixes prec :: "nat"   and x :: "real" shows "truncate_up prec (- x) = - truncate_down prec x"
poly_uminus_pCons_pCons: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "poly_uminus (pCons a (pCons b p)) = pCons a (pCons (- b) (poly_uminus p))"
empty_dom_iff_empty_subst: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "(subst_domain \<theta> = {}) = (\<theta> = Var)"
cr_scene_def: shows "cr_scene \<equiv> \<lambda>x y. x = Rep_scene y"
Units_m_closed: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "Group.monoid G"   and "x \<in> Units G"   and "y \<in> Units G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> Units G"
mult_le_cancel_iff2: fixes z :: "'b"   and x :: "'b"   and y :: "'b" assumes "(0::'b) < z" shows "(z * x \<le> z * y) = (x \<le> y)"
t_Lam: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and T1 :: "ty"   and t :: "lam"   and T2 :: "ty" assumes "x \<sharp> \<Gamma>"   and "(x, T1) # \<Gamma> \<turnstile> t : T2" shows "\<Gamma> \<turnstile> Lam [x].t : T1 \<rightarrow> T2"
dotprod_code: fixes xs :: "nat list"   and ys :: "nat list" shows "xs \<bullet> ys = sum_list (map2 (*) xs ys)"
uniformity_p_point_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
divisors_int_2: fixes x :: "int" shows "distinct (divisors_int x)"
rel_pred_neg_distr_imp: fixes B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f \<Rightarrow> 'd \<Rightarrow> bool" assumes "right_unique B \<and> left_total B \<or> left_unique A \<and> right_total A" shows "rel_pred_neg_distr_cond A B"
Myhill_Nerode2: fixes r :: "'b rexp" shows "finite (UNIV // \<approx>lang r)"
sign_changes_induct: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>xs. P xs \<Longrightarrow> P ((0::'a) # xs)"   and "\<And>x y xs. \<lbrakk>x \<noteq> (0::'a); P ((x + y) # xs)\<rbrakk> \<Longrightarrow> P (x # y # xs)" shows "P xs"
subcls1_def2: fixes P :: "'a prog" shows "subcls1 P = (SIGMA C:{C. is_class P C}. {D. C \<noteq> Object \<and> fst (the (class P C)) = D})"
wf_subcls1: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "wf_prog wf_md P" shows "wf ((subcls1 P)\<inverse>)"
path_connected_interval: fixes a :: "'b"   and b :: "'b" shows "path_connected {a..b}"
abs_le_square_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs x) (abs y) = less_eq (power.power one times x 2) (power.power one times y 2)"
subst_bvs_App: fixes args :: "Core.term list"   and s :: "Core.term"   and t :: "Core.term" shows "subst_bvs args (s $ t) = subst_bvs args s $ subst_bvs args t"
ext_special_cf_lo_zero: fixes n :: "nat"   and x :: "nat" assumes "0 < n"   and "x \<le> n - Suc 0" shows "snd (ext_cf S n (C\<^sub>0 S)) x = \<zero>\<^bsub>S\<^esub>"
antisymp_on_reflclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "antisymp_on P\<^sup>=\<^sup>= A = antisymp_on P A"
x0_hb_bound4: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "bs ! i / 2 < bs ! i + (hs ! i) x / x"
\<gamma>: fixes \<D> :: "('b \<times> 'b set) set"   and n :: "nat" assumes "\<D> tagged_division_of cbox a b"   and "\<gamma> n fine \<D>" shows "\<bar>\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R indicat_real S x\<bar> < e / 2 / ((real n + 1) * 2 ^ n)"
Extend_mem: fixes S :: "(('b, 'c) fm list \<times> 'c) set"   and A :: "'c set"   and f :: "nat \<Rightarrow> ('b, 'c) fm list \<times> 'c" shows "S \<subseteq> Extend A S f"
simps_3: fixes i :: "nat"   and j :: "nat"   and Q :: "pol" shows "isnorm (Pinj i (Pinj j Q)) = False"
leD: fixes y :: "'a list"   and x :: "'a list" assumes "prefix y x" shows "\<not> strict_prefix x y"
pos_imp_rmv_pos: fixes a :: "real"   and xs :: "real list" assumes "remove1 a xs \<noteq> []"   and "pos xs" shows "pos (remove1 a xs)"
step': fixes s :: "'a" assumes "guard s" shows "p' \<le> Sigma_Algebra.measure (measure_spmf (body s)) {s'. f s' < f s}"
eval_not: fixes a :: "atom fm"   and L :: "real list" shows "eval (neg a) L = (\<not> eval a L)"
simps: fixes Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list"   and w :: "nat" shows "Maxr Rr xs w = (let setx = {y. y \<le> w \<and> Rr (xs @ [y])} in if setx = {} then 0 else Max setx)"