hcju/isabelleps huggingface dataset:
check_eqv_sound: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" assumes "dual.check_eqvRE \<Sigma> n r s"   and "M2L_Equivalence_Checking.wf \<Sigma> n r"   and "M2L_Equivalence_Checking.wf \<Sigma> n s" shows "lang \<Sigma> n r = lang \<Sigma> n s"
F_to_idx_def: fixes F :: "'b set set" shows "F_to_idx F \<equiv> SPEC (\<lambda>Flist. distinct Flist \<and> set Flist = F) \<bind> (\<lambda>Flist. let num_acc = length Flist; acc = \<lambda>v. {i. i < num_acc \<and> v \<in> Flist ! i} in RETURN (num_acc, acc))"
hd_cs: shows "hd cs \<noteq> (0::'a)"
in_eqvt: fixes pi :: "(name \<times> name) list"   and x :: "'a"   and X :: "'a set" shows "pi \<bullet> (x \<in> X) = (pi \<bullet> x \<in> pi \<bullet> X)"
P: shows "pre @ [Broadcast (r, Store e i r)] prefix of k"
tauStepChainStatEq: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>' \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"
t_times_pow_pos: fixes n :: "nat" shows "0 < eint (2 ^ n) * t"
OstrowskiTr9: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 < v x" shows "0 < v (x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a x))\<^bsup>\<hyphen> K\<^esup>)"
diff_eq: fixes x :: "'b"   and y :: "'b" shows "x - y = x \<sqinter> - y"
invh_baldL_Black: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invh l"   and "invh r"   and "bheight l + 1 = bheight r"   and "color r = Black" shows "invh (baldL l a r) \<and> bheight (baldL l a r) = bheight r"
sum_SDG_edge_valid_SDG_node_2: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and b :: "bool"   and n' :: "'node SDG_node" assumes "sum_SDG_edge n Vopt popt b n'" shows "valid_SDG_node n'"
hyps_4: fixes \<Gamma> :: "binding list" assumes "TVarB X T1 # \<Gamma>'__ @ VarB x Q # \<Delta> = \<Gamma> @ VarB x Q # \<Delta>" shows "(\<Gamma> @ \<Delta>)\<turnstile>S2<:T2"
distrib_2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<sqinter> z \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
g: fixes x :: "'v" shows "(0::'a) \<le> g x"
simps_1: fixes s :: "'e" shows "last_state (s, []) = s"
nths_def: fixes xs :: "'m list"   and A :: "nat set" shows "nths xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<length xs]))"
power_abs: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times a n) = power.power one times (abs a) n"
gram_schmidt_code: fixes n :: "nat"   and ws :: "'a vec list" shows "gram_schmidt n ws = gram_schmidt_sub2 n [] ws"
b_init: fixes s :: "'a"   and T :: "('a, 'b) TS_scheme" assumes "s \<in> init T" shows "behp T [s]"
IS_TO_SORTED_LIST_def: fixes \<Omega> :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rs :: "'d set \<Rightarrow> 'e \<Rightarrow> assn"   and Rk :: "'d \<Rightarrow> 'f \<Rightarrow> assn"   and tsl :: "'e \<Rightarrow> 'f list Heap" shows "IS_TO_SORTED_LIST \<Omega> Rs Rk tsl \<equiv> (tsl, it_to_sorted_list \<Omega>) \<in> Rs\<^sup>k \<rightarrow>\<^sub>a list_assn Rk"
is_CSP_processE2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process P" obtains "CSP1 P = P" and "CSP2 P = P" and "R P = P"
deep_model_correct_params_y_def: fixes rs :: "nat list"   and y :: "nat" shows "deep_model_correct_params_y rs y \<equiv> deep_model_correct_params rs \<and> deep_model_correct_params_y_axioms rs y"
preserves_cod: fixes f :: "'a" assumes "C.arr f" shows "D.cod (G f) = G (C.cod f)"
ts_ord_refl: fixes x :: "'a" shows "n x \<sqsubseteq> n x"
integral_cnj: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> complex" shows "CLINT x|M. cnj (f x) = cnj (integral\<^sup>C M f)"
awalk_app_isoI: fixes u :: "'c"   and p :: "'d list"   and v :: "'c"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "H.awalk u p v"   and "H.digraph_isomorphism hom" shows "pre_digraph.awalk (app_iso hom (app_iso hom G)) (iso_verts hom u) (map (iso_arcs hom) p) (iso_verts hom v)"
