hcju/isabelleps huggingface dataset:
meet_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter> y \<in> carrier L"
semiring_simprules_9: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "semiring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
cnj: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla" shows "cnj \<phi>1 \<phi>2 \<in> fmla"
bind_def: fixes m :: "'b list \<Rightarrow> char list + 'c \<times> 'b list"   and f :: "'c \<Rightarrow> 'b list \<Rightarrow> char list + 'a \<times> 'b list"   and ts :: "'b list" shows "(m \<bind> f) ts = m ts \<bind> (\<lambda>(x, ts'). f x ts')"
lookup_env_restr: fixes x :: "'c"   and S :: "'c set"   and m :: "'c \<Rightarrow> 'd" assumes "x \<in> S" shows "(m f|` S) x = m x"
unionIntersectionEmpty: fixes X :: "'c set set"   and Y :: "'c set set" shows "(\<forall>x\<in>X. \<forall>y\<in>Y. x \<inter> y = {}) = (\<Union> X \<inter> \<Union> Y = {})"
stepsI: fixes x :: "'a"   and xs :: "'a list" assumes "x \<rightarrow> hd xs"   and "local.steps xs" shows "local.steps (x # xs)"
arr_cod: fixes f :: "'a" assumes "A.arr f" shows "A.arr (A.cod f)"
P_mod_diffxxx4: fixes t :: "'a"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and f :: "'a"   and gh :: "'a \<times> 'a"   and m :: "nat" assumes "Idomain S"   and "t \<in> carrier S"   and "t \<noteq> \<zero>\<^bsub>S\<^esub>"   and "maximal_ideal S (S \<diamondsuit>\<^sub>p t)"   and "PolynRg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y"   and "f \<in> carrier R"   and "gh \<in> carrier R \<times> carrier R"   and "deg R S X (fst gh) \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh))"   and "deg R S X (snd gh) + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh)) \<le> deg R S X f"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh))"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd gh))"   and "rel_prime_pols R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh)) (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd gh))"   and "P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (f \<plusminus> -\<^sub>a fst gh \<cdot>\<^sub>r snd gh)"   and "0 < m" shows "Hen\<^bsub> R S X t R' Y f\<^esub> m gh \<in> carrier R \<times> carrier R \<and> deg R S X (fst (Hen\<^bsub> R S X t R' Y f\<^esub> m gh)) \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (Hen\<^bsub> R S X t R' Y f\<^esub> m gh))) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (fst gh \<plusminus> -\<^sub>a fst (Hen\<^bsub> R S X t R' Y f\<^esub> m gh)) \<and> deg R S X (snd (Hen\<^bsub> R S X t R' Y f\<^esub> m gh)) + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst (Hen\<^bsub> R S X t R' Y f\<^esub> m gh))) \<le> deg R S X f \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (snd gh \<plusminus> -\<^sub>a snd (Hen\<^bsub> R S X t R' Y f\<^esub> m gh)) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) (f \<plusminus> -\<^sub>a fst (Hen\<^bsub> R S X t R' Y f\<^esub> m gh) \<cdot>\<^sub>r snd (Hen\<^bsub> R S X t R' Y f\<^esub> m gh))"
split_spec_def: fixes X :: "'e set" shows "split_spec X = SPEC (\<lambda>(A, B). X \<subseteq> A \<union> B)"
cobounded1: fixes a :: "'k multiset"   and b :: "'k multiset" shows "a \<inter># b \<subseteq># a"
map_of_from_function_graph_is_some_if: fixes xs :: "'c list"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "set xs \<noteq> {}"   and "x \<in> set xs" shows "map_of (map (\<lambda>x. (x, f x)) xs) x = Some (f x)"
is_run_def: fixes G :: "('h, 'i) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'h" shows "graph_defs.is_run G r \<equiv> r 0 \<in> g_V0 G \<and> ipath (g_E G) r"
b_least2_less_impl_eq2: fixes z :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" assumes "z < y"   and "f x z \<noteq> 0" shows "b_least2 f x y = (LEAST z. f x z \<noteq> 0)"