CharacterData_simp11: fixes character_data_ptr :: "(_) character_data_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('k \<Rightarrow> 'k) \<Rightarrow> 'h option RCharacterData_scheme \<Rightarrow> 'h option RCharacterData_scheme"   and v :: "'k"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r character_data_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
1: shows "(x \<oplus>\<^bsub>F\<^esub> (y \<oplus>\<^bsub>F\<^esub> z)) a = x a \<oplus> (y a \<oplus> z a)"
ODNmap_def: fixes D :: "'c Order"   and y :: "'c Order set" shows "ODNmap D y = (SOME z. z \<in> carrier D \<and> (\<forall>Y\<in>y. ord_equiv Y (Iod D (segment D z))))"
yup: fixes x :: "'a"   and y :: "'a"   and rest1 :: "'a list" shows "T\<^sub>p [x, y] (x # rest1) (OPT2 (x # rest1) [x, y]) = (let L = step [x, y] x (hd (OPT2 (x # rest1) [x, y])) in T\<^sub>p L rest1 (OPT2 rest1 L))"
glb_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and x :: "'b" shows "glb P B x = (lb P B x \<and> (\<forall>y. lb P B y \<longrightarrow> P\<^sup>=\<^sup>= y x))"
simps_1: fixes uu :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'e list" shows "merge_wrt uu xs [] = xs"
assms_3: fixes \<sigma> :: "'\<sigma>" assumes "I {} \<sigma>" shows "P \<sigma>"
Inr_eq_map_sum_iff: fixes x :: "'d"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and y :: "'e + 'f" shows "(Inr x = map_sum f g y) = (\<exists>z. y = Inr z \<and> x = g z)"
NotActualD: fixes \<phi> :: "\<o>" assumes "\<not> [\<^bold>\<A>\<phi> in dw]" shows "\<not> [\<phi> in dw]"
set_mono_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys" shows "set xs \<subseteq> set ys"
wt_subst_SMP_subset: fixes S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) Term.term set"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "trms\<^sub>s\<^sub>t S \<subseteq> SMP S'"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "trms\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> SMP S'"
cupcake_single_complete: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and res :: "(v, v) result" assumes "Big_Step_Unclocked_Single.evaluate env s e (s', res)"   and "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "cupcake_evaluate_single env e res"
isInitialState_def: fixes state :: "State"   and F0 :: "Literal list list" shows "isInitialState state F0 \<equiv> getF state = F0 \<and> getM state = [] \<and> getConflictFlag state = False \<and> getC state = []"
assms: shows "io \<in> language_state_for_input M q xs"
sum_atMost_remove_nat: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "k \<le> n" shows "sum f {..n} = f k + sum f ({..n} - {k})"
get_dom_component_get_scdom_component_is_l_get_dom_component_get_scdom_component: shows "l_get_dom_component_get_scdom_component Shadow_DOM.get_owner_document Shadow_DOM.heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM_SC_DOM_Components.get_scdom_component Shadow_DOM_DOM_Components.get_dom_component"
not_alw_not: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> alw (\<lambda>xs. \<not> \<phi> xs) xs) = ev \<phi>"
nn_integral_bij_count_space: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "bij_betw g A B" shows "\<integral>\<^sup>+ x. f (g x) \<partial>count_space A = integral\<^sup>N (count_space B) f"
wlsFSbSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "wlsFSbSw MOD \<equiv> wlsFSb MOD \<and> gSwapAllPresGWlsAll MOD \<and> gSwapCls MOD"
distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) \<parallel> c = a \<parallel> c + b \<parallel> c"
cont_proj: shows "continuous_on (UNIV - {0::'a}) proj"
ldropn_Suc_conv_ldropn: fixes n :: "nat"   and xs :: "'b llist" assumes "enat n < llength xs" shows "xs ?! n % ldropn (Suc n) xs = ldropn n xs"
alphaAbs_preserves_qGoodAbs1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs"   and A' :: "('a, 'b, 'varSort, 'var, 'c) qAbs" assumes "A $= A'"   and "qGoodAbs A" shows "qGoodAbs A'"
B: shows "offs_num (length ns) (x # xs) index key mi ma 0 = 0"
disc_8: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "\<not> isSend (Recv x31 x32 x33 x34 x35 x36)"
simps_2: fixes \<eta> :: "trm option" shows "Pluso undeft \<eta> = undeft"
cs_pop_rule: fixes x :: "'a"   and l :: "'a list"   and p :: "'a node ref option" shows "<cs_list (x # l) p> cs_pop p <\<lambda>(y, p'). cs_list l p' * true * \<up> (y = x)>"
free_vars_cexpr_comp: fixes e :: "cexpr"   and e' :: "cexpr" shows "free_vars (e \<circ>\<^sub>c e') \<subseteq> free_vars e - {0} \<union> free_vars e'"
arc_def: fixes e :: "'b"   and uv :: "'a \<times> 'a" shows "arc e uv \<equiv> e \<in> arcs G \<and> tail G e = fst uv \<and> head G e = snd uv"
keys_minus: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (p - q) \<subseteq> keys p \<union> keys q"
num_syms_ge_num_unique_vars: fixes t :: "('a, 'b) Term.term" shows "num_unique_vars t \<le> num_syms t"
altc_sample_writer_def: fixes altc_sample :: "'d \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c"   and p :: "'d"   and f :: "'e \<Rightarrow> ('a, 'b, 'c) writerT" shows "altc_sample_writer altc_sample p f = WriterT (altc_sample p (\<lambda>p. run_writer (f p)))"
UP_a_assoc: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "r \<in> carrier P" shows "p \<oplus>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
zero_adds_pp: fixes v :: "'a" shows "local.punit.adds_pp (0::'a) v"
summary_self: fixes loc :: "'loc" shows "summary loc loc = {}\<^sub>A"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "L.arr f"   and "L.cod f = b" shows "L.comp b f = f"
sig_red_rtrancl_monom_mult: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and s :: "'a" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q" shows "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* (monom_mult c s p) (monom_mult c s q)"
lemParallelSym: fixes lineA :: "'a Line"   and lineB :: "'a Line" assumes "lineA \<parallel>  lineB" shows "lineB \<parallel>  lineA"
klift_def: fixes f :: "'b \<Rightarrow> 'c set" shows "bd\<^sub>\<F> f = \<mu> \<circ> \<P> f"
LTL_prop_entailment_restrict_to_propos: fixes S :: "'c ltl set"   and \<phi> :: "'c ltl" shows "S \<Turnstile>\<^sub>P \<phi> = (S \<inter> propos \<phi>) \<Turnstile>\<^sub>P \<phi>"
insert_if: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "finite A" shows "prod g (insert x A) = (if x \<in> A then prod g A else g x * prod g A)"
finite_Atleast_Atmost_nat: fixes f :: "'b \<Rightarrow> 'a" shows "finite {f x |x. x \<in> UNIV}"
fold_random_permutation_foldl: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "finite A" shows "fold_random_permutation f x A = map_pmf (foldl (\<lambda>x y. f y x) x) (pmf_of_set (permutations_of_set A))"
strip_eq_Seq: fixes c :: "'a acom"   and c1 :: "com"   and c2 :: "com" shows "(strip c = c1;; c2) = (\<exists>d1 d2. c = d1;; d2 \<and> strip d1 = c1 \<and> strip d2 = c2)"
snd_Gauss_Jordan_iarrays_PA_eq: fixes A :: "'d iarray iarray" shows "snd (Gauss_Jordan_iarrays_PA A) = Gauss_Jordan_iarrays A"
intro: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and aOps :: "('k, 'j, 'd list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps" assumes "Algorithm jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps" shows "AlgorithmForAgent jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps"
one_step_implies_mult: fixes J :: "'a multiset"   and K :: "'a multiset"   and r :: "('a \<times> 'a) set"   and I :: "'a multiset" assumes "J \<noteq> {#}"   and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "(I + K, I + J) \<in> mult r"
finite: shows "emeasure M (\<Union> (range A)) \<noteq> \<infinity>"
min_galleryI_chamber_distance_betw: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "gallery (C # Cs @ [D])"   and "Suc (length Cs) = chamber_distance C D" shows "min_gallery (C # Cs @ [D])"