elims_2: fixes x :: "('b, 'c) Combinators list"   and xa :: "'b set list" assumes "all_in_list x xa" obtains l :: "'b set list" where "x = []" and "xa = l__"   | xb :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" and l :: "'b set list" where "x = xb # xs" and "xa = l" and "in_list xb l \<and> all_in_list xs l"
indep_not_basis: fixes X :: "'a set" assumes "indep_in carrier X"   and "\<not> \<E>.basis X" shows "\<exists>x\<in>carrier - X. indep_in carrier (insert x X)"
greaterThanAtMost_iff: fixes i :: "'a"   and l :: "'a"   and u :: "'a" shows "(i \<in> {l<..u}) = (l < i \<and> i \<le> u)"
insert_3: assumes "\<And>x. x \<in> S \<Longrightarrow> set (coeffs (f x)) \<subseteq> \<real>" shows "set (coeffs (prod f S)) \<subseteq> \<real>"
minusinfinity: fixes d :: "int"   and P1 :: "int \<Rightarrow> bool"   and P :: "int \<Rightarrow> bool" assumes "0 < d"   and "\<forall>x k. P1 x = P1 (x - k * d)"   and "\<exists>z. \<forall>x<z. P x = P1 x" shows "(\<exists>x. P1 x) \<longrightarrow> (\<exists>x. P x)"
map_prod_def: fixes f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
permutations_of_multisetD: fixes xs :: "'a list"   and A :: "'a multiset" assumes "xs \<in> permutations_of_multiset A" shows "mset xs = A"
inv_one: shows "inv\<^bsub>DirProds G I\<^esub> \<one>\<^bsub>DirProds G I\<^esub> = \<one>\<^bsub>DirProds G I\<^esub>"
in_set_left: fixes y :: "'c"   and l :: "'c tree"   and x :: "'c"   and r :: "'c tree" assumes "y \<in> set_of l" shows "y \<in> set_of (Node l x False r)"
assms_1: shows "p1 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<le> c}"
synthetic_div_correct: fixes p :: "'b poly"   and c :: "'b" shows "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
nipd_2: shows "\<forall>n'\<in>{k'..<l'}. \<pi>' n' \<noteq> ipd (\<pi>' k')"
jordan_block_def: fixes n :: "nat"   and a :: "'c" shows "jordan_block n a = mat n n (\<lambda>(i, j). if i = j then a else if Suc i = j then 1::'c else (0::'c))"
prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "ind is_square (p ^ k) = (if even k then 1::'b else (0::'b))"
comparables_def: fixes S :: "nat set"   and M :: "nat set" shows "comparables S M \<equiv> {T. finite T \<and> (init_segment T S \<or> init_segment S T \<and> T - S \<subseteq> M)}"
gauge_Int: fixes \<gamma>1 :: "'c \<Rightarrow> 'c set"   and \<gamma>2 :: "'c \<Rightarrow> 'c set" assumes "gauge \<gamma>1"   and "gauge \<gamma>2" shows "gauge (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x)"
keys_except: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" shows "keys (except p S) = keys p - S"
zero_divisor_mult_right: fixes b :: "'b"   and a :: "'b" assumes "zero_divisor b" shows "zero_divisor (a * b)"
not_invar2_ctd_init: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" assumes "prim_invar2_init Q \<pi>" shows "\<not> prim_invar2_ctd Q \<pi>"
scale_one: fixes x :: "'b" shows "(1::'a) *s x = x"
HPhase1or2ReadThen_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Inv2c s"   and "HPhase1or2ReadThen s s' p d q"   and "Inv2a s" shows "Inv2c s'"
add_diff_inverse_nat: fixes m :: "nat"   and n :: "nat" assumes "\<not> m < n" shows "n + (m - n) = m"
find_indices_def: fixes x :: "'e"   and xs :: "'e list" shows "find_indices x xs \<equiv> filter (\<lambda>i. xs ! i = x) [0..<length xs]"
assms_1: shows "is_parallel_solution_for_problem \<Pi> \<pi>"
a_def: shows "a = lead_coeff g ^ (Suc (degree f) - degree g)"
GF_suffix: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'a set"   and \<psi> :: "'a ltln" shows "suffix i w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)"
phiArg_path_ex': fixes g :: "'g"   and p :: "'val"   and q :: "'val" assumes "phiArg g p q"   and "p \<in> allVars g" obtains ns :: "'node list" and m :: "'node" where "g \<turnstile> defNode g q-ns\<rightarrow>m" and "old.EntryPath g ns" and "q \<in> phiUses g m" and "m \<in> set (old.predecessors g (defNode g p))"
assms_4: shows "\<guillemotleft>g : local.prod a b \<rightarrow> c\<guillemotright>"
SourcesS1_L2: shows "Sources level2 sS1 = {}"
atLeastAtMost_diff_ends: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b - {a, b} = ord.greaterThanLessThan less a b"
borel_measurable_cld: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "cld_measure M"   and "\<And>A a b. \<lbrakk>A \<in> sets M; 0 < emeasure M A; emeasure M A < \<infinity>; a < b\<rbrakk> \<Longrightarrow> min (outer_measure_of M {x \<in> A. f x \<le> a}) (outer_measure_of M {x \<in> A. b \<le> f x}) < emeasure M A" shows "f \<in> borel_measurable M"
elims: fixes x :: "int \<times> (real \<Rightarrow> real \<times> real)"   and y :: "real \<Rightarrow> real \<times> real" assumes "coeff_cube_to_path x = y" obtains k :: "int" and \<gamma> :: "real \<Rightarrow> real \<times> real" where "x = (k, \<gamma>)" and "y = (if k = 1 then \<gamma> else reversepath \<gamma>)"
approx_\<beta>_closure_\<alpha>: fixes Z :: "('c \<Rightarrow> real) set"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "Z \<subseteq> V"   and "Z = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n" shows "Approx\<^sub>\<beta> Z \<subseteq> Closure\<^sub>\<alpha> Z"
div_smult_left: fixes a :: "'a"   and x :: "'a poly"   and y :: "'a poly" shows "Polynomial.smult a x div y = Polynomial.smult a (x div y)"
equivpI: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "reflp R"   and "symp R"   and "transp R" shows "equivp R"
eq_mrexp3_imp_eq: fixes r :: "'c mrexp3"   and s :: "'c mrexp3" assumes "eq_mrexp3 r s"   and "ok3 r"   and "ok3 s" shows "r = s"
3: shows "rat_poly_times B (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) B) = B - B ^ 5"
degenerate_word: fixes x :: "'a word" assumes "LENGTH('a) = 1" shows "x = 0 \<or> x = 1"
swap_comp_involutory: fixes a :: "'b"   and b :: "'b" shows "Fun.swap a b \<circ> Fun.swap a b = id"
induct: fixes x1 :: "com \<times> (char list \<Rightarrow> int)"   and x2 :: "char list \<Rightarrow> int"   and P :: "com \<times> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "x1 \<Rightarrow> x2"   and "\<And>s. P (SKIP, s) s"   and "\<And>x a s. P (x ::= a, s) (s(x := aval a s))"   and "\<And>c\<^sub>1 s\<^sub>1 s\<^sub>2 c\<^sub>2 s\<^sub>3. \<lbrakk>(c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2; P (c\<^sub>1, s\<^sub>1) s\<^sub>2; (c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3; P (c\<^sub>2, s\<^sub>2) s\<^sub>3\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2, s\<^sub>1) s\<^sub>3"   and "\<And>b s c\<^sub>1 t c\<^sub>2. \<lbrakk>bval b s; (c\<^sub>1, s) \<Rightarrow> t; P (c\<^sub>1, s) t\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) t"   and "\<And>b s c\<^sub>2 t c\<^sub>1. \<lbrakk>\<not> bval b s; (c\<^sub>2, s) \<Rightarrow> t; P (c\<^sub>2, s) t\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) t"   and "\<And>b s c. \<not> bval b s \<Longrightarrow> P (WHILE b DO c, s) s"   and "\<And>b s\<^sub>1 c s\<^sub>2 s\<^sub>3. \<lbrakk>bval b s\<^sub>1; (c, s\<^sub>1) \<Rightarrow> s\<^sub>2; P (c, s\<^sub>1) s\<^sub>2; (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3; P (WHILE b DO c, s\<^sub>2) s\<^sub>3\<rbrakk> \<Longrightarrow> P (WHILE b DO c, s\<^sub>1) s\<^sub>3" shows "P x1 x2"
simps_9: fixes P1 :: "pol"   and x :: "nat"   and P2 :: "pol"   and Q1 :: "pol"   and y :: "nat"   and Q2 :: "pol" shows "PX P1 x P2 \<langle>*\<rangle> PX Q1 y Q2 = mkPX (P1 \<langle>*\<rangle> Q1) (x + y) (P2 \<langle>*\<rangle> Q2) \<langle>+\<rangle> (mkPX (P1 \<langle>*\<rangle> mkPinj 1 Q2) x (Pc 0) \<langle>+\<rangle> mkPX (Q1 \<langle>*\<rangle> mkPinj 1 P2) y (Pc 0))"
higher_differentiable_on_sum: fixes F :: "'b set"   and S :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and n :: "nat" assumes "\<And>i. \<lbrakk>i \<in> F; finite F\<rbrakk> \<Longrightarrow> higher_differentiable_on S (f i) n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. \<Sum>i\<in>F. f i x) n"