elims: fixes x :: "('c, 'd) SkewBinomialTree list"   and y :: "('c, 'd) SkewBinomialTree list" assumes "uniqify x = y" obtains "x = []" and "y = []"   | t :: "('c, 'd) SkewBinomialTree" and bq :: "('c, 'd) SkewBinomialTree list" where "x = t # bq" and "y = ins t bq"
1: shows "Right.in_hom \<mu> (local.dom \<mu>) (cod \<mu>)"
flrestriction_fcomp: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<circ>\<^sub>\<bullet> s) \<restriction>\<^sup>l\<^sub>\<bullet> A = r \<circ>\<^sub>\<bullet> s \<restriction>\<^sup>l\<^sub>\<bullet> A"
set_integrable_mult_left: fixes a :: "'a"   and M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "a \<noteq> (0::'a) \<Longrightarrow> set_integrable M A f" shows "set_integrable M A (\<lambda>t. f t * a)"
analz_insert_Key: fixes K :: "nat"   and H :: "msg set" assumes "K \<notin> keysFor (analz H)" shows "analz (insert (Key K) H) = insert (Key K) (analz H)"
get_host_ptr_in_heap: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h"
swap_def: fixes p :: "'f \<times> 'e" shows "prod.swap p = (snd p, fst p)"
Ag_ind_triangle1: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'f"   and B :: "'f set"   and j :: "'c" assumes "\<forall>j\<in>I. aGroup (A j)"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to f (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "j \<in> I" shows "compos (a\<Pi>\<^bsub>I\<^esub> A) (ProjInd I A f j) (Agii (a\<Pi>\<^bsub>I\<^esub> A) f) = \<pi>\<^bsub>I,A,j\<^esub>"
distance_attains_sup: fixes s :: "'a set"   and a :: "'a" assumes "compact s"   and "s \<noteq> {}" shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"
numeral_mult: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "num"   and n :: "num" assumes "class.semiring_numeral one times plus" shows "numeral.numeral one plus (m * n) = times (numeral.numeral one plus m) (numeral.numeral one plus n)"
const_typeof: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and v :: "v"   and t :: "t" assumes "\<S>\<bullet>\<C> \<turnstile> [$C v] : [] _> [t]" shows "typeof v = t"
eFresh_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var" shows "eFresh MOD ys y ERR"
disjoint_def: fixes A :: "'c set set" shows "disjoint A = (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
flow_initial_time_if: fixes t0 :: "real"   and x0 :: "real" shows "flow t0 x0 t0 = (if t0 \<in> T \<and> x0 \<in> X then x0 else 0)"
redex_simps: fixes e :: "('a, 'b, 'c) expr" shows "redex e = ((\<exists>x ea v. e = Apply (VE (Lambda x ea)) (VE v)) \<or> (\<exists>e1 e2. e = Ite (VE (CV T)) e1 e2) \<or> (\<exists>e1 e2. e = Ite (VE (CV F)) e1 e2) \<or> (\<exists>v. e = Ref (VE v)) \<or> (\<exists>l. e = Read (VE (Loc l))) \<or> (\<exists>l v. e = Assign (VE (Loc l)) (VE v)) \<or> (\<exists>ea. e = Rfork ea) \<or> (\<exists>r. e = Rjoin (VE (Rid r))))"
CCC: fixes r :: "'d rexp"   and r' :: "'d rexp"   and s :: "'d rexp"   and s' :: "'d rexp" assumes "(~)\<^sup>*\<^sup>* r r'"   and "(~)\<^sup>*\<^sup>* s s'" shows "(~)\<^sup>*\<^sup>* (Conc r s) (Conc r' s')"
drop_bin2bl: fixes m :: "nat"   and n :: "nat"   and bin :: "int" shows "drop m (bin_to_bl n bin) = bin_to_bl (n - m) bin"
encode_operator_precondition_valid: fixes sv :: "sat_plan_variable"   and t :: "nat"   and op :: "(nat \<times> nat) strips_operator" assumes "sv \<in> atoms (encode_operator_precondition (\<phi> prob_with_noop abs_prob ) t op)"   and "t < h"   and "op \<in> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)" shows "valid_state_var sv"
closure_p_eq_true: fixes P :: "'c \<Rightarrow> bool" shows "(\<forall>A. P A) = (P = true)"
assms_3: shows "polynomial (\<lambda>x. real (q x))"
card_multiset_only_sets: fixes A :: "'b set"   and k :: "nat" assumes "finite A" shows "card {M. M \<subseteq># mset_set A \<and> size M = k} = card A choose k"
setlocs_def: fixes s :: "state"   and l' :: "loc \<Rightarrow> nat" shows "setlocs s l' \<equiv> case s of st g l \<Rightarrow> st g l'"