that: fixes i :: "'b" assumes "finite I"   and "i \<in> I"   and "f i = (0::'a)" shows "thesis"
eq_iff_5: fixes "term" :: "Syntax.term"   and terma :: "Syntax.term" shows "(Inj1 term = Inj1 terma) = (term = terma)"
select_convs_1: fixes om :: "'b \<Rightarrow> 'c"   and am :: "'d \<Rightarrow> 'e"   and more :: "'f" shows "\<lparr>om = om, am = am, \<dots> = more\<rparr> \<^bsub>\<o>\<^esub> = om"
hyps: shows "edges_of_walk (xs @ x # xs') = edges_of_walk (xs @ [x]) \<union> edges_of_walk (x # xs')"
bot_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "\<bottom> \<equiv> Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e None"
costBIT_1y: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a" assumes "x \<noteq> y"   and "x \<in> {x0, y0}"   and "y \<in> {x0, y0}" shows "E (type1 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s y \<bind> (\<lambda>(a, is'). return_pmf (real (t\<^sub>p (fst s) y a))))) = 3 / 4"
Right_Coset_def: fixes H :: "'a set"   and x :: "'b" shows "H |\<cdot> x = {h \<cdot> x |h. h \<in> H}"
simps_71: fixes x1 :: "const" shows "LID\<^sub>V (CV x1) = {}"
list_update_id: fixes xs :: "'g list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
iso_runit: fixes f :: "'a" assumes "ide f" shows "local.iso (PB.runit f)"
hoare_complete': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
conv2: shows "\<I>2,\<I>2' \<turnstile>\<^sub>C conv2 \<sim> map_converter id id f2 g2 1\<^sub>C"
fps_of_poly_1: shows "fps_of_poly 1 = 1"
bit_or_iff: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "bit (a OR b) n = (bit a n \<or> bit b n)"
mod_div_relation: fixes a :: "nat"   and b :: "nat"   and c :: "nat" shows "a mod (b * c) div c = a div c mod b"
postrecording_event: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" shows "postrecording_event t i \<equiv> i < length t \<and> regular_event (t ! i) \<and> ps (S t i) (occurs_on (t ! i)) \<noteq> None"
f: shows "integrable (Pi\<^sub>M (insert i I) M) f"
Cd_range': fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Pds'.Cd d X" shows "x \<in> X \<inter> Field (Pd_shuffle_to_top ds' Y d)"
prems_1: shows "history_consistent \<theta> (hd_prog p ((r # sb\<^sub>1) @ sb')) ((r # sb\<^sub>1) @ sb')"
AB: shows "\<mu>.dom.leg0 \<cdot> doms.Prj\<^sub>1 = \<nu>.dom.leg1 \<cdot> doms.Prj\<^sub>1\<^sub>0"
wp_modularity_rule: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and Q' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT \<pi> P c Q"   and "P s"   and "\<And>s'. Q s s' \<Longrightarrow> Q' s'" shows "wp \<pi> c Q' s"
elims_2: fixes x :: "'b trm" assumes "is_a_constant x" obtains xa :: "'b" where "x = Const xa"
not_cong_2134: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Cong A B C D" shows "\<not> Cong B A C D"
fls_plus_subdegree: fixes f :: "'a fls"   and g :: "'a fls" assumes "f + g \<noteq> 0" shows "min (fls_subdegree f) (fls_subdegree g) \<le> fls_subdegree (f + g)"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_6: fixes c :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (\<langle>c: t \<in> s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
seq_inf_par: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "less_eq (seq a b) (times a b)"
ordered_fmap_sound: fixes S :: "('a \<times> 'b) fset"   and a :: "'a"   and b :: "'b" assumes "is_fmap S"   and "(a, b) \<in> set (ordered_fmap S)" shows "(a, b) |\<in>| S"
A_2: shows "normalize (\<Prod>\<^sub># A) = normalize x"
suffix_ConsI: fixes xs :: "'c list"   and ys :: "'c list"   and y :: "'c" assumes "suffix xs ys" shows "suffix xs (y # ys)"
has_integral_componentwise_iff: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and A :: "'b set" shows "(f has_integral y) A = (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y \<bullet> b) A)"
that: fixes C :: "'a set" assumes "comm_monoid_add_ow C pls zero"   and "f ` S \<subseteq> C"   and "Domainp (rel_set A) C" shows "thesis"
2: shows "Right_a.ide (f \<star> a) \<and> Right_a.ide (g \<star> a)"
unit_simps_1: fixes a :: "'c" assumes "C.obj a" shows "D.arr (G.unit a)"
infinite_set_less_Complement: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "finite B"   and "infinite UNIV" shows "ord.set_less less_eq A (- B)"
compl_structD1: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and d :: "'a \<Rightarrow> nat"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "compl_struct compl"   and "dickson_grading d"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps" shows "dgrad_p_set_le d (fst ` set (fst (compl gs bs (ps -- sps) sps data))) (args_to_set (gs, bs, ps))"
simps_1: fixes uu :: "String.literal \<Rightarrow> 'a"   and uv :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and uw :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool" shows "semantics uu uv uw Truth = True"
Bseq_add: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "Bseq f" shows "Bseq (\<lambda>x. f x + c)"
cupcake_list_correct_rval: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and vs :: "v list"   and s :: "'b state" assumes "list_all2_shortcircuit (\<lambda>e r. cupcake_evaluate_single env e r \<and> (is_cupcake_all_env env \<longrightarrow> is_cupcake_exp e \<longrightarrow> (\<forall>s. \<exists>s'. Big_Step_Unclocked_Single.evaluate env s e (s', r)))) es rs"   and "is_cupcake_all_env env"   and "list_all is_cupcake_exp es"   and "sequence_result rs = Rval vs" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s es (s', Rval vs)"
select_correct_factor_int_poly_def: fixes bnd_update :: "'a \<Rightarrow> 'a"   and bnd_get :: "'a \<Rightarrow> rat \<times> rat"   and init :: "'a"   and p :: "int poly" shows "select_correct_factor_int_poly bnd_update bnd_get init p \<equiv> let qs = factors_of_int_poly p; polys = map (\<lambda>q. (q, root_info q)) qs; ((q, ri), l, r) = select_correct_factor bnd_update bnd_get init polys in real_alg_2' ri q l r"
ord_equiv_trans: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and F :: "('g, 'h) Order_scheme" assumes "Order D"   and "Order E"   and "Order F"   and "ord_equiv D E"   and "ord_equiv E F" shows "ord_equiv D F"
map2_map_map: fixes h :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'b"   and xs :: "'d list"   and g :: "'d \<Rightarrow> 'c" shows "map2 h (map f xs) (map g xs) = map (\<lambda>x. h (f x) (g x)) xs"
sd_1r_eq: shows "(safe_distance_1r < s\<^sub>o - s\<^sub>e) = (u_max < s\<^sub>o)"
induct: fixes P :: "(event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> event_data option list set list \<times> event_data option list set list \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a1 :: "'a"   and a2 :: "event_data option list set list \<times> event_data option list set list"   and a3 :: "nat list" assumes "\<And>f z x xs y ys t ts. P f (f x y t z) (xs, ys) ts \<Longrightarrow> P f z (x # xs, y # ys) (t # ts)"   and "\<And>f z ys ts. P f z ([], ys) ts"   and "\<And>f z xs ts. P f z (xs, []) ts"   and "\<And>f z xs ys. P f z (xs, ys) []" shows "P a0 a1 a2 a3"
IP: assumes "c \<sigma>"   and "k \<in> dom m - (dom m - it)"   and "m k = Some v"   and "dom m - it \<subseteq> dom m"   and "I (dom m - it) \<sigma>"   and "\<forall>k' v'. k' \<in> dom m - (dom m - it) - {k} \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v')"   and "\<forall>k' v'. k' \<in> dom m - it \<and> m k' = Some v' \<longrightarrow> R (k', v') (k, v)" shows "I (insert k (dom m - it)) (f (k, v) \<sigma>)"
antecedent: shows "\<forall>y. O y z = (\<exists>x. F x \<and> O y x)"
cross_alt_clear_n: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" assumes "coprime p q" shows "cross_alt p q a b = cross_alt 1 (p * q) a b"
quotient_disj: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and X :: "'c set"   and Y :: "'c set" assumes "equiv A r"   and "X \<in> A // r"   and "Y \<in> A // r" shows "X = Y \<or> X \<inter> Y = {}"