evaln_IfI: fixes G :: "prog"   and s :: "state"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s1 :: "state"   and c1 :: "stmt"   and c2 :: "stmt"   and s2 :: "state" assumes "G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if the_Bool v then c1 else c2)\<midarrow>n\<rightarrow> s2" shows "G\<turnstile>s \<midarrow>If(e) c1 Else c2\<midarrow>n\<rightarrow> s2"
constr_rep_1: fixes x2 :: "'d"   and x1 :: "'d" shows "Rep_action (x2\<lparr>x1\<rparr>) = In_Rep x2 x1"
m_comm: fixes x :: "int"   and y :: "int" shows "x + y = y + x"
takeWhile_nth: fixes j :: "nat"   and P :: "'d \<Rightarrow> bool"   and xs :: "'d list" assumes "j < length (takeWhile P xs)" shows "takeWhile P xs ! j = xs ! j"
ex_Nleast: fixes A :: "nat set" assumes "A \<noteq> {}" shows "\<exists>!m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)"
cases: fixes a1 :: "'e"   and a2 :: "'f"   and r :: "('e \<times> 'g) set"   and s :: "('g \<times> 'f) set" assumes "(a1, a2) \<in> r O s" obtains a :: "'e" and b :: "'g" and c :: "'f" where "a1 = a" and "a2 = c" and "(a, b) \<in> r" and "(b, c) \<in> s"
assms_2: fixes x :: "'a" assumes "x0 \<le> x" shows "f x = g x"
absorb_iff2: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (min a b = b)"
positive_minus: fixes x :: "rat" assumes "\<not> Rat.positive x"   and "x \<noteq> 0" shows "Rat.positive (- x)"
DERIV_real_sqrt_generic: fixes x :: "real"   and D :: "real" assumes "x \<noteq> 0"   and "0 < x \<Longrightarrow> D = inverse (sqrt x) / 2"   and "x < 0 \<Longrightarrow> D = - inverse (sqrt x) / 2" shows "(sqrt has_real_derivative D) (at x)"
equivalence_map_is_ide: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.equivalence_map f" shows "Span.ide f"
arr_dom_iff_arr: fixes f :: "'a \<times> 'a" shows "VV.arr (VV.dom f) = VV.arr f"
exec_gets: fixes f :: "'c \<Rightarrow> 'd"   and m :: "'d \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) \<times> bool"   and s :: "'c" shows "(gets f >>= m) s = m (f s) s"
coeff_add: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
has: shows "P \<turnstile> D has F:T (fm) in D"
bijective_def: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" shows "bijective \<alpha> S T \<equiv> bij_betw \<alpha> S T"
continuous_on_Icc_at_leftD: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on {a..b} f"   and "a < b" shows "(f \<longlongrightarrow> f b) (at_left b)"
abs_eq: fixes x :: "'a \<Rightarrow> 'b option"   and xb :: "'a"   and xa :: "'b" assumes "eq_onp (\<lambda>m. finite (dom m)) x x" shows "fmupd xb xa (map_of_fmap x) = map_of_fmap (map_upd xb xa x)"
perm_append_single: fixes a :: "'b"   and xs :: "'b list" shows "a # xs <~~> xs @ [a]"
0: shows "prv (imp (\<phi>L \<phi>) (imp (PP \<langle>\<phi>L \<phi>\<rangle>) \<phi>))"
find_sub_False_proj_list_3_same_left: fixes xs :: "'b set list"   and ns :: "'b set list"   and rs :: "'b set list"   and w :: "'b set"   and ws :: "'b set list"   and ys :: "'b set list"   and z :: "'b set"   and zs :: "'b set list"   and xs' :: "'c list"   and rs' :: "'c list"   and w' :: "'c"   and ws' :: "'c list" assumes "find_sub_False xs ns = Some ((rs, w, ws), ys, z, zs)"   and "length xs = length xs'"   and "proj_list_3 xs' (rs, w, ws) = (rs', w', ws')" shows "xs' = rs' @ w' # ws'"
cexpr_sem_cexpr_comp_aux: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat"   and e :: "cexpr"   and e' :: "cexpr" shows "cexpr_sem \<sigma> (cexpr_comp_aux x e e') = cexpr_sem (\<sigma>(x := cexpr_sem \<sigma> e)) e'"
not_exact_match_in_doubt_allow_approx_match: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and a :: "action"   and m :: "'c match_expr"   and p :: "'d" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "a = action.Accept \<or> a = Reject \<or> a = action.Drop"   and "\<not> Semantics.matches \<gamma> m p" shows "a = action.Accept \<and> Matching_Ternary.matches (\<beta>, in_doubt_allow) m a p \<or> \<not> Matching_Ternary.matches (\<beta>, in_doubt_allow) m a p"